author | paulson <lp15@cam.ac.uk> |
Wed, 10 Apr 2019 21:29:32 +0100 | |
changeset 70113 | c8deb8ba6d05 |
parent 70097 | 4005298550a6 |
child 70365 | 4df0628e8545 |
permissions | -rw-r--r-- |
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(* Title: HOL/Computational_Algebra/Formal_Power_Series.thy |
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Author: Amine Chaieb, University of Cambridge |
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Author: Jeremy Sylvestre, University of Alberta (Augustana Campus) |
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Author: Manuel Eberl, TU München |
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*) |
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section \<open>A formalization of formal power series\<close> |
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|
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theory Formal_Power_Series |
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imports |
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Complex_Main |
|
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Euclidean_Algorithm |
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begin |
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subsection \<open>The type of formal power series\<close> |
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typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}" |
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morphisms fps_nth Abs_fps |
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by simp |
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notation fps_nth (infixl "$" 75) |
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lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p $ n = q $ n)" |
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by (simp add: fps_nth_inject [symmetric] fun_eq_iff) |
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lemmas fps_eq_iff = expand_fps_eq |
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lemma fps_ext: "(\<And>n. p $ n = q $ n) \<Longrightarrow> p = q" |
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by (simp add: expand_fps_eq) |
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lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n" |
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by (simp add: Abs_fps_inverse) |
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text \<open>Definition of the basic elements 0 and 1 and the basic operations of addition, |
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negation and multiplication.\<close> |
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instantiation fps :: (zero) zero |
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begin |
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definition fps_zero_def: "0 = Abs_fps (\<lambda>n. 0)" |
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instance .. |
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end |
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lemma fps_zero_nth [simp]: "0 $ n = 0" |
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unfolding fps_zero_def by simp |
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lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)" |
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by (simp add: expand_fps_eq) |
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lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m < n. f $ m = 0))" |
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(is "?lhs \<longleftrightarrow> ?rhs") |
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proof |
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let ?n = "LEAST n. f $ n \<noteq> 0" |
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show ?rhs if ?lhs |
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proof - |
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from that have "\<exists>n. f $ n \<noteq> 0" |
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by (simp add: fps_nonzero_nth) |
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then have "f $ ?n \<noteq> 0" |
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by (rule LeastI_ex) |
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moreover have "\<forall>m<?n. f $ m = 0" |
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by (auto dest: not_less_Least) |
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ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" .. |
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then show ?thesis .. |
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qed |
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show ?lhs if ?rhs |
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using that by (auto simp add: expand_fps_eq) |
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qed |
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lemma fps_nonzeroI: "f$n \<noteq> 0 \<Longrightarrow> f \<noteq> 0" |
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by auto |
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instantiation fps :: ("{one, zero}") one |
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begin |
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definition fps_one_def: "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)" |
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instance .. |
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end |
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lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)" |
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unfolding fps_one_def by simp |
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instantiation fps :: (plus) plus |
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begin |
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definition fps_plus_def: "(+) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))" |
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instance .. |
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end |
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lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n" |
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unfolding fps_plus_def by simp |
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instantiation fps :: (minus) minus |
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begin |
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definition fps_minus_def: "(-) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))" |
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instance .. |
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end |
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lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n" |
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unfolding fps_minus_def by simp |
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instantiation fps :: (uminus) uminus |
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begin |
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definition fps_uminus_def: "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))" |
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instance .. |
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end |
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lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)" |
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unfolding fps_uminus_def by simp |
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lemma fps_neg_0 [simp]: "-(0::'a::group_add fps) = 0" |
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by (rule iffD2, rule fps_eq_iff, auto) |
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instantiation fps :: ("{comm_monoid_add, times}") times |
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begin |
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definition fps_times_def: "(*) = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))" |
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instance .. |
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end |
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lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n - i))" |
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unfolding fps_times_def by simp |
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lemma fps_mult_nth_0 [simp]: "(f * g) $ 0 = f $ 0 * g $ 0" |
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unfolding fps_times_def by simp |
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lemma fps_mult_nth_1 [simp]: "(f * g) $ 1 = f$0 * g$1 + f$1 * g$0" |
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by (simp add: fps_mult_nth) |
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125 |
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lemmas mult_nth_0 = fps_mult_nth_0 |
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lemmas mult_nth_1 = fps_mult_nth_1 |
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128 |
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instance fps :: ("{comm_monoid_add, mult_zero}") mult_zero |
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proof |
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fix a :: "'a fps" |
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show "0 * a = 0" by (simp add: fps_ext fps_mult_nth) |
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show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth) |
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qed |
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declare atLeastAtMost_iff [presburger] |
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declare Bex_def [presburger] |
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declare Ball_def [presburger] |
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lemma mult_delta_left: |
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fixes x y :: "'a::mult_zero" |
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shows "(if b then x else 0) * y = (if b then x * y else 0)" |
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by simp |
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lemma mult_delta_right: |
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fixes x y :: "'a::mult_zero" |
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shows "x * (if b then y else 0) = (if b then x * y else 0)" |
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by simp |
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lemma fps_one_mult: |
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fixes f :: "'a::{comm_monoid_add, mult_zero, monoid_mult} fps" |
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shows "1 * f = f" |
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and "f * 1 = f" |
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by (simp_all add: fps_ext fps_mult_nth mult_delta_left mult_delta_right) |
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156 |
|
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subsection \<open>Subdegrees\<close> |
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158 |
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definition subdegree :: "('a::zero) fps \<Rightarrow> nat" where |
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160 |
"subdegree f = (if f = 0 then 0 else LEAST n. f$n \<noteq> 0)" |
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161 |
|
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162 |
lemma subdegreeI: |
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163 |
assumes "f $ d \<noteq> 0" and "\<And>i. i < d \<Longrightarrow> f $ i = 0" |
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164 |
shows "subdegree f = d" |
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165 |
proof- |
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166 |
from assms(1) have "f \<noteq> 0" by auto |
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167 |
moreover from assms(1) have "(LEAST i. f $ i \<noteq> 0) = d" |
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168 |
proof (rule Least_equality) |
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169 |
fix e assume "f $ e \<noteq> 0" |
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170 |
with assms(2) have "\<not>(e < d)" by blast |
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171 |
thus "e \<ge> d" by simp |
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172 |
qed |
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173 |
ultimately show ?thesis unfolding subdegree_def by simp |
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174 |
qed |
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175 |
|
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176 |
lemma nth_subdegree_nonzero [simp,intro]: "f \<noteq> 0 \<Longrightarrow> f $ subdegree f \<noteq> 0" |
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177 |
proof- |
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178 |
assume "f \<noteq> 0" |
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179 |
hence "subdegree f = (LEAST n. f $ n \<noteq> 0)" by (simp add: subdegree_def) |
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180 |
also from \<open>f \<noteq> 0\<close> have "\<exists>n. f$n \<noteq> 0" using fps_nonzero_nth by blast |
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181 |
from LeastI_ex[OF this] have "f $ (LEAST n. f $ n \<noteq> 0) \<noteq> 0" . |
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182 |
finally show ?thesis . |
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183 |
qed |
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184 |
|
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185 |
lemma nth_less_subdegree_zero [dest]: "n < subdegree f \<Longrightarrow> f $ n = 0" |
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186 |
proof (cases "f = 0") |
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187 |
assume "f \<noteq> 0" and less: "n < subdegree f" |
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188 |
note less |
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189 |
also from \<open>f \<noteq> 0\<close> have "subdegree f = (LEAST n. f $ n \<noteq> 0)" by (simp add: subdegree_def) |
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190 |
finally show "f $ n = 0" using not_less_Least by blast |
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191 |
qed simp_all |
62102
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192 |
|
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193 |
lemma subdegree_geI: |
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|
194 |
assumes "f \<noteq> 0" "\<And>i. i < n \<Longrightarrow> f$i = 0" |
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changeset
|
195 |
shows "subdegree f \<ge> n" |
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eberlm
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|
196 |
proof (rule ccontr) |
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|
197 |
assume "\<not>(subdegree f \<ge> n)" |
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|
198 |
with assms(2) have "f $ subdegree f = 0" by simp |
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|
199 |
moreover from assms(1) have "f $ subdegree f \<noteq> 0" by simp |
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|
200 |
ultimately show False by contradiction |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
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diff
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|
201 |
qed |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
changeset
|
202 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
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|
203 |
lemma subdegree_greaterI: |
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|
204 |
assumes "f \<noteq> 0" "\<And>i. i \<le> n \<Longrightarrow> f$i = 0" |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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changeset
|
205 |
shows "subdegree f > n" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
changeset
|
206 |
proof (rule ccontr) |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
changeset
|
207 |
assume "\<not>(subdegree f > n)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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changeset
|
208 |
with assms(2) have "f $ subdegree f = 0" by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
changeset
|
209 |
moreover from assms(1) have "f $ subdegree f \<noteq> 0" by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
changeset
|
210 |
ultimately show False by contradiction |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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diff
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|
211 |
qed |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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changeset
|
212 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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|
213 |
lemma subdegree_leI: |
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eberlm
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diff
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|
214 |
"f $ n \<noteq> 0 \<Longrightarrow> subdegree f \<le> n" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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changeset
|
215 |
by (rule leI) auto |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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|
216 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
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|
217 |
lemma subdegree_0 [simp]: "subdegree 0 = 0" |
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218 |
by (simp add: subdegree_def) |
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|
219 |
|
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220 |
lemma subdegree_1 [simp]: "subdegree 1 = 0" |
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221 |
by (cases "(1::'a) = 0") |
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222 |
(auto intro: subdegreeI fps_ext simp: subdegree_def) |
61608
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223 |
|
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
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|
224 |
lemma subdegree_eq_0_iff: "subdegree f = 0 \<longleftrightarrow> f = 0 \<or> f $ 0 \<noteq> 0" |
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|
225 |
proof (cases "f = 0") |
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subdegree/shift/cutoff and Euclidean ring instance for formal power series
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226 |
assume "f \<noteq> 0" |
a0487caabb4a
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|
227 |
thus ?thesis |
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|
228 |
using nth_subdegree_nonzero[OF \<open>f \<noteq> 0\<close>] by (fastforce intro!: subdegreeI) |
a0487caabb4a
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eberlm
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|
229 |
qed simp_all |
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|
230 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
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|
231 |
lemma subdegree_eq_0 [simp]: "f $ 0 \<noteq> 0 \<Longrightarrow> subdegree f = 0" |
a0487caabb4a
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|
232 |
by (simp add: subdegree_eq_0_iff) |
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233 |
|
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|
234 |
lemma nth_subdegree_zero_iff [simp]: "f $ subdegree f = 0 \<longleftrightarrow> f = 0" |
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235 |
by (cases "f = 0") auto |
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|
236 |
|
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237 |
lemma fps_nonzero_subdegree_nonzeroI: "subdegree f > 0 \<Longrightarrow> f \<noteq> 0" |
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238 |
by auto |
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|
239 |
|
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240 |
lemma subdegree_uminus [simp]: |
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241 |
"subdegree (-(f::('a::group_add) fps)) = subdegree f" |
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242 |
proof (cases "f=0") |
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243 |
case False thus ?thesis by (force intro: subdegreeI) |
195aeee8b30a
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244 |
qed simp |
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245 |
|
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246 |
lemma subdegree_minus_commute [simp]: |
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247 |
"subdegree (f-(g::('a::group_add) fps)) = subdegree (g - f)" |
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248 |
proof (-, cases "g-f=0") |
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249 |
case True |
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250 |
have "\<And>n. (f - g) $ n = -((g - f) $ n)" by simp |
195aeee8b30a
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251 |
with True have "f - g = 0" by (intro fps_ext) simp |
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252 |
with True show ?thesis by simp |
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253 |
next |
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Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
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|
254 |
case False show ?thesis |
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|
255 |
using nth_subdegree_nonzero[OF False] by (fastforce intro: subdegreeI) |
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256 |
qed |
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257 |
|
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258 |
lemma subdegree_add_ge': |
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259 |
fixes f g :: "'a::monoid_add fps" |
195aeee8b30a
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260 |
assumes "f + g \<noteq> 0" |
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261 |
shows "subdegree (f + g) \<ge> min (subdegree f) (subdegree g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
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|
262 |
using assms |
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263 |
by (force intro: subdegree_geI) |
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Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
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|
264 |
|
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Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
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diff
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|
265 |
lemma subdegree_add_ge: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
266 |
assumes "f \<noteq> -(g :: ('a :: group_add) fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
267 |
shows "subdegree (f + g) \<ge> min (subdegree f) (subdegree g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
268 |
proof (rule subdegree_add_ge') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
269 |
have "f + g = 0 \<Longrightarrow> False" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
270 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
271 |
assume fg: "f + g = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
272 |
have "\<And>n. f $ n = - g $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
273 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
274 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
275 |
from fg have "(f + g) $ n = 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
276 |
hence "f $ n + g $ n - g $ n = - g $ n" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
277 |
thus "f $ n = - g $ n" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
278 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
279 |
with assms show False by (auto intro: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
280 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
281 |
thus "f + g \<noteq> 0" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
282 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
283 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
284 |
lemma subdegree_add_eq1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
285 |
assumes "f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
286 |
and "subdegree f < subdegree (g :: 'a::monoid_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
287 |
shows "subdegree (f + g) = subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
288 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
289 |
by (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
290 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
291 |
lemma subdegree_add_eq2: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
292 |
assumes "g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
293 |
and "subdegree g < subdegree (f :: 'a :: monoid_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
294 |
shows "subdegree (f + g) = subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
295 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
296 |
by (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
297 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
298 |
lemma subdegree_diff_eq1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
299 |
assumes "f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
300 |
and "subdegree f < subdegree (g :: 'a :: group_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
301 |
shows "subdegree (f - g) = subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
302 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
303 |
by (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
304 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
305 |
lemma subdegree_diff_eq1_cancel: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
306 |
assumes "f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
307 |
and "subdegree f < subdegree (g :: 'a :: cancel_comm_monoid_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
308 |
shows "subdegree (f - g) = subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
309 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
310 |
by (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
311 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
312 |
lemma subdegree_diff_eq2: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
313 |
assumes "g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
314 |
and "subdegree g < subdegree (f :: 'a :: group_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
315 |
shows "subdegree (f - g) = subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
316 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
317 |
by (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
318 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
319 |
lemma subdegree_diff_ge [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
320 |
assumes "f \<noteq> (g :: 'a :: group_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
321 |
shows "subdegree (f - g) \<ge> min (subdegree f) (subdegree g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
322 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
323 |
from assms have "f = - (- g) \<Longrightarrow> False" using expand_fps_eq by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
324 |
hence "f \<noteq> - (- g)" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
325 |
moreover have "f + - g = f - g" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
326 |
ultimately show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
327 |
using subdegree_add_ge[of f "-g"] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
328 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
329 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
330 |
lemma subdegree_diff_ge': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
331 |
fixes f g :: "'a :: comm_monoid_diff fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
332 |
assumes "f - g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
333 |
shows "subdegree (f - g) \<ge> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
334 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
335 |
by (auto intro: subdegree_geI simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
336 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
337 |
lemma nth_subdegree_mult_left [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
338 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
339 |
shows "(f * g) $ (subdegree f) = f $ subdegree f * g $ 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
340 |
by (cases "subdegree f") (simp_all add: fps_mult_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
341 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
342 |
lemma nth_subdegree_mult_right [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
343 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
344 |
shows "(f * g) $ (subdegree g) = f $ 0 * g $ subdegree g" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
345 |
by (cases "subdegree g") (simp_all add: fps_mult_nth nth_less_subdegree_zero sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
346 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
347 |
lemma nth_subdegree_mult [simp]: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
348 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
349 |
shows "(f * g) $ (subdegree f + subdegree g) = f $ subdegree f * g $ subdegree g" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
350 |
proof- |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
351 |
let ?n = "subdegree f + subdegree g" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
352 |
have "(f * g) $ ?n = (\<Sum>i=0..?n. f$i * g$(?n-i))" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
353 |
by (simp add: fps_mult_nth) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
354 |
also have "... = (\<Sum>i=0..?n. if i = subdegree f then f$i * g$(?n-i) else 0)" |
64267 | 355 |
proof (intro sum.cong) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
356 |
fix x assume x: "x \<in> {0..?n}" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
357 |
hence "x = subdegree f \<or> x < subdegree f \<or> ?n - x < subdegree g" by auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
358 |
thus "f $ x * g $ (?n - x) = (if x = subdegree f then f $ x * g $ (?n - x) else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
359 |
by (elim disjE conjE) auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
360 |
qed auto |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
361 |
also have "... = f $ subdegree f * g $ subdegree g" by simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
362 |
finally show ?thesis . |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
363 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
364 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
365 |
lemma fps_mult_nth_eq0: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
366 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
367 |
assumes "n < subdegree f + subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
368 |
shows "(f*g) $ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
369 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
370 |
have "\<And>i. i\<in>{0..n} \<Longrightarrow> f$i * g$(n - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
371 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
372 |
fix i assume i: "i\<in>{0..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
373 |
show "f$i * g$(n - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
374 |
proof (cases "i < subdegree f \<or> n - i < subdegree g") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
375 |
case False with assms i show ?thesis by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
376 |
qed (auto simp: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
377 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
378 |
thus "(f * g) $ n = 0" by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
379 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
380 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
381 |
lemma fps_mult_subdegree_ge: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
382 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
383 |
assumes "f*g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
384 |
shows "subdegree (f*g) \<ge> subdegree f + subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
385 |
using assms fps_mult_nth_eq0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
386 |
by (intro subdegree_geI) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
387 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
388 |
lemma subdegree_mult': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
389 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
390 |
assumes "f $ subdegree f * g $ subdegree g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
391 |
shows "subdegree (f*g) = subdegree f + subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
392 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
393 |
from assms have "(f * g) $ (subdegree f + subdegree g) \<noteq> 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
394 |
hence "f*g \<noteq> 0" by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
395 |
hence "subdegree (f*g) \<ge> subdegree f + subdegree g" using fps_mult_subdegree_ge by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
396 |
moreover from assms have "subdegree (f*g) \<le> subdegree f + subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
397 |
by (intro subdegree_leI) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
398 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
399 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
400 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
401 |
lemma subdegree_mult [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
402 |
fixes f g :: "'a :: {semiring_no_zero_divisors} fps" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
403 |
assumes "f \<noteq> 0" "g \<noteq> 0" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
404 |
shows "subdegree (f * g) = subdegree f + subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
405 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
406 |
by (intro subdegree_mult') simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
407 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
408 |
lemma fps_mult_nth_conv_upto_subdegree_left: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
409 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
410 |
shows "(f * g) $ n = (\<Sum>i=subdegree f..n. f $ i * g $ (n - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
411 |
proof (cases "subdegree f \<le> n") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
412 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
413 |
hence "{0..n} = {0..<subdegree f} \<union> {subdegree f..n}" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
414 |
moreover have "{0..<subdegree f} \<inter> {subdegree f..n} = {}" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
415 |
ultimately show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
416 |
using nth_less_subdegree_zero[of _ f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
417 |
by (simp add: fps_mult_nth sum.union_disjoint) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
418 |
qed (simp add: fps_mult_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
419 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
420 |
lemma fps_mult_nth_conv_upto_subdegree_right: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
421 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
422 |
shows "(f * g) $ n = (\<Sum>i=0..n - subdegree g. f $ i * g $ (n - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
423 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
424 |
have "{0..n} = {0..n - subdegree g} \<union> {n - subdegree g<..n}" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
425 |
moreover have "{0..n - subdegree g} \<inter> {n - subdegree g<..n} = {}" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
426 |
moreover have "\<forall>i\<in>{n - subdegree g<..n}. g $ (n - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
427 |
using nth_less_subdegree_zero[of _ g] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
428 |
ultimately show ?thesis by (simp add: fps_mult_nth sum.union_disjoint) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
429 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
430 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
431 |
lemma fps_mult_nth_conv_inside_subdegrees: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
432 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
433 |
shows "(f * g) $ n = (\<Sum>i=subdegree f..n - subdegree g. f $ i * g $ (n - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
434 |
proof (cases "subdegree f \<le> n - subdegree g") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
435 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
436 |
hence "{subdegree f..n} = {subdegree f..n - subdegree g} \<union> {n - subdegree g<..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
437 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
438 |
moreover have "{subdegree f..n - subdegree g} \<inter> {n - subdegree g<..n} = {}" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
439 |
moreover have "\<forall>i\<in>{n - subdegree g<..n}. f $ i * g $ (n - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
440 |
using nth_less_subdegree_zero[of _ g] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
441 |
ultimately show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
442 |
using fps_mult_nth_conv_upto_subdegree_left[of f g n] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
443 |
by (simp add: sum.union_disjoint) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
444 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
445 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
446 |
hence 1: "subdegree f > n - subdegree g" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
447 |
show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
448 |
proof (cases "f*g = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
449 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
450 |
with 1 have "n < subdegree (f*g)" using fps_mult_subdegree_ge[of f g] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
451 |
with 1 show ?thesis by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
452 |
qed (simp add: 1) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
453 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
454 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
455 |
lemma fps_mult_nth_outside_subdegrees: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
456 |
fixes f g :: "('a :: {mult_zero,comm_monoid_add}) fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
457 |
shows "n < subdegree f \<Longrightarrow> (f * g) $ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
458 |
and "n < subdegree g \<Longrightarrow> (f * g) $ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
459 |
by (auto simp: fps_mult_nth_conv_inside_subdegrees) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
460 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
461 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
462 |
subsection \<open>Formal power series form a commutative ring with unity, if the range of sequences |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
463 |
they represent is a commutative ring with unity\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
464 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
465 |
instance fps :: (semigroup_add) semigroup_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
466 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
467 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
468 |
show "a + b + c = a + (b + c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
469 |
by (simp add: fps_ext add.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
470 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
471 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
472 |
instance fps :: (ab_semigroup_add) ab_semigroup_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
473 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
474 |
fix a b :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
475 |
show "a + b = b + a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
476 |
by (simp add: fps_ext add.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
477 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
478 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
479 |
instance fps :: (monoid_add) monoid_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
480 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
481 |
fix a :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
482 |
show "0 + a = a" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
483 |
show "a + 0 = a" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
484 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
485 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
486 |
instance fps :: (comm_monoid_add) comm_monoid_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
487 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
488 |
fix a :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
489 |
show "0 + a = a" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
490 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
491 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
492 |
instance fps :: (cancel_semigroup_add) cancel_semigroup_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
493 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
494 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
495 |
show "b = c" if "a + b = a + c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
496 |
using that by (simp add: expand_fps_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
497 |
show "b = c" if "b + a = c + a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
498 |
using that by (simp add: expand_fps_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
499 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
500 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
501 |
instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
502 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
503 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
504 |
show "a + b - a = b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
505 |
by (simp add: expand_fps_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
506 |
show "a - b - c = a - (b + c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
507 |
by (simp add: expand_fps_eq diff_diff_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
508 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
509 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
510 |
instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
511 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
512 |
instance fps :: (group_add) group_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
513 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
514 |
fix a b :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
515 |
show "- a + a = 0" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
516 |
show "a + - b = a - b" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
517 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
518 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
519 |
instance fps :: (ab_group_add) ab_group_add |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
520 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
521 |
fix a b :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
522 |
show "- a + a = 0" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
523 |
show "a - b = a + - b" by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
524 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
525 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
526 |
instance fps :: (zero_neq_one) zero_neq_one |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
527 |
by standard (simp add: expand_fps_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
528 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
529 |
lemma fps_mult_assoc_lemma: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
530 |
fixes k :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
531 |
and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
532 |
shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
533 |
(\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
534 |
by (induct k) (simp_all add: Suc_diff_le sum.distrib add.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
535 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
536 |
instance fps :: (semiring_0) semiring_0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
537 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
538 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
539 |
show "(a + b) * c = a * c + b * c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
540 |
by (simp add: expand_fps_eq fps_mult_nth distrib_right sum.distrib) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
541 |
show "a * (b + c) = a * b + a * c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
542 |
by (simp add: expand_fps_eq fps_mult_nth distrib_left sum.distrib) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
543 |
show "(a * b) * c = a * (b * c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
544 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
545 |
fix n :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
546 |
have "(\<Sum>j=0..n. \<Sum>i=0..j. a$i * b$(j - i) * c$(n - j)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
547 |
(\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
548 |
by (rule fps_mult_assoc_lemma) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
549 |
then show "((a * b) * c) $ n = (a * (b * c)) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
550 |
by (simp add: fps_mult_nth sum_distrib_left sum_distrib_right mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
551 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
552 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
553 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
554 |
instance fps :: (semiring_0_cancel) semiring_0_cancel .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
555 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
556 |
lemma fps_mult_commute_lemma: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
557 |
fixes n :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
558 |
and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
559 |
shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
560 |
by (rule sum.reindex_bij_witness[where i="(-) n" and j="(-) n"]) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
561 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
562 |
instance fps :: (comm_semiring_0) comm_semiring_0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
563 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
564 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
565 |
show "a * b = b * a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
566 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
567 |
fix n :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
568 |
have "(\<Sum>i=0..n. a$i * b$(n - i)) = (\<Sum>i=0..n. a$(n - i) * b$i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
569 |
by (rule fps_mult_commute_lemma) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
570 |
then show "(a * b) $ n = (b * a) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
571 |
by (simp add: fps_mult_nth mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
572 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
573 |
qed (simp add: distrib_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
574 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
575 |
instance fps :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
576 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
577 |
instance fps :: (semiring_1) semiring_1 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
578 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
579 |
fix a :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
580 |
show "1 * a = a" "a * 1 = a" by (simp_all add: fps_one_mult) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
581 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
582 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
583 |
instance fps :: (comm_semiring_1) comm_semiring_1 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
584 |
by standard simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
585 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
586 |
instance fps :: (semiring_1_cancel) semiring_1_cancel .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
587 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
588 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
589 |
subsection \<open>Selection of the nth power of the implicit variable in the infinite sum\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
590 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
591 |
lemma fps_square_nth: "(f^2) $ n = (\<Sum>k\<le>n. f $ k * f $ (n - k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
592 |
by (simp add: power2_eq_square fps_mult_nth atLeast0AtMost) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
593 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
594 |
lemma fps_sum_nth: "sum f S $ n = sum (\<lambda>k. (f k) $ n) S" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
595 |
proof (cases "finite S") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
596 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
597 |
then show ?thesis by (induct set: finite) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
598 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
599 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
600 |
then show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
601 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
602 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
603 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
604 |
subsection \<open>Injection of the basic ring elements and multiplication by scalars\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
605 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
606 |
definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
607 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
608 |
lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
609 |
unfolding fps_const_def by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
610 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
611 |
lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
612 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
613 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
614 |
lemma fps_const_nonzero_eq_nonzero: "c \<noteq> 0 \<Longrightarrow> fps_const c \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
615 |
using fps_nonzeroI[of "fps_const c" 0] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
616 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
617 |
lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
618 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
619 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
620 |
lemma subdegree_fps_const [simp]: "subdegree (fps_const c) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
621 |
by (cases "c = 0") (auto intro!: subdegreeI) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
622 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
623 |
lemma fps_const_neg [simp]: "- (fps_const (c::'a::group_add)) = fps_const (- c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
624 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
625 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
626 |
lemma fps_const_add [simp]: "fps_const (c::'a::monoid_add) + fps_const d = fps_const (c + d)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
627 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
628 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
629 |
lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
630 |
Abs_fps (\<lambda>n. if n = 0 then c + f$0 else f$n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
631 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
632 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
633 |
lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
634 |
Abs_fps (\<lambda>n. if n = 0 then f$0 + c else f$n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
635 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
636 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
637 |
lemma fps_const_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
638 |
by (simp add: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
639 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
640 |
lemmas fps_const_minus = fps_const_sub |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
641 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
642 |
lemma fps_const_mult[simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
643 |
fixes c d :: "'a::{comm_monoid_add,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
644 |
shows "fps_const c * fps_const d = fps_const (c * d)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
645 |
by (simp add: fps_eq_iff fps_mult_nth sum.neutral) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
646 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
647 |
lemma fps_const_mult_left: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
648 |
"fps_const (c::'a::{comm_monoid_add,mult_zero}) * f = Abs_fps (\<lambda>n. c * f$n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
649 |
unfolding fps_eq_iff fps_mult_nth |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
650 |
by (simp add: fps_const_def mult_delta_left) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
651 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
652 |
lemma fps_const_mult_right: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
653 |
"f * fps_const (c::'a::{comm_monoid_add,mult_zero}) = Abs_fps (\<lambda>n. f$n * c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
654 |
unfolding fps_eq_iff fps_mult_nth |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
655 |
by (simp add: fps_const_def mult_delta_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
656 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
657 |
lemma fps_mult_left_const_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
658 |
"(fps_const (c::'a::{comm_monoid_add,mult_zero}) * f)$n = c* f$n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
659 |
by (simp add: fps_mult_nth mult_delta_left) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
660 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
661 |
lemma fps_mult_right_const_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
662 |
"(f * fps_const (c::'a::{comm_monoid_add,mult_zero}))$n = f$n * c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
663 |
by (simp add: fps_mult_nth mult_delta_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
664 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
665 |
lemma fps_const_power [simp]: "fps_const c ^ n = fps_const (c^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
666 |
by (induct n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
667 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
668 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
669 |
subsection \<open>Formal power series form an integral domain\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
670 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
671 |
instance fps :: (ring) ring .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
672 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
673 |
instance fps :: (comm_ring) comm_ring .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
674 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
675 |
instance fps :: (ring_1) ring_1 .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
676 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
677 |
instance fps :: (comm_ring_1) comm_ring_1 .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
678 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
679 |
instance fps :: (semiring_no_zero_divisors) semiring_no_zero_divisors |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
680 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
681 |
fix a b :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
682 |
assume "a \<noteq> 0" and "b \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
683 |
hence "(a * b) $ (subdegree a + subdegree b) \<noteq> 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
684 |
thus "a * b \<noteq> 0" using fps_nonzero_nth by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
685 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
686 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
687 |
instance fps :: (semiring_1_no_zero_divisors) semiring_1_no_zero_divisors .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
688 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
689 |
instance fps :: ("{cancel_semigroup_add,semiring_no_zero_divisors_cancel}") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
690 |
semiring_no_zero_divisors_cancel |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
691 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
692 |
fix a b c :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
693 |
show "(a * c = b * c) = (c = 0 \<or> a = b)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
694 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
695 |
assume ab: "a * c = b * c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
696 |
have "c \<noteq> 0 \<Longrightarrow> a = b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
697 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
698 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
699 |
assume c: "c \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
700 |
show "a $ n = b $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
701 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
702 |
case (1 n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
703 |
with ab c show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
704 |
using fps_mult_nth_conv_upto_subdegree_right[of a c "subdegree c + n"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
705 |
fps_mult_nth_conv_upto_subdegree_right[of b c "subdegree c + n"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
706 |
by (cases n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
707 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
708 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
709 |
thus "c = 0 \<or> a = b" by fast |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
710 |
qed auto |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
711 |
show "(c * a = c * b) = (c = 0 \<or> a = b)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
712 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
713 |
assume ab: "c * a = c * b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
714 |
have "c \<noteq> 0 \<Longrightarrow> a = b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
715 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
716 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
717 |
assume c: "c \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
718 |
show "a $ n = b $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
719 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
720 |
case (1 n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
721 |
moreover have "\<forall>i\<in>{Suc (subdegree c)..subdegree c + n}. subdegree c + n - i < n" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
722 |
ultimately show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
723 |
using ab c fps_mult_nth_conv_upto_subdegree_left[of c a "subdegree c + n"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
724 |
fps_mult_nth_conv_upto_subdegree_left[of c b "subdegree c + n"] |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
725 |
by (simp add: sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
726 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
727 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
728 |
thus "c = 0 \<or> a = b" by fast |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
729 |
qed auto |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
730 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
731 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
732 |
instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
733 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
734 |
instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
735 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
736 |
instance fps :: (idom) idom .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
737 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
738 |
lemma fps_numeral_fps_const: "numeral k = fps_const (numeral k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
739 |
by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1 fps_const_add [symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
740 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
741 |
lemmas numeral_fps_const = fps_numeral_fps_const |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
742 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
743 |
lemma neg_numeral_fps_const: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
744 |
"(- numeral k :: 'a :: ring_1 fps) = fps_const (- numeral k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
745 |
by (simp add: numeral_fps_const) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
746 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
747 |
lemma fps_numeral_nth: "numeral n $ i = (if i = 0 then numeral n else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
748 |
by (simp add: numeral_fps_const) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
749 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
750 |
lemma fps_numeral_nth_0 [simp]: "numeral n $ 0 = numeral n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
751 |
by (simp add: numeral_fps_const) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
752 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
753 |
lemma subdegree_numeral [simp]: "subdegree (numeral n) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
754 |
by (simp add: numeral_fps_const) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
755 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
756 |
lemma fps_of_nat: "fps_const (of_nat c) = of_nat c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
757 |
by (induction c) (simp_all add: fps_const_add [symmetric] del: fps_const_add) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
758 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
759 |
lemma fps_of_int: "fps_const (of_int c) = of_int c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
760 |
by (induction c) (simp_all add: fps_const_minus [symmetric] fps_of_nat fps_const_neg [symmetric] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
761 |
del: fps_const_minus fps_const_neg) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
762 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
763 |
lemma fps_nth_of_nat [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
764 |
"(of_nat c) $ n = (if n=0 then of_nat c else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
765 |
by (simp add: fps_of_nat[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
766 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
767 |
lemma fps_nth_of_int [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
768 |
"(of_int c) $ n = (if n=0 then of_int c else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
769 |
by (simp add: fps_of_int[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
770 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
771 |
lemma fps_mult_of_nat_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
772 |
shows "(of_nat k * f) $ n = of_nat k * f$n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
773 |
and "(f * of_nat k ) $ n = f$n * of_nat k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
774 |
by (simp_all add: fps_of_nat[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
775 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
776 |
lemma fps_mult_of_int_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
777 |
shows "(of_int k * f) $ n = of_int k * f$n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
778 |
and "(f * of_int k ) $ n = f$n * of_int k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
779 |
by (simp_all add: fps_of_int[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
780 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
781 |
lemma numeral_neq_fps_zero [simp]: "(numeral f :: 'a :: field_char_0 fps) \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
782 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
783 |
assume "numeral f = (0 :: 'a fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
784 |
from arg_cong[of _ _ "\<lambda>F. F $ 0", OF this] show False by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
785 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
786 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
787 |
lemma subdegree_power_ge: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
788 |
"f^n \<noteq> 0 \<Longrightarrow> subdegree (f^n) \<ge> n * subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
789 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
790 |
case (Suc n) thus ?case using fps_mult_subdegree_ge by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
791 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
792 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
793 |
lemma fps_pow_nth_below_subdegree: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
794 |
"k < n * subdegree f \<Longrightarrow> (f^n) $ k = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
795 |
proof (cases "f^n = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
796 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
797 |
assume "k < n * subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
798 |
with False have "k < subdegree (f^n)" using subdegree_power_ge[of f n] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
799 |
thus "(f^n) $ k = 0" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
800 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
801 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
802 |
lemma fps_pow_base [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
803 |
"(f ^ n) $ (n * subdegree f) = (f $ subdegree f) ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
804 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
805 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
806 |
show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
807 |
proof (cases "Suc n * subdegree f < subdegree f + subdegree (f^n)") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
808 |
case True with Suc show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
809 |
by (auto simp: fps_mult_nth_eq0 distrib_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
810 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
811 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
812 |
hence "\<forall>i\<in>{Suc (subdegree f)..Suc n * subdegree f - subdegree (f ^ n)}. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
813 |
f ^ n $ (Suc n * subdegree f - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
814 |
by (auto simp: fps_pow_nth_below_subdegree) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
815 |
with False Suc show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
816 |
using fps_mult_nth_conv_inside_subdegrees[of f "f^n" "Suc n * subdegree f"] |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
817 |
sum.atLeast_Suc_atMost[of |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
818 |
"subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
819 |
"Suc n * subdegree f - subdegree (f ^ n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
820 |
"\<lambda>i. f $ i * f ^ n $ (Suc n * subdegree f - i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
821 |
] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
822 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
823 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
824 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
825 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
826 |
lemma subdegree_power_eqI: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
827 |
fixes f :: "'a::semiring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
828 |
shows "(f $ subdegree f) ^ n \<noteq> 0 \<Longrightarrow> subdegree (f ^ n) = n * subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
829 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
830 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
831 |
from Suc have 1: "subdegree (f ^ n) = n * subdegree f" by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
832 |
with Suc(2) have "f $ subdegree f * f ^ n $ subdegree (f ^ n) \<noteq> 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
833 |
with 1 show ?case using subdegree_mult'[of f "f^n"] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
834 |
qed simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
835 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
836 |
lemma subdegree_power [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
837 |
"subdegree ((f :: ('a :: semiring_1_no_zero_divisors) fps) ^ n) = n * subdegree f" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
838 |
by (cases "f = 0"; induction n) simp_all |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
839 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
840 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
841 |
subsection \<open>The efps_Xtractor series fps_X\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
842 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
843 |
lemma minus_one_power_iff: "(- (1::'a::ring_1)) ^ n = (if even n then 1 else - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
844 |
by (induct n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
845 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
846 |
definition "fps_X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
847 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
848 |
lemma subdegree_fps_X [simp]: "subdegree (fps_X :: ('a :: zero_neq_one) fps) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
849 |
by (auto intro!: subdegreeI simp: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
850 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
851 |
lemma fps_X_mult_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
852 |
fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
853 |
shows "(fps_X * f) $ n = (if n = 0 then 0 else f $ (n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
854 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
855 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
856 |
moreover have "(fps_X * f) $ Suc m = f $ (Suc m - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
857 |
proof (cases m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
858 |
case 0 thus ?thesis using fps_mult_nth_1[of "fps_X" f] by (simp add: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
859 |
next |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
860 |
case (Suc k) thus ?thesis by (simp add: fps_mult_nth fps_X_def sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
861 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
862 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
863 |
qed (simp add: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
864 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
865 |
lemma fps_X_mult_right_nth [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
866 |
fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
867 |
shows "(a * fps_X) $ n = (if n = 0 then 0 else a $ (n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
868 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
869 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
870 |
moreover have "(a * fps_X) $ Suc m = a $ (Suc m - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
871 |
proof (cases m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
872 |
case 0 thus ?thesis using fps_mult_nth_1[of a "fps_X"] by (simp add: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
873 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
874 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
875 |
hence "(a * fps_X) $ Suc m = (\<Sum>i=0..k. a$i * fps_X$(Suc m - i)) + a$(Suc k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
876 |
by (simp add: fps_mult_nth fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
877 |
moreover have "\<forall>i\<in>{0..k}. a$i * fps_X$(Suc m - i) = 0" by (auto simp: Suc fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
878 |
ultimately show ?thesis by (simp add: Suc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
879 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
880 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
881 |
qed (simp add: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
882 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
883 |
lemma fps_mult_fps_X_commute: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
884 |
fixes a :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
885 |
shows "fps_X * a = a * fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
886 |
by (simp add: fps_eq_iff) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
887 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
888 |
lemma fps_mult_fps_X_power_commute: "fps_X ^ k * a = a * fps_X ^ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
889 |
proof (induct k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
890 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
891 |
hence "fps_X ^ Suc k * a = a * fps_X * fps_X ^ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
892 |
by (simp add: mult.assoc fps_mult_fps_X_commute[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
893 |
thus ?case by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
894 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
895 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
896 |
lemma fps_subdegree_mult_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
897 |
fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
898 |
assumes "f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
899 |
shows "subdegree (fps_X * f) = subdegree f + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
900 |
and "subdegree (f * fps_X) = subdegree f + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
901 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
902 |
show "subdegree (fps_X * f) = subdegree f + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
903 |
proof (intro subdegreeI) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
904 |
fix i :: nat assume i: "i < subdegree f + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
905 |
show "(fps_X * f) $ i = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
906 |
proof (cases "i=0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
907 |
case False with i show ?thesis by (simp add: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
908 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
909 |
case True thus ?thesis using fps_X_mult_nth[of f i] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
910 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
911 |
qed (simp add: assms) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
912 |
thus "subdegree (f * fps_X) = subdegree f + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
913 |
by (simp add: fps_mult_fps_X_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
914 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
915 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
916 |
lemma fps_mult_fps_X_nonzero: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
917 |
fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
918 |
assumes "f \<noteq> 0" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
919 |
shows "fps_X * f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
920 |
and "f * fps_X \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
921 |
using assms fps_subdegree_mult_fps_X[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
922 |
fps_nonzero_subdegree_nonzeroI[of "fps_X * f"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
923 |
fps_nonzero_subdegree_nonzeroI[of "f * fps_X"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
924 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
925 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
926 |
lemma fps_mult_fps_X_power_nonzero: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
927 |
assumes "f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
928 |
shows "fps_X ^ n * f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
929 |
and "f * fps_X ^ n \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
930 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
931 |
show "fps_X ^ n * f \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
932 |
by (induct n) (simp_all add: assms mult.assoc fps_mult_fps_X_nonzero(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
933 |
thus "f * fps_X ^ n \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
934 |
by (simp add: fps_mult_fps_X_power_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
935 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
936 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
937 |
lemma fps_X_power_iff: "fps_X ^ n = Abs_fps (\<lambda>m. if m = n then 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
938 |
by (induction n) (auto simp: fps_eq_iff) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
939 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
940 |
lemma fps_X_nth[simp]: "fps_X$n = (if n = 1 then 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
941 |
by (simp add: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
942 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
943 |
lemma fps_X_power_nth[simp]: "(fps_X^k) $n = (if n = k then 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
944 |
by (simp add: fps_X_power_iff) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
945 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
946 |
lemma fps_X_power_subdegree: "subdegree (fps_X^n) = n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
947 |
by (auto intro: subdegreeI) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
948 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
949 |
lemma fps_X_power_mult_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
950 |
"(fps_X^k * f) $ n = (if n < k then 0 else f $ (n - k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
951 |
by (cases "n<k") |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
952 |
(simp_all add: fps_mult_nth_conv_upto_subdegree_left fps_X_power_subdegree sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
953 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
954 |
lemma fps_X_power_mult_right_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
955 |
"(f * fps_X^k) $ n = (if n < k then 0 else f $ (n - k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
956 |
using fps_mult_fps_X_power_commute[of k f] fps_X_power_mult_nth[of k f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
957 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
958 |
lemma fps_subdegree_mult_fps_X_power: |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
959 |
assumes "f \<noteq> 0" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
960 |
shows "subdegree (fps_X ^ n * f) = subdegree f + n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
961 |
and "subdegree (f * fps_X ^ n) = subdegree f + n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
962 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
963 |
from assms show "subdegree (fps_X ^ n * f) = subdegree f + n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
964 |
by (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
965 |
(simp_all add: algebra_simps fps_subdegree_mult_fps_X(1) fps_mult_fps_X_power_nonzero(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
966 |
thus "subdegree (f * fps_X ^ n) = subdegree f + n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
967 |
by (simp add: fps_mult_fps_X_power_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
968 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
969 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
970 |
lemma fps_mult_fps_X_plus_1_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
971 |
"((1+fps_X)*a) $n = (if n = 0 then (a$n :: 'a::semiring_1) else a$n + a$(n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
972 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
973 |
case 0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
974 |
then show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
975 |
by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
976 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
977 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
978 |
have "((1 + fps_X)*a) $ n = sum (\<lambda>i. (1 + fps_X) $ i * a $ (n - i)) {0..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
979 |
by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
980 |
also have "\<dots> = sum (\<lambda>i. (1+fps_X)$i * a$(n-i)) {0.. 1}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
981 |
unfolding Suc by (rule sum.mono_neutral_right) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
982 |
also have "\<dots> = (if n = 0 then a$n else a$n + a$(n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
983 |
by (simp add: Suc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
984 |
finally show ?thesis . |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
985 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
986 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
987 |
lemma fps_mult_right_fps_X_plus_1_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
988 |
fixes a :: "'a :: semiring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
989 |
shows "(a*(1+fps_X)) $ n = (if n = 0 then a$n else a$n + a$(n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
990 |
using fps_mult_fps_X_plus_1_nth |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
991 |
by (simp add: distrib_left fps_mult_fps_X_commute distrib_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
992 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
993 |
lemma fps_X_neq_fps_const [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> fps_const c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
994 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
995 |
assume "(fps_X::'a fps) = fps_const (c::'a)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
996 |
hence "fps_X$1 = (fps_const (c::'a))$1" by (simp only:) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
997 |
thus False by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
998 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
999 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1000 |
lemma fps_X_neq_zero [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1001 |
by (simp only: fps_const_0_eq_0[symmetric] fps_X_neq_fps_const) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1002 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1003 |
lemma fps_X_neq_one [simp]: "(fps_X :: 'a :: zero_neq_one fps) \<noteq> 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1004 |
by (simp only: fps_const_1_eq_1[symmetric] fps_X_neq_fps_const) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1005 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1006 |
lemma fps_X_neq_numeral [simp]: "fps_X \<noteq> numeral c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1007 |
by (simp only: numeral_fps_const fps_X_neq_fps_const) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1008 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1009 |
lemma fps_X_pow_eq_fps_X_pow_iff [simp]: "fps_X ^ m = fps_X ^ n \<longleftrightarrow> m = n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1010 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1011 |
assume "(fps_X :: 'a fps) ^ m = fps_X ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1012 |
hence "(fps_X :: 'a fps) ^ m $ m = fps_X ^ n $ m" by (simp only:) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1013 |
thus "m = n" by (simp split: if_split_asm) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1014 |
qed simp_all |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1015 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1016 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1017 |
subsection \<open>Shifting and slicing\<close> |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1018 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1019 |
definition fps_shift :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1020 |
"fps_shift n f = Abs_fps (\<lambda>i. f $ (i + n))" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1021 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1022 |
lemma fps_shift_nth [simp]: "fps_shift n f $ i = f $ (i + n)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1023 |
by (simp add: fps_shift_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1024 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1025 |
lemma fps_shift_0 [simp]: "fps_shift 0 f = f" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1026 |
by (intro fps_ext) (simp add: fps_shift_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1027 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1028 |
lemma fps_shift_zero [simp]: "fps_shift n 0 = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1029 |
by (intro fps_ext) (simp add: fps_shift_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1030 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1031 |
lemma fps_shift_one: "fps_shift n 1 = (if n = 0 then 1 else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1032 |
by (intro fps_ext) (simp add: fps_shift_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1033 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1034 |
lemma fps_shift_fps_const: "fps_shift n (fps_const c) = (if n = 0 then fps_const c else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1035 |
by (intro fps_ext) (simp add: fps_shift_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1036 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1037 |
lemma fps_shift_numeral: "fps_shift n (numeral c) = (if n = 0 then numeral c else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1038 |
by (simp add: numeral_fps_const fps_shift_fps_const) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1039 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1040 |
lemma fps_shift_fps_X [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1041 |
"n \<ge> 1 \<Longrightarrow> fps_shift n fps_X = (if n = 1 then 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1042 |
by (intro fps_ext) (auto simp: fps_X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1043 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
1044 |
lemma fps_shift_fps_X_power [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1045 |
"n \<le> m \<Longrightarrow> fps_shift n (fps_X ^ m) = fps_X ^ (m - n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1046 |
by (intro fps_ext) auto |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1047 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1048 |
lemma fps_shift_subdegree [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1049 |
"n \<le> subdegree f \<Longrightarrow> subdegree (fps_shift n f) = subdegree f - n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1050 |
by (cases "f=0") (auto intro: subdegreeI simp: nth_less_subdegree_zero) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1051 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1052 |
lemma fps_shift_fps_shift: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1053 |
"fps_shift (m + n) f = fps_shift m (fps_shift n f)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1054 |
by (rule fps_ext) (simp add: add_ac) |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1055 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1056 |
lemma fps_shift_fps_shift_reorder: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1057 |
"fps_shift m (fps_shift n f) = fps_shift n (fps_shift m f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1058 |
using fps_shift_fps_shift[of m n f] fps_shift_fps_shift[of n m f] by (simp add: add.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1059 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1060 |
lemma fps_shift_rev_shift: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1061 |
"m \<le> n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1062 |
"m > n \<Longrightarrow> fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1063 |
Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1064 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1065 |
assume "m \<le> n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1066 |
thus "fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = fps_shift (n-m) f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1067 |
by (intro fps_ext) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1068 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1069 |
assume mn: "m > n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1070 |
hence "\<And>k. k \<ge> m-n \<Longrightarrow> k+n-m = k - (m-n)" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1071 |
thus |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1072 |
"fps_shift n (Abs_fps (\<lambda>k. if k<m then 0 else f $ (k-m))) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1073 |
Abs_fps (\<lambda>k. if k<m-n then 0 else f $ (k-(m-n)))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1074 |
by (intro fps_ext) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1075 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1076 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1077 |
lemma fps_shift_add: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1078 |
"fps_shift n (f + g) = fps_shift n f + fps_shift n g" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1079 |
by (simp add: fps_eq_iff) |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1080 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1081 |
lemma fps_shift_diff: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1082 |
"fps_shift n (f - g) = fps_shift n f - fps_shift n g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1083 |
by (auto intro: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1084 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1085 |
lemma fps_shift_uminus: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1086 |
"fps_shift n (-f) = - fps_shift n f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1087 |
by (auto intro: fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1088 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1089 |
lemma fps_shift_mult: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1090 |
assumes "n \<le> subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1091 |
shows "fps_shift n (h*g) = h * fps_shift n g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1092 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1093 |
have case1: "\<And>a b::'b fps. 1 \<le> subdegree b \<Longrightarrow> fps_shift 1 (a*b) = a * fps_shift 1 b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1094 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1095 |
fix a b :: "'b fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1096 |
and n :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1097 |
assume b: "1 \<le> subdegree b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1098 |
have "\<And>i. i \<le> n \<Longrightarrow> n + 1 - i = (n-i) + 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1099 |
by (simp add: algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1100 |
with b show "fps_shift 1 (a*b) $ n = (a * fps_shift 1 b) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1101 |
by (simp add: fps_mult_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1102 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1103 |
have "n \<le> subdegree g \<Longrightarrow> fps_shift n (h*g) = h * fps_shift n g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1104 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1105 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1106 |
have "fps_shift (Suc n) (h*g) = fps_shift 1 (fps_shift n (h*g))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1107 |
by (simp add: fps_shift_fps_shift[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1108 |
also have "\<dots> = h * (fps_shift 1 (fps_shift n g))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1109 |
using Suc case1 by force |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1110 |
finally show ?case by (simp add: fps_shift_fps_shift[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1111 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1112 |
with assms show ?thesis by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1113 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1114 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1115 |
lemma fps_shift_mult_right_noncomm: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1116 |
assumes "n \<le> subdegree (g :: 'b :: {comm_monoid_add, mult_zero} fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1117 |
shows "fps_shift n (g*h) = fps_shift n g * h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1118 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1119 |
have case1: "\<And>a b::'b fps. 1 \<le> subdegree a \<Longrightarrow> fps_shift 1 (a*b) = fps_shift 1 a * b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1120 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1121 |
fix a b :: "'b fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1122 |
and n :: nat |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1123 |
assume "1 \<le> subdegree a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1124 |
hence "fps_shift 1 (a*b) $ n = (\<Sum>i=Suc 0..Suc n. a$i * b$(n+1-i))" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
1125 |
using sum.atLeast_Suc_atMost[of 0 "n+1" "\<lambda>i. a$i * b$(n+1-i)"] |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1126 |
by (simp add: fps_mult_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1127 |
thus "fps_shift 1 (a*b) $ n = (fps_shift 1 a * b) $ n" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
1128 |
using sum.shift_bounds_cl_Suc_ivl[of "\<lambda>i. a$i * b$(n+1-i)" 0 n] |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1129 |
by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1130 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1131 |
have "n \<le> subdegree g \<Longrightarrow> fps_shift n (g*h) = fps_shift n g * h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1132 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1133 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1134 |
have "fps_shift (Suc n) (g*h) = fps_shift 1 (fps_shift n (g*h))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1135 |
by (simp add: fps_shift_fps_shift[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1136 |
also have "\<dots> = (fps_shift 1 (fps_shift n g)) * h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1137 |
using Suc case1 by force |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1138 |
finally show ?case by (simp add: fps_shift_fps_shift[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1139 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1140 |
with assms show ?thesis by fast |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1141 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1142 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1143 |
lemma fps_shift_mult_right: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1144 |
assumes "n \<le> subdegree (g :: 'b :: comm_semiring_0 fps)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1145 |
shows "fps_shift n (g*h) = h * fps_shift n g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1146 |
by (simp add: assms fps_shift_mult_right_noncomm mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1147 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1148 |
lemma fps_shift_mult_both: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1149 |
fixes f g :: "'a::{comm_monoid_add, mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1150 |
assumes "m \<le> subdegree f" "n \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1151 |
shows "fps_shift m f * fps_shift n g = fps_shift (m+n) (f*g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1152 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1153 |
by (simp add: fps_shift_mult fps_shift_mult_right_noncomm fps_shift_fps_shift) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1154 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1155 |
lemma fps_shift_subdegree_zero_iff [simp]: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1156 |
"fps_shift (subdegree f) f = 0 \<longleftrightarrow> f = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1157 |
by (subst (1) nth_subdegree_zero_iff[symmetric], cases "f = 0") |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1158 |
(simp_all del: nth_subdegree_zero_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1159 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1160 |
lemma fps_shift_times_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1161 |
fixes f g :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1162 |
shows "1 \<le> subdegree f \<Longrightarrow> fps_shift 1 f * fps_X = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1163 |
by (intro fps_ext) (simp add: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1164 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1165 |
lemma fps_shift_times_fps_X' [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1166 |
fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1167 |
shows "fps_shift 1 (f * fps_X) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1168 |
by (intro fps_ext) (simp add: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1169 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1170 |
lemma fps_shift_times_fps_X'': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1171 |
fixes f :: "'a::{comm_monoid_add,mult_zero,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1172 |
shows "1 \<le> n \<Longrightarrow> fps_shift n (f * fps_X) = fps_shift (n - 1) f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1173 |
by (intro fps_ext) (simp add: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1174 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1175 |
lemma fps_shift_times_fps_X_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1176 |
"n \<le> subdegree f \<Longrightarrow> fps_shift n f * fps_X ^ n = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1177 |
by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1178 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1179 |
lemma fps_shift_times_fps_X_power' [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1180 |
"fps_shift n (f * fps_X^n) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1181 |
by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1182 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1183 |
lemma fps_shift_times_fps_X_power'': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1184 |
"m \<le> n \<Longrightarrow> fps_shift n (f * fps_X^m) = fps_shift (n - m) f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1185 |
by (intro fps_ext) (simp add: fps_X_power_mult_right_nth nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1186 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1187 |
lemma fps_shift_times_fps_X_power''': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1188 |
"m > n \<Longrightarrow> fps_shift n (f * fps_X^m) = f * fps_X^(m - n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1189 |
proof (cases "f=0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1190 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1191 |
assume m: "m>n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1192 |
hence "m = n + (m-n)" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1193 |
with False m show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1194 |
using power_add[of "fps_X::'a fps" n "m-n"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1195 |
fps_shift_mult_right_noncomm[of n "f * fps_X^n" "fps_X^(m-n)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1196 |
by (simp add: mult.assoc fps_subdegree_mult_fps_X_power(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1197 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1198 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1199 |
lemma subdegree_decompose: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1200 |
"f = fps_shift (subdegree f) f * fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1201 |
by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1202 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1203 |
lemma subdegree_decompose': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1204 |
"n \<le> subdegree f \<Longrightarrow> f = fps_shift n f * fps_X^n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1205 |
by (rule fps_ext) (auto simp: fps_X_power_mult_right_nth intro!: nth_less_subdegree_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1206 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1207 |
instantiation fps :: (zero) unit_factor |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1208 |
begin |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1209 |
definition fps_unit_factor_def [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1210 |
"unit_factor f = fps_shift (subdegree f) f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1211 |
instance .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1212 |
end |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1213 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1214 |
lemma fps_unit_factor_zero_iff: "unit_factor (f::'a::zero fps) = 0 \<longleftrightarrow> f = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1215 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1216 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1217 |
lemma fps_unit_factor_nth_0: "f \<noteq> 0 \<Longrightarrow> unit_factor f $ 0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1218 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1219 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1220 |
lemma fps_X_unit_factor: "unit_factor (fps_X :: 'a :: zero_neq_one fps) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1221 |
by (intro fps_ext) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1222 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1223 |
lemma fps_X_power_unit_factor: "unit_factor (fps_X ^ n) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1224 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1225 |
define X :: "'a fps" where "X \<equiv> fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1226 |
hence "unit_factor (X^n) = fps_shift n (X^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1227 |
by (simp add: fps_X_power_subdegree) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1228 |
moreover have "fps_shift n (X^n) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1229 |
by (auto intro: fps_ext simp: fps_X_power_iff X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1230 |
ultimately show ?thesis by (simp add: X_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1231 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1232 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1233 |
lemma fps_unit_factor_decompose: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1234 |
"f = unit_factor f * fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1235 |
by (simp add: subdegree_decompose) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1236 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1237 |
lemma fps_unit_factor_decompose': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1238 |
"f = fps_X ^ subdegree f * unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1239 |
using fps_unit_factor_decompose by (simp add: fps_mult_fps_X_power_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1240 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1241 |
lemma fps_unit_factor_uminus: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1242 |
"unit_factor (-f) = - unit_factor (f::'a::group_add fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1243 |
by (simp add: fps_shift_uminus) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1244 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1245 |
lemma fps_unit_factor_shift: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1246 |
assumes "n \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1247 |
shows "unit_factor (fps_shift n f) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1248 |
by (simp add: assms fps_shift_fps_shift[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1249 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1250 |
lemma fps_unit_factor_mult_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1251 |
fixes f :: "'a::{comm_monoid_add,monoid_mult,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1252 |
shows "unit_factor (fps_X * f) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1253 |
and "unit_factor (f * fps_X) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1254 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1255 |
show "unit_factor (fps_X * f) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1256 |
by (cases "f=0") (auto intro: fps_ext simp: fps_subdegree_mult_fps_X(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1257 |
thus "unit_factor (f * fps_X) = unit_factor f" by (simp add: fps_mult_fps_X_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1258 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1259 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1260 |
lemma fps_unit_factor_mult_fps_X_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1261 |
shows "unit_factor (fps_X ^ n * f) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1262 |
and "unit_factor (f * fps_X ^ n) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1263 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1264 |
show "unit_factor (fps_X ^ n * f) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1265 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1266 |
case (Suc m) thus ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1267 |
using fps_unit_factor_mult_fps_X(1)[of "fps_X ^ m * f"] by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1268 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1269 |
thus "unit_factor (f * fps_X ^ n) = unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1270 |
by (simp add: fps_mult_fps_X_power_commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1271 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1272 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1273 |
lemma fps_unit_factor_mult_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1274 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1275 |
shows "unit_factor (f * unit_factor g) = unit_factor (f * g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1276 |
and "unit_factor (unit_factor f * g) = unit_factor (f * g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1277 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1278 |
show "unit_factor (f * unit_factor g) = unit_factor (f * g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1279 |
proof (cases "f*g = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1280 |
case False thus ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1281 |
using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree g" "f*g"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1282 |
by (simp add: fps_shift_mult) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1283 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1284 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1285 |
moreover have "f * unit_factor g = fps_shift (subdegree g) (f*g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1286 |
by (simp add: fps_shift_mult) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1287 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1288 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1289 |
show "unit_factor (unit_factor f * g) = unit_factor (f * g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1290 |
proof (cases "f*g = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1291 |
case False thus ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1292 |
using fps_mult_subdegree_ge[of f g] fps_unit_factor_shift[of "subdegree f" "f*g"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1293 |
by (simp add: fps_shift_mult_right_noncomm) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1294 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1295 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1296 |
moreover have "unit_factor f * g = fps_shift (subdegree f) (f*g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1297 |
by (simp add: fps_shift_mult_right_noncomm) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1298 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1299 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1300 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1301 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1302 |
lemma fps_unit_factor_mult_both_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1303 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1304 |
shows "unit_factor (unit_factor f * unit_factor g) = unit_factor (f * g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1305 |
using fps_unit_factor_mult_unit_factor(1)[of "unit_factor f" g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1306 |
fps_unit_factor_mult_unit_factor(2)[of f g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1307 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1308 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1309 |
lemma fps_unit_factor_mult': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1310 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1311 |
assumes "f $ subdegree f * g $ subdegree g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1312 |
shows "unit_factor (f * g) = unit_factor f * unit_factor g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1313 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1314 |
by (simp add: subdegree_mult' fps_shift_mult_both) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1315 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1316 |
lemma fps_unit_factor_mult: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1317 |
fixes f g :: "'a::semiring_no_zero_divisors fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1318 |
shows "unit_factor (f * g) = unit_factor f * unit_factor g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1319 |
using fps_unit_factor_mult'[of f g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1320 |
by (cases "f=0 \<or> g=0") auto |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1321 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1322 |
definition "fps_cutoff n f = Abs_fps (\<lambda>i. if i < n then f$i else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1323 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1324 |
lemma fps_cutoff_nth [simp]: "fps_cutoff n f $ i = (if i < n then f$i else 0)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1325 |
unfolding fps_cutoff_def by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1326 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1327 |
lemma fps_cutoff_zero_iff: "fps_cutoff n f = 0 \<longleftrightarrow> (f = 0 \<or> n \<le> subdegree f)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1328 |
proof |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1329 |
assume A: "fps_cutoff n f = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1330 |
thus "f = 0 \<or> n \<le> subdegree f" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1331 |
proof (cases "f = 0") |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1332 |
assume "f \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1333 |
with A have "n \<le> subdegree f" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1334 |
by (intro subdegree_geI) (simp_all add: fps_eq_iff split: if_split_asm) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1335 |
thus ?thesis .. |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1336 |
qed simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1337 |
qed (auto simp: fps_eq_iff intro: nth_less_subdegree_zero) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1338 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1339 |
lemma fps_cutoff_0 [simp]: "fps_cutoff 0 f = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1340 |
by (simp add: fps_eq_iff) |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1341 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1342 |
lemma fps_cutoff_zero [simp]: "fps_cutoff n 0 = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1343 |
by (simp add: fps_eq_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1344 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1345 |
lemma fps_cutoff_one: "fps_cutoff n 1 = (if n = 0 then 0 else 1)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1346 |
by (simp add: fps_eq_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1347 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1348 |
lemma fps_cutoff_fps_const: "fps_cutoff n (fps_const c) = (if n = 0 then 0 else fps_const c)" |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1349 |
by (simp add: fps_eq_iff) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1350 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1351 |
lemma fps_cutoff_numeral: "fps_cutoff n (numeral c) = (if n = 0 then 0 else numeral c)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1352 |
by (simp add: numeral_fps_const fps_cutoff_fps_const) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1353 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1354 |
lemma fps_shift_cutoff: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1355 |
"fps_shift n f * fps_X^n + fps_cutoff n f = f" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
1356 |
by (simp add: fps_eq_iff fps_X_power_mult_right_nth) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1357 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1358 |
lemma fps_shift_cutoff': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1359 |
"fps_X^n * fps_shift n f + fps_cutoff n f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1360 |
by (simp add: fps_eq_iff fps_X_power_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1361 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1362 |
lemma fps_cutoff_left_mult_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1363 |
"k < n \<Longrightarrow> (fps_cutoff n f * g) $ k = (f * g) $ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1364 |
by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1365 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1366 |
lemma fps_cutoff_right_mult_nth: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1367 |
assumes "k < n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1368 |
shows "(f * fps_cutoff n g) $ k = (f * g) $ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1369 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1370 |
from assms have "\<forall>i\<in>{0..k}. fps_cutoff n g $ (k - i) = g $ (k - i)" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1371 |
thus ?thesis by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1372 |
qed |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1373 |
|
60501 | 1374 |
subsection \<open>Formal Power series form a metric space\<close> |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1375 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1376 |
instantiation fps :: ("{minus,zero}") dist |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1377 |
begin |
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1378 |
|
52891 | 1379 |
definition |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1380 |
dist_fps_def: "dist (a :: 'a fps) b = (if a = b then 0 else inverse (2 ^ subdegree (a - b)))" |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1381 |
|
54681 | 1382 |
lemma dist_fps_ge0: "dist (a :: 'a fps) b \<ge> 0" |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1383 |
by (simp add: dist_fps_def) |
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1384 |
|
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1385 |
instance .. |
48757 | 1386 |
|
30746 | 1387 |
end |
1388 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1389 |
instantiation fps :: (group_add) metric_space |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1390 |
begin |
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1391 |
|
62101 | 1392 |
definition uniformity_fps_def [code del]: |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
1393 |
"(uniformity :: ('a fps \<times> 'a fps) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})" |
62101 | 1394 |
|
1395 |
definition open_fps_def' [code del]: |
|
1396 |
"open (U :: 'a fps set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)" |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1397 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1398 |
lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1399 |
by (simp add: dist_fps_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1400 |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1401 |
instance |
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1402 |
proof |
60501 | 1403 |
show th: "dist a b = 0 \<longleftrightarrow> a = b" for a b :: "'a fps" |
62390 | 1404 |
by (simp add: dist_fps_def split: if_split_asm) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1405 |
then have th'[simp]: "dist a a = 0" for a :: "'a fps" by simp |
60501 | 1406 |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1407 |
fix a b c :: "'a fps" |
60501 | 1408 |
consider "a = b" | "c = a \<or> c = b" | "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" by blast |
1409 |
then show "dist a b \<le> dist a c + dist b c" |
|
1410 |
proof cases |
|
1411 |
case 1 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1412 |
then show ?thesis by (simp add: dist_fps_def) |
60501 | 1413 |
next |
1414 |
case 2 |
|
1415 |
then show ?thesis |
|
52891 | 1416 |
by (cases "c = a") (simp_all add: th dist_fps_sym) |
60501 | 1417 |
next |
60567 | 1418 |
case neq: 3 |
60558 | 1419 |
have False if "dist a b > dist a c + dist b c" |
1420 |
proof - |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1421 |
let ?n = "subdegree (a - b)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1422 |
from neq have "dist a b > 0" "dist b c > 0" and "dist a c > 0" by (simp_all add: dist_fps_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1423 |
with that have "dist a b > dist a c" and "dist a b > dist b c" by simp_all |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1424 |
with neq have "?n < subdegree (a - c)" and "?n < subdegree (b - c)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1425 |
by (simp_all add: dist_fps_def field_simps) |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1426 |
hence "(a - c) $ ?n = 0" and "(b - c) $ ?n = 0" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1427 |
by (simp_all only: nth_less_subdegree_zero) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1428 |
hence "(a - b) $ ?n = 0" by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1429 |
moreover from neq have "(a - b) $ ?n \<noteq> 0" by (intro nth_subdegree_nonzero) simp_all |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1430 |
ultimately show False by contradiction |
60558 | 1431 |
qed |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1432 |
thus ?thesis by (auto simp add: not_le[symmetric]) |
60501 | 1433 |
qed |
62101 | 1434 |
qed (rule open_fps_def' uniformity_fps_def)+ |
52891 | 1435 |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1436 |
end |
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1437 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1438 |
declare uniformity_Abort[where 'a="'a :: group_add fps", code] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1439 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1440 |
lemma open_fps_def: "open (S :: 'a::group_add fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> {y. dist y a < r} \<subseteq> S)" |
66373
56f8bfe1211c
Removed unnecessary constant 'ball' from Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
66311
diff
changeset
|
1441 |
unfolding open_dist subset_eq by simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1442 |
|
60558 | 1443 |
text \<open>The infinite sums and justification of the notation in textbooks.\<close> |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1444 |
|
52891 | 1445 |
lemma reals_power_lt_ex: |
54681 | 1446 |
fixes x y :: real |
1447 |
assumes xp: "x > 0" |
|
1448 |
and y1: "y > 1" |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1449 |
shows "\<exists>k>0. (1/y)^k < x" |
52891 | 1450 |
proof - |
54681 | 1451 |
have yp: "y > 0" |
1452 |
using y1 by simp |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1453 |
from reals_Archimedean2[of "max 0 (- log y x) + 1"] |
54681 | 1454 |
obtain k :: nat where k: "real k > max 0 (- log y x) + 1" |
1455 |
by blast |
|
1456 |
from k have kp: "k > 0" |
|
1457 |
by simp |
|
1458 |
from k have "real k > - log y x" |
|
1459 |
by simp |
|
1460 |
then have "ln y * real k > - ln x" |
|
1461 |
unfolding log_def |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1462 |
using ln_gt_zero_iff[OF yp] y1 |
54681 | 1463 |
by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric]) |
1464 |
then have "ln y * real k + ln x > 0" |
|
1465 |
by simp |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1466 |
then have "exp (real k * ln y + ln x) > exp 0" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1467 |
by (simp add: ac_simps) |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1468 |
then have "y ^ k * x > 1" |
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65435
diff
changeset
|
1469 |
unfolding exp_zero exp_add exp_of_nat_mult exp_ln [OF xp] exp_ln [OF yp] |
52891 | 1470 |
by simp |
1471 |
then have "x > (1 / y)^k" using yp |
|
60867 | 1472 |
by (simp add: field_simps) |
54681 | 1473 |
then show ?thesis |
1474 |
using kp by blast |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1475 |
qed |
52891 | 1476 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1477 |
lemma fps_sum_rep_nth: "(sum (\<lambda>i. fps_const(a$i)*fps_X^i) {0..m})$n = (if n \<le> m then a$n else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1478 |
by (simp add: fps_sum_nth if_distrib cong del: if_weak_cong) |
52891 | 1479 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
1480 |
lemma fps_notation: "(\<lambda>n. sum (\<lambda>i. fps_const(a$i) * fps_X^i) {0..n}) \<longlonglongrightarrow> a" |
61969 | 1481 |
(is "?s \<longlonglongrightarrow> a") |
52891 | 1482 |
proof - |
60558 | 1483 |
have "\<exists>n0. \<forall>n \<ge> n0. dist (?s n) a < r" if "r > 0" for r |
1484 |
proof - |
|
60501 | 1485 |
obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" |
1486 |
using reals_power_lt_ex[OF \<open>r > 0\<close>, of 2] by auto |
|
60558 | 1487 |
show ?thesis |
60501 | 1488 |
proof - |
60558 | 1489 |
have "dist (?s n) a < r" if nn0: "n \<ge> n0" for n |
1490 |
proof - |
|
1491 |
from that have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0" |
|
60501 | 1492 |
by (simp add: divide_simps) |
60558 | 1493 |
show ?thesis |
60501 | 1494 |
proof (cases "?s n = a") |
1495 |
case True |
|
1496 |
then show ?thesis |
|
1497 |
unfolding dist_eq_0_iff[of "?s n" a, symmetric] |
|
1498 |
using \<open>r > 0\<close> by (simp del: dist_eq_0_iff) |
|
1499 |
next |
|
1500 |
case False |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1501 |
from False have dth: "dist (?s n) a = (1/2)^subdegree (?s n - a)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1502 |
by (simp add: dist_fps_def field_simps) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1503 |
from False have kn: "subdegree (?s n - a) > n" |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1504 |
by (intro subdegree_greaterI) (simp_all add: fps_sum_rep_nth) |
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1505 |
then have "dist (?s n) a < (1/2)^n" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1506 |
by (simp add: field_simps dist_fps_def) |
60501 | 1507 |
also have "\<dots> \<le> (1/2)^n0" |
1508 |
using nn0 by (simp add: divide_simps) |
|
1509 |
also have "\<dots> < r" |
|
1510 |
using n0 by simp |
|
1511 |
finally show ?thesis . |
|
1512 |
qed |
|
60558 | 1513 |
qed |
60501 | 1514 |
then show ?thesis by blast |
1515 |
qed |
|
60558 | 1516 |
qed |
54681 | 1517 |
then show ?thesis |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1518 |
unfolding lim_sequentially by blast |
52891 | 1519 |
qed |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
1520 |
|
54681 | 1521 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1522 |
subsection \<open>Inverses and division of formal power series\<close> |
29687 | 1523 |
|
64267 | 1524 |
declare sum.cong[fundef_cong] |
29687 | 1525 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1526 |
fun fps_left_inverse_constructor :: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1527 |
"'a::{comm_monoid_add,times,uminus} fps \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1528 |
where |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1529 |
"fps_left_inverse_constructor f a 0 = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1530 |
| "fps_left_inverse_constructor f a (Suc n) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1531 |
- sum (\<lambda>i. fps_left_inverse_constructor f a i * f$(Suc n - i)) {0..n} * a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1532 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1533 |
\<comment> \<open>This will construct a left inverse for f in case that x * f$0 = 1\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1534 |
abbreviation "fps_left_inverse \<equiv> (\<lambda>f x. Abs_fps (fps_left_inverse_constructor f x))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1535 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1536 |
fun fps_right_inverse_constructor :: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1537 |
"'a::{comm_monoid_add,times,uminus} fps \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1538 |
where |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1539 |
"fps_right_inverse_constructor f a 0 = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1540 |
| "fps_right_inverse_constructor f a n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1541 |
- a * sum (\<lambda>i. f$i * fps_right_inverse_constructor f a (n - i)) {1..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1542 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1543 |
\<comment> \<open>This will construct a right inverse for f in case that f$0 * y = 1\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1544 |
abbreviation "fps_right_inverse \<equiv> (\<lambda>f y. Abs_fps (fps_right_inverse_constructor f y))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1545 |
|
60558 | 1546 |
instantiation fps :: ("{comm_monoid_add,inverse,times,uminus}") inverse |
29687 | 1547 |
begin |
1548 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1549 |
\<comment> \<open>For backwards compatibility.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1550 |
abbreviation natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1551 |
where "natfun_inverse f \<equiv> fps_right_inverse_constructor f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1552 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1553 |
definition fps_inverse_def: "inverse f = Abs_fps (natfun_inverse f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1554 |
\<comment> \<open> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1555 |
With scalars from a (possibly non-commutative) ring, this defines a right inverse. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1556 |
Furthermore, if scalars are of class @{class mult_zero} and satisfy |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1557 |
condition @{term "inverse 0 = 0"}, then this will evaluate to zero when |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1558 |
the zeroth term is zero. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1559 |
\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1560 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1561 |
definition fps_divide_def: "f div g = fps_shift (subdegree g) (f * inverse (unit_factor g))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1562 |
\<comment> \<open> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1563 |
If scalars are of class @{class mult_zero} and satisfy condition |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1564 |
@{term "inverse 0 = 0"}, then div by zero will equal zero. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1565 |
\<close> |
36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset
|
1566 |
|
29687 | 1567 |
instance .. |
36311
ed3a87a7f977
epheremal replacement of field_simps by field_eq_simps; dropped old division_by_zero instance
haftmann
parents:
36309
diff
changeset
|
1568 |
|
29687 | 1569 |
end |
1570 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1571 |
lemma fps_lr_inverse_0_iff: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1572 |
"(fps_left_inverse f x) $ 0 = 0 \<longleftrightarrow> x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1573 |
"(fps_right_inverse f x) $ 0 = 0 \<longleftrightarrow> x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1574 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1575 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1576 |
lemma fps_inverse_0_iff': "(inverse f) $ 0 = 0 \<longleftrightarrow> inverse (f $ 0) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1577 |
by (simp add: fps_inverse_def fps_lr_inverse_0_iff(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1578 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1579 |
lemma fps_inverse_0_iff[simp]: "(inverse f) $ 0 = (0::'a::division_ring) \<longleftrightarrow> f $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1580 |
by (simp add: fps_inverse_0_iff') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1581 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1582 |
lemma fps_lr_inverse_nth_0: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1583 |
"(fps_left_inverse f x) $ 0 = x" "(fps_right_inverse f x) $ 0 = x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1584 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1585 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1586 |
lemma fps_inverse_nth_0 [simp]: "(inverse f) $ 0 = inverse (f $ 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1587 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1588 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1589 |
lemma fps_lr_inverse_starting0: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1590 |
fixes f :: "'a::{comm_monoid_add,mult_zero,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1591 |
and g :: "'b::{ab_group_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1592 |
shows "fps_left_inverse f 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1593 |
and "fps_right_inverse g 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1594 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1595 |
show "fps_left_inverse f 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1596 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1597 |
fix n show "fps_left_inverse f 0 $ n = 0 $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1598 |
by (cases n) (simp_all add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1599 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1600 |
show "fps_right_inverse g 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1601 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1602 |
fix n show "fps_right_inverse g 0 $ n = 0 $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1603 |
by (cases n) (simp_all add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1604 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1605 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1606 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1607 |
lemma fps_lr_inverse_eq0_imp_starting0: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1608 |
"fps_left_inverse f x = 0 \<Longrightarrow> x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1609 |
"fps_right_inverse f x = 0 \<Longrightarrow> x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1610 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1611 |
assume A: "fps_left_inverse f x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1612 |
have "0 = fps_left_inverse f x $ 0" by (subst A) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1613 |
thus "x = 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1614 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1615 |
assume A: "fps_right_inverse f x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1616 |
have "0 = fps_right_inverse f x $ 0" by (subst A) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1617 |
thus "x = 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1618 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1619 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1620 |
lemma fps_lr_inverse_eq_0_iff: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1621 |
fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1622 |
and y :: "'b::{ab_group_add,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1623 |
shows "fps_left_inverse f x = 0 \<longleftrightarrow> x = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1624 |
and "fps_right_inverse g y = 0 \<longleftrightarrow> y = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1625 |
using fps_lr_inverse_starting0 fps_lr_inverse_eq0_imp_starting0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1626 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1627 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1628 |
lemma fps_inverse_eq_0_iff': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1629 |
fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1630 |
shows "inverse f = 0 \<longleftrightarrow> inverse (f $ 0) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1631 |
by (simp add: fps_inverse_def fps_lr_inverse_eq_0_iff(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1632 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1633 |
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::division_ring) fps) \<longleftrightarrow> f $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1634 |
using fps_inverse_eq_0_iff'[of f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1635 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1636 |
lemmas fps_inverse_eq_0' = iffD2[OF fps_inverse_eq_0_iff'] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1637 |
lemmas fps_inverse_eq_0 = iffD2[OF fps_inverse_eq_0_iff] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1638 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1639 |
lemma fps_const_lr_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1640 |
fixes a :: "'a::{ab_group_add,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1641 |
and b :: "'b::{comm_monoid_add,mult_zero,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1642 |
shows "fps_left_inverse (fps_const a) x = fps_const x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1643 |
and "fps_right_inverse (fps_const b) y = fps_const y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1644 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1645 |
show "fps_left_inverse (fps_const a) x = fps_const x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1646 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1647 |
fix n show "fps_left_inverse (fps_const a) x $ n = fps_const x $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1648 |
by (cases n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1649 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1650 |
show "fps_right_inverse (fps_const b) y = fps_const y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1651 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1652 |
fix n show "fps_right_inverse (fps_const b) y $ n = fps_const y $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1653 |
by (cases n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1654 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1655 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1656 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1657 |
lemma fps_const_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1658 |
fixes a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1659 |
shows "inverse (fps_const a) = fps_const (inverse a)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1660 |
unfolding fps_inverse_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1661 |
by (simp add: fps_const_lr_inverse(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1662 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1663 |
lemma fps_lr_inverse_zero: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1664 |
fixes x :: "'a::{ab_group_add,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1665 |
and y :: "'b::{comm_monoid_add,mult_zero,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1666 |
shows "fps_left_inverse 0 x = fps_const x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1667 |
and "fps_right_inverse 0 y = fps_const y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1668 |
using fps_const_lr_inverse[of 0] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1669 |
by simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1670 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1671 |
lemma fps_inverse_zero_conv_fps_const: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1672 |
"inverse (0::'a::{comm_monoid_add,mult_zero,uminus,inverse} fps) = fps_const (inverse 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1673 |
using fps_lr_inverse_zero(2)[of "inverse (0::'a)"] by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1674 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1675 |
lemma fps_inverse_zero': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1676 |
assumes "inverse (0::'a::{comm_monoid_add,inverse,mult_zero,uminus}) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1677 |
shows "inverse (0::'a fps) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1678 |
by (simp add: assms fps_inverse_zero_conv_fps_const) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1679 |
|
52891 | 1680 |
lemma fps_inverse_zero [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1681 |
"inverse (0::'a::division_ring fps) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1682 |
by (rule fps_inverse_zero'[OF inverse_zero]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1683 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1684 |
lemma fps_lr_inverse_one: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1685 |
fixes x :: "'a::{ab_group_add,mult_zero,one}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1686 |
and y :: "'b::{comm_monoid_add,mult_zero,uminus,one}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1687 |
shows "fps_left_inverse 1 x = fps_const x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1688 |
and "fps_right_inverse 1 y = fps_const y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1689 |
using fps_const_lr_inverse[of 1] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1690 |
by simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1691 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1692 |
lemma fps_lr_inverse_one_one: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1693 |
"fps_left_inverse 1 1 = (1::'a::{ab_group_add,mult_zero,one} fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1694 |
"fps_right_inverse 1 1 = (1::'b::{comm_monoid_add,mult_zero,uminus,one} fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1695 |
by (simp_all add: fps_lr_inverse_one) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1696 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1697 |
lemma fps_inverse_one': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1698 |
assumes "inverse (1::'a::{comm_monoid_add,inverse,mult_zero,uminus,one}) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1699 |
shows "inverse (1 :: 'a fps) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1700 |
using assms fps_lr_inverse_one_one(2) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1701 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1702 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1703 |
lemma fps_inverse_one [simp]: "inverse (1 :: 'a :: division_ring fps) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1704 |
by (rule fps_inverse_one'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1705 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1706 |
lemma fps_lr_inverse_minus: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1707 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1708 |
shows "fps_left_inverse (-f) (-x) = - fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1709 |
and "fps_right_inverse (-f) (-x) = - fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1710 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1711 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1712 |
show "fps_left_inverse (-f) (-x) = - fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1713 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1714 |
fix n show "fps_left_inverse (-f) (-x) $ n = - fps_left_inverse f x $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1715 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1716 |
case (1 n) thus ?case by (cases n) (simp_all add: sum_negf algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1717 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1718 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1719 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1720 |
show "fps_right_inverse (-f) (-x) = - fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1721 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1722 |
fix n show "fps_right_inverse (-f) (-x) $ n = - fps_right_inverse f x $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1723 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1724 |
case (1 n) show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1725 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1726 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1727 |
with 1 have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1728 |
"\<forall>i\<in>{1..Suc m}. fps_right_inverse (-f) (-x) $ (Suc m - i) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1729 |
- fps_right_inverse f x $ (Suc m - i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1730 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1731 |
with Suc show ?thesis by (simp add: sum_negf algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1732 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1733 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1734 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1735 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1736 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1737 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1738 |
lemma fps_inverse_minus [simp]: "inverse (-f) = -inverse (f :: 'a :: division_ring fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1739 |
by (simp add: fps_inverse_def fps_lr_inverse_minus(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1740 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1741 |
lemma fps_left_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1742 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1743 |
assumes f0: "x * f$0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1744 |
shows "fps_left_inverse f x * f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1745 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1746 |
fix n show "(fps_left_inverse f x * f) $ n = 1 $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1747 |
by (cases n) (simp_all add: f0 fps_mult_nth mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1748 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1749 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1750 |
lemma fps_right_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1751 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1752 |
assumes f0: "f$0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1753 |
shows "f * fps_right_inverse f y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1754 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1755 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1756 |
show "(f * fps_right_inverse f y) $ n = 1 $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1757 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1758 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1759 |
moreover from Suc have "fps_right_inverse f y $ n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1760 |
- y * sum (\<lambda>i. f$i * fps_right_inverse_constructor f y (n - i)) {1..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1761 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1762 |
hence |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1763 |
"(f * fps_right_inverse f y) $ n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1764 |
- 1 * sum (\<lambda>i. f$i * fps_right_inverse_constructor f y (n - i)) {1..n} + |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1765 |
sum (\<lambda>i. f$i * (fps_right_inverse_constructor f y (n - i))) {1..n}" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
1766 |
by (simp add: fps_mult_nth sum.atLeast_Suc_atMost mult.assoc f0[symmetric]) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1767 |
thus "(f * fps_right_inverse f y) $ n = 1 $ n" by (simp add: Suc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1768 |
qed (simp add: f0 fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1769 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1770 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1771 |
\<comment> \<open> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1772 |
It is possible in a ring for an element to have a left inverse but not a right inverse, or |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1773 |
vice versa. But when an element has both, they must be the same. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1774 |
\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1775 |
lemma fps_left_inverse_eq_fps_right_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1776 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1777 |
assumes f0: "x * f$0 = 1" "f $ 0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1778 |
\<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1779 |
shows "fps_left_inverse f x = fps_right_inverse f y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1780 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1781 |
from f0(2) have "f * fps_right_inverse f y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1782 |
by (simp add: fps_right_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1783 |
hence "fps_left_inverse f x * f * fps_right_inverse f y = fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1784 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1785 |
moreover from f0(1) have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1786 |
"fps_left_inverse f x * f * fps_right_inverse f y = fps_right_inverse f y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1787 |
by (simp add: fps_left_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1788 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1789 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1790 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1791 |
lemma fps_left_inverse_eq_fps_right_inverse_comm: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1792 |
fixes f :: "'a::comm_ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1793 |
assumes f0: "x * f$0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1794 |
shows "fps_left_inverse f x = fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1795 |
using assms fps_left_inverse_eq_fps_right_inverse[of x f x] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1796 |
by (simp add: mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1797 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1798 |
lemma fps_left_inverse': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1799 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1800 |
assumes "x * f$0 = 1" "f$0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1801 |
\<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1802 |
shows "fps_right_inverse f y * f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1803 |
using assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_left_inverse[of x f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1804 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1805 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1806 |
lemma fps_right_inverse': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1807 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1808 |
assumes "x * f$0 = 1" "f$0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1809 |
\<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1810 |
shows "f * fps_left_inverse f x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1811 |
using assms fps_left_inverse_eq_fps_right_inverse[of x f y] fps_right_inverse[of f y] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1812 |
by simp |
52891 | 1813 |
|
1814 |
lemma inverse_mult_eq_1 [intro]: |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1815 |
assumes "f$0 \<noteq> (0::'a::division_ring)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1816 |
shows "inverse f * f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1817 |
using fps_left_inverse'[of "inverse (f$0)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1818 |
by (simp add: assms fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1819 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1820 |
lemma inverse_mult_eq_1': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1821 |
assumes "f$0 \<noteq> (0::'a::division_ring)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1822 |
shows "f * inverse f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1823 |
using assms fps_right_inverse |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1824 |
by (force simp: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1825 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1826 |
lemma fps_mult_left_inverse_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1827 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1828 |
assumes "x * f $ subdegree f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1829 |
shows "fps_left_inverse (unit_factor f) x * f = fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1830 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1831 |
have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1832 |
"fps_left_inverse (unit_factor f) x * f = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1833 |
fps_left_inverse (unit_factor f) x * unit_factor f * fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1834 |
using fps_unit_factor_decompose[of f] by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1835 |
with assms show ?thesis by (simp add: fps_left_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1836 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1837 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1838 |
lemma fps_mult_right_inverse_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1839 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1840 |
assumes "f $ subdegree f * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1841 |
shows "f * fps_right_inverse (unit_factor f) y = fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1842 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1843 |
have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1844 |
"f * fps_right_inverse (unit_factor f) y = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1845 |
fps_X ^ subdegree f * (unit_factor f * fps_right_inverse (unit_factor f) y)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1846 |
using fps_unit_factor_decompose'[of f] by (simp add: mult.assoc[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1847 |
with assms show ?thesis by (simp add: fps_right_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1848 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1849 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1850 |
lemma fps_mult_right_inverse_unit_factor_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1851 |
"(f :: 'a::division_ring fps) \<noteq> 0 \<Longrightarrow> f * inverse (unit_factor f) = fps_X ^ subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1852 |
using fps_mult_right_inverse_unit_factor[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1853 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1854 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1855 |
lemma fps_left_inverse_idempotent_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1856 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1857 |
assumes "x * f$0 = 1" "y * x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1858 |
\<comment> \<open>These assumptions imply y equals f$0, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1859 |
shows "fps_left_inverse (fps_left_inverse f x) y = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1860 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1861 |
from assms(1) have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1862 |
"fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x * f = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1863 |
fps_left_inverse (fps_left_inverse f x) y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1864 |
by (simp add: fps_left_inverse mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1865 |
moreover from assms(2) have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1866 |
"fps_left_inverse (fps_left_inverse f x) y * fps_left_inverse f x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1867 |
by (simp add: fps_left_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1868 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1869 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1870 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1871 |
lemma fps_left_inverse_idempotent_comm_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1872 |
fixes f :: "'a::comm_ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1873 |
assumes "x * f$0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1874 |
shows "fps_left_inverse (fps_left_inverse f x) (f$0) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1875 |
using assms fps_left_inverse_idempotent_ring1[of x f "f$0"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1876 |
by (simp add: mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1877 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1878 |
lemma fps_right_inverse_idempotent_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1879 |
fixes f :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1880 |
assumes "f$0 * x = 1" "x * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1881 |
\<comment> \<open>These assumptions imply y equals f$0, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1882 |
shows "fps_right_inverse (fps_right_inverse f x) y = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1883 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1884 |
from assms(1) have "f * (fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1885 |
fps_right_inverse (fps_right_inverse f x) y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1886 |
by (simp add: fps_right_inverse mult.assoc[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1887 |
moreover from assms(2) have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1888 |
"fps_right_inverse f x * fps_right_inverse (fps_right_inverse f x) y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1889 |
by (simp add: fps_right_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1890 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1891 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1892 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1893 |
lemma fps_right_inverse_idempotent_comm_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1894 |
fixes f :: "'a::comm_ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1895 |
assumes "f$0 * x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1896 |
shows "fps_right_inverse (fps_right_inverse f x) (f$0) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1897 |
using assms fps_right_inverse_idempotent_ring1[of f x "f$0"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1898 |
by (simp add: mult.commute) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1899 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1900 |
lemma fps_inverse_idempotent[intro, simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1901 |
"f$0 \<noteq> (0::'a::division_ring) \<Longrightarrow> inverse (inverse f) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1902 |
using fps_right_inverse_idempotent_ring1[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1903 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1904 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1905 |
lemma fps_lr_inverse_unique_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1906 |
fixes f g :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1907 |
assumes fg: "f * g = 1" "g$0 * f$0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1908 |
shows "fps_left_inverse g (f$0) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1909 |
and "fps_right_inverse f (g$0) = g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1910 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1911 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1912 |
show "fps_left_inverse g (f$0) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1913 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1914 |
fix n show "fps_left_inverse g (f$0) $ n = f $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1915 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1916 |
case (1 n) show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1917 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1918 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1919 |
hence "\<forall>i\<in>{0..k}. fps_left_inverse g (f$0) $ i = f $ i" using 1 by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1920 |
hence "fps_left_inverse g (f$0) $ Suc k = f $ Suc k - 1 $ Suc k * f$0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1921 |
by (simp add: fps_mult_nth fg(1)[symmetric] distrib_right mult.assoc fg(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1922 |
with Suc show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1923 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1924 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1925 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1926 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1927 |
show "fps_right_inverse f (g$0) = g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1928 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1929 |
fix n show "fps_right_inverse f (g$0) $ n = g $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1930 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1931 |
case (1 n) show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1932 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1933 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1934 |
hence "\<forall>i\<in>{1..Suc k}. fps_right_inverse f (g$0) $ (Suc k - i) = g $ (Suc k - i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1935 |
using 1 by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1936 |
hence |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1937 |
"fps_right_inverse f (g$0) $ Suc k = 1 * g $ Suc k - g$0 * 1 $ Suc k" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
1938 |
by (simp add: fps_mult_nth fg(1)[symmetric] algebra_simps fg(2)[symmetric] sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1939 |
with Suc show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1940 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1941 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1942 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1943 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1944 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1945 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1946 |
lemma fps_lr_inverse_unique_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1947 |
fixes f g :: "'a ::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1948 |
assumes fg: "f * g = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1949 |
shows "fps_left_inverse g (f$0) = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1950 |
and "fps_right_inverse f (g$0) = g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1951 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1952 |
from fg have "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1953 |
hence "g$0 * f$0 = 1" using inverse_unique[of "f$0"] left_inverse[of "f$0"] by force |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1954 |
thus "fps_left_inverse g (f$0) = f" "fps_right_inverse f (g$0) = g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1955 |
using fg fps_lr_inverse_unique_ring1 by auto |
29687 | 1956 |
qed |
1957 |
||
48757 | 1958 |
lemma fps_inverse_unique: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1959 |
fixes f g :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1960 |
assumes fg: "f * g = 1" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
1961 |
shows "inverse f = g" |
52891 | 1962 |
proof - |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1963 |
from fg have if0: "inverse (f$0) = g$0" "f$0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1964 |
using inverse_unique[of "f$0"] fps_mult_nth_0[of f g] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1965 |
with fg have "fps_right_inverse f (g$0) = g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1966 |
using left_inverse[of "f$0"] by (intro fps_lr_inverse_unique_ring1(2)) simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1967 |
with if0(1) show ?thesis by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1968 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1969 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1970 |
lemma inverse_fps_numeral: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1971 |
"inverse (numeral n :: ('a :: field_char_0) fps) = fps_const (inverse (numeral n))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1972 |
by (intro fps_inverse_unique fps_ext) (simp_all add: fps_numeral_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1973 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1974 |
lemma inverse_fps_of_nat: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1975 |
"inverse (of_nat n :: 'a :: {semiring_1,times,uminus,inverse} fps) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1976 |
fps_const (inverse (of_nat n))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1977 |
by (simp add: fps_of_nat fps_const_inverse[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
1978 |
|
64267 | 1979 |
lemma sum_zero_lemma: |
60162 | 1980 |
fixes n::nat |
1981 |
assumes "0 < n" |
|
1982 |
shows "(\<Sum>i = 0..n. if n = i then 1 else if n - i = 1 then - 1 else 0) = (0::'a::field)" |
|
54681 | 1983 |
proof - |
60162 | 1984 |
let ?f = "\<lambda>i. if n = i then 1 else if n - i = 1 then - 1 else 0" |
1985 |
let ?g = "\<lambda>i. if i = n then 1 else if i = n - 1 then - 1 else 0" |
|
29687 | 1986 |
let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0" |
64267 | 1987 |
have th1: "sum ?f {0..n} = sum ?g {0..n}" |
1988 |
by (rule sum.cong) auto |
|
1989 |
have th2: "sum ?g {0..n - 1} = sum ?h {0..n - 1}" |
|
1990 |
apply (rule sum.cong) |
|
60162 | 1991 |
using assms |
54681 | 1992 |
apply auto |
1993 |
done |
|
1994 |
have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" |
|
1995 |
by auto |
|
60162 | 1996 |
from assms have d: "{0.. n - 1} \<inter> {n} = {}" |
54681 | 1997 |
by auto |
1998 |
have f: "finite {0.. n - 1}" "finite {n}" |
|
1999 |
by auto |
|
60162 | 2000 |
show ?thesis |
30488 | 2001 |
unfolding th1 |
64267 | 2002 |
apply (simp add: sum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def) |
29687 | 2003 |
unfolding th2 |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2004 |
apply simp |
52891 | 2005 |
done |
29687 | 2006 |
qed |
2007 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2008 |
lemma fps_lr_inverse_mult_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2009 |
fixes f g :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2010 |
assumes x: "x * f$0 = 1" "f$0 * x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2011 |
and y: "y * g$0 = 1" "g$0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2012 |
shows "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2013 |
and "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2014 |
proof - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2015 |
define h where "h \<equiv> fps_left_inverse g y * fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2016 |
hence h0: "h$0 = y*x" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2017 |
have "fps_left_inverse (f*g) (h$0) = h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2018 |
proof (intro fps_lr_inverse_unique_ring1(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2019 |
from h_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2020 |
have "h * (f * g) = fps_left_inverse g y * (fps_left_inverse f x * f) * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2021 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2022 |
thus "h * (f * g) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2023 |
using fps_left_inverse[OF x(1)] fps_left_inverse[OF y(1)] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2024 |
from h_def have "(f*g)$0 * h$0 = f$0 * 1 * x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2025 |
by (simp add: mult.assoc y(2)[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2026 |
with x(2) show "(f * g) $ 0 * h $ 0 = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2027 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2028 |
with h_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2029 |
show "fps_left_inverse (f * g) (y*x) = fps_left_inverse g y * fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2030 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2031 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2032 |
define h where "h \<equiv> fps_right_inverse g y * fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2033 |
hence h0: "h$0 = y*x" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2034 |
have "fps_right_inverse (f*g) (h$0) = h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2035 |
proof (intro fps_lr_inverse_unique_ring1(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2036 |
from h_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2037 |
have "f * g * h = f * (g * fps_right_inverse g y) * fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2038 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2039 |
thus "f * g * h = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2040 |
using fps_right_inverse[OF x(2)] fps_right_inverse[OF y(2)] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2041 |
from h_def have "h$0 * (f*g)$0 = y * 1 * g$0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2042 |
by (simp add: mult.assoc x(1)[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2043 |
with y(1) show "h$0 * (f*g)$0 = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2044 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2045 |
with h_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2046 |
show "fps_right_inverse (f * g) (y*x) = fps_right_inverse g y * fps_right_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2047 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2048 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2049 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2050 |
lemma fps_lr_inverse_mult_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2051 |
fixes f g :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2052 |
shows "fps_left_inverse (f * g) (inverse ((f*g)$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2053 |
fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2054 |
and "fps_right_inverse (f * g) (inverse ((f*g)$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2055 |
fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2056 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2057 |
show "fps_left_inverse (f * g) (inverse ((f*g)$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2058 |
fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2059 |
proof (cases "f$0 = 0 \<or> g$0 = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2060 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2061 |
hence "fps_left_inverse (f * g) (inverse ((f*g)$0)) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2062 |
by (simp add: fps_lr_inverse_eq_0_iff(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2063 |
moreover from True have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2064 |
"fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0)) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2065 |
by (auto simp: fps_lr_inverse_eq_0_iff(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2066 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2067 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2068 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2069 |
hence "fps_left_inverse (f * g) (inverse (g$0) * inverse (f$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2070 |
fps_left_inverse g (inverse (g$0)) * fps_left_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2071 |
by (intro fps_lr_inverse_mult_ring1(1)) simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2072 |
with False show ?thesis by (simp add: nonzero_inverse_mult_distrib) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2073 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2074 |
show "fps_right_inverse (f * g) (inverse ((f*g)$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2075 |
fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2076 |
proof (cases "f$0 = 0 \<or> g$0 = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2077 |
case True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2078 |
from True have "fps_right_inverse (f * g) (inverse ((f*g)$0)) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2079 |
by (simp add: fps_lr_inverse_eq_0_iff(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2080 |
moreover from True have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2081 |
"fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0)) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2082 |
by (auto simp: fps_lr_inverse_eq_0_iff(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2083 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2084 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2085 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2086 |
hence "fps_right_inverse (f * g) (inverse (g$0) * inverse (f$0)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2087 |
fps_right_inverse g (inverse (g$0)) * fps_right_inverse f (inverse (f$0))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2088 |
by (intro fps_lr_inverse_mult_ring1(2)) simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2089 |
with False show ?thesis by (simp add: nonzero_inverse_mult_distrib) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2090 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2091 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2092 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2093 |
lemma fps_inverse_mult_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2094 |
"inverse (f * g) = inverse g * inverse (f :: 'a::division_ring fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2095 |
using fps_lr_inverse_mult_divring(2) by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2096 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2097 |
lemma fps_inverse_mult: "inverse (f * g :: 'a::field fps) = inverse f * inverse g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2098 |
by (simp add: fps_inverse_mult_divring) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2099 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2100 |
lemma fps_lr_inverse_gp_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2101 |
fixes ones ones_inv :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2102 |
defines "ones \<equiv> Abs_fps (\<lambda>n. 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2103 |
and "ones_inv \<equiv> Abs_fps (\<lambda>n. if n=0 then 1 else if n=1 then - 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2104 |
shows "fps_left_inverse ones 1 = ones_inv" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2105 |
and "fps_right_inverse ones 1 = ones_inv" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2106 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2107 |
show "fps_left_inverse ones 1 = ones_inv" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2108 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2109 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2110 |
show "fps_left_inverse ones 1 $ n = ones_inv $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2111 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2112 |
case (1 n) show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2113 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2114 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2115 |
have m: "n = Suc m" by fact |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2116 |
moreover have "fps_left_inverse ones 1 $ Suc m = ones_inv $ Suc m" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2117 |
proof (cases m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2118 |
case (Suc k) thus ?thesis |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
2119 |
using Suc m 1 by (simp add: ones_def ones_inv_def sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2120 |
qed (simp add: ones_def ones_inv_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2121 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2122 |
qed (simp add: ones_inv_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2123 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2124 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2125 |
moreover have "fps_right_inverse ones 1 = fps_left_inverse ones 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2126 |
by (auto intro: fps_left_inverse_eq_fps_right_inverse[symmetric] simp: ones_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2127 |
ultimately show "fps_right_inverse ones 1 = ones_inv" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2128 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2129 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2130 |
lemma fps_lr_inverse_gp_ring1': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2131 |
fixes ones :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2132 |
defines "ones \<equiv> Abs_fps (\<lambda>n. 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2133 |
shows "fps_left_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2134 |
and "fps_right_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2135 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2136 |
define ones_inv :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2137 |
where "ones_inv \<equiv> Abs_fps (\<lambda>n. if n=0 then 1 else if n=1 then - 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2138 |
hence "fps_left_inverse ones 1 = ones_inv" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2139 |
and "fps_right_inverse ones 1 = ones_inv" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2140 |
using ones_def fps_lr_inverse_gp_ring1 by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2141 |
thus "fps_left_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2142 |
and "fps_right_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2143 |
by (auto intro: fps_ext simp: ones_inv_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2144 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2145 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2146 |
lemma fps_inverse_gp: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2147 |
"inverse (Abs_fps(\<lambda>n. (1::'a::division_ring))) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2148 |
Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2149 |
using fps_lr_inverse_gp_ring1(2) by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2150 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2151 |
lemma fps_inverse_gp': "inverse (Abs_fps (\<lambda>n. 1::'a::division_ring)) = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2152 |
by (simp add: fps_inverse_def fps_lr_inverse_gp_ring1'(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2153 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2154 |
lemma fps_lr_inverse_one_minus_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2155 |
fixes ones :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2156 |
defines "ones \<equiv> Abs_fps (\<lambda>n. 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2157 |
shows "fps_left_inverse (1 - fps_X) 1 = ones" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2158 |
and "fps_right_inverse (1 - fps_X) 1 = ones" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2159 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2160 |
have "fps_left_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2161 |
using fps_lr_inverse_gp_ring1'(1) by (simp add: ones_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2162 |
thus "fps_left_inverse (1 - fps_X) 1 = ones" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2163 |
using fps_left_inverse_idempotent_ring1[of 1 ones 1] by (simp add: ones_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2164 |
have "fps_right_inverse ones 1 = 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2165 |
using fps_lr_inverse_gp_ring1'(2) by (simp add: ones_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2166 |
thus "fps_right_inverse (1 - fps_X) 1 = ones" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2167 |
using fps_right_inverse_idempotent_ring1[of ones 1 1] by (simp add: ones_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2168 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2169 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2170 |
lemma fps_inverse_one_minus_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2171 |
fixes ones :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2172 |
defines "ones \<equiv> Abs_fps (\<lambda>n. 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2173 |
shows "inverse (1 - fps_X) = ones" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2174 |
by (simp add: fps_inverse_def assms fps_lr_inverse_one_minus_fps_X(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2175 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2176 |
lemma fps_lr_one_over_one_minus_fps_X_squared: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2177 |
shows "fps_left_inverse ((1 - fps_X)^2) (1::'a::ring_1) = Abs_fps (\<lambda>n. of_nat (n+1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2178 |
"fps_right_inverse ((1 - fps_X)^2) (1::'a) = Abs_fps (\<lambda>n. of_nat (n+1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2179 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2180 |
define f invf2 :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2181 |
where "f \<equiv> (1 - fps_X)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2182 |
and "invf2 \<equiv> Abs_fps (\<lambda>n. of_nat (n+1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2183 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2184 |
have f2_nth_simps: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2185 |
"f^2 $ 1 = - of_nat 2" "f^2 $ 2 = 1" "\<And>n. n>2 \<Longrightarrow> f^2 $ n = 0" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
2186 |
by (simp_all add: power2_eq_square f_def fps_mult_nth sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2187 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2188 |
show "fps_left_inverse (f^2) 1 = invf2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2189 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2190 |
fix n show "fps_left_inverse (f^2) 1 $ n = invf2 $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2191 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2192 |
case (1 t) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2193 |
hence induct_assm: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2194 |
"\<And>m. m < t \<Longrightarrow> fps_left_inverse (f\<^sup>2) 1 $ m = invf2 $ m" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2195 |
by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2196 |
show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2197 |
proof (cases t) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2198 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2199 |
have m: "t = Suc m" by fact |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2200 |
moreover have "fps_left_inverse (f^2) 1 $ Suc m = invf2 $ Suc m" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2201 |
proof (cases m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2202 |
case 0 thus ?thesis using f2_nth_simps(1) by (simp add: invf2_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2203 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2204 |
case (Suc l) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2205 |
have l: "m = Suc l" by fact |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2206 |
moreover have "fps_left_inverse (f^2) 1 $ Suc (Suc l) = invf2 $ Suc (Suc l)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2207 |
proof (cases l) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2208 |
case 0 thus ?thesis using f2_nth_simps(1,2) by (simp add: Suc_1[symmetric] invf2_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2209 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2210 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2211 |
from Suc l m |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2212 |
have A: "fps_left_inverse (f\<^sup>2) 1 $ Suc (Suc k) = invf2 $ Suc (Suc k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2213 |
and B: "fps_left_inverse (f\<^sup>2) 1 $ Suc k = invf2 $ Suc k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2214 |
using induct_assm[of "Suc k"] induct_assm[of "Suc (Suc k)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2215 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2216 |
have times2: "\<And>a::nat. 2*a = a + a" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2217 |
have "\<forall>i\<in>{0..k}. (f^2)$(Suc (Suc (Suc k)) - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2218 |
using f2_nth_simps(3) by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2219 |
hence |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2220 |
"fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2221 |
fps_left_inverse (f\<^sup>2) 1 $ Suc (Suc k) * of_nat 2 - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2222 |
fps_left_inverse (f\<^sup>2) 1 $ Suc k" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
2223 |
using sum.ub_add_nat f2_nth_simps(1,2) by simp |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2224 |
also have "\<dots> = of_nat (2 * Suc (Suc (Suc k))) - of_nat (Suc (Suc k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2225 |
by (subst A, subst B) (simp add: invf2_def mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2226 |
also have "\<dots> = of_nat (Suc (Suc (Suc k)) + 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2227 |
by (subst times2[of "Suc (Suc (Suc k))"]) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2228 |
finally have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2229 |
"fps_left_inverse (f^2) 1 $ Suc (Suc (Suc k)) = invf2 $ Suc (Suc (Suc k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2230 |
by (simp add: invf2_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2231 |
with Suc show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2232 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2233 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2234 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2235 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2236 |
qed (simp add: invf2_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2237 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2238 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2239 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2240 |
moreover have "fps_right_inverse (f^2) 1 = fps_left_inverse (f^2) 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2241 |
by (auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2242 |
intro: fps_left_inverse_eq_fps_right_inverse[symmetric] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2243 |
simp: f_def power2_eq_square |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2244 |
) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2245 |
ultimately show "fps_right_inverse (f^2) 1 = invf2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2246 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2247 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2248 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2249 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2250 |
lemma fps_one_over_one_minus_fps_X_squared': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2251 |
assumes "inverse (1::'a::{ring_1,inverse}) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2252 |
shows "inverse ((1 - fps_X)^2 :: 'a fps) = Abs_fps (\<lambda>n. of_nat (n+1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2253 |
using assms fps_lr_one_over_one_minus_fps_X_squared(2) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2254 |
by (simp add: fps_inverse_def power2_eq_square) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2255 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2256 |
lemma fps_one_over_one_minus_fps_X_squared: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2257 |
"inverse ((1 - fps_X)^2 :: 'a :: division_ring fps) = Abs_fps (\<lambda>n. of_nat (n+1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2258 |
by (rule fps_one_over_one_minus_fps_X_squared'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2259 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2260 |
lemma fps_lr_inverse_fps_X_plus1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2261 |
"fps_left_inverse (1 + fps_X) (1::'a::ring_1) = Abs_fps (\<lambda>n. (-1)^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2262 |
"fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2263 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2264 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2265 |
show "fps_left_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2266 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2267 |
fix n show "fps_left_inverse (1 + fps_X) (1::'a) $ n = Abs_fps (\<lambda>n. (-1)^n) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2268 |
proof (induct n rule: nat_less_induct) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2269 |
case (1 n) show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2270 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2271 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2272 |
have m: "n = Suc m" by fact |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2273 |
from Suc 1 have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2274 |
A: "fps_left_inverse (1 + fps_X) (1::'a) $ n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2275 |
- (\<Sum>i=0..m. (- 1)^i * (1 + fps_X) $ (Suc m - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2276 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2277 |
show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2278 |
proof (cases m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2279 |
case (Suc l) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2280 |
have "\<forall>i\<in>{0..l}. ((1::'a fps) + fps_X) $ (Suc (Suc l) - i) = 0" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2281 |
with Suc A m show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2282 |
qed (simp add: m A) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2283 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2284 |
qed |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2285 |
qed |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2286 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2287 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2288 |
"fps_right_inverse (1 + fps_X) (1::'a) = fps_left_inverse (1 + fps_X) 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2289 |
by (intro fps_left_inverse_eq_fps_right_inverse[symmetric]) simp_all |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2290 |
ultimately show "fps_right_inverse (1 + fps_X) (1::'a) = Abs_fps (\<lambda>n. (-1)^n)" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2291 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2292 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2293 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2294 |
lemma fps_inverse_fps_X_plus1': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2295 |
assumes "inverse (1::'a::{ring_1,inverse}) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2296 |
shows "inverse (1 + fps_X) = Abs_fps (\<lambda>n. (- (1::'a)) ^ n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2297 |
using assms fps_lr_inverse_fps_X_plus1(2) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2298 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2299 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2300 |
lemma fps_inverse_fps_X_plus1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2301 |
"inverse (1 + fps_X) = Abs_fps (\<lambda>n. (- (1::'a::division_ring)) ^ n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2302 |
by (rule fps_inverse_fps_X_plus1'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2303 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2304 |
lemma subdegree_lr_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2305 |
fixes x :: "'a::{comm_monoid_add,mult_zero,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2306 |
and y :: "'b::{ab_group_add,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2307 |
shows "subdegree (fps_left_inverse f x) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2308 |
and "subdegree (fps_right_inverse g y) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2309 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2310 |
show "subdegree (fps_left_inverse f x) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2311 |
using fps_lr_inverse_eq_0_iff(1) subdegree_eq_0_iff by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2312 |
show "subdegree (fps_right_inverse g y) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2313 |
using fps_lr_inverse_eq_0_iff(2) subdegree_eq_0_iff by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2314 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2315 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2316 |
lemma subdegree_inverse [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2317 |
fixes f :: "'a::{ab_group_add,inverse,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2318 |
shows "subdegree (inverse f) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2319 |
using subdegree_lr_inverse(2) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2320 |
by (simp add: fps_inverse_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2321 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2322 |
lemma fps_div_zero [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2323 |
"0 div (g :: 'a :: {comm_monoid_add,inverse,mult_zero,uminus} fps) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2324 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2325 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2326 |
lemma fps_div_by_zero': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2327 |
fixes g :: "'a::{comm_monoid_add,inverse,mult_zero,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2328 |
assumes "inverse (0::'a) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2329 |
shows "g div 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2330 |
by (simp add: fps_divide_def assms fps_inverse_zero') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2331 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2332 |
lemma fps_div_by_zero [simp]: "(g::'a::division_ring fps) div 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2333 |
by (rule fps_div_by_zero'[OF inverse_zero]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2334 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2335 |
lemma fps_divide_unit': "subdegree g = 0 \<Longrightarrow> f div g = f * inverse g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2336 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2337 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2338 |
lemma fps_divide_unit: "g$0 \<noteq> 0 \<Longrightarrow> f div g = f * inverse g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2339 |
by (intro fps_divide_unit') (simp add: subdegree_eq_0_iff) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2340 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2341 |
lemma fps_divide_nth_0': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2342 |
"subdegree (g::'a::division_ring fps) = 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2343 |
by (simp add: fps_divide_unit' divide_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2344 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2345 |
lemma fps_divide_nth_0 [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2346 |
"g $ 0 \<noteq> 0 \<Longrightarrow> (f div g) $ 0 = f $ 0 / (g $ 0 :: _ :: division_ring)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2347 |
by (simp add: fps_divide_nth_0') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2348 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2349 |
lemma fps_divide_nth_below: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2350 |
fixes f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2351 |
shows "n < subdegree f - subdegree g \<Longrightarrow> (f div g) $ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2352 |
by (simp add: fps_divide_def fps_mult_nth_eq0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2353 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2354 |
lemma fps_divide_nth_base: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2355 |
fixes f g :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2356 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2357 |
shows "(f div g) $ (subdegree f - subdegree g) = f $ subdegree f * inverse (g $ subdegree g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2358 |
by (simp add: assms fps_divide_def fps_divide_unit') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2359 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2360 |
lemma fps_divide_subdegree_ge: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2361 |
fixes f g :: "'a::{comm_monoid_add,uminus,mult_zero,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2362 |
assumes "f / g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2363 |
shows "subdegree (f / g) \<ge> subdegree f - subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2364 |
by (intro subdegree_geI) (simp_all add: assms fps_divide_nth_below) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2365 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2366 |
lemma fps_divide_subdegree: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2367 |
fixes f g :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2368 |
assumes "f \<noteq> 0" "g \<noteq> 0" "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2369 |
shows "subdegree (f / g) = subdegree f - subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2370 |
proof (intro antisym) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2371 |
from assms have 1: "(f div g) $ (subdegree f - subdegree g) \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2372 |
using fps_divide_nth_base[of g f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2373 |
thus "subdegree (f / g) \<le> subdegree f - subdegree g" by (intro subdegree_leI) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2374 |
from 1 have "f / g \<noteq> 0" by (auto intro: fps_nonzeroI) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2375 |
thus "subdegree f - subdegree g \<le> subdegree (f / g)" by (rule fps_divide_subdegree_ge) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2376 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2377 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2378 |
lemma fps_divide_shift_numer: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2379 |
fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2380 |
assumes "n \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2381 |
shows "fps_shift n f / g = fps_shift n (f/g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2382 |
using assms fps_shift_mult_right_noncomm[of n f "inverse (unit_factor g)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2383 |
fps_shift_fps_shift_reorder[of "subdegree g" n "f * inverse (unit_factor g)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2384 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2385 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2386 |
lemma fps_divide_shift_denom: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2387 |
fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2388 |
assumes "n \<le> subdegree g" "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2389 |
shows "f / fps_shift n g = Abs_fps (\<lambda>k. if k<n then 0 else (f/g) $ (k-n))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2390 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2391 |
fix k |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2392 |
from assms(1) have LHS: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2393 |
"(f / fps_shift n g) $ k = (f * inverse (unit_factor g)) $ (k + (subdegree g - n))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2394 |
using fps_unit_factor_shift[of n g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2395 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2396 |
show "(f / fps_shift n g) $ k = Abs_fps (\<lambda>k. if k<n then 0 else (f/g) $ (k-n)) $ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2397 |
proof (cases "k<n") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2398 |
case True with assms LHS show ?thesis using fps_mult_nth_eq0[of _ f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2399 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2400 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2401 |
hence "(f/g) $ (k-n) = (f * inverse (unit_factor g)) $ ((k-n) + subdegree g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2402 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2403 |
with False LHS assms(1) show ?thesis by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2404 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2405 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2406 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2407 |
lemma fps_divide_unit_factor_numer: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2408 |
fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2409 |
shows "unit_factor f / g = fps_shift (subdegree f) (f/g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2410 |
by (simp add: fps_divide_shift_numer) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2411 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2412 |
lemma fps_divide_unit_factor_denom: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2413 |
fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2414 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2415 |
shows |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2416 |
"f / unit_factor g = Abs_fps (\<lambda>k. if k<subdegree g then 0 else (f/g) $ (k-subdegree g))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2417 |
by (simp add: assms fps_divide_shift_denom) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2418 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2419 |
lemma fps_divide_unit_factor_both': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2420 |
fixes f g :: "'a::{inverse,comm_monoid_add,uminus,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2421 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2422 |
shows "unit_factor f / unit_factor g = fps_shift (subdegree f - subdegree g) (f / g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2423 |
using assms fps_divide_unit_factor_numer[of f "unit_factor g"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2424 |
fps_divide_unit_factor_denom[of g f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2425 |
fps_shift_rev_shift(1)[of "subdegree g" "subdegree f" "f/g"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2426 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2427 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2428 |
lemma fps_divide_unit_factor_both: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2429 |
fixes f g :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2430 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2431 |
shows "unit_factor f / unit_factor g = unit_factor (f / g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2432 |
using assms fps_divide_unit_factor_both'[of g f] fps_divide_subdegree[of f g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2433 |
by (cases "f=0 \<or> g=0") auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2434 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2435 |
lemma fps_divide_self: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2436 |
"(f::'a::division_ring fps) \<noteq> 0 \<Longrightarrow> f / f = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2437 |
using fps_mult_right_inverse_unit_factor_divring[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2438 |
by (simp add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2439 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2440 |
lemma fps_divide_add: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2441 |
fixes f g h :: "'a::{semiring_0,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2442 |
shows "(f + g) / h = f / h + g / h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2443 |
by (simp add: fps_divide_def algebra_simps fps_shift_add) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2444 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2445 |
lemma fps_divide_diff: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2446 |
fixes f g h :: "'a::{ring,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2447 |
shows "(f - g) / h = f / h - g / h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2448 |
by (simp add: fps_divide_def algebra_simps fps_shift_diff) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2449 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2450 |
lemma fps_divide_uminus: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2451 |
fixes f g h :: "'a::{ring,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2452 |
shows "(- f) / g = - (f / g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2453 |
by (simp add: fps_divide_def algebra_simps fps_shift_uminus) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2454 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2455 |
lemma fps_divide_uminus': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2456 |
fixes f g h :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2457 |
shows "f / (- g) = - (f / g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2458 |
by (simp add: fps_divide_def fps_unit_factor_uminus fps_shift_uminus) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2459 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2460 |
lemma fps_divide_times: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2461 |
fixes f g h :: "'a::{semiring_0,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2462 |
assumes "subdegree h \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2463 |
shows "(f * g) / h = f * (g / h)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2464 |
using assms fps_mult_subdegree_ge[of g "inverse (unit_factor h)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2465 |
fps_shift_mult[of "subdegree h" "g * inverse (unit_factor h)" f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2466 |
by (fastforce simp add: fps_divide_def mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2467 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2468 |
lemma fps_divide_times2: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2469 |
fixes f g h :: "'a::{comm_semiring_0,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2470 |
assumes "subdegree h \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2471 |
shows "(f * g) / h = (f / h) * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2472 |
using assms fps_divide_times[of h f g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2473 |
by (simp add: mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2474 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2475 |
lemma fps_times_divide_eq: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2476 |
fixes f g :: "'a::field fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2477 |
assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2478 |
shows "f div g * g = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2479 |
using assms fps_divide_times2[of g f g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2480 |
by (simp add: fps_divide_times fps_divide_self) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2481 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2482 |
lemma fps_divide_times_eq: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2483 |
"(g :: 'a::division_ring fps) \<noteq> 0 \<Longrightarrow> (f * g) div g = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2484 |
by (simp add: fps_divide_times fps_divide_self) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2485 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2486 |
lemma fps_divide_by_mult': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2487 |
fixes f g h :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2488 |
assumes "subdegree h \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2489 |
shows "f / (g * h) = f / h / g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2490 |
proof (cases "f=0 \<or> g=0 \<or> h=0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2491 |
case False with assms show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2492 |
using fps_unit_factor_mult[of g h] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2493 |
by (auto simp: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2494 |
fps_divide_def fps_shift_fps_shift fps_inverse_mult_divring mult.assoc |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2495 |
fps_shift_mult_right_noncomm |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2496 |
) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2497 |
qed auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2498 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2499 |
lemma fps_divide_by_mult: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2500 |
fixes f g h :: "'a :: field fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2501 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2502 |
shows "f / (g * h) = f / g / h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2503 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2504 |
have "f / (g * h) = f / (h * g)" by (simp add: mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2505 |
also have "\<dots> = f / g / h" using fps_divide_by_mult'[OF assms] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2506 |
finally show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2507 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2508 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2509 |
lemma fps_divide_cancel: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2510 |
fixes f g h :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2511 |
shows "h \<noteq> 0 \<Longrightarrow> (f * h) div (g * h) = f div g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2512 |
by (cases "f=0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2513 |
(auto simp: fps_divide_by_mult' fps_divide_times_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2514 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2515 |
lemma fps_divide_1': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2516 |
fixes a :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2517 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2518 |
shows "a / 1 = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2519 |
using assms fps_inverse_one' fps_one_mult(2)[of a] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2520 |
by (force simp: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2521 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2522 |
lemma fps_divide_1 [simp]: "(a :: 'a::division_ring fps) / 1 = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2523 |
by (rule fps_divide_1'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2524 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2525 |
lemma fps_divide_X': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2526 |
fixes f :: "'a::{comm_monoid_add,inverse,mult_zero,uminus,zero_neq_one,monoid_mult} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2527 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2528 |
shows "f / fps_X = fps_shift 1 f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2529 |
using assms fps_one_mult(2)[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2530 |
by (simp add: fps_divide_def fps_X_unit_factor fps_inverse_one') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2531 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2532 |
lemma fps_divide_X [simp]: "a / fps_X = fps_shift 1 (a::'a::division_ring fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2533 |
by (rule fps_divide_X'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2534 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2535 |
lemma fps_divide_X_power': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2536 |
fixes f :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2537 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2538 |
shows "f / (fps_X ^ n) = fps_shift n f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2539 |
using fps_inverse_one'[OF assms] fps_one_mult(2)[of f] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2540 |
by (simp add: fps_divide_def fps_X_power_subdegree) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2541 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2542 |
lemma fps_divide_X_power [simp]: "a / (fps_X ^ n) = fps_shift n (a::'a::division_ring fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2543 |
by (rule fps_divide_X_power'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2544 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2545 |
lemma fps_divide_shift_denom_conv_times_fps_X_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2546 |
fixes f g :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2547 |
assumes "n \<le> subdegree g" "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2548 |
shows "f / fps_shift n g = f / g * fps_X ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2549 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2550 |
by (intro fps_ext) (simp_all add: fps_divide_shift_denom fps_X_power_mult_right_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2551 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2552 |
lemma fps_divide_unit_factor_denom_conv_times_fps_X_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2553 |
fixes f g :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2554 |
assumes "subdegree g \<le> subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2555 |
shows "f / unit_factor g = f / g * fps_X ^ subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2556 |
by (simp add: assms fps_divide_shift_denom_conv_times_fps_X_power) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2557 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2558 |
lemma fps_shift_altdef': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2559 |
fixes f :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2560 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2561 |
shows "fps_shift n f = f div fps_X^n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2562 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2563 |
by (simp add: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2564 |
fps_divide_def fps_X_power_subdegree fps_X_power_unit_factor fps_inverse_one' |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2565 |
) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2566 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2567 |
lemma fps_shift_altdef: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2568 |
"fps_shift n f = (f :: 'a :: division_ring fps) div fps_X^n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2569 |
by (rule fps_shift_altdef'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2570 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2571 |
lemma fps_div_fps_X_power_nth': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2572 |
fixes f :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2573 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2574 |
shows "(f div fps_X^n) $ k = f $ (k + n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2575 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2576 |
by (simp add: fps_shift_altdef' [symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2577 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2578 |
lemma fps_div_fps_X_power_nth: "((f :: 'a :: division_ring fps) div fps_X^n) $ k = f $ (k + n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2579 |
by (rule fps_div_fps_X_power_nth'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2580 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2581 |
lemma fps_div_fps_X_nth': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2582 |
fixes f :: "'a::{semiring_1,inverse,uminus} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2583 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2584 |
shows "(f div fps_X) $ k = f $ Suc k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2585 |
using assms fps_div_fps_X_power_nth'[of f 1] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2586 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2587 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2588 |
lemma fps_div_fps_X_nth: "((f :: 'a :: division_ring fps) div fps_X) $ k = f $ Suc k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2589 |
by (rule fps_div_fps_X_nth'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2590 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2591 |
lemma divide_fps_const': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2592 |
fixes c :: "'a :: {inverse,comm_monoid_add,uminus,mult_zero}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2593 |
shows "f / fps_const c = f * fps_const (inverse c)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2594 |
by (simp add: fps_divide_def fps_const_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2595 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2596 |
lemma divide_fps_const [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2597 |
fixes c :: "'a :: {comm_semiring_0,inverse,uminus}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2598 |
shows "f / fps_const c = fps_const (inverse c) * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2599 |
by (simp add: divide_fps_const' mult.commute) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2600 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2601 |
lemma fps_const_divide: "fps_const (x :: _ :: division_ring) / fps_const y = fps_const (x / y)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2602 |
by (simp add: fps_divide_def fps_const_inverse divide_inverse) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2603 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2604 |
lemma fps_numeral_divide_divide: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2605 |
"x / numeral b / numeral c = (x / numeral (b * c) :: 'a :: field fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2606 |
by (simp add: fps_divide_by_mult[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2607 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2608 |
lemma fps_numeral_mult_divide: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2609 |
"numeral b * x / numeral c = (numeral b / numeral c * x :: 'a :: field fps)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2610 |
by (simp add: fps_divide_times2) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2611 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2612 |
lemmas fps_numeral_simps = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2613 |
fps_numeral_divide_divide fps_numeral_mult_divide inverse_fps_numeral neg_numeral_fps_const |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2614 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2615 |
lemma fps_is_left_unit_iff_zeroth_is_left_unit: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2616 |
fixes f :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2617 |
shows "(\<exists>g. 1 = f * g) \<longleftrightarrow> (\<exists>k. 1 = f$0 * k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2618 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2619 |
assume "\<exists>g. 1 = f * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2620 |
then obtain g where "1 = f * g" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2621 |
hence "1 = f$0 * g$0" using fps_mult_nth_0[of f g] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2622 |
thus "\<exists>k. 1 = f$0 * k" by auto |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2623 |
next |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2624 |
assume "\<exists>k. 1 = f$0 * k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2625 |
then obtain k where "1 = f$0 * k" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2626 |
hence "1 = f * fps_right_inverse f k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2627 |
using fps_right_inverse by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2628 |
thus "\<exists>g. 1 = f * g" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2629 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2630 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2631 |
lemma fps_is_right_unit_iff_zeroth_is_right_unit: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2632 |
fixes f :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2633 |
shows "(\<exists>g. 1 = g * f) \<longleftrightarrow> (\<exists>k. 1 = k * f$0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2634 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2635 |
assume "\<exists>g. 1 = g * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2636 |
then obtain g where "1 = g * f" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2637 |
hence "1 = g$0 * f$0" using fps_mult_nth_0[of g f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2638 |
thus "\<exists>k. 1 = k * f$0" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2639 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2640 |
assume "\<exists>k. 1 = k * f$0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2641 |
then obtain k where "1 = k * f$0" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2642 |
hence "1 = fps_left_inverse f k * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2643 |
using fps_left_inverse by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2644 |
thus "\<exists>g. 1 = g * f" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2645 |
qed |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2646 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2647 |
lemma fps_is_unit_iff [simp]: "(f :: 'a :: field fps) dvd 1 \<longleftrightarrow> f $ 0 \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2648 |
proof |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2649 |
assume "f dvd 1" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2650 |
then obtain g where "1 = f * g" by (elim dvdE) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2651 |
from this[symmetric] have "(f*g) $ 0 = 1" by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2652 |
thus "f $ 0 \<noteq> 0" by auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2653 |
next |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2654 |
assume A: "f $ 0 \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2655 |
thus "f dvd 1" by (simp add: inverse_mult_eq_1[OF A, symmetric]) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2656 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2657 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2658 |
lemma subdegree_eq_0_left: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2659 |
fixes f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2660 |
assumes "\<exists>g. 1 = f * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2661 |
shows "subdegree f = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2662 |
proof (intro subdegree_eq_0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2663 |
from assms obtain g where "1 = f * g" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2664 |
hence "f$0 * g$0 = 1" using fps_mult_nth_0[of f g] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2665 |
thus "f$0 \<noteq> 0" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2666 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2667 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2668 |
lemma subdegree_eq_0_right: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2669 |
fixes f :: "'a::{comm_monoid_add,zero_neq_one,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2670 |
assumes "\<exists>g. 1 = g * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2671 |
shows "subdegree f = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2672 |
proof (intro subdegree_eq_0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2673 |
from assms obtain g where "1 = g * f" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2674 |
hence "g$0 * f$0 = 1" using fps_mult_nth_0[of g f] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2675 |
thus "f$0 \<noteq> 0" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2676 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2677 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2678 |
lemma subdegree_eq_0' [simp]: "(f :: 'a :: field fps) dvd 1 \<Longrightarrow> subdegree f = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2679 |
by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2680 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2681 |
lemma fps_dvd1_left_trivial_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2682 |
fixes f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2683 |
assumes "\<exists>g. 1 = f * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2684 |
shows "unit_factor f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2685 |
using assms subdegree_eq_0_left |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2686 |
by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2687 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2688 |
lemma fps_dvd1_right_trivial_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2689 |
fixes f :: "'a::{comm_monoid_add, zero_neq_one, mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2690 |
assumes "\<exists>g. 1 = g * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2691 |
shows "unit_factor f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2692 |
using assms subdegree_eq_0_right |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2693 |
by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2694 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2695 |
lemma fps_dvd1_trivial_unit_factor: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2696 |
"(f :: 'a::comm_semiring_1 fps) dvd 1 \<Longrightarrow> unit_factor f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2697 |
unfolding dvd_def by (rule fps_dvd1_left_trivial_unit_factor) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2698 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2699 |
lemma fps_unit_dvd_left: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2700 |
fixes f :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2701 |
assumes "f $ 0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2702 |
shows "\<exists>g. 1 = f * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2703 |
using assms fps_is_left_unit_iff_zeroth_is_left_unit right_inverse |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2704 |
by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2705 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2706 |
lemma fps_unit_dvd_right: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2707 |
fixes f :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2708 |
assumes "f $ 0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2709 |
shows "\<exists>g. 1 = g * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2710 |
using assms fps_is_right_unit_iff_zeroth_is_right_unit left_inverse |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2711 |
by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2712 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2713 |
lemma fps_unit_dvd [simp]: "(f $ 0 :: 'a :: field) \<noteq> 0 \<Longrightarrow> f dvd g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2714 |
using fps_unit_dvd_left dvd_trans[of f 1] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2715 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2716 |
lemma dvd_left_imp_subdegree_le: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2717 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2718 |
assumes "\<exists>k. g = f * k" "g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2719 |
shows "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2720 |
using assms fps_mult_subdegree_ge |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2721 |
by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2722 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2723 |
lemma dvd_right_imp_subdegree_le: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2724 |
fixes f g :: "'a::{comm_monoid_add,mult_zero} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2725 |
assumes "\<exists>k. g = k * f" "g \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2726 |
shows "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2727 |
using assms fps_mult_subdegree_ge |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2728 |
by fastforce |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2729 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
2730 |
lemma dvd_imp_subdegree_le: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2731 |
"f dvd g \<Longrightarrow> g \<noteq> 0 \<Longrightarrow> subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2732 |
using dvd_left_imp_subdegree_le by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2733 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2734 |
lemma subdegree_le_imp_dvd_left_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2735 |
fixes f g :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2736 |
assumes "\<exists>y. f $ subdegree f * y = 1" "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2737 |
shows "\<exists>k. g = f * k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2738 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2739 |
define h :: "'a fps" where "h \<equiv> fps_X ^ (subdegree g - subdegree f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2740 |
from assms(1) obtain y where "f $ subdegree f * y = 1" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2741 |
hence "unit_factor f $ 0 * y = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2742 |
from this obtain k where "1 = unit_factor f * k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2743 |
using fps_is_left_unit_iff_zeroth_is_left_unit[of "unit_factor f"] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2744 |
hence "fps_X ^ subdegree f = fps_X ^ subdegree f * unit_factor f * k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2745 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2746 |
moreover have "fps_X ^ subdegree f * unit_factor f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2747 |
by (rule fps_unit_factor_decompose'[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2748 |
ultimately have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2749 |
"fps_X ^ (subdegree f + (subdegree g - subdegree f)) = f * k * h" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2750 |
by (simp add: power_add h_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2751 |
hence "g = f * (k * h * unit_factor g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2752 |
using fps_unit_factor_decompose'[of g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2753 |
by (simp add: assms(2) mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2754 |
thus ?thesis by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2755 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2756 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2757 |
lemma subdegree_le_imp_dvd_left_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2758 |
fixes f g :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2759 |
assumes "f \<noteq> 0" "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2760 |
shows "\<exists>k. g = f * k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2761 |
proof (intro subdegree_le_imp_dvd_left_ring1) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2762 |
from assms(1) have "f $ subdegree f \<noteq> 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2763 |
thus "\<exists>y. f $ subdegree f * y = 1" using right_inverse by blast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2764 |
qed (rule assms(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2765 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2766 |
lemma subdegree_le_imp_dvd_right_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2767 |
fixes f g :: "'a :: ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2768 |
assumes "\<exists>x. x * f $ subdegree f = 1" "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2769 |
shows "\<exists>k. g = k * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2770 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2771 |
define h :: "'a fps" where "h \<equiv> fps_X ^ (subdegree g - subdegree f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2772 |
from assms(1) obtain x where "x * f $ subdegree f = 1" by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2773 |
hence "x * unit_factor f $ 0 = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2774 |
from this obtain k where "1 = k * unit_factor f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2775 |
using fps_is_right_unit_iff_zeroth_is_right_unit[of "unit_factor f"] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2776 |
hence "fps_X ^ subdegree f = k * (unit_factor f * fps_X ^ subdegree f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2777 |
by (simp add: mult.assoc[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2778 |
moreover have "unit_factor f * fps_X ^ subdegree f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2779 |
by (rule fps_unit_factor_decompose[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2780 |
ultimately have "fps_X ^ (subdegree g - subdegree f + subdegree f) = h * k * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2781 |
by (simp add: power_add h_def mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2782 |
hence "g = unit_factor g * h * k * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2783 |
using fps_unit_factor_decompose[of g] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2784 |
by (simp add: assms(2) mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2785 |
thus ?thesis by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2786 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2787 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2788 |
lemma subdegree_le_imp_dvd_right_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2789 |
fixes f g :: "'a :: division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2790 |
assumes "f \<noteq> 0" "subdegree f \<le> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2791 |
shows "\<exists>k. g = k * f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2792 |
proof (intro subdegree_le_imp_dvd_right_ring1) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2793 |
from assms(1) have "f $ subdegree f \<noteq> 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2794 |
thus "\<exists>x. x * f $ subdegree f = 1" using left_inverse by blast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2795 |
qed (rule assms(2)) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2796 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
2797 |
lemma fps_dvd_iff: |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2798 |
assumes "(f :: 'a :: field fps) \<noteq> 0" "g \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2799 |
shows "f dvd g \<longleftrightarrow> subdegree f \<le> subdegree g" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2800 |
proof |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2801 |
assume "subdegree f \<le> subdegree g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2802 |
with assms show "f dvd g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2803 |
using subdegree_le_imp_dvd_left_divring |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2804 |
by (auto intro: dvdI) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2805 |
qed (simp add: assms dvd_imp_subdegree_le) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2806 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2807 |
lemma subdegree_div': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2808 |
fixes p q :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2809 |
assumes "\<exists>k. p = k * q" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2810 |
shows "subdegree (p div q) = subdegree p - subdegree q" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2811 |
proof (cases "p = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2812 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2813 |
from assms(1) obtain k where k: "p = k * q" by blast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2814 |
with False have "subdegree (p div q) = subdegree k" by (simp add: fps_divide_times_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2815 |
moreover have "k $ subdegree k * q $ subdegree q \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2816 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2817 |
assume "k $ subdegree k * q $ subdegree q = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2818 |
hence "k $ subdegree k * q $ subdegree q * inverse (q $ subdegree q) = 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2819 |
with False k show False by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2820 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2821 |
ultimately show ?thesis by (simp add: k subdegree_mult') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2822 |
qed simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2823 |
|
66550
e5d82cf3c387
Some small lemmas about polynomials and FPSs
eberlm <eberlm@in.tum.de>
parents:
66480
diff
changeset
|
2824 |
lemma subdegree_div: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2825 |
fixes p q :: "'a :: field fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2826 |
assumes "q dvd p" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2827 |
shows "subdegree (p div q) = subdegree p - subdegree q" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2828 |
using assms |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2829 |
unfolding dvd_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2830 |
by (auto intro: subdegree_div') |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2831 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2832 |
lemma subdegree_div_unit': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2833 |
fixes p q :: "'a :: {ab_group_add,mult_zero,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2834 |
assumes "q $ 0 \<noteq> 0" "p $ subdegree p * inverse (q $ 0) \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2835 |
shows "subdegree (p div q) = subdegree p" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2836 |
using assms subdegree_mult'[of p "inverse q"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2837 |
by (auto simp add: fps_divide_unit) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2838 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2839 |
lemma subdegree_div_unit'': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2840 |
fixes p q :: "'a :: {ring_no_zero_divisors,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2841 |
assumes "q $ 0 \<noteq> 0" "inverse (q $ 0) \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2842 |
shows "subdegree (p div q) = subdegree p" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2843 |
by (cases "p = 0") (auto intro: subdegree_div_unit' simp: assms) |
66550
e5d82cf3c387
Some small lemmas about polynomials and FPSs
eberlm <eberlm@in.tum.de>
parents:
66480
diff
changeset
|
2844 |
|
e5d82cf3c387
Some small lemmas about polynomials and FPSs
eberlm <eberlm@in.tum.de>
parents:
66480
diff
changeset
|
2845 |
lemma subdegree_div_unit: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2846 |
fixes p q :: "'a :: division_ring fps" |
66550
e5d82cf3c387
Some small lemmas about polynomials and FPSs
eberlm <eberlm@in.tum.de>
parents:
66480
diff
changeset
|
2847 |
assumes "q $ 0 \<noteq> 0" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2848 |
shows "subdegree (p div q) = subdegree p" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2849 |
by (intro subdegree_div_unit'') (simp_all add: assms) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2850 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2851 |
instantiation fps :: ("{comm_semiring_1,inverse,uminus}") modulo |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2852 |
begin |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2853 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2854 |
definition fps_mod_def: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2855 |
"f mod g = (if g = 0 then f else |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2856 |
let h = unit_factor g in fps_cutoff (subdegree g) (f * inverse h) * h)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2857 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2858 |
instance .. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2859 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2860 |
end |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2861 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2862 |
lemma fps_mod_zero [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2863 |
"(f::'a::{comm_semiring_1,inverse,uminus} fps) mod 0 = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2864 |
by (simp add: fps_mod_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2865 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2866 |
lemma fps_mod_eq_zero: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2867 |
assumes "g \<noteq> 0" and "subdegree f \<ge> subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2868 |
shows "f mod g = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2869 |
proof (cases "f * inverse (unit_factor g) = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2870 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2871 |
have "fps_cutoff (subdegree g) (f * inverse (unit_factor g)) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2872 |
using False assms(2) fps_mult_subdegree_ge fps_cutoff_zero_iff by force |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2873 |
with assms(1) show ?thesis by (simp add: fps_mod_def Let_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2874 |
qed (simp add: assms fps_mod_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2875 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2876 |
lemma fps_mod_unit [simp]: "g$0 \<noteq> 0 \<Longrightarrow> f mod g = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2877 |
by (intro fps_mod_eq_zero) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2878 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2879 |
lemma subdegree_mod: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2880 |
assumes "subdegree (f::'a::field fps) < subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2881 |
shows "subdegree (f mod g) = subdegree f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2882 |
proof (cases "f = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2883 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2884 |
with assms show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2885 |
by (intro subdegreeI) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2886 |
(auto simp: inverse_mult_eq_1 fps_mod_def Let_def fps_cutoff_left_mult_nth mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2887 |
qed (simp add: fps_mod_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2888 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2889 |
instance fps :: (field) idom_modulo |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2890 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2891 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2892 |
fix f g :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2893 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2894 |
define n where "n = subdegree g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2895 |
define h where "h = f * inverse (unit_factor g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2896 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2897 |
show "f div g * g + f mod g = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2898 |
proof (cases "g = 0") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2899 |
case False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2900 |
with n_def h_def have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2901 |
"f div g * g + f mod g = (fps_shift n h * fps_X ^ n + fps_cutoff n h) * unit_factor g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2902 |
by (simp add: fps_divide_def fps_mod_def Let_def subdegree_decompose algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2903 |
with False show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2904 |
by (simp add: fps_shift_cutoff h_def inverse_mult_eq_1) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2905 |
qed auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2906 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2907 |
qed (rule fps_divide_times_eq, simp_all add: fps_divide_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2908 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2909 |
instantiation fps :: (field) normalization_semidom |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2910 |
begin |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2911 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2912 |
definition fps_normalize_def [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2913 |
"normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2914 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2915 |
instance proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2916 |
fix f g :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2917 |
show "unit_factor (f * g) = unit_factor f * unit_factor g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2918 |
using fps_unit_factor_mult by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2919 |
show "unit_factor f * normalize f = f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2920 |
by (simp add: fps_shift_times_fps_X_power) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2921 |
qed (simp_all add: fps_divide_def Let_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2922 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2923 |
end |
66550
e5d82cf3c387
Some small lemmas about polynomials and FPSs
eberlm <eberlm@in.tum.de>
parents:
66480
diff
changeset
|
2924 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2925 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2926 |
subsection \<open>Formal power series form a Euclidean ring\<close> |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2927 |
|
64784
5cb5e7ecb284
reshaped euclidean semiring into hierarchy of euclidean semirings culminating in uniquely determined euclidean divion
haftmann
parents:
64592
diff
changeset
|
2928 |
instantiation fps :: (field) euclidean_ring_cancel |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2929 |
begin |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2930 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
2931 |
definition fps_euclidean_size_def: |
62422 | 2932 |
"euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2933 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2934 |
instance proof |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2935 |
fix f g :: "'a fps" assume [simp]: "g \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2936 |
show "euclidean_size f \<le> euclidean_size (f * g)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2937 |
by (cases "f = 0") (simp_all add: fps_euclidean_size_def) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2938 |
show "euclidean_size (f mod g) < euclidean_size g" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2939 |
apply (cases "f = 0", simp add: fps_euclidean_size_def) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2940 |
apply (rule disjE[OF le_less_linear[of "subdegree g" "subdegree f"]]) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2941 |
apply (simp_all add: fps_mod_eq_zero fps_euclidean_size_def subdegree_mod) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2942 |
done |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2943 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2944 |
fix f g h :: "'a fps" assume [simp]: "h \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2945 |
show "(h * f) div (h * g) = f div g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2946 |
by (simp add: fps_divide_cancel mult.commute) |
66806
a4e82b58d833
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
haftmann
parents:
66804
diff
changeset
|
2947 |
show "(f + g * h) div h = g + f div h" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2948 |
by (simp add: fps_divide_add fps_divide_times_eq) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2949 |
qed (simp add: fps_euclidean_size_def) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2950 |
|
66806
a4e82b58d833
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
haftmann
parents:
66804
diff
changeset
|
2951 |
end |
a4e82b58d833
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
haftmann
parents:
66804
diff
changeset
|
2952 |
|
66817 | 2953 |
instance fps :: (field) normalization_euclidean_semiring .. |
2954 |
||
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2955 |
instantiation fps :: (field) euclidean_ring_gcd |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2956 |
begin |
64786
340db65fd2c1
reworked to provide auxiliary operations Euclidean_Algorithm.* to instantiate gcd etc. for euclidean rings
haftmann
parents:
64784
diff
changeset
|
2957 |
definition fps_gcd_def: "(gcd :: 'a fps \<Rightarrow> _) = Euclidean_Algorithm.gcd" |
340db65fd2c1
reworked to provide auxiliary operations Euclidean_Algorithm.* to instantiate gcd etc. for euclidean rings
haftmann
parents:
64784
diff
changeset
|
2958 |
definition fps_lcm_def: "(lcm :: 'a fps \<Rightarrow> _) = Euclidean_Algorithm.lcm" |
340db65fd2c1
reworked to provide auxiliary operations Euclidean_Algorithm.* to instantiate gcd etc. for euclidean rings
haftmann
parents:
64784
diff
changeset
|
2959 |
definition fps_Gcd_def: "(Gcd :: 'a fps set \<Rightarrow> _) = Euclidean_Algorithm.Gcd" |
340db65fd2c1
reworked to provide auxiliary operations Euclidean_Algorithm.* to instantiate gcd etc. for euclidean rings
haftmann
parents:
64784
diff
changeset
|
2960 |
definition fps_Lcm_def: "(Lcm :: 'a fps set \<Rightarrow> _) = Euclidean_Algorithm.Lcm" |
62422 | 2961 |
instance by standard (simp_all add: fps_gcd_def fps_lcm_def fps_Gcd_def fps_Lcm_def) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2962 |
end |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2963 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2964 |
lemma fps_gcd: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2965 |
assumes [simp]: "f \<noteq> 0" "g \<noteq> 0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2966 |
shows "gcd f g = fps_X ^ min (subdegree f) (subdegree g)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2967 |
proof - |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2968 |
let ?m = "min (subdegree f) (subdegree g)" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2969 |
show "gcd f g = fps_X ^ ?m" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2970 |
proof (rule sym, rule gcdI) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2971 |
fix d assume "d dvd f" "d dvd g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2972 |
thus "d dvd fps_X ^ ?m" by (cases "d = 0") (simp_all add: fps_dvd_iff) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2973 |
qed (simp_all add: fps_dvd_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2974 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2975 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2976 |
lemma fps_gcd_altdef: "gcd f g = |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2977 |
(if f = 0 \<and> g = 0 then 0 else |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2978 |
if f = 0 then fps_X ^ subdegree g else |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2979 |
if g = 0 then fps_X ^ subdegree f else |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2980 |
fps_X ^ min (subdegree f) (subdegree g))" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2981 |
by (simp add: fps_gcd) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2982 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2983 |
lemma fps_lcm: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2984 |
assumes [simp]: "f \<noteq> 0" "g \<noteq> 0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2985 |
shows "lcm f g = fps_X ^ max (subdegree f) (subdegree g)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2986 |
proof - |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2987 |
let ?m = "max (subdegree f) (subdegree g)" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2988 |
show "lcm f g = fps_X ^ ?m" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2989 |
proof (rule sym, rule lcmI) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2990 |
fix d assume "f dvd d" "g dvd d" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2991 |
thus "fps_X ^ ?m dvd d" by (cases "d = 0") (simp_all add: fps_dvd_iff) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2992 |
qed (simp_all add: fps_dvd_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2993 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2994 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
2995 |
lemma fps_lcm_altdef: "lcm f g = |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
2996 |
(if f = 0 \<or> g = 0 then 0 else fps_X ^ max (subdegree f) (subdegree g))" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2997 |
by (simp add: fps_lcm) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2998 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
2999 |
lemma fps_Gcd: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3000 |
assumes "A - {0} \<noteq> {}" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3001 |
shows "Gcd A = fps_X ^ (INF f\<in>A-{0}. subdegree f)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3002 |
proof (rule sym, rule GcdI) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3003 |
fix f assume "f \<in> A" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3004 |
thus "fps_X ^ (INF f\<in>A - {0}. subdegree f) dvd f" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3005 |
by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cINF_lower) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3006 |
next |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3007 |
fix d assume d: "\<And>f. f \<in> A \<Longrightarrow> d dvd f" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3008 |
from assms obtain f where "f \<in> A - {0}" by auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3009 |
with d[of f] have [simp]: "d \<noteq> 0" by auto |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3010 |
from d assms have "subdegree d \<le> (INF f\<in>A-{0}. subdegree f)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3011 |
by (intro cINF_greatest) (simp_all add: fps_dvd_iff[symmetric]) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3012 |
with d assms show "d dvd fps_X ^ (INF f\<in>A-{0}. subdegree f)" by (simp add: fps_dvd_iff) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3013 |
qed simp_all |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3014 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3015 |
lemma fps_Gcd_altdef: "Gcd A = |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3016 |
(if A \<subseteq> {0} then 0 else fps_X ^ (INF f\<in>A-{0}. subdegree f))" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3017 |
using fps_Gcd by auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3018 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3019 |
lemma fps_Lcm: |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3020 |
assumes "A \<noteq> {}" "0 \<notin> A" "bdd_above (subdegree`A)" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3021 |
shows "Lcm A = fps_X ^ (SUP f\<in>A. subdegree f)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3022 |
proof (rule sym, rule LcmI) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3023 |
fix f assume "f \<in> A" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3024 |
moreover from assms(3) have "bdd_above (subdegree ` A)" by auto |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3025 |
ultimately show "f dvd fps_X ^ (SUP f\<in>A. subdegree f)" using assms(2) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3026 |
by (cases "f = 0") (auto simp: fps_dvd_iff intro!: cSUP_upper) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3027 |
next |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3028 |
fix d assume d: "\<And>f. f \<in> A \<Longrightarrow> f dvd d" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3029 |
from assms obtain f where f: "f \<in> A" "f \<noteq> 0" by auto |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3030 |
show "fps_X ^ (SUP f\<in>A. subdegree f) dvd d" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3031 |
proof (cases "d = 0") |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3032 |
assume "d \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3033 |
moreover from d have "\<And>f. f \<in> A \<Longrightarrow> f \<noteq> 0 \<Longrightarrow> f dvd d" by blast |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3034 |
ultimately have "subdegree d \<ge> (SUP f\<in>A. subdegree f)" using assms |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3035 |
by (intro cSUP_least) (auto simp: fps_dvd_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3036 |
with \<open>d \<noteq> 0\<close> show ?thesis by (simp add: fps_dvd_iff) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3037 |
qed simp_all |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3038 |
qed simp_all |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3039 |
|
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3040 |
lemma fps_Lcm_altdef: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3041 |
"Lcm A = |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3042 |
(if 0 \<in> A \<or> \<not>bdd_above (subdegree`A) then 0 else |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69085
diff
changeset
|
3043 |
if A = {} then 1 else fps_X ^ (SUP f\<in>A. subdegree f))" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3044 |
proof (cases "bdd_above (subdegree`A)") |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3045 |
assume unbounded: "\<not>bdd_above (subdegree`A)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3046 |
have "Lcm A = 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3047 |
proof (rule ccontr) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3048 |
assume "Lcm A \<noteq> 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3049 |
from unbounded obtain f where f: "f \<in> A" "subdegree (Lcm A) < subdegree f" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3050 |
unfolding bdd_above_def by (auto simp: not_le) |
63539 | 3051 |
moreover from f and \<open>Lcm A \<noteq> 0\<close> have "subdegree f \<le> subdegree (Lcm A)" |
62422 | 3052 |
by (intro dvd_imp_subdegree_le dvd_Lcm) simp_all |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3053 |
ultimately show False by simp |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3054 |
qed |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3055 |
with unbounded show ?thesis by simp |
62422 | 3056 |
qed (simp_all add: fps_Lcm Lcm_eq_0_I) |
3057 |
||
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3058 |
|
54681 | 3059 |
|
60500 | 3060 |
subsection \<open>Formal Derivatives, and the MacLaurin theorem around 0\<close> |
29687 | 3061 |
|
3062 |
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))" |
|
3063 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3064 |
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n + 1) * f $ (n + 1)" |
48757 | 3065 |
by (simp add: fps_deriv_def) |
3066 |
||
65398 | 3067 |
lemma fps_0th_higher_deriv: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3068 |
"(fps_deriv ^^ n) f $ 0 = fact n * f $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3069 |
by (induction n arbitrary: f) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3070 |
(simp_all add: funpow_Suc_right mult_of_nat_commute algebra_simps del: funpow.simps) |
29687 | 3071 |
|
30488 | 3072 |
lemma fps_deriv_mult[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3073 |
"fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3074 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3075 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3076 |
have LHS: "fps_deriv (f * g) $ n = (\<Sum>i=0..Suc n. of_nat (n+1) * f$i * g$(Suc n - i))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3077 |
by (simp add: fps_mult_nth sum_distrib_left algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3078 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3079 |
have "\<forall>i\<in>{1..n}. n - (i - 1) = n - i + 1" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3080 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3081 |
"(\<Sum>i=0..n. of_nat (i+1) * f$(i+1) * g$(n - i)) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3082 |
(\<Sum>i=1..Suc n. of_nat i * f$i * g$(n - (i - 1)))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3083 |
by (intro sum.reindex_bij_witness[where i="\<lambda>x. x-1" and j="\<lambda>x. x+1"]) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3084 |
ultimately have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3085 |
"(f * fps_deriv g + fps_deriv f * g) $ n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3086 |
of_nat (Suc n) * f$0 * g$(Suc n) + |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3087 |
(\<Sum>i=1..n. (of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1)) + |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3088 |
of_nat (Suc n) * f$(Suc n) * g$0" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3089 |
by (simp add: fps_mult_nth algebra_simps mult_of_nat_commute sum.atLeast_Suc_atMost sum.distrib) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3090 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3091 |
"\<forall>i\<in>{1..n}. |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3092 |
(of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3093 |
of_nat (n + 1) * f $ i * g $ (Suc n - i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3094 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3095 |
fix i assume i: "i \<in> {1..n}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3096 |
from i have "of_nat (n - i + 1) + (of_nat i :: 'a) = of_nat (n + 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3097 |
using of_nat_add[of "n-i+1" i,symmetric] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3098 |
moreover from i have "Suc n - i = n - i + 1" by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3099 |
ultimately show "(of_nat (n - i + 1) + of_nat i) * f $ i * g $ (n - i + 1) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3100 |
of_nat (n + 1) * f $ i * g $ (Suc n - i)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3101 |
by simp |
60558 | 3102 |
qed |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3103 |
ultimately have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3104 |
"(f * fps_deriv g + fps_deriv f * g) $ n = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3105 |
(\<Sum>i=0..Suc n. of_nat (Suc n) * f $ i * g $ (Suc n - i))" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3106 |
by (simp add: sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3107 |
with LHS show "fps_deriv (f * g) $ n = (f * fps_deriv g + fps_deriv f * g) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3108 |
by simp |
29687 | 3109 |
qed |
3110 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3111 |
lemma fps_deriv_fps_X[simp]: "fps_deriv fps_X = 1" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3112 |
by (simp add: fps_deriv_def fps_X_def fps_eq_iff) |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
3113 |
|
54681 | 3114 |
lemma fps_deriv_neg[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3115 |
"fps_deriv (- (f:: 'a::ring_1 fps)) = - (fps_deriv f)" |
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset
|
3116 |
by (simp add: fps_eq_iff fps_deriv_def) |
52891 | 3117 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3118 |
lemma fps_deriv_add[simp]: "fps_deriv (f + g) = fps_deriv f + fps_deriv g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3119 |
by (auto intro: fps_ext simp: algebra_simps) |
29687 | 3120 |
|
54681 | 3121 |
lemma fps_deriv_sub[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3122 |
"fps_deriv ((f:: 'a::ring_1 fps) - g) = fps_deriv f - fps_deriv g" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3123 |
using fps_deriv_add [of f "- g"] by simp |
29687 | 3124 |
|
3125 |
lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0" |
|
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset
|
3126 |
by (simp add: fps_ext fps_deriv_def fps_const_def) |
29687 | 3127 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3128 |
lemma fps_deriv_of_nat [simp]: "fps_deriv (of_nat n) = 0" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3129 |
by (simp add: fps_of_nat [symmetric]) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3130 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3131 |
lemma fps_deriv_of_int [simp]: "fps_deriv (of_int n) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3132 |
by (simp add: fps_of_int [symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3133 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3134 |
lemma fps_deriv_numeral [simp]: "fps_deriv (numeral n) = 0" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3135 |
by (simp add: numeral_fps_const) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
3136 |
|
48757 | 3137 |
lemma fps_deriv_mult_const_left[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3138 |
"fps_deriv (fps_const c * f) = fps_const c * fps_deriv f" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3139 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3140 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3141 |
lemma fps_deriv_linear[simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3142 |
"fps_deriv (fps_const a * f + fps_const b * g) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3143 |
fps_const a * fps_deriv f + fps_const b * fps_deriv g" |
29687 | 3144 |
by simp |
3145 |
||
3146 |
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0" |
|
3147 |
by (simp add: fps_deriv_def fps_eq_iff) |
|
3148 |
||
3149 |
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3150 |
by (simp add: fps_deriv_def fps_eq_iff) |
29687 | 3151 |
|
48757 | 3152 |
lemma fps_deriv_mult_const_right[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3153 |
"fps_deriv (f * fps_const c) = fps_deriv f * fps_const c" |
29687 | 3154 |
by simp |
3155 |
||
64267 | 3156 |
lemma fps_deriv_sum: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3157 |
"fps_deriv (sum f S) = sum (\<lambda>i. fps_deriv (f i)) S" |
53195 | 3158 |
proof (cases "finite S") |
3159 |
case False |
|
3160 |
then show ?thesis by simp |
|
3161 |
next |
|
3162 |
case True |
|
3163 |
show ?thesis by (induct rule: finite_induct [OF True]) simp_all |
|
29687 | 3164 |
qed |
3165 |
||
52902 | 3166 |
lemma fps_deriv_eq_0_iff [simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3167 |
"fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f$0 :: 'a::{semiring_no_zero_divisors,semiring_char_0})" |
60501 | 3168 |
proof |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3169 |
assume f: "fps_deriv f = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3170 |
show "f = fps_const (f$0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3171 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3172 |
fix n show "f $ n = fps_const (f$0) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3173 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3174 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3175 |
have "(of_nat (Suc m) :: 'a) \<noteq> 0" by (rule of_nat_neq_0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3176 |
with f Suc show ?thesis using fps_deriv_nth[of f] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3177 |
qed simp |
60501 | 3178 |
qed |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3179 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3180 |
show "f = fps_const (f$0) \<Longrightarrow> fps_deriv f = 0" using fps_deriv_const[of "f$0"] by simp |
29687 | 3181 |
qed |
3182 |
||
30488 | 3183 |
lemma fps_deriv_eq_iff: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3184 |
fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps" |
29687 | 3185 |
shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)" |
52891 | 3186 |
proof - |
52903 | 3187 |
have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3188 |
using fps_deriv_sub[of f g] |
52903 | 3189 |
by simp |
54681 | 3190 |
also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f - g) $ 0)" |
52903 | 3191 |
unfolding fps_deriv_eq_0_iff .. |
60501 | 3192 |
finally show ?thesis |
3193 |
by (simp add: field_simps) |
|
29687 | 3194 |
qed |
3195 |
||
48757 | 3196 |
lemma fps_deriv_eq_iff_ex: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3197 |
fixes f g :: "'a::{ring_1_no_zero_divisors,semiring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3198 |
shows "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>c. f = fps_const c + g)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3199 |
by (auto simp: fps_deriv_eq_iff) |
48757 | 3200 |
|
3201 |
||
54681 | 3202 |
fun fps_nth_deriv :: "nat \<Rightarrow> 'a::semiring_1 fps \<Rightarrow> 'a fps" |
48757 | 3203 |
where |
29687 | 3204 |
"fps_nth_deriv 0 f = f" |
3205 |
| "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)" |
|
3206 |
||
3207 |
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)" |
|
48757 | 3208 |
by (induct n arbitrary: f) auto |
3209 |
||
3210 |
lemma fps_nth_deriv_linear[simp]: |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3211 |
"fps_nth_deriv n (fps_const a * f + fps_const b * g) = |
48757 | 3212 |
fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3213 |
by (induct n arbitrary: f g) auto |
48757 | 3214 |
|
3215 |
lemma fps_nth_deriv_neg[simp]: |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3216 |
"fps_nth_deriv n (- (f :: 'a::ring_1 fps)) = - (fps_nth_deriv n f)" |
48757 | 3217 |
by (induct n arbitrary: f) simp_all |
3218 |
||
3219 |
lemma fps_nth_deriv_add[simp]: |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3220 |
"fps_nth_deriv n ((f :: 'a::ring_1 fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g" |
29687 | 3221 |
using fps_nth_deriv_linear[of n 1 f 1 g] by simp |
3222 |
||
48757 | 3223 |
lemma fps_nth_deriv_sub[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3224 |
"fps_nth_deriv n ((f :: 'a::ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
3225 |
using fps_nth_deriv_add [of n f "- g"] by simp |
29687 | 3226 |
|
3227 |
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0" |
|
48757 | 3228 |
by (induct n) simp_all |
29687 | 3229 |
|
3230 |
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)" |
|
48757 | 3231 |
by (induct n) simp_all |
3232 |
||
3233 |
lemma fps_nth_deriv_const[simp]: |
|
3234 |
"fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)" |
|
3235 |
by (cases n) simp_all |
|
3236 |
||
3237 |
lemma fps_nth_deriv_mult_const_left[simp]: |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3238 |
"fps_nth_deriv n (fps_const c * f) = fps_const c * fps_nth_deriv n f" |
29687 | 3239 |
using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp |
3240 |
||
48757 | 3241 |
lemma fps_nth_deriv_mult_const_right[simp]: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3242 |
"fps_nth_deriv n (f * fps_const c) = fps_nth_deriv n f * fps_const c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3243 |
by (induct n arbitrary: f) auto |
29687 | 3244 |
|
64267 | 3245 |
lemma fps_nth_deriv_sum: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3246 |
"fps_nth_deriv n (sum f S) = sum (\<lambda>i. fps_nth_deriv n (f i :: 'a::ring_1 fps)) S" |
52903 | 3247 |
proof (cases "finite S") |
3248 |
case True |
|
3249 |
show ?thesis by (induct rule: finite_induct [OF True]) simp_all |
|
3250 |
next |
|
3251 |
case False |
|
3252 |
then show ?thesis by simp |
|
29687 | 3253 |
qed |
3254 |
||
48757 | 3255 |
lemma fps_deriv_maclauren_0: |
54681 | 3256 |
"(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) $ 0 = of_nat (fact k) * f $ k" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3257 |
by (induct k arbitrary: f) (simp_all add: field_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3258 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3259 |
lemma fps_deriv_lr_inverse: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3260 |
fixes x y :: "'a::ring_1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3261 |
assumes "x * f$0 = 1" "f$0 * y = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3262 |
\<comment> \<open>These assumptions imply x equals y, but no need to assume that.\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3263 |
shows "fps_deriv (fps_left_inverse f x) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3264 |
- fps_left_inverse f x * fps_deriv f * fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3265 |
and "fps_deriv (fps_right_inverse f y) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3266 |
- fps_right_inverse f y * fps_deriv f * fps_right_inverse f y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3267 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3268 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3269 |
define L where "L \<equiv> fps_left_inverse f x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3270 |
hence "fps_deriv (L * f) = 0" using fps_left_inverse[OF assms(1)] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3271 |
with assms show "fps_deriv L = - L * fps_deriv f * L" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3272 |
using fps_right_inverse'[OF assms] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3273 |
by (simp add: minus_unique mult.assoc L_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3274 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3275 |
define R where "R \<equiv> fps_right_inverse f y" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3276 |
hence "fps_deriv (f * R) = 0" using fps_right_inverse[OF assms(2)] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3277 |
hence 1: "f * fps_deriv R + fps_deriv f * R = 0" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3278 |
have "R * f * fps_deriv R = - R * fps_deriv f * R" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3279 |
using iffD2[OF eq_neg_iff_add_eq_0, OF 1] by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3280 |
thus "fps_deriv R = - R * fps_deriv f * R" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3281 |
using fps_left_inverse'[OF assms] by (simp add: R_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3282 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3283 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3284 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3285 |
lemma fps_deriv_lr_inverse_comm: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3286 |
fixes x :: "'a::comm_ring_1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3287 |
assumes "x * f$0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3288 |
shows "fps_deriv (fps_left_inverse f x) = - fps_deriv f * (fps_left_inverse f x)\<^sup>2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3289 |
and "fps_deriv (fps_right_inverse f x) = - fps_deriv f * (fps_right_inverse f x)\<^sup>2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3290 |
using assms fps_deriv_lr_inverse[of x f x] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3291 |
by (simp_all add: mult.commute power2_eq_square) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3292 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3293 |
lemma fps_inverse_deriv_divring: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3294 |
fixes a :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3295 |
assumes "a$0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3296 |
shows "fps_deriv (inverse a) = - inverse a * fps_deriv a * inverse a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3297 |
using assms fps_deriv_lr_inverse(2)[of "inverse (a$0)" a "inverse (a$0)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3298 |
by (simp add: fps_inverse_def) |
29687 | 3299 |
|
30488 | 3300 |
lemma fps_inverse_deriv: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3301 |
fixes a :: "'a::field fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3302 |
assumes "a$0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3303 |
shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3304 |
using assms fps_deriv_lr_inverse_comm(2)[of "inverse (a$0)" a] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3305 |
by (simp add: fps_inverse_def) |
29687 | 3306 |
|
30488 | 3307 |
lemma fps_inverse_deriv': |
54681 | 3308 |
fixes a :: "'a::field fps" |
60501 | 3309 |
assumes a0: "a $ 0 \<noteq> 0" |
53077 | 3310 |
shows "fps_deriv (inverse a) = - fps_deriv a / a\<^sup>2" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3311 |
using fps_inverse_deriv[OF a0] a0 |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3312 |
by (simp add: fps_divide_unit power2_eq_square fps_inverse_mult) |
29687 | 3313 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3314 |
(* FIXME: The last part of this proof should go through by simp once we have a proper |
61804
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3315 |
theorem collection for simplifying division on rings *) |
52902 | 3316 |
lemma fps_divide_deriv: |
61804
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3317 |
assumes "b dvd (a :: 'a :: field fps)" |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3318 |
shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b^2" |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3319 |
proof - |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3320 |
have eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b div c" for a b c :: "'a :: field fps" |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3321 |
by (drule sym) (simp add: mult.assoc) |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3322 |
from assms have "a = a / b * b" by simp |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3323 |
also have "fps_deriv (a / b * b) = fps_deriv (a / b) * b + a / b * fps_deriv b" by simp |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3324 |
finally have "fps_deriv (a / b) * b^2 = fps_deriv a * b - a * fps_deriv b" using assms |
67381557cee8
Generalised derivative rule for division on formal power series
eberlm
parents:
61799
diff
changeset
|
3325 |
by (simp add: power2_eq_square algebra_simps) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3326 |
thus ?thesis by (cases "b = 0") (simp_all add: eq_divide_imp) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
3327 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
3328 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3329 |
lemma fps_nth_deriv_fps_X[simp]: "fps_nth_deriv n fps_X = (if n = 0 then fps_X else if n=1 then 1 else 0)" |
52902 | 3330 |
by (cases n) simp_all |
29687 | 3331 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3332 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3333 |
subsection \<open>Powers\<close> |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3334 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3335 |
lemma fps_power_zeroth: "(a^n) $ 0 = (a$0)^n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3336 |
by (induct n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3337 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3338 |
lemma fps_power_zeroth_eq_one: "a$0 = 1 \<Longrightarrow> a^n $ 0 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3339 |
by (simp add: fps_power_zeroth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3340 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3341 |
lemma fps_power_first: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3342 |
fixes a :: "'a::comm_semiring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3343 |
shows "(a^n) $ 1 = of_nat n * (a$0)^(n-1) * a$1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3344 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3345 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3346 |
have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1) * a$1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3347 |
proof (induct m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3348 |
case (Suc k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3349 |
hence "(a ^ Suc (Suc k)) $ 1 = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3350 |
a$0 * of_nat (Suc k) * (a $ 0)^k * a$1 + a$1 * ((a$0)^(Suc k))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3351 |
using fps_mult_nth_1[of a] by (simp add: fps_power_zeroth[symmetric] mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3352 |
thus ?case by (simp add: algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3353 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3354 |
with Suc show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3355 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3356 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3357 |
lemma fps_power_first_eq: "a $ 0 = 1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3358 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3359 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3360 |
show ?case unfolding power_Suc fps_mult_nth |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3361 |
using Suc.hyps[OF \<open>a$0 = 1\<close>] \<open>a$0 = 1\<close> fps_power_zeroth_eq_one[OF \<open>a$0=1\<close>] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3362 |
by (simp add: algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3363 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3364 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3365 |
lemma fps_power_first_eq': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3366 |
assumes "a $ 1 = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3367 |
shows "a^n $ 1 = of_nat n * (a$0)^(n-1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3368 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3369 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3370 |
from assms have "(a ^ Suc m) $ 1 = of_nat (Suc m) * (a$0)^(Suc m - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3371 |
using fps_mult_nth_1[of a] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3372 |
by (induct m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3373 |
(simp_all add: algebra_simps mult_of_nat_commute fps_power_zeroth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3374 |
with Suc show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3375 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3376 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3377 |
lemmas startsby_one_power = fps_power_zeroth_eq_one |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3378 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3379 |
lemma startsby_zero_power: "a $ 0 = 0 \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3380 |
by (simp add: fps_power_zeroth zero_power) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3381 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3382 |
lemma startsby_power: "a $0 = v \<Longrightarrow> a^n $0 = v^n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3383 |
by (simp add: fps_power_zeroth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3384 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3385 |
lemma startsby_nonzero_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3386 |
fixes a :: "'a::semiring_1_no_zero_divisors fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3387 |
shows "a $ 0 \<noteq> 0 \<Longrightarrow> a^n $ 0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3388 |
by (simp add: startsby_power) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3389 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3390 |
lemma startsby_zero_power_iff[simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3391 |
"a^n $0 = (0::'a::semiring_1_no_zero_divisors) \<longleftrightarrow> n \<noteq> 0 \<and> a$0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3392 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3393 |
show "a ^ n $ 0 = 0 \<Longrightarrow> n \<noteq> 0 \<and> a $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3394 |
proof |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3395 |
assume a: "a^n $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3396 |
thus "a $ 0 = 0" using startsby_nonzero_power by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3397 |
have "n = 0 \<Longrightarrow> a^n $ 0 = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3398 |
with a show "n \<noteq> 0" by fastforce |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3399 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3400 |
show "n \<noteq> 0 \<and> a $ 0 = 0 \<Longrightarrow> a ^ n $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3401 |
by (cases n) auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3402 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3403 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3404 |
lemma startsby_zero_power_prefix: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3405 |
assumes a0: "a $ 0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3406 |
shows "\<forall>n < k. a ^ k $ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3407 |
proof (induct k rule: nat_less_induct, clarify) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3408 |
case (1 k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3409 |
fix j :: nat assume j: "j < k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3410 |
show "a ^ k $ j = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3411 |
proof (cases k) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3412 |
case 0 with j show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3413 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3414 |
case (Suc i) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3415 |
with 1 j have "\<forall>m\<in>{0<..j}. a ^ i $ (j - m) = 0" by auto |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3416 |
with Suc a0 show ?thesis by (simp add: fps_mult_nth sum.atLeast_Suc_atMost) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3417 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3418 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3419 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3420 |
lemma startsby_zero_sum_depends: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3421 |
assumes a0: "a $0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3422 |
and kn: "n \<ge> k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3423 |
shows "sum (\<lambda>i. (a ^ i)$k) {0 .. n} = sum (\<lambda>i. (a ^ i)$k) {0 .. k}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3424 |
apply (rule sum.mono_neutral_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3425 |
using kn |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3426 |
apply auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3427 |
apply (rule startsby_zero_power_prefix[rule_format, OF a0]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3428 |
apply arith |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3429 |
done |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3430 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3431 |
lemma startsby_zero_power_nth_same: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3432 |
assumes a0: "a$0 = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3433 |
shows "a^n $ n = (a$1) ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3434 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3435 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3436 |
have "\<forall>i\<in>{Suc 1..Suc n}. a ^ n $ (Suc n - i) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3437 |
using a0 startsby_zero_power_prefix[of a n] by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3438 |
thus ?case |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3439 |
using a0 Suc sum.atLeast_Suc_atMost[of 0 "Suc n" "\<lambda>i. a $ i * a ^ n $ (Suc n - i)"] |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3440 |
sum.atLeast_Suc_atMost[of 1 "Suc n" "\<lambda>i. a $ i * a ^ n $ (Suc n - i)"] |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3441 |
by (simp add: fps_mult_nth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3442 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3443 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3444 |
lemma fps_lr_inverse_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3445 |
fixes a :: "'a::ring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3446 |
assumes "x * a$0 = 1" "a$0 * x = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3447 |
shows "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3448 |
and "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3449 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3450 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3451 |
from assms have xn: "\<And>n. x^n * (a^n $ 0) = 1" "\<And>n. (a^n $ 0) * x^n = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3452 |
by (simp_all add: left_right_inverse_power fps_power_zeroth) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3453 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3454 |
show "fps_left_inverse (a^n) (x^n) = fps_left_inverse a x ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3455 |
proof (induct n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3456 |
case 0 |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3457 |
then show ?case by (simp add: fps_lr_inverse_one_one(1)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3458 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3459 |
case (Suc n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3460 |
with assms show ?case |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3461 |
using xn fps_lr_inverse_mult_ring1(1)[of x a "x^n" "a^n"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3462 |
by (simp add: power_Suc2[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3463 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3464 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3465 |
moreover have "fps_right_inverse (a^n) (x^n) = fps_left_inverse (a^n) (x^n)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3466 |
using xn by (intro fps_left_inverse_eq_fps_right_inverse[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3467 |
moreover have "fps_right_inverse a x = fps_left_inverse a x" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3468 |
using assms by (intro fps_left_inverse_eq_fps_right_inverse[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3469 |
ultimately show "fps_right_inverse (a^n) (x^n) = fps_right_inverse a x ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3470 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3471 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3472 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3473 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3474 |
lemma fps_inverse_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3475 |
fixes a :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3476 |
shows "inverse (a^n) = inverse a ^ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3477 |
proof (cases "n=0" "a$0 = 0" rule: case_split[case_product case_split]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3478 |
case False_True |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3479 |
hence LHS: "inverse (a^n) = 0" and RHS: "inverse a ^ n = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3480 |
by (simp_all add: startsby_zero_power) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3481 |
show ?thesis using trans_sym[OF LHS RHS] by fast |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3482 |
next |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3483 |
case False_False |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3484 |
from False_False(2) show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3485 |
by (simp add: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3486 |
fps_inverse_def fps_power_zeroth power_inverse fps_lr_inverse_power(2)[symmetric] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3487 |
) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3488 |
qed auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3489 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3490 |
lemma fps_deriv_power': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3491 |
fixes a :: "'a::comm_semiring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3492 |
shows "fps_deriv (a ^ n) = (of_nat n) * fps_deriv a * a ^ (n - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3493 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3494 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3495 |
moreover have "fps_deriv (a^Suc m) = of_nat (Suc m) * fps_deriv a * a^m" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3496 |
by (induct m) (simp_all add: algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3497 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3498 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3499 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3500 |
lemma fps_deriv_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3501 |
fixes a :: "'a::comm_semiring_1 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3502 |
shows "fps_deriv (a ^ n) = fps_const (of_nat n) * fps_deriv a * a ^ (n - 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3503 |
by (simp add: fps_deriv_power' fps_of_nat) |
29687 | 3504 |
|
30488 | 3505 |
|
60501 | 3506 |
subsection \<open>Integration\<close> |
31273 | 3507 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3508 |
definition fps_integral :: "'a::{semiring_1,inverse} fps \<Rightarrow> 'a \<Rightarrow> 'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3509 |
where "fps_integral a a0 = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3510 |
Abs_fps (\<lambda>n. if n=0 then a0 else inverse (of_nat n) * a$(n - 1))" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3511 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3512 |
abbreviation "fps_integral0 a \<equiv> fps_integral a 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3513 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3514 |
lemma fps_integral_nth_0_Suc [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3515 |
fixes a :: "'a::{semiring_1,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3516 |
shows "fps_integral a a0 $ 0 = a0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3517 |
and "fps_integral a a0 $ Suc n = inverse (of_nat (Suc n)) * a $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3518 |
by (auto simp: fps_integral_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3519 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3520 |
lemma fps_integral_conv_plus_const: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3521 |
"fps_integral a a0 = fps_integral a 0 + fps_const a0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3522 |
unfolding fps_integral_def by (intro fps_ext) simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3523 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3524 |
lemma fps_deriv_fps_integral: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3525 |
fixes a :: "'a::{division_ring,ring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3526 |
shows "fps_deriv (fps_integral a a0) = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3527 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3528 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3529 |
have "(of_nat (Suc n) :: 'a) \<noteq> 0" by (rule of_nat_neq_0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3530 |
hence "of_nat (Suc n) * inverse (of_nat (Suc n) :: 'a) = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3531 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3532 |
"fps_deriv (fps_integral a a0) $ n = of_nat (Suc n) * inverse (of_nat (Suc n)) * a $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3533 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3534 |
ultimately show "fps_deriv (fps_integral a a0) $ n = a $ n" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3535 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3536 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3537 |
lemma fps_integral0_deriv: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3538 |
fixes a :: "'a::{division_ring,ring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3539 |
shows "fps_integral0 (fps_deriv a) = a - fps_const (a$0)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3540 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3541 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3542 |
show "fps_integral0 (fps_deriv a) $ n = (a - fps_const (a$0)) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3543 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3544 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3545 |
have "(of_nat (Suc m) :: 'a) \<noteq> 0" by (rule of_nat_neq_0) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3546 |
hence "inverse (of_nat (Suc m) :: 'a) * of_nat (Suc m) = 1" by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3547 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3548 |
"fps_integral0 (fps_deriv a) $ Suc m = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3549 |
inverse (of_nat (Suc m)) * of_nat (Suc m) * a $ (Suc m)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3550 |
by (simp add: mult.assoc) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3551 |
ultimately show ?thesis using Suc by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3552 |
qed simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3553 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3554 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3555 |
lemma fps_integral_deriv: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3556 |
fixes a :: "'a::{division_ring,ring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3557 |
shows "fps_integral (fps_deriv a) (a$0) = a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3558 |
using fps_integral_conv_plus_const[of "fps_deriv a" "a$0"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3559 |
by (simp add: fps_integral0_deriv) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3560 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3561 |
lemma fps_integral0_zero: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3562 |
"fps_integral0 (0::'a::{semiring_1,inverse} fps) = 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3563 |
by (intro fps_ext) (simp add: fps_integral_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3564 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3565 |
lemma fps_integral0_fps_const': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3566 |
fixes c :: "'a::{semiring_1,inverse}" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3567 |
assumes "inverse (1::'a) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3568 |
shows "fps_integral0 (fps_const c) = fps_const c * fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3569 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3570 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3571 |
show "fps_integral0 (fps_const c) $ n = (fps_const c * fps_X) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3572 |
by (cases n) (simp_all add: assms mult_delta_right) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3573 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3574 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3575 |
lemma fps_integral0_fps_const: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3576 |
fixes c :: "'a::division_ring" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3577 |
shows "fps_integral0 (fps_const c) = fps_const c * fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3578 |
by (rule fps_integral0_fps_const'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3579 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3580 |
lemma fps_integral0_one': |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3581 |
assumes "inverse (1::'a::{semiring_1,inverse}) = 1" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3582 |
shows "fps_integral0 (1::'a fps) = fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3583 |
using assms fps_integral0_fps_const'[of "1::'a"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3584 |
by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3585 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3586 |
lemma fps_integral0_one: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3587 |
"fps_integral0 (1::'a::division_ring fps) = fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3588 |
by (rule fps_integral0_one'[OF inverse_1]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3589 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3590 |
lemma fps_integral0_fps_const_mult_left: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3591 |
fixes a :: "'a::division_ring fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3592 |
shows "fps_integral0 (fps_const c * a) = fps_const c * fps_integral0 a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3593 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3594 |
fix n |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3595 |
show "fps_integral0 (fps_const c * a) $ n = (fps_const c * fps_integral0 a) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3596 |
using mult_inverse_of_nat_commute[of n c, symmetric] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3597 |
mult.assoc[of "inverse (of_nat n)" c "a$(n-1)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3598 |
mult.assoc[of c "inverse (of_nat n)" "a$(n-1)"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3599 |
by (simp add: fps_integral_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3600 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3601 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3602 |
lemma fps_integral0_fps_const_mult_right: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3603 |
fixes a :: "'a::{semiring_1,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3604 |
shows "fps_integral0 (a * fps_const c) = fps_integral0 a * fps_const c" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3605 |
by (intro fps_ext) (simp add: fps_integral_def algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3606 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3607 |
lemma fps_integral0_neg: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3608 |
fixes a :: "'a::{ring_1,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3609 |
shows "fps_integral0 (-a) = - fps_integral0 a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3610 |
using fps_integral0_fps_const_mult_right[of a "-1"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3611 |
by (simp add: fps_const_neg[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3612 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3613 |
lemma fps_integral0_add: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3614 |
"fps_integral0 (a+b) = fps_integral0 a + fps_integral0 b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3615 |
by (intro fps_ext) (simp add: fps_integral_def algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3616 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3617 |
lemma fps_integral0_linear: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3618 |
fixes a b :: "'a::division_ring" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3619 |
shows "fps_integral0 (fps_const a * f + fps_const b * g) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3620 |
fps_const a * fps_integral0 f + fps_const b * fps_integral0 g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3621 |
by (simp add: fps_integral0_add fps_integral0_fps_const_mult_left) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3622 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3623 |
lemma fps_integral0_linear2: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3624 |
"fps_integral0 (f * fps_const a + g * fps_const b) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3625 |
fps_integral0 f * fps_const a + fps_integral0 g * fps_const b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3626 |
by (simp add: fps_integral0_add fps_integral0_fps_const_mult_right) |
29687 | 3627 |
|
31273 | 3628 |
lemma fps_integral_linear: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3629 |
fixes a b a0 b0 :: "'a::division_ring" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3630 |
shows |
31273 | 3631 |
"fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) = |
3632 |
fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3633 |
using fps_integral_conv_plus_const[of |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3634 |
"fps_const a * f + fps_const b * g" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3635 |
"a*a0 + b*b0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3636 |
] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3637 |
fps_integral_conv_plus_const[of f a0] fps_integral_conv_plus_const[of g b0] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3638 |
by (simp add: fps_integral0_linear algebra_simps) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3639 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3640 |
lemma fps_integral0_sub: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3641 |
fixes a b :: "'a::{ring_1,inverse} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3642 |
shows "fps_integral0 (a-b) = fps_integral0 a - fps_integral0 b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3643 |
using fps_integral0_linear2[of a 1 b "-1"] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3644 |
by (simp add: fps_const_neg[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3645 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3646 |
lemma fps_integral0_of_nat: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3647 |
"fps_integral0 (of_nat n :: 'a::division_ring fps) = of_nat n * fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3648 |
using fps_integral0_fps_const[of "of_nat n :: 'a"] by (simp add: fps_of_nat) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3649 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3650 |
lemma fps_integral0_sum: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3651 |
"fps_integral0 (sum f S) = sum (\<lambda>i. fps_integral0 (f i)) S" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3652 |
proof (cases "finite S") |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3653 |
case True show ?thesis |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3654 |
by (induct rule: finite_induct [OF True]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3655 |
(simp_all add: fps_integral0_zero fps_integral0_add) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3656 |
qed (simp add: fps_integral0_zero) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3657 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3658 |
lemma fps_integral0_by_parts: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3659 |
fixes a b :: "'a::{division_ring,ring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3660 |
shows |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3661 |
"fps_integral0 (a * b) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3662 |
a * fps_integral0 b - fps_integral0 (fps_deriv a * fps_integral0 b)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3663 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3664 |
have "fps_integral0 (fps_deriv (a * fps_integral0 b)) = a * fps_integral0 b" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3665 |
using fps_integral0_deriv[of "(a * fps_integral0 b)"] by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3666 |
moreover have |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3667 |
"fps_integral0 (a * b) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3668 |
fps_integral0 (fps_deriv (a * fps_integral0 b)) - |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3669 |
fps_integral0 (fps_deriv a * fps_integral0 b)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3670 |
by (auto simp: fps_deriv_fps_integral fps_integral0_sub[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3671 |
ultimately show ?thesis by simp |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3672 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3673 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3674 |
lemma fps_integral0_fps_X: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3675 |
"fps_integral0 (fps_X::'a::{semiring_1,inverse} fps) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3676 |
fps_const (inverse (of_nat 2)) * fps_X\<^sup>2" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3677 |
by (intro fps_ext) (auto simp: fps_integral_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3678 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3679 |
lemma fps_integral0_fps_X_power: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3680 |
"fps_integral0 ((fps_X::'a::{semiring_1,inverse} fps) ^ n) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3681 |
fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3682 |
proof (intro fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3683 |
fix k show |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3684 |
"fps_integral0 ((fps_X::'a fps) ^ n) $ k = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3685 |
(fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n) $ k" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3686 |
by (cases k) simp_all |
29687 | 3687 |
qed |
30488 | 3688 |
|
53195 | 3689 |
|
60500 | 3690 |
subsection \<open>Composition of FPSs\<close> |
53195 | 3691 |
|
60501 | 3692 |
definition fps_compose :: "'a::semiring_1 fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) |
64267 | 3693 |
where "a oo b = Abs_fps (\<lambda>n. sum (\<lambda>i. a$i * (b^i$n)) {0..n})" |
3694 |
||
3695 |
lemma fps_compose_nth: "(a oo b)$n = sum (\<lambda>i. a$i * (b^i$n)) {0..n}" |
|
48757 | 3696 |
by (simp add: fps_compose_def) |
29687 | 3697 |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3698 |
lemma fps_compose_nth_0 [simp]: "(f oo g) $ 0 = f $ 0" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3699 |
by (simp add: fps_compose_nth) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
3700 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3701 |
lemma fps_compose_fps_X[simp]: "a oo fps_X = (a :: 'a::comm_ring_1 fps)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3702 |
by (simp add: fps_ext fps_compose_def mult_delta_right) |
30488 | 3703 |
|
60501 | 3704 |
lemma fps_const_compose[simp]: "fps_const (a::'a::comm_ring_1) oo b = fps_const a" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3705 |
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left) |
29687 | 3706 |
|
54681 | 3707 |
lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset
|
3708 |
unfolding numeral_fps_const by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset
|
3709 |
|
54681 | 3710 |
lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset
|
3711 |
unfolding neg_numeral_fps_const by simp |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
3712 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3713 |
lemma fps_X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> fps_X oo a = (a :: 'a::comm_ring_1 fps)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3714 |
by (simp add: fps_eq_iff fps_compose_def mult_delta_left not_le) |
29687 | 3715 |
|
3716 |
||
60500 | 3717 |
subsection \<open>Rules from Herbert Wilf's Generatingfunctionology\<close> |
3718 |
||
3719 |
subsubsection \<open>Rule 1\<close> |
|
64267 | 3720 |
(* {a_{n+k}}_0^infty Corresponds to (f - sum (\<lambda>i. a_i * x^i))/x^h, for h>0*) |
29687 | 3721 |
|
30488 | 3722 |
lemma fps_power_mult_eq_shift: |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3723 |
"fps_X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3724 |
Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a::comm_ring_1) * fps_X^i) {0 .. k}" |
52902 | 3725 |
(is "?lhs = ?rhs") |
3726 |
proof - |
|
60501 | 3727 |
have "?lhs $ n = ?rhs $ n" for n :: nat |
3728 |
proof - |
|
30488 | 3729 |
have "?lhs $ n = (if n < Suc k then 0 else a n)" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3730 |
unfolding fps_X_power_mult_nth by auto |
29687 | 3731 |
also have "\<dots> = ?rhs $ n" |
52902 | 3732 |
proof (induct k) |
3733 |
case 0 |
|
60501 | 3734 |
then show ?case |
64267 | 3735 |
by (simp add: fps_sum_nth) |
29687 | 3736 |
next |
3737 |
case (Suc k) |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3738 |
have "(Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a) * fps_X^i) {0 .. Suc k})$n = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3739 |
(Abs_fps a - sum (\<lambda>i. fps_const (a i :: 'a) * fps_X^i) {0 .. k} - |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3740 |
fps_const (a (Suc k)) * fps_X^ Suc k) $ n" |
52902 | 3741 |
by (simp add: field_simps) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3742 |
also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * fps_X^ Suc k)$n" |
60501 | 3743 |
using Suc.hyps[symmetric] unfolding fps_sub_nth by simp |
29687 | 3744 |
also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3745 |
unfolding fps_X_power_mult_right_nth |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
3746 |
apply (auto simp add: not_less fps_const_def) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
3747 |
apply (rule cong[of a a, OF refl]) |
52902 | 3748 |
apply arith |
3749 |
done |
|
60501 | 3750 |
finally show ?case |
3751 |
by simp |
|
29687 | 3752 |
qed |
60501 | 3753 |
finally show ?thesis . |
3754 |
qed |
|
3755 |
then show ?thesis |
|
3756 |
by (simp add: fps_eq_iff) |
|
29687 | 3757 |
qed |
3758 |
||
53195 | 3759 |
|
60500 | 3760 |
subsubsection \<open>Rule 2\<close> |
29687 | 3761 |
|
3762 |
(* We can not reach the form of Wilf, but still near to it using rewrite rules*) |
|
30488 | 3763 |
(* If f reprents {a_n} and P is a polynomial, then |
29687 | 3764 |
P(xD) f represents {P(n) a_n}*) |
3765 |
||
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68975
diff
changeset
|
3766 |
definition "fps_XD = (*) fps_X \<circ> fps_deriv" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3767 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3768 |
lemma fps_XD_add[simp]:"fps_XD (a + b) = fps_XD a + fps_XD (b :: 'a::comm_ring_1 fps)" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3769 |
by (simp add: fps_XD_def field_simps) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3770 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3771 |
lemma fps_XD_mult_const[simp]:"fps_XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * fps_XD a" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3772 |
by (simp add: fps_XD_def field_simps) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3773 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3774 |
lemma fps_XD_linear[simp]: "fps_XD (fps_const c * a + fps_const d * b) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3775 |
fps_const c * fps_XD a + fps_const d * fps_XD (b :: 'a::comm_ring_1 fps)" |
29687 | 3776 |
by simp |
3777 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3778 |
lemma fps_XDN_linear: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3779 |
"(fps_XD ^^ n) (fps_const c * a + fps_const d * b) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3780 |
fps_const c * (fps_XD ^^ n) a + fps_const d * (fps_XD ^^ n) (b :: 'a::comm_ring_1 fps)" |
48757 | 3781 |
by (induct n) simp_all |
29687 | 3782 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3783 |
lemma fps_mult_fps_X_deriv_shift: "fps_X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" |
52902 | 3784 |
by (simp add: fps_eq_iff) |
29687 | 3785 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3786 |
lemma fps_mult_fps_XD_shift: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3787 |
"(fps_XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3788 |
by (induct k arbitrary: a) (simp_all add: fps_XD_def fps_eq_iff field_simps del: One_nat_def) |
29687 | 3789 |
|
53195 | 3790 |
|
60501 | 3791 |
subsubsection \<open>Rule 3\<close> |
3792 |
||
61585 | 3793 |
text \<open>Rule 3 is trivial and is given by \<open>fps_times_def\<close>.\<close> |
60501 | 3794 |
|
60500 | 3795 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3796 |
subsubsection \<open>Rule 5 --- summation and "division" by (1 - fps_X)\<close> |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3797 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3798 |
lemma fps_divide_fps_X_minus1_sum_lemma: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3799 |
"a = ((1::'a::ring_1 fps) - fps_X) * Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3800 |
proof (rule fps_ext) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3801 |
define f g :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3802 |
where "f \<equiv> 1 - fps_X" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3803 |
and "g \<equiv> Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3804 |
fix n show "a $ n= (f * g) $ n" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3805 |
proof (cases n) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3806 |
case (Suc m) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3807 |
hence "(f * g) $ n = g $ Suc m - g $ m" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3808 |
using fps_mult_nth[of f g "Suc m"] |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3809 |
sum.atLeast_Suc_atMost[of 0 "Suc m" "\<lambda>i. f $ i * g $ (Suc m - i)"] |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
3810 |
sum.atLeast_Suc_atMost[of 1 "Suc m" "\<lambda>i. f $ i * g $ (Suc m - i)"] |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3811 |
by (simp add: f_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3812 |
with Suc show ?thesis by (simp add: g_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3813 |
qed (simp add: f_def g_def) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3814 |
qed |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3815 |
|
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3816 |
lemma fps_divide_fps_X_minus1_sum_ring1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3817 |
assumes "inverse 1 = (1::'a::{ring_1,inverse})" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3818 |
shows "a /((1::'a fps) - fps_X) = Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3819 |
proof- |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3820 |
from assms have "a /((1::'a fps) - fps_X) = a * Abs_fps (\<lambda>n. 1)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3821 |
by (simp add: fps_divide_def fps_inverse_def fps_lr_inverse_one_minus_fps_X(2)) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3822 |
thus ?thesis by (auto intro: fps_ext simp: fps_mult_nth) |
29687 | 3823 |
qed |
3824 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
3825 |
lemma fps_divide_fps_X_minus1_sum: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3826 |
"a /((1::'a::division_ring fps) - fps_X) = Abs_fps (\<lambda>n. sum (\<lambda>i. a $ i) {0..n})" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
3827 |
using fps_divide_fps_X_minus1_sum_ring1[of a] by simp |
29687 | 3828 |
|
53195 | 3829 |
|
60501 | 3830 |
subsubsection \<open>Rule 4 in its more general form: generalizes Rule 3 for an arbitrary |
60500 | 3831 |
finite product of FPS, also the relvant instance of powers of a FPS\<close> |
29687 | 3832 |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3833 |
definition "natpermute n k = {l :: nat list. length l = k \<and> sum_list l = n}" |
29687 | 3834 |
|
3835 |
lemma natlist_trivial_1: "natpermute n 1 = {[n]}" |
|
3836 |
apply (auto simp add: natpermute_def) |
|
52902 | 3837 |
apply (case_tac x) |
3838 |
apply auto |
|
29687 | 3839 |
done |
3840 |
||
3841 |
lemma append_natpermute_less_eq: |
|
54452 | 3842 |
assumes "xs @ ys \<in> natpermute n k" |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3843 |
shows "sum_list xs \<le> n" |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3844 |
and "sum_list ys \<le> n" |
52902 | 3845 |
proof - |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3846 |
from assms have "sum_list (xs @ ys) = n" |
54452 | 3847 |
by (simp add: natpermute_def) |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3848 |
then have "sum_list xs + sum_list ys = n" |
54452 | 3849 |
by simp |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3850 |
then show "sum_list xs \<le> n" and "sum_list ys \<le> n" |
54452 | 3851 |
by simp_all |
29687 | 3852 |
qed |
3853 |
||
3854 |
lemma natpermute_split: |
|
54452 | 3855 |
assumes "h \<le> k" |
52902 | 3856 |
shows "natpermute n k = |
3857 |
(\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})" |
|
60558 | 3858 |
(is "?L = ?R" is "_ = (\<Union>m \<in>{0..n}. ?S m)") |
3859 |
proof |
|
3860 |
show "?R \<subseteq> ?L" |
|
3861 |
proof |
|
52902 | 3862 |
fix l |
3863 |
assume l: "l \<in> ?R" |
|
3864 |
from l obtain m xs ys where h: "m \<in> {0..n}" |
|
3865 |
and xs: "xs \<in> natpermute m h" |
|
3866 |
and ys: "ys \<in> natpermute (n - m) (k - h)" |
|
3867 |
and leq: "l = xs@ys" by blast |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3868 |
from xs have xs': "sum_list xs = m" |
52902 | 3869 |
by (simp add: natpermute_def) |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3870 |
from ys have ys': "sum_list ys = n - m" |
52902 | 3871 |
by (simp add: natpermute_def) |
60558 | 3872 |
show "l \<in> ?L" using leq xs ys h |
46131 | 3873 |
apply (clarsimp simp add: natpermute_def) |
29687 | 3874 |
unfolding xs' ys' |
54452 | 3875 |
using assms xs ys |
48757 | 3876 |
unfolding natpermute_def |
3877 |
apply simp |
|
3878 |
done |
|
60558 | 3879 |
qed |
3880 |
show "?L \<subseteq> ?R" |
|
3881 |
proof |
|
52902 | 3882 |
fix l |
3883 |
assume l: "l \<in> natpermute n k" |
|
29687 | 3884 |
let ?xs = "take h l" |
3885 |
let ?ys = "drop h l" |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3886 |
let ?m = "sum_list ?xs" |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3887 |
from l have ls: "sum_list (?xs @ ?ys) = n" |
52902 | 3888 |
by (simp add: natpermute_def) |
54452 | 3889 |
have xs: "?xs \<in> natpermute ?m h" using l assms |
52902 | 3890 |
by (simp add: natpermute_def) |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
3891 |
have l_take_drop: "sum_list l = sum_list (take h l @ drop h l)" |
52902 | 3892 |
by simp |
3893 |
then have ys: "?ys \<in> natpermute (n - ?m) (k - h)" |
|
54452 | 3894 |
using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id) |
52902 | 3895 |
from ls have m: "?m \<in> {0..n}" |
3896 |
by (simp add: l_take_drop del: append_take_drop_id) |
|
60558 | 3897 |
from xs ys ls show "l \<in> ?R" |
29687 | 3898 |
apply auto |
52902 | 3899 |
apply (rule bexI [where x = "?m"]) |
3900 |
apply (rule exI [where x = "?xs"]) |
|
3901 |
apply (rule exI [where x = "?ys"]) |
|
52891 | 3902 |
using ls l |
46131 | 3903 |
apply (auto simp add: natpermute_def l_take_drop simp del: append_take_drop_id) |
48757 | 3904 |
apply simp |
3905 |
done |
|
60558 | 3906 |
qed |
29687 | 3907 |
qed |
3908 |
||
3909 |
lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})" |
|
3910 |
by (auto simp add: natpermute_def) |
|
52902 | 3911 |
|
29687 | 3912 |
lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})" |
3913 |
apply (auto simp add: set_replicate_conv_if natpermute_def) |
|
3914 |
apply (rule nth_equalityI) |
|
48757 | 3915 |
apply simp_all |
3916 |
done |
|
29687 | 3917 |
|
3918 |
lemma natpermute_finite: "finite (natpermute n k)" |
|
52902 | 3919 |
proof (induct k arbitrary: n) |
3920 |
case 0 |
|
3921 |
then show ?case |
|
29687 | 3922 |
apply (subst natpermute_split[of 0 0, simplified]) |
52902 | 3923 |
apply (simp add: natpermute_0) |
3924 |
done |
|
29687 | 3925 |
next |
3926 |
case (Suc k) |
|
52902 | 3927 |
then show ?case unfolding natpermute_split [of k "Suc k", simplified] |
29687 | 3928 |
apply - |
3929 |
apply (rule finite_UN_I) |
|
3930 |
apply simp |
|
3931 |
unfolding One_nat_def[symmetric] natlist_trivial_1 |
|
3932 |
apply simp |
|
3933 |
done |
|
3934 |
qed |
|
3935 |
||
3936 |
lemma natpermute_contain_maximal: |
|
60558 | 3937 |
"{xs \<in> natpermute n (k + 1). n \<in> set xs} = (\<Union>i\<in>{0 .. k}. {(replicate (k + 1) 0) [i:=n]})" |
29687 | 3938 |
(is "?A = ?B") |
60558 | 3939 |
proof |
3940 |
show "?A \<subseteq> ?B" |
|
3941 |
proof |
|
52902 | 3942 |
fix xs |
60558 | 3943 |
assume "xs \<in> ?A" |
3944 |
then have H: "xs \<in> natpermute n (k + 1)" and n: "n \<in> set xs" |
|
3945 |
by blast+ |
|
3946 |
then obtain i where i: "i \<in> {0.. k}" "xs!i = n" |
|
30488 | 3947 |
unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def) |
52902 | 3948 |
have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" |
3949 |
using i by auto |
|
3950 |
have f: "finite({0..k} - {i})" "finite {i}" |
|
3951 |
by auto |
|
3952 |
have d: "({0..k} - {i}) \<inter> {i} = {}" |
|
3953 |
using i by auto |
|
64267 | 3954 |
from H have "n = sum (nth xs) {0..k}" |
52902 | 3955 |
apply (simp add: natpermute_def) |
64267 | 3956 |
apply (auto simp add: atLeastLessThanSuc_atLeastAtMost sum_list_sum_nth) |
52902 | 3957 |
done |
64267 | 3958 |
also have "\<dots> = n + sum (nth xs) ({0..k} - {i})" |
3959 |
unfolding sum.union_disjoint[OF f d, unfolded eqs] using i by simp |
|
52902 | 3960 |
finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" |
3961 |
by auto |
|
3962 |
from H have xsl: "length xs = k+1" |
|
3963 |
by (simp add: natpermute_def) |
|
29687 | 3964 |
from i have i': "i < length (replicate (k+1) 0)" "i < k+1" |
52902 | 3965 |
unfolding length_replicate by presburger+ |
69085 | 3966 |
have "xs = (replicate (k+1) 0) [i := n]" |
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3967 |
proof (rule nth_equalityI) |
69085 | 3968 |
show "length xs = length ((replicate (k + 1) 0)[i := n])" |
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3969 |
by (metis length_list_update length_replicate xsl) |
69085 | 3970 |
show "xs ! j = (replicate (k + 1) 0)[i := n] ! j" if "j < length xs" for j |
68975
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3971 |
proof (cases "j = i") |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3972 |
case True |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3973 |
then show ?thesis |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3974 |
by (metis i'(1) i(2) nth_list_update) |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3975 |
next |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3976 |
case False |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3977 |
with that show ?thesis |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3978 |
by (simp add: xsl zxs del: replicate.simps split: nat.split) |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3979 |
qed |
5ce4d117cea7
A few new results, elimination of duplicates and more use of "pairwise"
paulson <lp15@cam.ac.uk>
parents:
68442
diff
changeset
|
3980 |
qed |
60558 | 3981 |
then show "xs \<in> ?B" using i by blast |
3982 |
qed |
|
3983 |
show "?B \<subseteq> ?A" |
|
3984 |
proof |
|
3985 |
fix xs |
|
3986 |
assume "xs \<in> ?B" |
|
69085 | 3987 |
then obtain i where i: "i \<in> {0..k}" and xs: "xs = (replicate (k + 1) 0) [i:=n]" |
60558 | 3988 |
by auto |
3989 |
have nxs: "n \<in> set xs" |
|
3990 |
unfolding xs |
|
52902 | 3991 |
apply (rule set_update_memI) |
3992 |
using i apply simp |
|
3993 |
done |
|
60558 | 3994 |
have xsl: "length xs = k + 1" |
3995 |
by (simp only: xs length_replicate length_list_update) |
|
64267 | 3996 |
have "sum_list xs = sum (nth xs) {0..<k+1}" |
3997 |
unfolding sum_list_sum_nth xsl .. |
|
3998 |
also have "\<dots> = sum (\<lambda>j. if j = i then n else 0) {0..< k+1}" |
|
3999 |
by (rule sum.cong) (simp_all add: xs del: replicate.simps) |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4000 |
also have "\<dots> = n" using i by simp |
60558 | 4001 |
finally have "xs \<in> natpermute n (k + 1)" |
52902 | 4002 |
using xsl unfolding natpermute_def mem_Collect_eq by blast |
60558 | 4003 |
then show "xs \<in> ?A" |
4004 |
using nxs by blast |
|
4005 |
qed |
|
29687 | 4006 |
qed |
4007 |
||
60558 | 4008 |
text \<open>The general form.\<close> |
64272 | 4009 |
lemma fps_prod_nth: |
52902 | 4010 |
fixes m :: nat |
54681 | 4011 |
and a :: "nat \<Rightarrow> 'a::comm_ring_1 fps" |
64272 | 4012 |
shows "(prod a {0 .. m}) $ n = |
4013 |
sum (\<lambda>v. prod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))" |
|
29687 | 4014 |
(is "?P m n") |
52902 | 4015 |
proof (induct m arbitrary: n rule: nat_less_induct) |
29687 | 4016 |
fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n" |
53196 | 4017 |
show "?P m n" |
4018 |
proof (cases m) |
|
4019 |
case 0 |
|
4020 |
then show ?thesis |
|
4021 |
apply simp |
|
4022 |
unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] |
|
4023 |
apply simp |
|
4024 |
done |
|
4025 |
next |
|
4026 |
case (Suc k) |
|
4027 |
then have km: "k < m" by arith |
|
52902 | 4028 |
have u0: "{0 .. k} \<union> {m} = {0..m}" |
54452 | 4029 |
using Suc by (simp add: set_eq_iff) presburger |
29687 | 4030 |
have f0: "finite {0 .. k}" "finite {m}" by auto |
53196 | 4031 |
have d0: "{0 .. k} \<inter> {m} = {}" using Suc by auto |
64272 | 4032 |
have "(prod a {0 .. m}) $ n = (prod a {0 .. k} * a m) $ n" |
4033 |
unfolding prod.union_disjoint[OF f0 d0, unfolded u0] by simp |
|
29687 | 4034 |
also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j $ v ! j) * a m $ (n - i))" |
4035 |
unfolding fps_mult_nth H[rule_format, OF km] .. |
|
4036 |
also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)" |
|
53196 | 4037 |
apply (simp add: Suc) |
48757 | 4038 |
unfolding natpermute_split[of m "m + 1", simplified, of n, |
53196 | 4039 |
unfolded natlist_trivial_1[unfolded One_nat_def] Suc] |
64267 | 4040 |
apply (subst sum.UNION_disjoint) |
30488 | 4041 |
apply simp |
29687 | 4042 |
apply simp |
4043 |
unfolding image_Collect[symmetric] |
|
4044 |
apply clarsimp |
|
4045 |
apply (rule finite_imageI) |
|
4046 |
apply (rule natpermute_finite) |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
4047 |
apply (clarsimp simp add: set_eq_iff) |
29687 | 4048 |
apply auto |
64267 | 4049 |
apply (rule sum.cong) |
57418 | 4050 |
apply (rule refl) |
64267 | 4051 |
unfolding sum_distrib_right |
29687 | 4052 |
apply (rule sym) |
64267 | 4053 |
apply (rule_tac l = "\<lambda>xs. xs @ [n - x]" in sum.reindex_cong) |
29687 | 4054 |
apply (simp add: inj_on_def) |
4055 |
apply auto |
|
64272 | 4056 |
unfolding prod.union_disjoint[OF f0 d0, unfolded u0, unfolded Suc] |
29687 | 4057 |
apply (clarsimp simp add: natpermute_def nth_append) |
4058 |
done |
|
53196 | 4059 |
finally show ?thesis . |
4060 |
qed |
|
29687 | 4061 |
qed |
4062 |
||
60558 | 4063 |
text \<open>The special form for powers.\<close> |
29687 | 4064 |
lemma fps_power_nth_Suc: |
52903 | 4065 |
fixes m :: nat |
54681 | 4066 |
and a :: "'a::comm_ring_1 fps" |
64272 | 4067 |
shows "(a ^ Suc m)$n = sum (\<lambda>v. prod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))" |
52902 | 4068 |
proof - |
64272 | 4069 |
have th0: "a^Suc m = prod (\<lambda>i. a) {0..m}" |
4070 |
by (simp add: prod_constant) |
|
4071 |
show ?thesis unfolding th0 fps_prod_nth .. |
|
29687 | 4072 |
qed |
52902 | 4073 |
|
29687 | 4074 |
lemma fps_power_nth: |
54452 | 4075 |
fixes m :: nat |
54681 | 4076 |
and a :: "'a::comm_ring_1 fps" |
53196 | 4077 |
shows "(a ^m)$n = |
64272 | 4078 |
(if m=0 then 1$n else sum (\<lambda>v. prod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))" |
52902 | 4079 |
by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc) |
29687 | 4080 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4081 |
lemmas fps_nth_power_0 = fps_power_zeroth |
29687 | 4082 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4083 |
lemma natpermute_max_card: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4084 |
assumes n0: "n \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4085 |
shows "card {xs \<in> natpermute n (k + 1). n \<in> set xs} = k + 1" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4086 |
unfolding natpermute_contain_maximal |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4087 |
proof - |
69085 | 4088 |
let ?A = "\<lambda>i. {(replicate (k + 1) 0)[i := n]}" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4089 |
let ?K = "{0 ..k}" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4090 |
have fK: "finite ?K" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4091 |
by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4092 |
have fAK: "\<forall>i\<in>?K. finite (?A i)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4093 |
by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4094 |
have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow> |
69085 | 4095 |
{(replicate (k + 1) 0)[i := n]} \<inter> {(replicate (k + 1) 0)[j := n]} = {}" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4096 |
proof clarify |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4097 |
fix i j |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4098 |
assume i: "i \<in> ?K" and j: "j \<in> ?K" and ij: "i \<noteq> j" |
69085 | 4099 |
have False if eq: "(replicate (k+1) 0)[i:=n] = (replicate (k+1) 0)[j:= n]" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4100 |
proof - |
69085 | 4101 |
have "(replicate (k+1) 0) [i:=n] ! i = n" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4102 |
using i by (simp del: replicate.simps) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4103 |
moreover |
69085 | 4104 |
have "(replicate (k+1) 0) [j:=n] ! i = 0" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4105 |
using i ij by (simp del: replicate.simps) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4106 |
ultimately show ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4107 |
using eq n0 by (simp del: replicate.simps) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4108 |
qed |
69085 | 4109 |
then show "{(replicate (k + 1) 0)[i := n]} \<inter> {(replicate (k + 1) 0)[j := n]} = {}" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4110 |
by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4111 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4112 |
from card_UN_disjoint[OF fK fAK d] |
69085 | 4113 |
show "card (\<Union>i\<in>{0..k}. {(replicate (k + 1) 0)[i := n]}) = k + 1" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4114 |
by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4115 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4116 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4117 |
lemma fps_power_Suc_nth: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4118 |
fixes f :: "'a :: comm_ring_1 fps" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4119 |
assumes k: "k > 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4120 |
shows "(f ^ Suc m) $ k = |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4121 |
of_nat (Suc m) * (f $ k * (f $ 0) ^ m) + |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4122 |
(\<Sum>v\<in>{v\<in>natpermute k (m+1). k \<notin> set v}. \<Prod>j = 0..m. f $ v ! j)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4123 |
proof - |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4124 |
define A B |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4125 |
where "A = {v\<in>natpermute k (m+1). k \<in> set v}" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4126 |
and "B = {v\<in>natpermute k (m+1). k \<notin> set v}" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4127 |
have [simp]: "finite A" "finite B" "A \<inter> B = {}" by (auto simp: A_def B_def natpermute_finite) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4128 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4129 |
from natpermute_max_card[of k m] k have card_A: "card A = m + 1" by (simp add: A_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4130 |
{ |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4131 |
fix v assume v: "v \<in> A" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4132 |
from v have [simp]: "length v = Suc m" by (simp add: A_def natpermute_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4133 |
from v have "\<exists>j. j \<le> m \<and> v ! j = k" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4134 |
by (auto simp: set_conv_nth A_def natpermute_def less_Suc_eq_le) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4135 |
then guess j by (elim exE conjE) note j = this |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4136 |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
4137 |
from v have "k = sum_list v" by (simp add: A_def natpermute_def) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4138 |
also have "\<dots> = (\<Sum>i=0..m. v ! i)" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
4139 |
by (simp add: sum_list_sum_nth atLeastLessThanSuc_atLeastAtMost del: sum.op_ivl_Suc) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4140 |
also from j have "{0..m} = insert j ({0..m}-{j})" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4141 |
also from j have "(\<Sum>i\<in>\<dots>. v ! i) = k + (\<Sum>i\<in>{0..m}-{j}. v ! i)" |
64267 | 4142 |
by (subst sum.insert) simp_all |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4143 |
finally have "(\<Sum>i\<in>{0..m}-{j}. v ! i) = 0" by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4144 |
hence zero: "v ! i = 0" if "i \<in> {0..m}-{j}" for i using that |
64267 | 4145 |
by (subst (asm) sum_eq_0_iff) auto |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4146 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4147 |
from j have "{0..m} = insert j ({0..m} - {j})" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4148 |
also from j have "(\<Prod>i\<in>\<dots>. f $ (v ! i)) = f $ k * (\<Prod>i\<in>{0..m} - {j}. f $ (v ! i))" |
64272 | 4149 |
by (subst prod.insert) auto |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4150 |
also have "(\<Prod>i\<in>{0..m} - {j}. f $ (v ! i)) = (\<Prod>i\<in>{0..m} - {j}. f $ 0)" |
64272 | 4151 |
by (intro prod.cong) (simp_all add: zero) |
4152 |
also from j have "\<dots> = (f $ 0) ^ m" by (subst prod_constant) simp_all |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4153 |
finally have "(\<Prod>j = 0..m. f $ (v ! j)) = f $ k * (f $ 0) ^ m" . |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4154 |
} note A = this |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4155 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4156 |
have "(f ^ Suc m) $ k = (\<Sum>v\<in>natpermute k (m + 1). \<Prod>j = 0..m. f $ v ! j)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4157 |
by (rule fps_power_nth_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4158 |
also have "natpermute k (m+1) = A \<union> B" unfolding A_def B_def by blast |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4159 |
also have "(\<Sum>v\<in>\<dots>. \<Prod>j = 0..m. f $ (v ! j)) = |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4160 |
(\<Sum>v\<in>A. \<Prod>j = 0..m. f $ (v ! j)) + (\<Sum>v\<in>B. \<Prod>j = 0..m. f $ (v ! j))" |
64267 | 4161 |
by (intro sum.union_disjoint) simp_all |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4162 |
also have "(\<Sum>v\<in>A. \<Prod>j = 0..m. f $ (v ! j)) = of_nat (Suc m) * (f $ k * (f $ 0) ^ m)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4163 |
by (simp add: A card_A) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4164 |
finally show ?thesis by (simp add: B_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4165 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4166 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4167 |
lemma fps_power_Suc_eqD: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4168 |
fixes f g :: "'a :: {idom,semiring_char_0} fps" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4169 |
assumes "f ^ Suc m = g ^ Suc m" "f $ 0 = g $ 0" "f $ 0 \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4170 |
shows "f = g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4171 |
proof (rule fps_ext) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4172 |
fix k :: nat |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4173 |
show "f $ k = g $ k" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4174 |
proof (induction k rule: less_induct) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4175 |
case (less k) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4176 |
show ?case |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4177 |
proof (cases "k = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4178 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4179 |
let ?h = "\<lambda>f. (\<Sum>v | v \<in> natpermute k (m + 1) \<and> k \<notin> set v. \<Prod>j = 0..m. f $ v ! j)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4180 |
from False fps_power_Suc_nth[of k f m] fps_power_Suc_nth[of k g m] |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4181 |
have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h f = |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4182 |
g $ k * (of_nat (Suc m) * (f $ 0) ^ m) + ?h g" using assms |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4183 |
by (simp add: mult_ac del: power_Suc of_nat_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4184 |
also have "v ! i < k" if "v \<in> {v\<in>natpermute k (m+1). k \<notin> set v}" "i \<le> m" for v i |
66311 | 4185 |
using that elem_le_sum_list[of i v] unfolding natpermute_def |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4186 |
by (auto simp: set_conv_nth dest!: spec[of _ i]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4187 |
hence "?h f = ?h g" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4188 |
by (intro sum.cong refl prod.cong less lessI) (simp add: natpermute_def) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4189 |
finally have "f $ k * (of_nat (Suc m) * (f $ 0) ^ m) = g $ k * (of_nat (Suc m) * (f $ 0) ^ m)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4190 |
by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4191 |
with assms show "f $ k = g $ k" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4192 |
by (subst (asm) mult_right_cancel) (auto simp del: of_nat_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4193 |
qed (simp_all add: assms) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4194 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4195 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4196 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4197 |
lemma fps_power_Suc_eqD': |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4198 |
fixes f g :: "'a :: {idom,semiring_char_0} fps" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4199 |
assumes "f ^ Suc m = g ^ Suc m" "f $ subdegree f = g $ subdegree g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4200 |
shows "f = g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4201 |
proof (cases "f = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4202 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4203 |
have "Suc m * subdegree f = subdegree (f ^ Suc m)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4204 |
by (rule subdegree_power [symmetric]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4205 |
also have "f ^ Suc m = g ^ Suc m" by fact |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4206 |
also have "subdegree \<dots> = Suc m * subdegree g" by (rule subdegree_power) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4207 |
finally have [simp]: "subdegree f = subdegree g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4208 |
by (subst (asm) Suc_mult_cancel1) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4209 |
have "fps_shift (subdegree f) f * fps_X ^ subdegree f = f" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4210 |
by (rule subdegree_decompose [symmetric]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4211 |
also have "\<dots> ^ Suc m = g ^ Suc m" by fact |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4212 |
also have "g = fps_shift (subdegree g) g * fps_X ^ subdegree g" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4213 |
by (rule subdegree_decompose) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4214 |
also have "subdegree f = subdegree g" by fact |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4215 |
finally have "fps_shift (subdegree g) f ^ Suc m = fps_shift (subdegree g) g ^ Suc m" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4216 |
by (simp add: algebra_simps power_mult_distrib del: power_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4217 |
hence "fps_shift (subdegree g) f = fps_shift (subdegree g) g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4218 |
by (rule fps_power_Suc_eqD) (insert assms False, auto) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4219 |
with subdegree_decompose[of f] subdegree_decompose[of g] show ?thesis by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4220 |
qed (insert assms, simp_all) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4221 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4222 |
lemma fps_power_eqD': |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4223 |
fixes f g :: "'a :: {idom,semiring_char_0} fps" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4224 |
assumes "f ^ m = g ^ m" "f $ subdegree f = g $ subdegree g" "m > 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4225 |
shows "f = g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4226 |
using fps_power_Suc_eqD'[of f "m-1" g] assms by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4227 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4228 |
lemma fps_power_eqD: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4229 |
fixes f g :: "'a :: {idom,semiring_char_0} fps" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4230 |
assumes "f ^ m = g ^ m" "f $ 0 = g $ 0" "f $ 0 \<noteq> 0" "m > 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4231 |
shows "f = g" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4232 |
by (rule fps_power_eqD'[of f m g]) (insert assms, simp_all) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
4233 |
|
30488 | 4234 |
lemma fps_compose_inj_right: |
54681 | 4235 |
assumes a0: "a$0 = (0::'a::idom)" |
52902 | 4236 |
and a1: "a$1 \<noteq> 0" |
54681 | 4237 |
shows "(b oo a = c oo a) \<longleftrightarrow> b = c" |
4238 |
(is "?lhs \<longleftrightarrow>?rhs") |
|
53196 | 4239 |
proof |
60501 | 4240 |
show ?lhs if ?rhs using that by simp |
4241 |
show ?rhs if ?lhs |
|
4242 |
proof - |
|
4243 |
have "b$n = c$n" for n |
|
53196 | 4244 |
proof (induct n rule: nat_less_induct) |
4245 |
fix n |
|
4246 |
assume H: "\<forall>m<n. b$m = c$m" |
|
60501 | 4247 |
show "b$n = c$n" |
4248 |
proof (cases n) |
|
4249 |
case 0 |
|
4250 |
from \<open>?lhs\<close> have "(b oo a)$n = (c oo a)$n" |
|
4251 |
by simp |
|
4252 |
then show ?thesis |
|
4253 |
using 0 by (simp add: fps_compose_nth) |
|
4254 |
next |
|
4255 |
case (Suc n1) |
|
53196 | 4256 |
have f: "finite {0 .. n1}" "finite {n}" by simp_all |
60501 | 4257 |
have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using Suc by auto |
4258 |
have d: "{0 .. n1} \<inter> {n} = {}" using Suc by auto |
|
53196 | 4259 |
have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)" |
64267 | 4260 |
apply (rule sum.cong) |
60501 | 4261 |
using H Suc |
53196 | 4262 |
apply auto |
4263 |
done |
|
4264 |
have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n" |
|
64267 | 4265 |
unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] seq |
53196 | 4266 |
using startsby_zero_power_nth_same[OF a0] |
4267 |
by simp |
|
4268 |
have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n" |
|
64267 | 4269 |
unfolding fps_compose_nth sum.union_disjoint[OF f d, unfolded eq] |
53196 | 4270 |
using startsby_zero_power_nth_same[OF a0] |
4271 |
by simp |
|
60501 | 4272 |
from \<open>?lhs\<close>[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1 |
4273 |
show ?thesis by auto |
|
4274 |
qed |
|
4275 |
qed |
|
4276 |
then show ?rhs by (simp add: fps_eq_iff) |
|
4277 |
qed |
|
29687 | 4278 |
qed |
4279 |
||
4280 |
||
60500 | 4281 |
subsection \<open>Radicals\<close> |
29687 | 4282 |
|
64272 | 4283 |
declare prod.cong [fundef_cong] |
52903 | 4284 |
|
54681 | 4285 |
function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a::field fps \<Rightarrow> nat \<Rightarrow> 'a" |
52902 | 4286 |
where |
29687 | 4287 |
"radical r 0 a 0 = 1" |
4288 |
| "radical r 0 a (Suc n) = 0" |
|
4289 |
| "radical r (Suc k) a 0 = r (Suc k) (a$0)" |
|
48757 | 4290 |
| "radical r (Suc k) a (Suc n) = |
64272 | 4291 |
(a$ Suc n - sum (\<lambda>xs. prod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k}) |
48757 | 4292 |
{xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) / |
4293 |
(of_nat (Suc k) * (radical r (Suc k) a 0)^k)" |
|
52902 | 4294 |
by pat_completeness auto |
29687 | 4295 |
|
4296 |
termination radical |
|
4297 |
proof |
|
4298 |
let ?R = "measure (\<lambda>(r, k, a, n). n)" |
|
4299 |
{ |
|
52902 | 4300 |
show "wf ?R" by auto |
4301 |
next |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4302 |
fix r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4303 |
and a :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4304 |
and k n xs i |
29687 | 4305 |
assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}" |
60558 | 4306 |
have False if c: "Suc n \<le> xs ! i" |
4307 |
proof - |
|
52902 | 4308 |
from xs i have "xs !i \<noteq> Suc n" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4309 |
by (simp add: in_set_conv_nth natpermute_def) |
29687 | 4310 |
with c have c': "Suc n < xs!i" by arith |
52902 | 4311 |
have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" |
4312 |
by simp_all |
|
4313 |
have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" |
|
4314 |
by auto |
|
4315 |
have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" |
|
4316 |
using i by auto |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
4317 |
from xs have "Suc n = sum_list xs" |
52902 | 4318 |
by (simp add: natpermute_def) |
64267 | 4319 |
also have "\<dots> = sum (nth xs) {0..<Suc k}" using xs |
4320 |
by (simp add: natpermute_def sum_list_sum_nth) |
|
4321 |
also have "\<dots> = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}" |
|
4322 |
unfolding eqs sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] |
|
4323 |
unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)] |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4324 |
by simp |
60558 | 4325 |
finally show ?thesis using c' by simp |
4326 |
qed |
|
52902 | 4327 |
then show "((r, Suc k, a, xs!i), r, Suc k, a, Suc n) \<in> ?R" |
4328 |
apply auto |
|
4329 |
apply (metis not_less) |
|
4330 |
done |
|
4331 |
next |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4332 |
fix r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4333 |
and a :: "'a fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4334 |
and k n |
52902 | 4335 |
show "((r, Suc k, a, 0), r, Suc k, a, Suc n) \<in> ?R" by simp |
4336 |
} |
|
29687 | 4337 |
qed |
4338 |
||
4339 |
definition "fps_radical r n a = Abs_fps (radical r n a)" |
|
4340 |
||
4341 |
lemma fps_radical0[simp]: "fps_radical r 0 a = 1" |
|
52902 | 4342 |
apply (auto simp add: fps_eq_iff fps_radical_def) |
4343 |
apply (case_tac n) |
|
4344 |
apply auto |
|
4345 |
done |
|
29687 | 4346 |
|
60501 | 4347 |
lemma fps_radical_nth_0[simp]: "fps_radical r n a $ 0 = (if n = 0 then 1 else r n (a$0))" |
52902 | 4348 |
by (cases n) (simp_all add: fps_radical_def) |
29687 | 4349 |
|
30488 | 4350 |
lemma fps_radical_power_nth[simp]: |
29687 | 4351 |
assumes r: "(r k (a$0)) ^ k = a$0" |
4352 |
shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)" |
|
53196 | 4353 |
proof (cases k) |
4354 |
case 0 |
|
4355 |
then show ?thesis by simp |
|
4356 |
next |
|
4357 |
case (Suc h) |
|
4358 |
have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)" |
|
4359 |
unfolding fps_power_nth Suc by simp |
|
4360 |
also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))" |
|
64272 | 4361 |
apply (rule prod.cong) |
53196 | 4362 |
apply simp |
4363 |
using Suc |
|
54681 | 4364 |
apply (subgoal_tac "replicate k 0 ! x = 0") |
53196 | 4365 |
apply (auto intro: nth_replicate simp del: replicate.simps) |
4366 |
done |
|
60501 | 4367 |
also have "\<dots> = a$0" |
64272 | 4368 |
using r Suc by (simp add: prod_constant) |
60501 | 4369 |
finally show ?thesis |
4370 |
using Suc by simp |
|
30488 | 4371 |
qed |
29687 | 4372 |
|
30488 | 4373 |
lemma power_radical: |
31273 | 4374 |
fixes a:: "'a::field_char_0 fps" |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4375 |
assumes a0: "a$0 \<noteq> 0" |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4376 |
shows "(r (Suc k) (a$0)) ^ Suc k = a$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a" |
60558 | 4377 |
(is "?lhs \<longleftrightarrow> ?rhs") |
4378 |
proof |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4379 |
let ?r = "fps_radical r (Suc k) a" |
60558 | 4380 |
show ?rhs if r0: ?lhs |
4381 |
proof - |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4382 |
from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto |
60501 | 4383 |
have "?r ^ Suc k $ z = a$z" for z |
4384 |
proof (induct z rule: nat_less_induct) |
|
4385 |
fix n |
|
4386 |
assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m" |
|
4387 |
show "?r ^ Suc k $ n = a $n" |
|
4388 |
proof (cases n) |
|
4389 |
case 0 |
|
4390 |
then show ?thesis |
|
4391 |
using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp |
|
4392 |
next |
|
4393 |
case (Suc n1) |
|
4394 |
then have "n \<noteq> 0" by simp |
|
4395 |
let ?Pnk = "natpermute n (k + 1)" |
|
4396 |
let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}" |
|
4397 |
let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}" |
|
4398 |
have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast |
|
4399 |
have d: "?Pnkn \<inter> ?Pnknn = {}" by blast |
|
4400 |
have f: "finite ?Pnkn" "finite ?Pnknn" |
|
4401 |
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] |
|
4402 |
by (metis natpermute_finite)+ |
|
4403 |
let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j" |
|
64267 | 4404 |
have "sum ?f ?Pnkn = sum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" |
4405 |
proof (rule sum.cong) |
|
60501 | 4406 |
fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}" |
4407 |
let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = |
|
4408 |
fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k" |
|
69085 | 4409 |
from v obtain i where i: "i \<in> {0..k}" "v = (replicate (k+1) 0) [i:= n]" |
60501 | 4410 |
unfolding natpermute_contain_maximal by auto |
4411 |
have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = |
|
4412 |
(\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))" |
|
64272 | 4413 |
apply (rule prod.cong, simp) |
60501 | 4414 |
using i r0 |
4415 |
apply (simp del: replicate.simps) |
|
4416 |
done |
|
4417 |
also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k" |
|
64272 | 4418 |
using i r0 by (simp add: prod_gen_delta) |
60501 | 4419 |
finally show ?ths . |
4420 |
qed rule |
|
64267 | 4421 |
then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k" |
60501 | 4422 |
by (simp add: natpermute_max_card[OF \<open>n \<noteq> 0\<close>, simplified]) |
64267 | 4423 |
also have "\<dots> = a$n - sum ?f ?Pnknn" |
60501 | 4424 |
unfolding Suc using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc) |
64267 | 4425 |
finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" . |
4426 |
have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn" |
|
4427 |
unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] .. |
|
60501 | 4428 |
also have "\<dots> = a$n" unfolding fn by simp |
4429 |
finally show ?thesis . |
|
52903 | 4430 |
qed |
60501 | 4431 |
qed |
60558 | 4432 |
then show ?thesis using r0 by (simp add: fps_eq_iff) |
4433 |
qed |
|
4434 |
show ?lhs if ?rhs |
|
4435 |
proof - |
|
4436 |
from that have "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0" |
|
4437 |
by simp |
|
4438 |
then show ?thesis |
|
52903 | 4439 |
unfolding fps_power_nth_Suc |
64272 | 4440 |
by (simp add: prod_constant del: replicate.simps) |
60558 | 4441 |
qed |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4442 |
qed |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4443 |
|
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4444 |
(* |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4445 |
lemma power_radical: |
31273 | 4446 |
fixes a:: "'a::field_char_0 fps" |
29687 | 4447 |
assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0" |
30488 | 4448 |
shows "(fps_radical r (Suc k) a) ^ (Suc k) = a" |
29687 | 4449 |
proof- |
4450 |
let ?r = "fps_radical r (Suc k) a" |
|
4451 |
from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto |
|
4452 |
{fix z have "?r ^ Suc k $ z = a$z" |
|
4453 |
proof(induct z rule: nat_less_induct) |
|
4454 |
fix n assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m" |
|
54452 | 4455 |
{assume "n = 0" then have "?r ^ Suc k $ n = a $n" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4456 |
using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp} |
29687 | 4457 |
moreover |
4458 |
{fix n1 assume n1: "n = Suc n1" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4459 |
have fK: "finite {0..k}" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4460 |
have nz: "n \<noteq> 0" using n1 by arith |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4461 |
let ?Pnk = "natpermute n (k + 1)" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4462 |
let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4463 |
let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4464 |
have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4465 |
have d: "?Pnkn \<inter> ?Pnknn = {}" by blast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4466 |
have f: "finite ?Pnkn" "finite ?Pnknn" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4467 |
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4468 |
by (metis natpermute_finite)+ |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4469 |
let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j" |
64267 | 4470 |
have "sum ?f ?Pnkn = sum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn" |
4471 |
proof(rule sum.cong2) |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4472 |
fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4473 |
let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = fps_radical r (Suc k) a $ n * r (Suc k) (a $ 0) ^ k" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4474 |
from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4475 |
unfolding natpermute_contain_maximal by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4476 |
have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a $ n else r (Suc k) (a$0))" |
64272 | 4477 |
apply (rule prod.cong, simp) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4478 |
using i r0 by (simp del: replicate.simps) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4479 |
also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k" |
64272 | 4480 |
unfolding prod_gen_delta[OF fK] using i r0 by simp |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4481 |
finally show ?ths . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4482 |
qed |
64267 | 4483 |
then have "sum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4484 |
by (simp add: natpermute_max_card[OF nz, simplified]) |
64267 | 4485 |
also have "\<dots> = a$n - sum ?f ?Pnknn" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4486 |
unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc ) |
64267 | 4487 |
finally have fn: "sum ?f ?Pnkn = a$n - sum ?f ?Pnknn" . |
4488 |
have "(?r ^ Suc k)$n = sum ?f ?Pnkn + sum ?f ?Pnknn" |
|
4489 |
unfolding fps_power_nth_Suc sum.union_disjoint[OF f d, unfolded eq] .. |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4490 |
also have "\<dots> = a$n" unfolding fn by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4491 |
finally have "?r ^ Suc k $ n = a $n" .} |
29687 | 4492 |
ultimately show "?r ^ Suc k $ n = a $n" by (cases n, auto) |
4493 |
qed } |
|
4494 |
then show ?thesis by (simp add: fps_eq_iff) |
|
4495 |
qed |
|
4496 |
||
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4497 |
*) |
52903 | 4498 |
lemma eq_divide_imp': |
60501 | 4499 |
fixes c :: "'a::field" |
4500 |
shows "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c" |
|
56480
093ea91498e6
field_simps: better support for negation and division, and power
hoelzl
parents:
56479
diff
changeset
|
4501 |
by (simp add: field_simps) |
29687 | 4502 |
|
30488 | 4503 |
lemma radical_unique: |
4504 |
assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0" |
|
52903 | 4505 |
and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0" |
4506 |
and b0: "b$0 \<noteq> 0" |
|
29687 | 4507 |
shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b" |
60501 | 4508 |
(is "?lhs \<longleftrightarrow> ?rhs" is "_ \<longleftrightarrow> a = ?r") |
4509 |
proof |
|
4510 |
show ?lhs if ?rhs |
|
4511 |
using that using power_radical[OF b0, of r k, unfolded r0] by simp |
|
4512 |
show ?rhs if ?lhs |
|
4513 |
proof - |
|
4514 |
have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto |
|
29687 | 4515 |
have ceq: "card {0..k} = Suc k" by simp |
4516 |
from a0 have a0r0: "a$0 = ?r$0" by simp |
|
60501 | 4517 |
have "a $ n = ?r $ n" for n |
4518 |
proof (induct n rule: nat_less_induct) |
|
52903 | 4519 |
fix n |
60501 | 4520 |
assume h: "\<forall>m<n. a$m = ?r $m" |
4521 |
show "a$n = ?r $ n" |
|
4522 |
proof (cases n) |
|
4523 |
case 0 |
|
4524 |
then show ?thesis using a0 by simp |
|
4525 |
next |
|
4526 |
case (Suc n1) |
|
4527 |
have fK: "finite {0..k}" by simp |
|
4528 |
have nz: "n \<noteq> 0" using Suc by simp |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4529 |
let ?Pnk = "natpermute n (Suc k)" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4530 |
let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4531 |
let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4532 |
have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4533 |
have d: "?Pnkn \<inter> ?Pnknn = {}" by blast |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4534 |
have f: "finite ?Pnkn" "finite ?Pnknn" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4535 |
using finite_Un[of ?Pnkn ?Pnknn, unfolded eq] |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4536 |
by (metis natpermute_finite)+ |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4537 |
let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4538 |
let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a $ v ! j" |
64267 | 4539 |
have "sum ?g ?Pnkn = sum (\<lambda>v. a $ n * (?r$0)^k) ?Pnkn" |
4540 |
proof (rule sum.cong) |
|
52903 | 4541 |
fix v |
4542 |
assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4543 |
let ?ths = "(\<Prod>j\<in>{0..k}. a $ v ! j) = a $ n * (?r$0)^k" |
69085 | 4544 |
from v obtain i where i: "i \<in> {0..k}" "v = (replicate (k+1) 0) [i:= n]" |
52903 | 4545 |
unfolding Suc_eq_plus1 natpermute_contain_maximal |
4546 |
by (auto simp del: replicate.simps) |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4547 |
have "(\<Prod>j\<in>{0..k}. a $ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then a $ n else r (Suc k) (b$0))" |
64272 | 4548 |
apply (rule prod.cong, simp) |
54452 | 4549 |
using i a0 |
4550 |
apply (simp del: replicate.simps) |
|
52903 | 4551 |
done |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4552 |
also have "\<dots> = a $ n * (?r $ 0)^k" |
64272 | 4553 |
using i by (simp add: prod_gen_delta) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4554 |
finally show ?ths . |
57418 | 4555 |
qed rule |
64267 | 4556 |
then have th0: "sum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4557 |
by (simp add: natpermute_max_card[OF nz, simplified]) |
64267 | 4558 |
have th1: "sum ?g ?Pnknn = sum ?f ?Pnknn" |
64272 | 4559 |
proof (rule sum.cong, rule refl, rule prod.cong, simp) |
52903 | 4560 |
fix xs i |
4561 |
assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}" |
|
60501 | 4562 |
have False if c: "n \<le> xs ! i" |
4563 |
proof - |
|
4564 |
from xs i have "xs ! i \<noteq> n" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4565 |
by (simp add: in_set_conv_nth natpermute_def) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4566 |
with c have c': "n < xs!i" by arith |
52903 | 4567 |
have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" |
4568 |
by simp_all |
|
4569 |
have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" |
|
4570 |
by auto |
|
4571 |
have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" |
|
4572 |
using i by auto |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63589
diff
changeset
|
4573 |
from xs have "n = sum_list xs" |
52903 | 4574 |
by (simp add: natpermute_def) |
64267 | 4575 |
also have "\<dots> = sum (nth xs) {0..<Suc k}" |
4576 |
using xs by (simp add: natpermute_def sum_list_sum_nth) |
|
4577 |
also have "\<dots> = xs!i + sum (nth xs) {0..<i} + sum (nth xs) {i+1..<Suc k}" |
|
4578 |
unfolding eqs sum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)] |
|
4579 |
unfolding sum.union_disjoint[OF fths(2) fths(3) d(2)] |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4580 |
by simp |
60501 | 4581 |
finally show ?thesis using c' by simp |
4582 |
qed |
|
52902 | 4583 |
then have thn: "xs!i < n" by presburger |
52903 | 4584 |
from h[rule_format, OF thn] show "a$(xs !i) = ?r$(xs!i)" . |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4585 |
qed |
54681 | 4586 |
have th00: "\<And>x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x" |
36350 | 4587 |
by (simp add: field_simps del: of_nat_Suc) |
60501 | 4588 |
from \<open>?lhs\<close> have "b$n = a^Suc k $ n" |
52903 | 4589 |
by (simp add: fps_eq_iff) |
64267 | 4590 |
also have "a ^ Suc k$n = sum ?g ?Pnkn + sum ?g ?Pnknn" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4591 |
unfolding fps_power_nth_Suc |
64267 | 4592 |
using sum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric], |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4593 |
unfolded eq, of ?g] by simp |
64267 | 4594 |
also have "\<dots> = of_nat (k+1) * a $ n * (?r $ 0)^k + sum ?f ?Pnknn" |
52903 | 4595 |
unfolding th0 th1 .. |
64267 | 4596 |
finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - sum ?f ?Pnknn" |
52903 | 4597 |
by simp |
64267 | 4598 |
then have "a$n = (b$n - sum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4599 |
apply - |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4600 |
apply (rule eq_divide_imp') |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4601 |
using r00 |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4602 |
apply (simp del: of_nat_Suc) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
4603 |
apply (simp add: ac_simps) |
52903 | 4604 |
done |
60501 | 4605 |
then show ?thesis |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
4606 |
apply (simp del: of_nat_Suc) |
60501 | 4607 |
unfolding fps_radical_def Suc |
4608 |
apply (simp add: field_simps Suc th00 del: of_nat_Suc) |
|
52903 | 4609 |
done |
4610 |
qed |
|
60501 | 4611 |
qed |
4612 |
then show ?rhs by (simp add: fps_eq_iff) |
|
4613 |
qed |
|
29687 | 4614 |
qed |
4615 |
||
4616 |
||
30488 | 4617 |
lemma radical_power: |
4618 |
assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0" |
|
54681 | 4619 |
and a0: "(a$0 :: 'a::field_char_0) \<noteq> 0" |
29687 | 4620 |
shows "(fps_radical r (Suc k) (a ^ Suc k)) = a" |
52903 | 4621 |
proof - |
29687 | 4622 |
let ?ak = "a^ Suc k" |
52903 | 4623 |
have ak0: "?ak $ 0 = (a$0) ^ Suc k" |
4624 |
by (simp add: fps_nth_power_0 del: power_Suc) |
|
4625 |
from r0 have th0: "r (Suc k) (a ^ Suc k $ 0) ^ Suc k = a ^ Suc k $ 0" |
|
4626 |
using ak0 by auto |
|
4627 |
from r0 ak0 have th1: "r (Suc k) (a ^ Suc k $ 0) = a $ 0" |
|
4628 |
by auto |
|
4629 |
from ak0 a0 have ak00: "?ak $ 0 \<noteq>0 " |
|
4630 |
by auto |
|
4631 |
from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis |
|
4632 |
by metis |
|
29687 | 4633 |
qed |
4634 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4635 |
lemma fps_deriv_radical': |
54681 | 4636 |
fixes a :: "'a::field_char_0 fps" |
52903 | 4637 |
assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" |
4638 |
and a0: "a$0 \<noteq> 0" |
|
53196 | 4639 |
shows "fps_deriv (fps_radical r (Suc k) a) = |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4640 |
fps_deriv a / ((of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)" |
52903 | 4641 |
proof - |
4642 |
let ?r = "fps_radical r (Suc k) a" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4643 |
let ?w = "(of_nat (Suc k)) * ?r ^ k" |
52903 | 4644 |
from a0 r0 have r0': "r (Suc k) (a$0) \<noteq> 0" |
4645 |
by auto |
|
4646 |
from r0' have w0: "?w $ 0 \<noteq> 0" |
|
4647 |
by (simp del: of_nat_Suc) |
|
29687 | 4648 |
note th0 = inverse_mult_eq_1[OF w0] |
4649 |
let ?iw = "inverse ?w" |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4650 |
from iffD1[OF power_radical[of a r], OF a0 r0] |
52903 | 4651 |
have "fps_deriv (?r ^ Suc k) = fps_deriv a" |
4652 |
by simp |
|
54452 | 4653 |
then have "fps_deriv ?r * ?w = fps_deriv a" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4654 |
by (simp add: fps_deriv_power' ac_simps del: power_Suc) |
54452 | 4655 |
then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" |
52903 | 4656 |
by simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4657 |
with a0 r0 have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4658 |
by (subst fps_divide_unit) (auto simp del: of_nat_Suc) |
30488 | 4659 |
then show ?thesis unfolding th0 by simp |
29687 | 4660 |
qed |
4661 |
||
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4662 |
lemma fps_deriv_radical: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4663 |
fixes a :: "'a::field_char_0 fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4664 |
assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4665 |
and a0: "a$0 \<noteq> 0" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4666 |
shows "fps_deriv (fps_radical r (Suc k) a) = |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4667 |
fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4668 |
using fps_deriv_radical'[of r k a, OF r0 a0] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4669 |
by (simp add: fps_of_nat[symmetric]) |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4670 |
|
30488 | 4671 |
lemma radical_mult_distrib: |
54681 | 4672 |
fixes a :: "'a::field_char_0 fps" |
48757 | 4673 |
assumes k: "k > 0" |
4674 |
and ra0: "r k (a $ 0) ^ k = a $ 0" |
|
4675 |
and rb0: "r k (b $ 0) ^ k = b $ 0" |
|
60558 | 4676 |
and a0: "a $ 0 \<noteq> 0" |
4677 |
and b0: "b $ 0 \<noteq> 0" |
|
48757 | 4678 |
shows "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0) \<longleftrightarrow> |
60558 | 4679 |
fps_radical r k (a * b) = fps_radical r k a * fps_radical r k b" |
4680 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
4681 |
proof |
|
4682 |
show ?rhs if r0': ?lhs |
|
4683 |
proof - |
|
4684 |
from r0' have r0: "(r k ((a * b) $ 0)) ^ k = (a * b) $ 0" |
|
52903 | 4685 |
by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib) |
60558 | 4686 |
show ?thesis |
60501 | 4687 |
proof (cases k) |
4688 |
case 0 |
|
4689 |
then show ?thesis using r0' by simp |
|
4690 |
next |
|
4691 |
case (Suc h) |
|
52903 | 4692 |
let ?ra = "fps_radical r (Suc h) a" |
4693 |
let ?rb = "fps_radical r (Suc h) b" |
|
4694 |
have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" |
|
60501 | 4695 |
using r0' Suc by (simp add: fps_mult_nth) |
52903 | 4696 |
have ab0: "(a*b) $ 0 \<noteq> 0" |
4697 |
using a0 b0 by (simp add: fps_mult_nth) |
|
60501 | 4698 |
from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded Suc] th0 ab0, symmetric] |
4699 |
iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded Suc]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded Suc]] Suc r0' |
|
4700 |
show ?thesis |
|
4701 |
by (auto simp add: power_mult_distrib simp del: power_Suc) |
|
4702 |
qed |
|
60558 | 4703 |
qed |
4704 |
show ?lhs if ?rhs |
|
4705 |
proof - |
|
4706 |
from that have "(fps_radical r k (a * b)) $ 0 = (fps_radical r k a * fps_radical r k b) $ 0" |
|
52903 | 4707 |
by simp |
60558 | 4708 |
then show ?thesis |
52903 | 4709 |
using k by (simp add: fps_mult_nth) |
60558 | 4710 |
qed |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4711 |
qed |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4712 |
|
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4713 |
(* |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4714 |
lemma radical_mult_distrib: |
31273 | 4715 |
fixes a:: "'a::field_char_0 fps" |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4716 |
assumes |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4717 |
ra0: "r k (a $ 0) ^ k = a $ 0" |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4718 |
and rb0: "r k (b $ 0) ^ k = b $ 0" |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4719 |
and r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)" |
29687 | 4720 |
and a0: "a$0 \<noteq> 0" |
4721 |
and b0: "b$0 \<noteq> 0" |
|
4722 |
shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)" |
|
4723 |
proof- |
|
4724 |
from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0" |
|
4725 |
by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib) |
|
54452 | 4726 |
{assume "k=0" then have ?thesis by simp} |
29687 | 4727 |
moreover |
4728 |
{fix h assume k: "k = Suc h" |
|
4729 |
let ?ra = "fps_radical r (Suc h) a" |
|
4730 |
let ?rb = "fps_radical r (Suc h) b" |
|
30488 | 4731 |
have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0" |
29687 | 4732 |
using r0' k by (simp add: fps_mult_nth) |
4733 |
have ab0: "(a*b) $ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth) |
|
30488 | 4734 |
from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric] |
29687 | 4735 |
power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
29915
diff
changeset
|
4736 |
have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)} |
29687 | 4737 |
ultimately show ?thesis by (cases k, auto) |
4738 |
qed |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4739 |
*) |
29687 | 4740 |
|
4741 |
lemma radical_divide: |
|
31273 | 4742 |
fixes a :: "'a::field_char_0 fps" |
52903 | 4743 |
assumes kp: "k > 0" |
4744 |
and ra0: "(r k (a $ 0)) ^ k = a $ 0" |
|
4745 |
and rb0: "(r k (b $ 0)) ^ k = b $ 0" |
|
4746 |
and a0: "a$0 \<noteq> 0" |
|
4747 |
and b0: "b$0 \<noteq> 0" |
|
4748 |
shows "r k ((a $ 0) / (b$0)) = r k (a$0) / r k (b $ 0) \<longleftrightarrow> |
|
4749 |
fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b" |
|
4750 |
(is "?lhs = ?rhs") |
|
60501 | 4751 |
proof |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4752 |
let ?r = "fps_radical r k" |
60558 | 4753 |
from kp obtain h where k: "k = Suc h" |
4754 |
by (cases k) auto |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4755 |
have ra0': "r k (a$0) \<noteq> 0" using a0 ra0 k by auto |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4756 |
have rb0': "r k (b$0) \<noteq> 0" using b0 rb0 k by auto |
30488 | 4757 |
|
60501 | 4758 |
show ?lhs if ?rhs |
4759 |
proof - |
|
4760 |
from that have "?r (a/b) $ 0 = (?r a / ?r b)$0" |
|
4761 |
by simp |
|
4762 |
then show ?thesis |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4763 |
using k a0 b0 rb0' by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse) |
60501 | 4764 |
qed |
4765 |
show ?rhs if ?lhs |
|
4766 |
proof - |
|
52891 | 4767 |
from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0" |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4768 |
by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def) |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4769 |
have th0: "r k ((a/b)$0) ^ k = (a/b)$0" |
60867 | 4770 |
by (simp add: \<open>?lhs\<close> power_divide ra0 rb0) |
60501 | 4771 |
from a0 b0 ra0' rb0' kp \<open>?lhs\<close> |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4772 |
have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4773 |
by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def divide_inverse) |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4774 |
from a0 b0 ra0' rb0' kp have ab0': "(a / b) $ 0 \<noteq> 0" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4775 |
by (simp add: fps_divide_unit fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero) |
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4776 |
note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]] |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4777 |
note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]] |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4778 |
from b0 rb0' have th2: "(?r a / ?r b)^k = a/b" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
4779 |
by (simp add: fps_divide_unit power_mult_distrib fps_inverse_power[symmetric]) |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
4780 |
|
52902 | 4781 |
from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2] |
60501 | 4782 |
show ?thesis . |
4783 |
qed |
|
29687 | 4784 |
qed |
4785 |
||
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4786 |
lemma radical_inverse: |
31273 | 4787 |
fixes a :: "'a::field_char_0 fps" |
52903 | 4788 |
assumes k: "k > 0" |
4789 |
and ra0: "r k (a $ 0) ^ k = a $ 0" |
|
4790 |
and r1: "(r k 1)^k = 1" |
|
4791 |
and a0: "a$0 \<noteq> 0" |
|
53196 | 4792 |
shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow> |
4793 |
fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a" |
|
31073
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4794 |
using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0 |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4795 |
by (simp add: divide_inverse fps_divide_def) |
4b44c4d08aa6
Generalized distributivity theorems of radicals over multiplication, division and inverses
chaieb
parents:
31021
diff
changeset
|
4796 |
|
60501 | 4797 |
|
4798 |
subsection \<open>Derivative of composition\<close> |
|
29687 | 4799 |
|
30488 | 4800 |
lemma fps_compose_deriv: |
54681 | 4801 |
fixes a :: "'a::idom fps" |
29687 | 4802 |
assumes b0: "b$0 = 0" |
54681 | 4803 |
shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b" |
52903 | 4804 |
proof - |
60501 | 4805 |
have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n |
4806 |
proof - |
|
64267 | 4807 |
have "(fps_deriv (a oo b))$n = sum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}" |
4808 |
by (simp add: fps_compose_def field_simps sum_distrib_left del: of_nat_Suc) |
|
4809 |
also have "\<dots> = sum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}" |
|
36350 | 4810 |
by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc) |
64267 | 4811 |
also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}" |
52903 | 4812 |
unfolding fps_mult_left_const_nth by (simp add: field_simps) |
64267 | 4813 |
also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (sum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}" |
52903 | 4814 |
unfolding fps_mult_nth .. |
64267 | 4815 |
also have "\<dots> = sum (\<lambda>i. of_nat i * a$i * (sum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}" |
4816 |
apply (rule sum.mono_neutral_right) |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4817 |
apply (auto simp add: mult_delta_left not_le) |
52903 | 4818 |
done |
64267 | 4819 |
also have "\<dots> = sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" |
52903 | 4820 |
unfolding fps_deriv_nth |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4821 |
by (rule sum.reindex_cong [of Suc]) (simp_all add: mult.assoc) |
52903 | 4822 |
finally have th0: "(fps_deriv (a oo b))$n = |
64267 | 4823 |
sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" . |
4824 |
||
4825 |
have "(((fps_deriv a) oo b) * (fps_deriv b))$n = sum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
4826 |
unfolding fps_mult_nth by (simp add: ac_simps) |
64267 | 4827 |
also have "\<dots> = sum (\<lambda>i. sum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}" |
4828 |
unfolding fps_deriv_nth fps_compose_nth sum_distrib_left mult.assoc |
|
4829 |
apply (rule sum.cong) |
|
57418 | 4830 |
apply (rule refl) |
64267 | 4831 |
apply (rule sum.mono_neutral_left) |
52903 | 4832 |
apply (simp_all add: subset_eq) |
4833 |
apply clarify |
|
4834 |
apply (subgoal_tac "b^i$x = 0") |
|
4835 |
apply simp |
|
4836 |
apply (rule startsby_zero_power_prefix[OF b0, rule_format]) |
|
4837 |
apply simp |
|
4838 |
done |
|
64267 | 4839 |
also have "\<dots> = sum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (sum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" |
4840 |
unfolding sum_distrib_left |
|
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66550
diff
changeset
|
4841 |
apply (subst sum.swap) |
64267 | 4842 |
apply (rule sum.cong, rule refl)+ |
52903 | 4843 |
apply simp |
4844 |
done |
|
60501 | 4845 |
finally show ?thesis |
52903 | 4846 |
unfolding th0 by simp |
60501 | 4847 |
qed |
52903 | 4848 |
then show ?thesis by (simp add: fps_eq_iff) |
29687 | 4849 |
qed |
4850 |
||
54681 | 4851 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4852 |
subsection \<open>Finite FPS (i.e. polynomials) and fps_X\<close> |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4853 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4854 |
lemma fps_poly_sum_fps_X: |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4855 |
assumes "\<forall>i > n. a$i = 0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4856 |
shows "a = sum (\<lambda>i. fps_const (a$i) * fps_X^i) {0..n}" (is "a = ?r") |
52903 | 4857 |
proof - |
60501 | 4858 |
have "a$i = ?r$i" for i |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4859 |
unfolding fps_sum_nth fps_mult_left_const_nth fps_X_power_nth |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4860 |
by (simp add: mult_delta_right assms) |
60501 | 4861 |
then show ?thesis |
4862 |
unfolding fps_eq_iff by blast |
|
29687 | 4863 |
qed |
4864 |
||
52903 | 4865 |
|
60501 | 4866 |
subsection \<open>Compositional inverses\<close> |
29687 | 4867 |
|
54681 | 4868 |
fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::field" |
52903 | 4869 |
where |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4870 |
"compinv a 0 = fps_X$0" |
52903 | 4871 |
| "compinv a (Suc n) = |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4872 |
(fps_X$ Suc n - sum (\<lambda>i. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" |
29687 | 4873 |
|
4874 |
definition "fps_inv a = Abs_fps (compinv a)" |
|
4875 |
||
52903 | 4876 |
lemma fps_inv: |
4877 |
assumes a0: "a$0 = 0" |
|
4878 |
and a1: "a$1 \<noteq> 0" |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4879 |
shows "fps_inv a oo a = fps_X" |
52903 | 4880 |
proof - |
29687 | 4881 |
let ?i = "fps_inv a oo a" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4882 |
have "?i $n = fps_X$n" for n |
60501 | 4883 |
proof (induct n rule: nat_less_induct) |
52903 | 4884 |
fix n |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4885 |
assume h: "\<forall>m<n. ?i$m = fps_X$m" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4886 |
show "?i $ n = fps_X$n" |
60501 | 4887 |
proof (cases n) |
4888 |
case 0 |
|
4889 |
then show ?thesis using a0 |
|
4890 |
by (simp add: fps_compose_nth fps_inv_def) |
|
4891 |
next |
|
4892 |
case (Suc n1) |
|
64267 | 4893 |
have "?i $ n = sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1" |
60501 | 4894 |
by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc) |
64267 | 4895 |
also have "\<dots> = sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4896 |
(fps_X$ Suc n1 - sum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})" |
60501 | 4897 |
using a0 a1 Suc by (simp add: fps_inv_def) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4898 |
also have "\<dots> = fps_X$n" using Suc by simp |
60501 | 4899 |
finally show ?thesis . |
52903 | 4900 |
qed |
60501 | 4901 |
qed |
4902 |
then show ?thesis |
|
4903 |
by (simp add: fps_eq_iff) |
|
29687 | 4904 |
qed |
4905 |
||
4906 |
||
54681 | 4907 |
fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::field" |
52903 | 4908 |
where |
29687 | 4909 |
"gcompinv b a 0 = b$0" |
52903 | 4910 |
| "gcompinv b a (Suc n) = |
64267 | 4911 |
(b$ Suc n - sum (\<lambda>i. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n" |
29687 | 4912 |
|
4913 |
definition "fps_ginv b a = Abs_fps (gcompinv b a)" |
|
4914 |
||
52903 | 4915 |
lemma fps_ginv: |
4916 |
assumes a0: "a$0 = 0" |
|
4917 |
and a1: "a$1 \<noteq> 0" |
|
29687 | 4918 |
shows "fps_ginv b a oo a = b" |
52903 | 4919 |
proof - |
29687 | 4920 |
let ?i = "fps_ginv b a oo a" |
60501 | 4921 |
have "?i $n = b$n" for n |
4922 |
proof (induct n rule: nat_less_induct) |
|
52903 | 4923 |
fix n |
60501 | 4924 |
assume h: "\<forall>m<n. ?i$m = b$m" |
4925 |
show "?i $ n = b$n" |
|
4926 |
proof (cases n) |
|
4927 |
case 0 |
|
4928 |
then show ?thesis using a0 |
|
4929 |
by (simp add: fps_compose_nth fps_ginv_def) |
|
4930 |
next |
|
4931 |
case (Suc n1) |
|
64267 | 4932 |
have "?i $ n = sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + fps_ginv b a $ Suc n1 * (a $ 1)^ Suc n1" |
60501 | 4933 |
by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc) |
64267 | 4934 |
also have "\<dots> = sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + |
4935 |
(b$ Suc n1 - sum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})" |
|
60501 | 4936 |
using a0 a1 Suc by (simp add: fps_ginv_def) |
4937 |
also have "\<dots> = b$n" using Suc by simp |
|
4938 |
finally show ?thesis . |
|
52903 | 4939 |
qed |
60501 | 4940 |
qed |
4941 |
then show ?thesis |
|
4942 |
by (simp add: fps_eq_iff) |
|
29687 | 4943 |
qed |
4944 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
4945 |
lemma fps_inv_ginv: "fps_inv = fps_ginv fps_X" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
4946 |
apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def) |
48757 | 4947 |
apply (induct_tac n rule: nat_less_induct) |
4948 |
apply auto |
|
29687 | 4949 |
apply (case_tac na) |
4950 |
apply simp |
|
4951 |
apply simp |
|
4952 |
done |
|
4953 |
||
4954 |
lemma fps_compose_1[simp]: "1 oo a = 1" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4955 |
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left) |
29687 | 4956 |
|
4957 |
lemma fps_compose_0[simp]: "0 oo a = 0" |
|
29913 | 4958 |
by (simp add: fps_eq_iff fps_compose_nth) |
29687 | 4959 |
|
60867 | 4960 |
lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a $ 0)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4961 |
by (simp add: fps_eq_iff fps_compose_nth power_0_left sum.neutral) |
29687 | 4962 |
|
4963 |
lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)" |
|
64267 | 4964 |
by (simp add: fps_eq_iff fps_compose_nth field_simps sum.distrib) |
4965 |
||
4966 |
lemma fps_compose_sum_distrib: "(sum f S) oo a = sum (\<lambda>i. f i oo a) S" |
|
52903 | 4967 |
proof (cases "finite S") |
4968 |
case True |
|
4969 |
show ?thesis |
|
4970 |
proof (rule finite_induct[OF True]) |
|
64267 | 4971 |
show "sum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" |
60501 | 4972 |
by simp |
52903 | 4973 |
next |
4974 |
fix x F |
|
4975 |
assume fF: "finite F" |
|
4976 |
and xF: "x \<notin> F" |
|
64267 | 4977 |
and h: "sum f F oo a = sum (\<lambda>i. f i oo a) F" |
4978 |
show "sum f (insert x F) oo a = sum (\<lambda>i. f i oo a) (insert x F)" |
|
52903 | 4979 |
using fF xF h by (simp add: fps_compose_add_distrib) |
4980 |
qed |
|
4981 |
next |
|
4982 |
case False |
|
4983 |
then show ?thesis by simp |
|
29687 | 4984 |
qed |
4985 |
||
30488 | 4986 |
lemma convolution_eq: |
64267 | 4987 |
"sum (\<lambda>i. a (i :: nat) * b (n - i)) {0 .. n} = |
4988 |
sum (\<lambda>(i,j). a i * b j) {(i,j). i \<le> n \<and> j \<le> n \<and> i + j = n}" |
|
4989 |
by (rule sum.reindex_bij_witness[where i=fst and j="\<lambda>i. (i, n - i)"]) auto |
|
29687 | 4990 |
|
4991 |
lemma product_composition_lemma: |
|
52903 | 4992 |
assumes c0: "c$0 = (0::'a::idom)" |
4993 |
and d0: "d$0 = 0" |
|
4994 |
shows "((a oo c) * (b oo d))$n = |
|
64267 | 4995 |
sum (\<lambda>(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m \<le> n}" (is "?l = ?r") |
52903 | 4996 |
proof - |
54681 | 4997 |
let ?S = "{(k::nat, m::nat). k + m \<le> n}" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
4998 |
have s: "?S \<subseteq> {0..n} \<times> {0..n}" by (simp add: subset_eq) |
54681 | 4999 |
have f: "finite {(k::nat, m::nat). k + m \<le> n}" |
29687 | 5000 |
apply (rule finite_subset[OF s]) |
52903 | 5001 |
apply auto |
5002 |
done |
|
64267 | 5003 |
have "?r = sum (\<lambda>i. sum (\<lambda>(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m \<le> n}) {0..n}" |
5004 |
apply (simp add: fps_mult_nth sum_distrib_left) |
|
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66550
diff
changeset
|
5005 |
apply (subst sum.swap) |
64267 | 5006 |
apply (rule sum.cong) |
52903 | 5007 |
apply (auto simp add: field_simps) |
5008 |
done |
|
30488 | 5009 |
also have "\<dots> = ?l" |
64267 | 5010 |
apply (simp add: fps_mult_nth fps_compose_nth sum_product) |
5011 |
apply (rule sum.cong) |
|
57418 | 5012 |
apply (rule refl) |
64267 | 5013 |
apply (simp add: sum.cartesian_product mult.assoc) |
5014 |
apply (rule sum.mono_neutral_right[OF f]) |
|
54452 | 5015 |
apply (simp add: subset_eq) |
5016 |
apply presburger |
|
29687 | 5017 |
apply clarsimp |
5018 |
apply (rule ccontr) |
|
5019 |
apply (clarsimp simp add: not_le) |
|
5020 |
apply (case_tac "x < aa") |
|
5021 |
apply simp |
|
5022 |
apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0]) |
|
5023 |
apply blast |
|
5024 |
apply simp |
|
5025 |
apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0]) |
|
5026 |
apply blast |
|
5027 |
done |
|
5028 |
finally show ?thesis by simp |
|
5029 |
qed |
|
5030 |
||
5031 |
lemma product_composition_lemma': |
|
52903 | 5032 |
assumes c0: "c$0 = (0::'a::idom)" |
5033 |
and d0: "d$0 = 0" |
|
5034 |
shows "((a oo c) * (b oo d))$n = |
|
64267 | 5035 |
sum (\<lambda>k. sum (\<lambda>m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}" (is "?l = ?r") |
29687 | 5036 |
unfolding product_composition_lemma[OF c0 d0] |
64267 | 5037 |
unfolding sum.cartesian_product |
5038 |
apply (rule sum.mono_neutral_left) |
|
29687 | 5039 |
apply simp |
5040 |
apply (clarsimp simp add: subset_eq) |
|
5041 |
apply clarsimp |
|
5042 |
apply (rule ccontr) |
|
5043 |
apply (subgoal_tac "(c^aa * d^ba) $ n = 0") |
|
5044 |
apply simp |
|
5045 |
unfolding fps_mult_nth |
|
64267 | 5046 |
apply (rule sum.neutral) |
29687 | 5047 |
apply (clarsimp simp add: not_le) |
51489 | 5048 |
apply (case_tac "x < aa") |
29687 | 5049 |
apply (rule startsby_zero_power_prefix[OF c0, rule_format]) |
5050 |
apply simp |
|
51489 | 5051 |
apply (subgoal_tac "n - x < ba") |
29687 | 5052 |
apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format]) |
5053 |
apply simp |
|
5054 |
apply arith |
|
5055 |
done |
|
30488 | 5056 |
|
29687 | 5057 |
|
64267 | 5058 |
lemma sum_pair_less_iff: |
5059 |
"sum (\<lambda>((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} = |
|
5060 |
sum (\<lambda>s. sum (\<lambda>i. a i * b (s - i) * c s) {0..s}) {0..n}" |
|
52903 | 5061 |
(is "?l = ?r") |
5062 |
proof - |
|
5063 |
let ?KM = "{(k,m). k + m \<le> n}" |
|
69325 | 5064 |
let ?f = "\<lambda>s. \<Union>i\<in>{0..s}. {(i, s - i)}" |
5065 |
have th0: "?KM = \<Union> (?f ` {0..n})" |
|
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62102
diff
changeset
|
5066 |
by auto |
29687 | 5067 |
show "?l = ?r " |
5068 |
unfolding th0 |
|
64267 | 5069 |
apply (subst sum.UNION_disjoint) |
29687 | 5070 |
apply auto |
64267 | 5071 |
apply (subst sum.UNION_disjoint) |
29687 | 5072 |
apply auto |
5073 |
done |
|
5074 |
qed |
|
5075 |
||
5076 |
lemma fps_compose_mult_distrib_lemma: |
|
5077 |
assumes c0: "c$0 = (0::'a::idom)" |
|
64267 | 5078 |
shows "((a oo c) * (b oo c))$n = sum (\<lambda>s. sum (\<lambda>i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}" |
29687 | 5079 |
unfolding product_composition_lemma[OF c0 c0] power_add[symmetric] |
64267 | 5080 |
unfolding sum_pair_less_iff[where a = "\<lambda>k. a$k" and b="\<lambda>m. b$m" and c="\<lambda>s. (c ^ s)$n" and n = n] .. |
29687 | 5081 |
|
30488 | 5082 |
lemma fps_compose_mult_distrib: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54452
diff
changeset
|
5083 |
assumes c0: "c $ 0 = (0::'a::idom)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54452
diff
changeset
|
5084 |
shows "(a * b) oo c = (a oo c) * (b oo c)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54452
diff
changeset
|
5085 |
apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0]) |
64267 | 5086 |
apply (simp add: fps_compose_nth fps_mult_nth sum_distrib_right) |
52903 | 5087 |
done |
5088 |
||
64272 | 5089 |
lemma fps_compose_prod_distrib: |
29687 | 5090 |
assumes c0: "c$0 = (0::'a::idom)" |
64272 | 5091 |
shows "prod a S oo c = prod (\<lambda>k. a k oo c) S" |
29687 | 5092 |
apply (cases "finite S") |
5093 |
apply simp_all |
|
5094 |
apply (induct S rule: finite_induct) |
|
5095 |
apply simp |
|
5096 |
apply (simp add: fps_compose_mult_distrib[OF c0]) |
|
5097 |
done |
|
5098 |
||
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5099 |
lemma fps_compose_divide: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5100 |
assumes [simp]: "g dvd f" "h $ 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5101 |
shows "fps_compose f h = fps_compose (f / g :: 'a :: field fps) h * fps_compose g h" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5102 |
proof - |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5103 |
have "f = (f / g) * g" by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5104 |
also have "fps_compose \<dots> h = fps_compose (f / g) h * fps_compose g h" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5105 |
by (subst fps_compose_mult_distrib) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5106 |
finally show ?thesis . |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5107 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5108 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5109 |
lemma fps_compose_divide_distrib: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5110 |
assumes "g dvd f" "h $ 0 = 0" "fps_compose g h \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5111 |
shows "fps_compose (f / g :: 'a :: field fps) h = fps_compose f h / fps_compose g h" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5112 |
using fps_compose_divide[OF assms(1,2)] assms(3) by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5113 |
|
53195 | 5114 |
lemma fps_compose_power: |
5115 |
assumes c0: "c$0 = (0::'a::idom)" |
|
5116 |
shows "(a oo c)^n = a^n oo c" |
|
52903 | 5117 |
proof (cases n) |
5118 |
case 0 |
|
5119 |
then show ?thesis by simp |
|
5120 |
next |
|
5121 |
case (Suc m) |
|
67970 | 5122 |
have "(\<Prod>n = 0..m. a) oo c = (\<Prod>n = 0..m. a oo c)" |
5123 |
using c0 fps_compose_prod_distrib by blast |
|
5124 |
moreover have th0: "a^n = prod (\<lambda>k. a) {0..m}" "(a oo c) ^ n = prod (\<lambda>k. a oo c) {0..m}" |
|
64272 | 5125 |
by (simp_all add: prod_constant Suc) |
67970 | 5126 |
ultimately show ?thesis |
5127 |
by presburger |
|
29687 | 5128 |
qed |
5129 |
||
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5130 |
lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)" |
64267 | 5131 |
by (simp add: fps_eq_iff fps_compose_nth field_simps sum_negf[symmetric]) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5132 |
|
52903 | 5133 |
lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53374
diff
changeset
|
5134 |
using fps_compose_add_distrib [of a "- b" c] by (simp add: fps_compose_uminus) |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5135 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5136 |
lemma fps_X_fps_compose: "fps_X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a$n)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5137 |
by (simp add: fps_eq_iff fps_compose_nth mult_delta_left) |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5138 |
|
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5139 |
lemma fps_inverse_compose: |
52903 | 5140 |
assumes b0: "(b$0 :: 'a::field) = 0" |
5141 |
and a0: "a$0 \<noteq> 0" |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5142 |
shows "inverse a oo b = inverse (a oo b)" |
52903 | 5143 |
proof - |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5144 |
let ?ia = "inverse a" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5145 |
let ?ab = "a oo b" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5146 |
let ?iab = "inverse ?ab" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5147 |
|
52903 | 5148 |
from a0 have ia0: "?ia $ 0 \<noteq> 0" by simp |
5149 |
from a0 have ab0: "?ab $ 0 \<noteq> 0" by (simp add: fps_compose_def) |
|
5150 |
have "(?ia oo b) * (a oo b) = 1" |
|
5151 |
unfolding fps_compose_mult_distrib[OF b0, symmetric] |
|
5152 |
unfolding inverse_mult_eq_1[OF a0] |
|
5153 |
fps_compose_1 .. |
|
54452 | 5154 |
|
52903 | 5155 |
then have "(?ia oo b) * (a oo b) * ?iab = 1 * ?iab" by simp |
5156 |
then have "(?ia oo b) * (?iab * (a oo b)) = ?iab" by simp |
|
5157 |
then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5158 |
qed |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5159 |
|
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5160 |
lemma fps_divide_compose: |
52903 | 5161 |
assumes c0: "(c$0 :: 'a::field) = 0" |
5162 |
and b0: "b$0 \<noteq> 0" |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5163 |
shows "(a/b) oo c = (a oo c) / (b oo c)" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
5164 |
using b0 c0 by (simp add: fps_divide_unit fps_inverse_compose fps_compose_mult_distrib) |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5165 |
|
52903 | 5166 |
lemma gp: |
5167 |
assumes a0: "a$0 = (0::'a::field)" |
|
5168 |
shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)" |
|
5169 |
(is "?one oo a = _") |
|
5170 |
proof - |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5171 |
have o0: "?one $ 0 \<noteq> 0" by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5172 |
have th0: "(1 - fps_X) $ 0 \<noteq> (0::'a)" by simp |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5173 |
from fps_inverse_gp[where ?'a = 'a] |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5174 |
have "inverse ?one = 1 - fps_X" by (simp add: fps_eq_iff) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5175 |
then have "inverse (inverse ?one) = inverse (1 - fps_X)" by simp |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5176 |
then have th: "?one = 1/(1 - fps_X)" unfolding fps_inverse_idempotent[OF o0] |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5177 |
by (simp add: fps_divide_def) |
52903 | 5178 |
show ?thesis |
5179 |
unfolding th |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5180 |
unfolding fps_divide_compose[OF a0 th0] |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5181 |
fps_compose_1 fps_compose_sub_distrib fps_X_fps_compose_startby0[OF a0] .. |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5182 |
qed |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5183 |
|
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5184 |
lemma fps_compose_radical: |
31273 | 5185 |
assumes b0: "b$0 = (0::'a::field_char_0)" |
52903 | 5186 |
and ra0: "r (Suc k) (a$0) ^ Suc k = a$0" |
5187 |
and a0: "a$0 \<noteq> 0" |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5188 |
shows "fps_radical r (Suc k) a oo b = fps_radical r (Suc k) (a oo b)" |
52903 | 5189 |
proof - |
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5190 |
let ?r = "fps_radical r (Suc k)" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5191 |
let ?ab = "a oo b" |
52903 | 5192 |
have ab0: "?ab $ 0 = a$0" |
5193 |
by (simp add: fps_compose_def) |
|
5194 |
from ab0 a0 ra0 have rab0: "?ab $ 0 \<noteq> 0" "r (Suc k) (?ab $ 0) ^ Suc k = ?ab $ 0" |
|
5195 |
by simp_all |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5196 |
have th00: "r (Suc k) ((a oo b) $ 0) = (fps_radical r (Suc k) a oo b) $ 0" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5197 |
by (simp add: ab0 fps_compose_def) |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5198 |
have th0: "(?r a oo b) ^ (Suc k) = a oo b" |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5199 |
unfolding fps_compose_power[OF b0] |
52891 | 5200 |
unfolding iffD1[OF power_radical[of a r k], OF a0 ra0] .. |
52903 | 5201 |
from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0] |
5202 |
show ?thesis . |
|
31199
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5203 |
qed |
10d413b08fa7
FPS composition distributes over inverses, division and arbitrary nth roots. General geometric series theorem
chaieb
parents:
31148
diff
changeset
|
5204 |
|
52903 | 5205 |
lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b" |
64267 | 5206 |
by (simp add: fps_eq_iff fps_compose_nth sum_distrib_left mult.assoc) |
29687 | 5207 |
|
5208 |
lemma fps_const_mult_apply_right: |
|
5209 |
"(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5210 |
by (simp add: fps_const_mult_apply_left mult.commute) |
29687 | 5211 |
|
30488 | 5212 |
lemma fps_compose_assoc: |
52903 | 5213 |
assumes c0: "c$0 = (0::'a::idom)" |
5214 |
and b0: "b$0 = 0" |
|
29687 | 5215 |
shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r") |
52903 | 5216 |
proof - |
60501 | 5217 |
have "?l$n = ?r$n" for n |
5218 |
proof - |
|
64267 | 5219 |
have "?l$n = (sum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n" |
52903 | 5220 |
by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left |
64267 | 5221 |
sum_distrib_left mult.assoc fps_sum_nth) |
5222 |
also have "\<dots> = ((sum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n" |
|
5223 |
by (simp add: fps_compose_sum_distrib) |
|
29687 | 5224 |
also have "\<dots> = ?r$n" |
64267 | 5225 |
apply (simp add: fps_compose_nth fps_sum_nth sum_distrib_right mult.assoc) |
5226 |
apply (rule sum.cong) |
|
57418 | 5227 |
apply (rule refl) |
64267 | 5228 |
apply (rule sum.mono_neutral_right) |
29687 | 5229 |
apply (auto simp add: not_le) |
52903 | 5230 |
apply (erule startsby_zero_power_prefix[OF b0, rule_format]) |
5231 |
done |
|
60501 | 5232 |
finally show ?thesis . |
5233 |
qed |
|
5234 |
then show ?thesis |
|
5235 |
by (simp add: fps_eq_iff) |
|
29687 | 5236 |
qed |
5237 |
||
5238 |
||
5239 |
lemma fps_X_power_compose: |
|
52903 | 5240 |
assumes a0: "a$0=0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5241 |
shows "fps_X^k oo a = (a::'a::idom fps)^k" |
54681 | 5242 |
(is "?l = ?r") |
52903 | 5243 |
proof (cases k) |
5244 |
case 0 |
|
5245 |
then show ?thesis by simp |
|
5246 |
next |
|
53196 | 5247 |
case (Suc h) |
60501 | 5248 |
have "?l $ n = ?r $n" for n |
5249 |
proof - |
|
5250 |
consider "k > n" | "k \<le> n" by arith |
|
5251 |
then show ?thesis |
|
5252 |
proof cases |
|
5253 |
case 1 |
|
5254 |
then show ?thesis |
|
5255 |
using a0 startsby_zero_power_prefix[OF a0] Suc |
|
52903 | 5256 |
by (simp add: fps_compose_nth del: power_Suc) |
60501 | 5257 |
next |
5258 |
case 2 |
|
5259 |
then show ?thesis |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5260 |
by (simp add: fps_compose_nth mult_delta_left) |
60501 | 5261 |
qed |
5262 |
qed |
|
5263 |
then show ?thesis |
|
5264 |
unfolding fps_eq_iff by blast |
|
29687 | 5265 |
qed |
5266 |
||
52903 | 5267 |
lemma fps_inv_right: |
5268 |
assumes a0: "a$0 = 0" |
|
5269 |
and a1: "a$1 \<noteq> 0" |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5270 |
shows "a oo fps_inv a = fps_X" |
52903 | 5271 |
proof - |
29687 | 5272 |
let ?ia = "fps_inv a" |
5273 |
let ?iaa = "a oo fps_inv a" |
|
60501 | 5274 |
have th0: "?ia $ 0 = 0" |
5275 |
by (simp add: fps_inv_def) |
|
5276 |
have th1: "?iaa $ 0 = 0" |
|
5277 |
using a0 a1 by (simp add: fps_inv_def fps_compose_nth) |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5278 |
have th2: "fps_X$0 = 0" |
60501 | 5279 |
by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5280 |
from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo fps_X" |
60501 | 5281 |
by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5282 |
then have "(a oo fps_inv a) oo a = fps_X oo a" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5283 |
by (simp add: fps_compose_assoc[OF a0 th0] fps_X_fps_compose_startby0[OF a0]) |
60501 | 5284 |
with fps_compose_inj_right[OF a0 a1] show ?thesis |
5285 |
by simp |
|
29687 | 5286 |
qed |
5287 |
||
5288 |
lemma fps_inv_deriv: |
|
60501 | 5289 |
assumes a0: "a$0 = (0::'a::field)" |
52903 | 5290 |
and a1: "a$1 \<noteq> 0" |
29687 | 5291 |
shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)" |
52903 | 5292 |
proof - |
29687 | 5293 |
let ?ia = "fps_inv a" |
5294 |
let ?d = "fps_deriv a oo ?ia" |
|
5295 |
let ?dia = "fps_deriv ?ia" |
|
60501 | 5296 |
have ia0: "?ia$0 = 0" |
5297 |
by (simp add: fps_inv_def) |
|
5298 |
have th0: "?d$0 \<noteq> 0" |
|
5299 |
using a1 by (simp add: fps_compose_nth) |
|
29687 | 5300 |
from fps_inv_right[OF a0 a1] have "?d * ?dia = 1" |
5301 |
by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] ) |
|
60501 | 5302 |
then have "inverse ?d * ?d * ?dia = inverse ?d * 1" |
5303 |
by simp |
|
5304 |
with inverse_mult_eq_1 [OF th0] show "?dia = inverse ?d" |
|
5305 |
by simp |
|
29687 | 5306 |
qed |
5307 |
||
52891 | 5308 |
lemma fps_inv_idempotent: |
52903 | 5309 |
assumes a0: "a$0 = 0" |
5310 |
and a1: "a$1 \<noteq> 0" |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5311 |
shows "fps_inv (fps_inv a) = a" |
52903 | 5312 |
proof - |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5313 |
let ?r = "fps_inv" |
60501 | 5314 |
have ra0: "?r a $ 0 = 0" |
5315 |
by (simp add: fps_inv_def) |
|
5316 |
from a1 have ra1: "?r a $ 1 \<noteq> 0" |
|
5317 |
by (simp add: fps_inv_def field_simps) |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5318 |
have fps_X0: "fps_X$0 = 0" |
60501 | 5319 |
by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5320 |
from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = fps_X" . |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5321 |
then have "?r (?r a) oo ?r a oo a = fps_X oo a" |
60501 | 5322 |
by simp |
52891 | 5323 |
then have "?r (?r a) oo (?r a oo a) = a" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5324 |
unfolding fps_X_fps_compose_startby0[OF a0] |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5325 |
unfolding fps_compose_assoc[OF a0 ra0, symmetric] . |
60501 | 5326 |
then show ?thesis |
5327 |
unfolding fps_inv[OF a0 a1] by simp |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5328 |
qed |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5329 |
|
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5330 |
lemma fps_ginv_ginv: |
52903 | 5331 |
assumes a0: "a$0 = 0" |
5332 |
and a1: "a$1 \<noteq> 0" |
|
5333 |
and c0: "c$0 = 0" |
|
5334 |
and c1: "c$1 \<noteq> 0" |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5335 |
shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c" |
52903 | 5336 |
proof - |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5337 |
let ?r = "fps_ginv" |
60501 | 5338 |
from c0 have rca0: "?r c a $0 = 0" |
5339 |
by (simp add: fps_ginv_def) |
|
5340 |
from a1 c1 have rca1: "?r c a $ 1 \<noteq> 0" |
|
5341 |
by (simp add: fps_ginv_def field_simps) |
|
52891 | 5342 |
from fps_ginv[OF rca0 rca1] |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5343 |
have "?r b (?r c a) oo ?r c a = b" . |
60501 | 5344 |
then have "?r b (?r c a) oo ?r c a oo a = b oo a" |
5345 |
by simp |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5346 |
then have "?r b (?r c a) oo (?r c a oo a) = b oo a" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5347 |
apply (subst fps_compose_assoc) |
53195 | 5348 |
using a0 c0 |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5349 |
apply (simp_all add: fps_ginv_def) |
52903 | 5350 |
done |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5351 |
then have "?r b (?r c a) oo c = b oo a" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5352 |
unfolding fps_ginv[OF a0 a1] . |
60501 | 5353 |
then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c" |
5354 |
by simp |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5355 |
then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5356 |
apply (subst fps_compose_assoc) |
53195 | 5357 |
using a0 c0 |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5358 |
apply (simp_all add: fps_inv_def) |
52903 | 5359 |
done |
60501 | 5360 |
then show ?thesis |
5361 |
unfolding fps_inv_right[OF c0 c1] by simp |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5362 |
qed |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5363 |
|
32410 | 5364 |
lemma fps_ginv_deriv: |
54681 | 5365 |
assumes a0:"a$0 = (0::'a::field)" |
52903 | 5366 |
and a1: "a$1 \<noteq> 0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5367 |
shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv fps_X a" |
52903 | 5368 |
proof - |
32410 | 5369 |
let ?ia = "fps_ginv b a" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5370 |
let ?ifps_Xa = "fps_ginv fps_X a" |
32410 | 5371 |
let ?d = "fps_deriv" |
5372 |
let ?dia = "?d ?ia" |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5373 |
have ifps_Xa0: "?ifps_Xa $ 0 = 0" |
60501 | 5374 |
by (simp add: fps_ginv_def) |
5375 |
have da0: "?d a $ 0 \<noteq> 0" |
|
5376 |
using a1 by simp |
|
5377 |
from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" |
|
5378 |
by simp |
|
5379 |
then have "(?d ?ia oo a) * ?d a = ?d b" |
|
5380 |
unfolding fps_compose_deriv[OF a0] . |
|
5381 |
then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" |
|
5382 |
by simp |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
5383 |
with a1 have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
5384 |
by (simp add: fps_divide_unit) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5385 |
then have "(?d ?ia oo a) oo ?ifps_Xa = (?d b / ?d a) oo ?ifps_Xa" |
32410 | 5386 |
unfolding inverse_mult_eq_1[OF da0] by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5387 |
then have "?d ?ia oo (a oo ?ifps_Xa) = (?d b / ?d a) oo ?ifps_Xa" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5388 |
unfolding fps_compose_assoc[OF ifps_Xa0 a0] . |
32410 | 5389 |
then show ?thesis unfolding fps_inv_ginv[symmetric] |
5390 |
unfolding fps_inv_right[OF a0 a1] by simp |
|
5391 |
qed |
|
5392 |
||
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5393 |
lemma fps_compose_linear: |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5394 |
"fps_compose (f :: 'a :: comm_ring_1 fps) (fps_const c * fps_X) = Abs_fps (\<lambda>n. c^n * f $ n)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5395 |
by (simp add: fps_eq_iff fps_compose_def power_mult_distrib |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5396 |
if_distrib cong: if_cong) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5397 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5398 |
lemma fps_compose_uminus': |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5399 |
"fps_compose f (-fps_X :: 'a :: comm_ring_1 fps) = Abs_fps (\<lambda>n. (-1)^n * f $ n)" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5400 |
using fps_compose_linear[of f "-1"] |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5401 |
by (simp only: fps_const_neg [symmetric] fps_const_1_eq_1) simp |
60501 | 5402 |
|
5403 |
subsection \<open>Elementary series\<close> |
|
5404 |
||
5405 |
subsubsection \<open>Exponential series\<close> |
|
53195 | 5406 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5407 |
definition "fps_exp x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5408 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5409 |
lemma fps_exp_deriv[simp]: "fps_deriv (fps_exp a) = fps_const (a::'a::field_char_0) * fps_exp a" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5410 |
(is "?l = ?r") |
52903 | 5411 |
proof - |
60501 | 5412 |
have "?l$n = ?r $ n" for n |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5413 |
apply (auto simp add: fps_exp_def field_simps power_Suc[symmetric] |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63317
diff
changeset
|
5414 |
simp del: fact_Suc of_nat_Suc power_Suc) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5415 |
apply (simp add: field_simps) |
60501 | 5416 |
done |
5417 |
then show ?thesis |
|
5418 |
by (simp add: fps_eq_iff) |
|
29687 | 5419 |
qed |
5420 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5421 |
lemma fps_exp_unique_ODE: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5422 |
"fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * fps_exp (c::'a::field_char_0)" |
29687 | 5423 |
(is "?lhs \<longleftrightarrow> ?rhs") |
52903 | 5424 |
proof |
60501 | 5425 |
show ?rhs if ?lhs |
5426 |
proof - |
|
5427 |
from that have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)" |
|
5428 |
by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc) |
|
5429 |
have th': "a$n = a$0 * c ^ n/ (fact n)" for n |
|
5430 |
proof (induct n) |
|
5431 |
case 0 |
|
5432 |
then show ?case by simp |
|
5433 |
next |
|
5434 |
case Suc |
|
5435 |
then show ?case |
|
5436 |
unfolding th |
|
5437 |
using fact_gt_zero |
|
5438 |
apply (simp add: field_simps del: of_nat_Suc fact_Suc) |
|
5439 |
apply simp |
|
5440 |
done |
|
5441 |
qed |
|
5442 |
show ?thesis |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5443 |
by (auto simp add: fps_eq_iff fps_const_mult_left fps_exp_def intro: th') |
60501 | 5444 |
qed |
5445 |
show ?lhs if ?rhs |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5446 |
using that by (metis fps_exp_deriv fps_deriv_mult_const_left mult.left_commute) |
29687 | 5447 |
qed |
5448 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5449 |
lemma fps_exp_add_mult: "fps_exp (a + b) = fps_exp (a::'a::field_char_0) * fps_exp b" (is "?l = ?r") |
52903 | 5450 |
proof - |
60501 | 5451 |
have "fps_deriv ?r = fps_const (a + b) * ?r" |
36350 | 5452 |
by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add) |
60501 | 5453 |
then have "?r = ?l" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5454 |
by (simp only: fps_exp_unique_ODE) (simp add: fps_mult_nth fps_exp_def) |
29687 | 5455 |
then show ?thesis .. |
5456 |
qed |
|
5457 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5458 |
lemma fps_exp_nth[simp]: "fps_exp a $ n = a^n / of_nat (fact n)" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5459 |
by (simp add: fps_exp_def) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5460 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5461 |
lemma fps_exp_0[simp]: "fps_exp (0::'a::field) = 1" |
29687 | 5462 |
by (simp add: fps_eq_iff power_0_left) |
5463 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5464 |
lemma fps_exp_neg: "fps_exp (- a) = inverse (fps_exp (a::'a::field_char_0))" |
52903 | 5465 |
proof - |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5466 |
from fps_exp_add_mult[of a "- a"] have th0: "fps_exp a * fps_exp (- a) = 1" by simp |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
5467 |
from fps_inverse_unique[OF th0] show ?thesis by simp |
29687 | 5468 |
qed |
5469 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5470 |
lemma fps_exp_nth_deriv[simp]: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5471 |
"fps_nth_deriv n (fps_exp (a::'a::field_char_0)) = (fps_const a)^n * (fps_exp a)" |
52902 | 5472 |
by (induct n) auto |
29687 | 5473 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5474 |
lemma fps_X_compose_fps_exp[simp]: "fps_X oo fps_exp (a::'a::field) = fps_exp a - 1" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5475 |
by (simp add: fps_eq_iff fps_X_fps_compose) |
29687 | 5476 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5477 |
lemma fps_inv_fps_exp_compose: |
60501 | 5478 |
assumes a: "a \<noteq> 0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5479 |
shows "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5480 |
and "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" |
53195 | 5481 |
proof - |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5482 |
let ?b = "fps_exp a - 1" |
60501 | 5483 |
have b0: "?b $ 0 = 0" |
5484 |
by simp |
|
5485 |
have b1: "?b $ 1 \<noteq> 0" |
|
5486 |
by (simp add: a) |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5487 |
from fps_inv[OF b0 b1] show "fps_inv (fps_exp a - 1) oo (fps_exp a - 1) = fps_X" . |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5488 |
from fps_inv_right[OF b0 b1] show "(fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_X" . |
29687 | 5489 |
qed |
5490 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5491 |
lemma fps_exp_power_mult: "(fps_exp (c::'a::field_char_0))^n = fps_exp (of_nat n * c)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5492 |
by (induct n) (simp_all add: field_simps fps_exp_add_mult) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5493 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5494 |
lemma radical_fps_exp: |
52891 | 5495 |
assumes r: "r (Suc k) 1 = 1" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5496 |
shows "fps_radical r (Suc k) (fps_exp (c::'a::field_char_0)) = fps_exp (c / of_nat (Suc k))" |
52903 | 5497 |
proof - |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5498 |
let ?ck = "(c / of_nat (Suc k))" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5499 |
let ?r = "fps_radical r (Suc k)" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5500 |
have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c" |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5501 |
by (simp_all del: of_nat_Suc) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5502 |
have th0: "fps_exp ?ck ^ (Suc k) = fps_exp c" unfolding fps_exp_power_mult eq0 .. |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5503 |
have th: "r (Suc k) (fps_exp c $0) ^ Suc k = fps_exp c $ 0" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5504 |
"r (Suc k) (fps_exp c $ 0) = fps_exp ?ck $ 0" "fps_exp c $ 0 \<noteq> 0" using r by simp_all |
60501 | 5505 |
from th0 radical_unique[where r=r and k=k, OF th] show ?thesis |
5506 |
by auto |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5507 |
qed |
29687 | 5508 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5509 |
lemma fps_exp_compose_linear [simp]: |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5510 |
"fps_exp (d::'a::field_char_0) oo (fps_const c * fps_X) = fps_exp (c * d)" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5511 |
by (simp add: fps_compose_linear fps_exp_def fps_eq_iff power_mult_distrib) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5512 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5513 |
lemma fps_fps_exp_compose_minus [simp]: |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5514 |
"fps_compose (fps_exp c) (-fps_X) = fps_exp (-c :: 'a :: field_char_0)" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5515 |
using fps_exp_compose_linear[of c "-1 :: 'a"] |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5516 |
unfolding fps_const_neg [symmetric] fps_const_1_eq_1 by simp |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5517 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5518 |
lemma fps_exp_eq_iff [simp]: "fps_exp c = fps_exp d \<longleftrightarrow> c = (d :: 'a :: field_char_0)" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5519 |
proof |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5520 |
assume "fps_exp c = fps_exp d" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5521 |
from arg_cong[of _ _ "\<lambda>F. F $ 1", OF this] show "c = d" by simp |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5522 |
qed simp_all |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5523 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5524 |
lemma fps_exp_eq_fps_const_iff [simp]: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5525 |
"fps_exp (c :: 'a :: field_char_0) = fps_const c' \<longleftrightarrow> c = 0 \<and> c' = 1" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5526 |
proof |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5527 |
assume "c = 0 \<and> c' = 1" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5528 |
thus "fps_exp c = fps_const c'" by (simp add: fps_eq_iff) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5529 |
next |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5530 |
assume "fps_exp c = fps_const c'" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5531 |
from arg_cong[of _ _ "\<lambda>F. F $ 1", OF this] arg_cong[of _ _ "\<lambda>F. F $ 0", OF this] |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5532 |
show "c = 0 \<and> c' = 1" by simp_all |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5533 |
qed |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5534 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5535 |
lemma fps_exp_neq_0 [simp]: "\<not>fps_exp (c :: 'a :: field_char_0) = 0" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5536 |
unfolding fps_const_0_eq_0 [symmetric] fps_exp_eq_fps_const_iff by simp |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5537 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5538 |
lemma fps_exp_eq_1_iff [simp]: "fps_exp (c :: 'a :: field_char_0) = 1 \<longleftrightarrow> c = 0" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5539 |
unfolding fps_const_1_eq_1 [symmetric] fps_exp_eq_fps_const_iff by simp |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5540 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5541 |
lemma fps_exp_neq_numeral_iff [simp]: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5542 |
"fps_exp (c :: 'a :: field_char_0) = numeral n \<longleftrightarrow> c = 0 \<and> n = Num.One" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5543 |
unfolding numeral_fps_const fps_exp_eq_fps_const_iff by simp |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5544 |
|
53195 | 5545 |
|
60501 | 5546 |
subsubsection \<open>Logarithmic series\<close> |
29687 | 5547 |
|
52891 | 5548 |
lemma Abs_fps_if_0: |
60501 | 5549 |
"Abs_fps (\<lambda>n. if n = 0 then (v::'a::ring_1) else f n) = |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5550 |
fps_const v + fps_X * Abs_fps (\<lambda>n. f (Suc n))" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5551 |
by (simp add: fps_eq_iff) |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5552 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5553 |
definition fps_ln :: "'a::field_char_0 \<Rightarrow> 'a fps" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5554 |
where "fps_ln c = fps_const (1/c) * Abs_fps (\<lambda>n. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5555 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5556 |
lemma fps_ln_deriv: "fps_deriv (fps_ln c) = fps_const (1/c) * inverse (1 + fps_X)" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5557 |
unfolding fps_inverse_fps_X_plus1 |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5558 |
by (simp add: fps_ln_def fps_eq_iff del: of_nat_Suc) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5559 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5560 |
lemma fps_ln_nth: "fps_ln c $ n = (if n = 0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5561 |
by (simp add: fps_ln_def field_simps) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5562 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5563 |
lemma fps_ln_0 [simp]: "fps_ln c $ 0 = 0" by (simp add: fps_ln_def) |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5564 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5565 |
lemma fps_ln_fps_exp_inv: |
54452 | 5566 |
fixes a :: "'a::field_char_0" |
5567 |
assumes a: "a \<noteq> 0" |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5568 |
shows "fps_ln a = fps_inv (fps_exp a - 1)" (is "?l = ?r") |
52903 | 5569 |
proof - |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5570 |
let ?b = "fps_exp a - 1" |
29687 | 5571 |
have b0: "?b $ 0 = 0" by simp |
5572 |
have b1: "?b $ 1 \<noteq> 0" by (simp add: a) |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5573 |
have "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5574 |
(fps_const a * (fps_exp a - 1) + fps_const a) oo fps_inv (fps_exp a - 1)" |
36350 | 5575 |
by (simp add: field_simps) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5576 |
also have "\<dots> = fps_const a * (fps_X + 1)" |
52903 | 5577 |
apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1]) |
5578 |
apply (simp add: field_simps) |
|
5579 |
done |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5580 |
finally have eq: "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)" . |
29687 | 5581 |
from fps_inv_deriv[OF b0 b1, unfolded eq] |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5582 |
have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (fps_X + 1)" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5583 |
using a by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult) |
54452 | 5584 |
then have "fps_deriv ?l = fps_deriv ?r" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5585 |
by (simp add: fps_ln_deriv add.commute fps_divide_def divide_inverse) |
29687 | 5586 |
then show ?thesis unfolding fps_deriv_eq_iff |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5587 |
by (simp add: fps_ln_nth fps_inv_def) |
29687 | 5588 |
qed |
5589 |
||
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5590 |
lemma fps_ln_mult_add: |
52903 | 5591 |
assumes c0: "c\<noteq>0" |
5592 |
and d0: "d\<noteq>0" |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5593 |
shows "fps_ln c + fps_ln d = fps_const (c+d) * fps_ln (c*d)" |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5594 |
(is "?r = ?l") |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5595 |
proof- |
36350 | 5596 |
from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5597 |
have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + fps_X)" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5598 |
by (simp add: fps_ln_deriv fps_const_add[symmetric] algebra_simps del: fps_const_add) |
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5599 |
also have "\<dots> = fps_deriv ?l" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5600 |
apply (simp add: fps_ln_deriv) |
52903 | 5601 |
apply (simp add: fps_eq_iff eq) |
5602 |
done |
|
31369
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5603 |
finally show ?thesis |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5604 |
unfolding fps_deriv_eq_iff by simp |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5605 |
qed |
8b460fd12100
Reverses idempotent; radical of E; generalized logarithm;
chaieb
parents:
31199
diff
changeset
|
5606 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5607 |
lemma fps_X_dvd_fps_ln [simp]: "fps_X dvd fps_ln c" |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5608 |
proof - |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5609 |
have "fps_ln c = fps_X * Abs_fps (\<lambda>n. (-1) ^ n / (of_nat (Suc n) * c))" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
5610 |
by (intro fps_ext) (simp add: fps_ln_def of_nat_diff) |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5611 |
thus ?thesis by simp |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5612 |
qed |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
5613 |
|
53196 | 5614 |
|
60501 | 5615 |
subsubsection \<open>Binomial series\<close> |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5616 |
|
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5617 |
definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5618 |
|
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5619 |
lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5620 |
by (simp add: fps_binomial_def) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5621 |
|
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5622 |
lemma fps_binomial_ODE_unique: |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5623 |
fixes c :: "'a::field_char_0" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5624 |
shows "fps_deriv a = (fps_const c * a) / (1 + fps_X) \<longleftrightarrow> a = fps_const (a$0) * fps_binomial c" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5625 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60501 | 5626 |
proof |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5627 |
let ?da = "fps_deriv a" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5628 |
let ?x1 = "(1 + fps_X):: 'a fps" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5629 |
let ?l = "?x1 * ?da" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5630 |
let ?r = "fps_const c * a" |
60501 | 5631 |
|
5632 |
have eq: "?l = ?r \<longleftrightarrow> ?lhs" |
|
5633 |
proof - |
|
5634 |
have x10: "?x1 $ 0 \<noteq> 0" by simp |
|
5635 |
have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp |
|
5636 |
also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1" |
|
5637 |
apply (simp only: fps_divide_def mult.assoc[symmetric] inverse_mult_eq_1[OF x10]) |
|
5638 |
apply (simp add: field_simps) |
|
5639 |
done |
|
5640 |
finally show ?thesis . |
|
5641 |
qed |
|
5642 |
||
5643 |
show ?rhs if ?lhs |
|
5644 |
proof - |
|
5645 |
from eq that have h: "?l = ?r" .. |
|
5646 |
have th0: "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" for n |
|
5647 |
proof - |
|
5648 |
from h have "?l $ n = ?r $ n" by simp |
|
5649 |
then show ?thesis |
|
36350 | 5650 |
apply (simp add: field_simps del: of_nat_Suc) |
60501 | 5651 |
apply (cases n) |
5652 |
apply (simp_all add: field_simps del: of_nat_Suc) |
|
5653 |
done |
|
5654 |
qed |
|
5655 |
have th1: "a $ n = (c gchoose n) * a $ 0" for n |
|
5656 |
proof (induct n) |
|
5657 |
case 0 |
|
5658 |
then show ?case by simp |
|
5659 |
next |
|
5660 |
case (Suc m) |
|
5661 |
then show ?case |
|
5662 |
unfolding th0 |
|
5663 |
apply (simp add: field_simps del: of_nat_Suc) |
|
5664 |
unfolding mult.assoc[symmetric] gbinomial_mult_1 |
|
5665 |
apply (simp add: field_simps) |
|
5666 |
done |
|
5667 |
qed |
|
5668 |
show ?thesis |
|
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5669 |
apply (simp add: fps_eq_iff) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5670 |
apply (subst th1) |
53196 | 5671 |
apply (simp add: field_simps) |
5672 |
done |
|
60501 | 5673 |
qed |
5674 |
||
5675 |
show ?lhs if ?rhs |
|
5676 |
proof - |
|
5677 |
have th00: "x * (a $ 0 * y) = a $ 0 * (x * y)" for x y |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
5678 |
by (simp add: mult.commute) |
52891 | 5679 |
have "?l = ?r" |
60501 | 5680 |
apply (subst \<open>?rhs\<close>) |
5681 |
apply (subst (2) \<open>?rhs\<close>) |
|
36350 | 5682 |
apply (clarsimp simp add: fps_eq_iff field_simps) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
5683 |
unfolding mult.assoc[symmetric] th00 gbinomial_mult_1 |
53196 | 5684 |
apply (simp add: field_simps gbinomial_mult_1) |
5685 |
done |
|
60501 | 5686 |
with eq show ?thesis .. |
5687 |
qed |
|
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5688 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5689 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5690 |
lemma fps_binomial_ODE_unique': |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5691 |
"(fps_deriv a = fps_const c * a / (1 + fps_X) \<and> a $ 0 = 1) \<longleftrightarrow> (a = fps_binomial c)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5692 |
by (subst fps_binomial_ODE_unique) auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5693 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5694 |
lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + fps_X)" |
53196 | 5695 |
proof - |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5696 |
let ?a = "fps_binomial c" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5697 |
have th0: "?a = fps_const (?a$0) * ?a" by (simp) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5698 |
from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis . |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5699 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5700 |
|
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5701 |
lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r") |
53196 | 5702 |
proof - |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5703 |
let ?P = "?r - ?l" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5704 |
let ?b = "fps_binomial" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5705 |
let ?db = "\<lambda>x. fps_deriv (?b x)" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5706 |
have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)" by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5707 |
also have "\<dots> = inverse (1 + fps_X) * |
53196 | 5708 |
(fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5709 |
unfolding fps_binomial_deriv |
36350 | 5710 |
by (simp add: fps_divide_def field_simps) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5711 |
also have "\<dots> = (fps_const (c + d)/ (1 + fps_X)) * ?P" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
5712 |
by (simp add: field_simps fps_divide_unit fps_const_add[symmetric] del: fps_const_add) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5713 |
finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + fps_X)" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5714 |
by (simp add: fps_divide_def) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5715 |
have "?P = fps_const (?P$0) * ?b (c + d)" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5716 |
unfolding fps_binomial_ODE_unique[symmetric] |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5717 |
using th0 by simp |
54452 | 5718 |
then have "?P = 0" by (simp add: fps_mult_nth) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5719 |
then show ?thesis by simp |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5720 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5721 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5722 |
lemma fps_binomial_minus_one: "fps_binomial (- 1) = inverse (1 + fps_X)" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5723 |
(is "?l = inverse ?r") |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5724 |
proof- |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5725 |
have th: "?r$0 \<noteq> 0" by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5726 |
have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + fps_X)" |
53196 | 5727 |
by (simp add: fps_inverse_deriv[OF th] fps_divide_def |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
5728 |
power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5729 |
have eq: "inverse ?r $ 0 = 1" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5730 |
by (simp add: fps_inverse_def) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5731 |
from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + fps_X)" "- 1"] th'] eq |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5732 |
show ?thesis by (simp add: fps_inverse_def) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5733 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5734 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5735 |
lemma fps_binomial_of_nat: "fps_binomial (of_nat n) = (1 + fps_X :: 'a :: field_char_0 fps) ^ n" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5736 |
proof (cases "n = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5737 |
case [simp]: True |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5738 |
have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = 0" by simp |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5739 |
also have "\<dots> = fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" by (simp add: fps_binomial_def) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5740 |
finally show ?thesis by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5741 |
next |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5742 |
case False |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5743 |
have "fps_deriv ((1 + fps_X) ^ n :: 'a fps) = fps_const (of_nat n) * (1 + fps_X) ^ (n - 1)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5744 |
by (simp add: fps_deriv_power) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5745 |
also have "(1 + fps_X :: 'a fps) $ 0 \<noteq> 0" by simp |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5746 |
hence "(1 + fps_X :: 'a fps) \<noteq> 0" by (intro notI) (simp only: , simp) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5747 |
with False have "(1 + fps_X :: 'a fps) ^ (n - 1) = (1 + fps_X) ^ n / (1 + fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5748 |
by (cases n) (simp_all ) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5749 |
also have "fps_const (of_nat n :: 'a) * ((1 + fps_X) ^ n / (1 + fps_X)) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5750 |
fps_const (of_nat n) * (1 + fps_X) ^ n / (1 + fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5751 |
by (simp add: unit_div_mult_swap) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5752 |
finally show ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5753 |
by (subst sym, subst fps_binomial_ODE_unique' [symmetric]) (simp_all add: fps_power_nth) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5754 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5755 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5756 |
lemma fps_binomial_0 [simp]: "fps_binomial 0 = 1" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5757 |
using fps_binomial_of_nat[of 0] by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5758 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5759 |
lemma fps_binomial_power: "fps_binomial a ^ n = fps_binomial (of_nat n * a)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5760 |
by (induction n) (simp_all add: fps_binomial_add_mult ring_distribs) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5761 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5762 |
lemma fps_binomial_1: "fps_binomial 1 = 1 + fps_X" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5763 |
using fps_binomial_of_nat[of 1] by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5764 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5765 |
lemma fps_binomial_minus_of_nat: |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5766 |
"fps_binomial (- of_nat n) = inverse ((1 + fps_X :: 'a :: field_char_0 fps) ^ n)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5767 |
by (rule sym, rule fps_inverse_unique) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5768 |
(simp add: fps_binomial_of_nat [symmetric] fps_binomial_add_mult [symmetric]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5769 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5770 |
lemma one_minus_const_fps_X_power: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5771 |
"c \<noteq> 0 \<Longrightarrow> (1 - fps_const c * fps_X) ^ n = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5772 |
fps_compose (fps_binomial (of_nat n)) (-fps_const c * fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5773 |
by (subst fps_binomial_of_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5774 |
(simp add: fps_compose_power [symmetric] fps_compose_add_distrib fps_const_neg [symmetric] |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5775 |
del: fps_const_neg) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5776 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5777 |
lemma one_minus_fps_X_const_neg_power: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5778 |
"inverse ((1 - fps_const c * fps_X) ^ n) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5779 |
fps_compose (fps_binomial (-of_nat n)) (-fps_const c * fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5780 |
proof (cases "c = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5781 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5782 |
thus ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5783 |
by (subst fps_binomial_minus_of_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5784 |
(simp add: fps_compose_power [symmetric] fps_inverse_compose fps_compose_add_distrib |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5785 |
fps_const_neg [symmetric] del: fps_const_neg) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5786 |
qed simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5787 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5788 |
lemma fps_X_plus_const_power: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5789 |
"c \<noteq> 0 \<Longrightarrow> (fps_X + fps_const c) ^ n = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5790 |
fps_const (c^n) * fps_compose (fps_binomial (of_nat n)) (fps_const (inverse c) * fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5791 |
by (subst fps_binomial_of_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5792 |
(simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5793 |
fps_const_power [symmetric] power_mult_distrib [symmetric] |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5794 |
algebra_simps inverse_mult_eq_1' del: fps_const_power) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5795 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5796 |
lemma fps_X_plus_const_neg_power: |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5797 |
"c \<noteq> 0 \<Longrightarrow> inverse ((fps_X + fps_const c) ^ n) = |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5798 |
fps_const (inverse c^n) * fps_compose (fps_binomial (-of_nat n)) (fps_const (inverse c) * fps_X)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5799 |
by (subst fps_binomial_minus_of_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5800 |
(simp add: fps_compose_power [symmetric] fps_binomial_of_nat fps_compose_add_distrib |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5801 |
fps_const_power [symmetric] power_mult_distrib [symmetric] fps_inverse_compose |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5802 |
algebra_simps fps_const_inverse [symmetric] fps_inverse_mult [symmetric] |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5803 |
fps_inverse_power [symmetric] inverse_mult_eq_1' |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5804 |
del: fps_const_power) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5805 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5806 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5807 |
lemma one_minus_const_fps_X_neg_power': |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5808 |
"n > 0 \<Longrightarrow> inverse ((1 - fps_const (c :: 'a :: field_char_0) * fps_X) ^ n) = |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5809 |
Abs_fps (\<lambda>k. of_nat ((n + k - 1) choose k) * c^k)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5810 |
apply (rule fps_ext) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
5811 |
apply (subst one_minus_fps_X_const_neg_power, subst fps_const_neg, subst fps_compose_linear) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5812 |
apply (simp add: power_mult_distrib [symmetric] mult.assoc [symmetric] |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5813 |
gbinomial_minus binomial_gbinomial of_nat_diff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5814 |
done |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63040
diff
changeset
|
5815 |
|
60558 | 5816 |
text \<open>Vandermonde's Identity as a consequence.\<close> |
53196 | 5817 |
lemma gbinomial_Vandermonde: |
64267 | 5818 |
"sum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n" |
53196 | 5819 |
proof - |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5820 |
let ?ba = "fps_binomial a" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5821 |
let ?bb = "fps_binomial b" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5822 |
let ?bab = "fps_binomial (a + b)" |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5823 |
from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5824 |
then show ?thesis by (simp add: fps_mult_nth) |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5825 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5826 |
|
53196 | 5827 |
lemma binomial_Vandermonde: |
64267 | 5828 |
"sum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5829 |
using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n] |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
5830 |
by (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] |
64267 | 5831 |
of_nat_sum[symmetric] of_nat_add[symmetric] of_nat_eq_iff) |
5832 |
||
5833 |
lemma binomial_Vandermonde_same: "sum (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2 * n) choose n" |
|
60501 | 5834 |
using binomial_Vandermonde[of n n n, symmetric] |
53195 | 5835 |
unfolding mult_2 |
5836 |
apply (simp add: power2_eq_square) |
|
64267 | 5837 |
apply (rule sum.cong) |
53195 | 5838 |
apply (auto intro: binomial_symmetric) |
5839 |
done |
|
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
5840 |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5841 |
lemma Vandermonde_pochhammer_lemma: |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5842 |
fixes a :: "'a::field_char_0" |
60504 | 5843 |
assumes b: "\<forall>j\<in>{0 ..<n}. b \<noteq> of_nat j" |
64267 | 5844 |
shows "sum (\<lambda>k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) / |
53196 | 5845 |
(of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} = |
54452 | 5846 |
pochhammer (- (a + b)) n / pochhammer (- b) n" |
53196 | 5847 |
(is "?l = ?r") |
5848 |
proof - |
|
54452 | 5849 |
let ?m1 = "\<lambda>m. (- 1 :: 'a) ^ m" |
5850 |
let ?f = "\<lambda>m. of_nat (fact m)" |
|
5851 |
let ?p = "\<lambda>(x::'a). pochhammer (- x)" |
|
60501 | 5852 |
from b have bn0: "?p b n \<noteq> 0" |
5853 |
unfolding pochhammer_eq_0_iff by simp |
|
60558 | 5854 |
have th00: |
5855 |
"b gchoose (n - k) = |
|
5856 |
(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" |
|
5857 |
(is ?gchoose) |
|
5858 |
"pochhammer (1 + b - of_nat n) k \<noteq> 0" |
|
5859 |
(is ?pochhammer) |
|
5860 |
if kn: "k \<in> {0..n}" for k |
|
5861 |
proof - |
|
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5862 |
from kn have "k \<le> n" by simp |
60501 | 5863 |
have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0" |
5864 |
proof |
|
5865 |
assume "pochhammer (1 + b - of_nat n) n = 0" |
|
5866 |
then have c: "pochhammer (b - of_nat n + 1) n = 0" |
|
5867 |
by (simp add: algebra_simps) |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5868 |
then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5869 |
unfolding pochhammer_eq_0_iff by blast |
52891 | 5870 |
from j have "b = of_nat n - of_nat j - of_nat 1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5871 |
by (simp add: algebra_simps) |
52891 | 5872 |
then have "b = of_nat (n - j - 1)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5873 |
using j kn by (simp add: of_nat_diff) |
60501 | 5874 |
with b show False using j by auto |
5875 |
qed |
|
52891 | 5876 |
|
5877 |
from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0" |
|
35175 | 5878 |
by (rule pochhammer_neq_0_mono) |
60504 | 5879 |
|
60567 | 5880 |
consider "k = 0 \<or> n = 0" | "k \<noteq> 0" "n \<noteq> 0" |
5881 |
by blast |
|
60504 | 5882 |
then have "b gchoose (n - k) = |
5883 |
(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" |
|
5884 |
proof cases |
|
5885 |
case 1 |
|
5886 |
then show ?thesis |
|
5887 |
using kn by (cases "k = 0") (simp_all add: gbinomial_pochhammer) |
|
5888 |
next |
|
60567 | 5889 |
case neq: 2 |
60501 | 5890 |
then obtain m where m: "n = Suc m" |
5891 |
by (cases n) auto |
|
60567 | 5892 |
from neq(1) obtain h where h: "k = Suc h" |
60501 | 5893 |
by (cases k) auto |
60504 | 5894 |
show ?thesis |
60501 | 5895 |
proof (cases "k = n") |
5896 |
case True |
|
5897 |
then show ?thesis |
|
59862 | 5898 |
using pochhammer_minus'[where k=k and b=b] |
5899 |
apply (simp add: pochhammer_same) |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5900 |
using bn0 |
53195 | 5901 |
apply (simp add: field_simps power_add[symmetric]) |
5902 |
done |
|
60501 | 5903 |
next |
5904 |
case False |
|
5905 |
with kn have kn': "k < n" |
|
5906 |
by simp |
|
64272 | 5907 |
have m1nk: "?m1 n = prod (\<lambda>i. - 1) {..m}" "?m1 k = prod (\<lambda>i. - 1) {0..h}" |
5908 |
by (simp_all add: prod_constant m h) |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5909 |
have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0" |
52891 | 5910 |
using bn0 kn |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5911 |
unfolding pochhammer_eq_0_iff |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5912 |
apply auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5913 |
apply (erule_tac x= "n - ka - 1" in allE) |
53196 | 5914 |
apply (auto simp add: algebra_simps of_nat_diff) |
5915 |
done |
|
64272 | 5916 |
have eq1: "prod (\<lambda>k. (1::'a) + of_nat m - of_nat k) {..h} = |
5917 |
prod of_nat {Suc (m - h) .. Suc m}" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5918 |
using kn' h m |
64272 | 5919 |
by (intro prod.reindex_bij_witness[where i="\<lambda>k. Suc m - k" and j="\<lambda>k. Suc m - k"]) |
57129
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56480
diff
changeset
|
5920 |
(auto simp: of_nat_diff) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5921 |
have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5922 |
apply (simp add: pochhammer_minus field_simps) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5923 |
using \<open>k \<le> n\<close> apply (simp add: fact_split [of k n]) |
64272 | 5924 |
apply (simp add: pochhammer_prod) |
67411
3f4b0c84630f
restored naming of lemmas after corresponding constants
haftmann
parents:
67399
diff
changeset
|
5925 |
using prod.atLeastLessThan_shift_bounds [where ?'a = 'a, of "\<lambda>i. 1 + of_nat i" 0 "n - k" k] |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5926 |
apply (auto simp add: of_nat_diff field_simps) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5927 |
done |
64272 | 5928 |
have th20: "?m1 n * ?p b n = prod (\<lambda>i. b - of_nat i) {0..m}" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5929 |
apply (simp add: pochhammer_minus field_simps m) |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
5930 |
apply (auto simp add: pochhammer_prod_rev of_nat_diff prod.atLeast_Suc_atMost_Suc_shift simp del: prod.cl_ivl_Suc) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5931 |
done |
64272 | 5932 |
have th21:"pochhammer (b - of_nat n + 1) k = prod (\<lambda>i. b - of_nat i) {n - k .. n - 1}" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
5933 |
using kn apply (simp add: pochhammer_prod_rev m h prod.atLeast_Suc_atMost_Suc_shift del: prod.op_ivl_Suc del: prod.cl_ivl_Suc) |
67411
3f4b0c84630f
restored naming of lemmas after corresponding constants
haftmann
parents:
67399
diff
changeset
|
5934 |
using prod.atLeastAtMost_shift_0 [of "m - h" m, where ?'a = 'a] |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5935 |
apply (auto simp add: of_nat_diff field_simps) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5936 |
done |
53196 | 5937 |
have "?m1 n * ?p b n = |
64272 | 5938 |
prod (\<lambda>i. b - of_nat i) {0.. n - k - 1} * pochhammer (b - of_nat n + 1) k" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5939 |
using kn' m h unfolding th20 th21 apply simp |
64272 | 5940 |
apply (subst prod.union_disjoint [symmetric]) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5941 |
apply auto |
64272 | 5942 |
apply (rule prod.cong) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5943 |
apply auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5944 |
done |
53196 | 5945 |
then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k = |
64272 | 5946 |
prod (\<lambda>i. b - of_nat i) {0.. n - k - 1}" |
36350 | 5947 |
using nz' by (simp add: field_simps) |
53196 | 5948 |
have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) = |
5949 |
((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5950 |
using bnz0 |
36350 | 5951 |
by (simp add: field_simps) |
52891 | 5952 |
also have "\<dots> = b gchoose (n - k)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
5953 |
unfolding th1 th2 |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5954 |
using kn' m h |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5955 |
apply (simp add: field_simps gbinomial_mult_fact) |
64272 | 5956 |
apply (rule prod.cong) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5957 |
apply auto |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
5958 |
done |
60501 | 5959 |
finally show ?thesis by simp |
5960 |
qed |
|
60504 | 5961 |
qed |
60558 | 5962 |
then show ?gchoose and ?pochhammer |
53195 | 5963 |
apply (cases "n = 0") |
52891 | 5964 |
using nz' |
53195 | 5965 |
apply auto |
5966 |
done |
|
60558 | 5967 |
qed |
60504 | 5968 |
have "?r = ((a + b) gchoose n) * (of_nat (fact n) / (?m1 n * pochhammer (- b) n))" |
52891 | 5969 |
unfolding gbinomial_pochhammer |
36350 | 5970 |
using bn0 by (auto simp add: field_simps) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5971 |
also have "\<dots> = ?l" |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5972 |
unfolding gbinomial_Vandermonde[symmetric] |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5973 |
apply (simp add: th00) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5974 |
unfolding gbinomial_pochhammer |
53196 | 5975 |
using bn0 |
64267 | 5976 |
apply (simp add: sum_distrib_right sum_distrib_left field_simps) |
53196 | 5977 |
done |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5978 |
finally show ?thesis by simp |
52891 | 5979 |
qed |
5980 |
||
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5981 |
lemma Vandermonde_pochhammer: |
53195 | 5982 |
fixes a :: "'a::field_char_0" |
54452 | 5983 |
assumes c: "\<forall>i \<in> {0..< n}. c \<noteq> - of_nat i" |
64267 | 5984 |
shows "sum (\<lambda>k. (pochhammer a k * pochhammer (- (of_nat n)) k) / |
53195 | 5985 |
(of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n" |
5986 |
proof - |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5987 |
let ?a = "- a" |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5988 |
let ?b = "c + of_nat n - 1" |
60558 | 5989 |
have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j" |
5990 |
using c |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5991 |
apply (auto simp add: algebra_simps of_nat_diff) |
60501 | 5992 |
apply (erule_tac x = "n - j - 1" in ballE) |
53195 | 5993 |
apply (auto simp add: of_nat_diff algebra_simps) |
5994 |
done |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5995 |
have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n" |
59862 | 5996 |
unfolding pochhammer_minus |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5997 |
by (simp add: algebra_simps) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
5998 |
have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n" |
59862 | 5999 |
unfolding pochhammer_minus |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6000 |
by simp |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6001 |
have nz: "pochhammer c n \<noteq> 0" using c |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6002 |
by (simp add: pochhammer_eq_0_iff) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6003 |
from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1] |
60501 | 6004 |
show ?thesis |
64267 | 6005 |
using nz by (simp add: field_simps sum_distrib_left) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6006 |
qed |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6007 |
|
53195 | 6008 |
|
60501 | 6009 |
subsubsection \<open>Formal trigonometric functions\<close> |
29687 | 6010 |
|
31273 | 6011 |
definition "fps_sin (c::'a::field_char_0) = |
29687 | 6012 |
Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))" |
6013 |
||
31273 | 6014 |
definition "fps_cos (c::'a::field_char_0) = |
6015 |
Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)" |
|
29687 | 6016 |
|
66466
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6017 |
lemma fps_sin_0 [simp]: "fps_sin 0 = 0" |
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6018 |
by (intro fps_ext) (auto simp: fps_sin_def elim!: oddE) |
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6019 |
|
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6020 |
lemma fps_cos_0 [simp]: "fps_cos 0 = 1" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6021 |
by (intro fps_ext) (simp add: fps_cos_def) |
66466
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6022 |
|
30488 | 6023 |
lemma fps_sin_deriv: |
29687 | 6024 |
"fps_deriv (fps_sin c) = fps_const c * fps_cos c" |
6025 |
(is "?lhs = ?rhs") |
|
31273 | 6026 |
proof (rule fps_ext) |
53195 | 6027 |
fix n :: nat |
60501 | 6028 |
show "?lhs $ n = ?rhs $ n" |
6029 |
proof (cases "even n") |
|
6030 |
case True |
|
52902 | 6031 |
have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp |
6032 |
also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))" |
|
60501 | 6033 |
using True by (simp add: fps_sin_def) |
52902 | 6034 |
also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))" |
6035 |
unfolding fact_Suc of_nat_mult |
|
6036 |
by (simp add: field_simps del: of_nat_add of_nat_Suc) |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6037 |
also have "\<dots> = (- 1)^(n div 2) * c^Suc n / of_nat (fact n)" |
52902 | 6038 |
by (simp add: field_simps del: of_nat_add of_nat_Suc) |
60501 | 6039 |
finally show ?thesis |
6040 |
using True by (simp add: fps_cos_def field_simps) |
|
6041 |
next |
|
6042 |
case False |
|
6043 |
then show ?thesis |
|
6044 |
by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def) |
|
6045 |
qed |
|
29687 | 6046 |
qed |
6047 |
||
52902 | 6048 |
lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)" |
29687 | 6049 |
(is "?lhs = ?rhs") |
31273 | 6050 |
proof (rule fps_ext) |
60501 | 6051 |
have th0: "- ((- 1::'a) ^ n) = (- 1)^Suc n" for n |
6052 |
by simp |
|
6053 |
show "?lhs $ n = ?rhs $ n" for n |
|
6054 |
proof (cases "even n") |
|
6055 |
case False |
|
6056 |
then have n0: "n \<noteq> 0" by presburger |
|
6057 |
from False have th1: "Suc ((n - 1) div 2) = Suc n div 2" |
|
6058 |
by (cases n) simp_all |
|
52902 | 6059 |
have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp |
6060 |
also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))" |
|
60501 | 6061 |
using False by (simp add: fps_cos_def) |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6062 |
also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))" |
52902 | 6063 |
unfolding fact_Suc of_nat_mult |
6064 |
by (simp add: field_simps del: of_nat_add of_nat_Suc) |
|
6065 |
also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)" |
|
6066 |
by (simp add: field_simps del: of_nat_add of_nat_Suc) |
|
6067 |
also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)" |
|
60501 | 6068 |
unfolding th0 unfolding th1 by simp |
6069 |
finally show ?thesis |
|
6070 |
using False by (simp add: fps_sin_def field_simps) |
|
6071 |
next |
|
6072 |
case True |
|
6073 |
then show ?thesis |
|
6074 |
by (simp_all add: fps_deriv_def fps_sin_def fps_cos_def) |
|
6075 |
qed |
|
29687 | 6076 |
qed |
6077 |
||
60501 | 6078 |
lemma fps_sin_cos_sum_of_squares: "(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1" |
6079 |
(is "?lhs = _") |
|
53077 | 6080 |
proof - |
29687 | 6081 |
have "fps_deriv ?lhs = 0" |
52902 | 6082 |
apply (simp add: fps_deriv_power fps_sin_deriv fps_cos_deriv) |
6083 |
apply (simp add: field_simps fps_const_neg[symmetric] del: fps_const_neg) |
|
6084 |
done |
|
29687 | 6085 |
then have "?lhs = fps_const (?lhs $ 0)" |
6086 |
unfolding fps_deriv_eq_0_iff . |
|
6087 |
also have "\<dots> = 1" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6088 |
by (simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def) |
29687 | 6089 |
finally show ?thesis . |
6090 |
qed |
|
6091 |
||
31274 | 6092 |
lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0" |
53195 | 6093 |
unfolding fps_sin_def by simp |
31274 | 6094 |
|
6095 |
lemma fps_sin_nth_1 [simp]: "fps_sin c $ 1 = c" |
|
53195 | 6096 |
unfolding fps_sin_def by simp |
31274 | 6097 |
|
6098 |
lemma fps_sin_nth_add_2: |
|
60501 | 6099 |
"fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat (n + 1) * of_nat (n + 2)))" |
53195 | 6100 |
unfolding fps_sin_def |
60501 | 6101 |
apply (cases n) |
6102 |
apply simp |
|
60162 | 6103 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq del: of_nat_Suc fact_Suc) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
6104 |
apply simp |
53195 | 6105 |
done |
31274 | 6106 |
|
6107 |
lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1" |
|
53195 | 6108 |
unfolding fps_cos_def by simp |
31274 | 6109 |
|
6110 |
lemma fps_cos_nth_1 [simp]: "fps_cos c $ 1 = 0" |
|
53195 | 6111 |
unfolding fps_cos_def by simp |
31274 | 6112 |
|
6113 |
lemma fps_cos_nth_add_2: |
|
60501 | 6114 |
"fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat (n + 1) * of_nat (n + 2)))" |
52902 | 6115 |
unfolding fps_cos_def |
60162 | 6116 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq del: of_nat_Suc fact_Suc) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
6117 |
apply simp |
52902 | 6118 |
done |
6119 |
||
6120 |
lemma nat_induct2: "P 0 \<Longrightarrow> P 1 \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (n + 2)) \<Longrightarrow> P (n::nat)" |
|
6121 |
unfolding One_nat_def numeral_2_eq_2 |
|
6122 |
apply (induct n rule: nat_less_induct) |
|
53196 | 6123 |
apply (case_tac n) |
6124 |
apply simp |
|
6125 |
apply (rename_tac m) |
|
6126 |
apply (case_tac m) |
|
6127 |
apply simp |
|
6128 |
apply (rename_tac k) |
|
6129 |
apply (case_tac k) |
|
6130 |
apply simp_all |
|
52902 | 6131 |
done |
31274 | 6132 |
|
6133 |
lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2" |
|
52902 | 6134 |
by simp |
31274 | 6135 |
|
6136 |
lemma eq_fps_sin: |
|
52902 | 6137 |
assumes 0: "a $ 0 = 0" |
6138 |
and 1: "a $ 1 = c" |
|
6139 |
and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)" |
|
31274 | 6140 |
shows "a = fps_sin c" |
52902 | 6141 |
apply (rule fps_ext) |
6142 |
apply (induct_tac n rule: nat_induct2) |
|
6143 |
apply (simp add: 0) |
|
6144 |
apply (simp add: 1 del: One_nat_def) |
|
6145 |
apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2]) |
|
6146 |
apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2 |
|
6147 |
del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc') |
|
6148 |
apply (subst minus_divide_left) |
|
60162 | 6149 |
apply (subst nonzero_eq_divide_eq) |
52902 | 6150 |
apply (simp del: of_nat_add of_nat_Suc) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
6151 |
apply (simp only: ac_simps) |
52902 | 6152 |
done |
31274 | 6153 |
|
6154 |
lemma eq_fps_cos: |
|
52902 | 6155 |
assumes 0: "a $ 0 = 1" |
6156 |
and 1: "a $ 1 = 0" |
|
6157 |
and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)" |
|
31274 | 6158 |
shows "a = fps_cos c" |
52902 | 6159 |
apply (rule fps_ext) |
6160 |
apply (induct_tac n rule: nat_induct2) |
|
6161 |
apply (simp add: 0) |
|
6162 |
apply (simp add: 1 del: One_nat_def) |
|
6163 |
apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2]) |
|
6164 |
apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2 |
|
6165 |
del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc') |
|
6166 |
apply (subst minus_divide_left) |
|
60162 | 6167 |
apply (subst nonzero_eq_divide_eq) |
52902 | 6168 |
apply (simp del: of_nat_add of_nat_Suc) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
6169 |
apply (simp only: ac_simps) |
52902 | 6170 |
done |
31274 | 6171 |
|
52902 | 6172 |
lemma fps_sin_add: "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b" |
6173 |
apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def) |
|
6174 |
apply (simp del: fps_const_neg fps_const_add fps_const_mult |
|
6175 |
add: fps_const_add [symmetric] fps_const_neg [symmetric] |
|
6176 |
fps_sin_deriv fps_cos_deriv algebra_simps) |
|
6177 |
done |
|
6178 |
||
6179 |
lemma fps_cos_add: "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b" |
|
6180 |
apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def) |
|
6181 |
apply (simp del: fps_const_neg fps_const_add fps_const_mult |
|
6182 |
add: fps_const_add [symmetric] fps_const_neg [symmetric] |
|
6183 |
fps_sin_deriv fps_cos_deriv algebra_simps) |
|
6184 |
done |
|
31274 | 6185 |
|
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
6186 |
lemma fps_sin_even: "fps_sin (- c) = - fps_sin c" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6187 |
by (simp add: fps_eq_iff fps_sin_def) |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
6188 |
|
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
6189 |
lemma fps_cos_odd: "fps_cos (- c) = fps_cos c" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6190 |
by (simp add: fps_eq_iff fps_cos_def) |
31968
0314441a53a6
FPS form a metric space, which justifies the infinte sum notation
chaieb
parents:
31790
diff
changeset
|
6191 |
|
29687 | 6192 |
definition "fps_tan c = fps_sin c / fps_cos c" |
6193 |
||
66466
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6194 |
lemma fps_tan_0 [simp]: "fps_tan 0 = 0" |
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6195 |
by (simp add: fps_tan_def) |
aec5d9c88d69
More lemmas for HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
66373
diff
changeset
|
6196 |
|
53077 | 6197 |
lemma fps_tan_deriv: "fps_deriv (fps_tan c) = fps_const c / (fps_cos c)\<^sup>2" |
52902 | 6198 |
proof - |
29687 | 6199 |
have th0: "fps_cos c $ 0 \<noteq> 0" by (simp add: fps_cos_def) |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6200 |
from this have "fps_cos c \<noteq> 0" by (intro notI) simp |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
6201 |
hence "fps_deriv (fps_tan c) = |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6202 |
fps_const c * (fps_cos c^2 + fps_sin c^2) / (fps_cos c^2)" |
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
6203 |
by (simp add: fps_tan_def fps_divide_deriv power2_eq_square algebra_simps |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6204 |
fps_sin_deriv fps_cos_deriv fps_const_neg[symmetric] div_mult_swap |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6205 |
del: fps_const_neg) |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6206 |
also note fps_sin_cos_sum_of_squares |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6207 |
finally show ?thesis by simp |
29687 | 6208 |
qed |
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset
|
6209 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6210 |
text \<open>Connection to @{const "fps_exp"} over the complex numbers --- Euler and de Moivre.\<close> |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6211 |
|
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6212 |
lemma fps_exp_ii_sin_cos: "fps_exp (\<i> * c) = fps_cos c + fps_const \<i> * fps_sin c" |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6213 |
(is "?l = ?r") |
52902 | 6214 |
proof - |
60501 | 6215 |
have "?l $ n = ?r $ n" for n |
6216 |
proof (cases "even n") |
|
6217 |
case True |
|
6218 |
then obtain m where m: "n = 2 * m" .. |
|
6219 |
show ?thesis |
|
6220 |
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"]) |
|
6221 |
next |
|
6222 |
case False |
|
6223 |
then obtain m where m: "n = 2 * m + 1" .. |
|
6224 |
show ?thesis |
|
6225 |
by (simp add: m fps_sin_def fps_cos_def power_mult_distrib |
|
6226 |
power_mult power_minus [of "c ^ 2"]) |
|
6227 |
qed |
|
6228 |
then show ?thesis |
|
6229 |
by (simp add: fps_eq_iff) |
|
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6230 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6231 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6232 |
lemma fps_exp_minus_ii_sin_cos: "fps_exp (- (\<i> * c)) = fps_cos c - fps_const \<i> * fps_sin c" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6233 |
unfolding minus_mult_right fps_exp_ii_sin_cos by (simp add: fps_sin_even fps_cos_odd) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6234 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6235 |
lemma fps_cos_fps_exp_ii: "fps_cos c = (fps_exp (\<i> * c) + fps_exp (- \<i> * c)) / fps_const 2" |
52902 | 6236 |
proof - |
52891 | 6237 |
have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset
|
6238 |
by (simp add: numeral_fps_const) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6239 |
show ?thesis |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6240 |
unfolding fps_exp_ii_sin_cos minus_mult_commute |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6241 |
by (simp add: fps_sin_even fps_cos_odd numeral_fps_const fps_divide_unit fps_const_inverse th) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6242 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6243 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6244 |
lemma fps_sin_fps_exp_ii: "fps_sin c = (fps_exp (\<i> * c) - fps_exp (- \<i> * c)) / fps_const (2*\<i>)" |
52902 | 6245 |
proof - |
63589 | 6246 |
have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * \<i>)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46757
diff
changeset
|
6247 |
by (simp add: fps_eq_iff numeral_fps_const) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6248 |
show ?thesis |
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6249 |
unfolding fps_exp_ii_sin_cos minus_mult_commute |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6250 |
by (simp add: fps_sin_even fps_cos_odd fps_divide_unit fps_const_inverse th) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6251 |
qed |
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6252 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6253 |
lemma fps_tan_fps_exp_ii: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6254 |
"fps_tan c = (fps_exp (\<i> * c) - fps_exp (- \<i> * c)) / |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6255 |
(fps_const \<i> * (fps_exp (\<i> * c) + fps_exp (- \<i> * c)))" |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6256 |
unfolding fps_tan_def fps_sin_fps_exp_ii fps_cos_fps_exp_ii mult_minus_left fps_exp_neg |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6257 |
apply (simp add: fps_divide_unit fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult) |
52902 | 6258 |
apply simp |
6259 |
done |
|
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6260 |
|
60501 | 6261 |
lemma fps_demoivre: |
63589 | 6262 |
"(fps_cos a + fps_const \<i> * fps_sin a)^n = |
6263 |
fps_cos (of_nat n * a) + fps_const \<i> * fps_sin (of_nat n * a)" |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6264 |
unfolding fps_exp_ii_sin_cos[symmetric] fps_exp_power_mult |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
6265 |
by (simp add: ac_simps) |
32157
adea7a729c7a
Moved important theorems from FPS_Examples to FPS --- they are not
chaieb
parents:
31968
diff
changeset
|
6266 |
|
52902 | 6267 |
|
60500 | 6268 |
subsection \<open>Hypergeometric series\<close> |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6269 |
|
68442 | 6270 |
definition "fps_hypergeo as bs (c::'a::field_char_0) = |
54452 | 6271 |
Abs_fps (\<lambda>n. (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) / |
6272 |
(foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n)))" |
|
52902 | 6273 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6274 |
lemma fps_hypergeo_nth[simp]: "fps_hypergeo as bs c $ n = |
52902 | 6275 |
(foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) / |
6276 |
(foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n))" |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6277 |
by (simp add: fps_hypergeo_def) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6278 |
|
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6279 |
lemma foldl_mult_start: |
54452 | 6280 |
fixes v :: "'a::comm_ring_1" |
6281 |
shows "foldl (\<lambda>r x. r * f x) v as * x = foldl (\<lambda>r x. r * f x) (v * x) as " |
|
48757 | 6282 |
by (induct as arbitrary: x v) (auto simp add: algebra_simps) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6283 |
|
53196 | 6284 |
lemma foldr_mult_foldl: |
54452 | 6285 |
fixes v :: "'a::comm_ring_1" |
6286 |
shows "foldr (\<lambda>x r. r * f x) as v = foldl (\<lambda>r x. r * f x) v as" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6287 |
by (induct as arbitrary: v) (simp_all add: foldl_mult_start) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6288 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6289 |
lemma fps_hypergeo_nth_alt: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6290 |
"fps_hypergeo as bs c $ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) / |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6291 |
foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))" |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6292 |
by (simp add: foldl_mult_start foldr_mult_foldl) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6293 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6294 |
lemma fps_hypergeo_fps_exp[simp]: "fps_hypergeo [] [] c = fps_exp c" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6295 |
by (simp add: fps_eq_iff) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6296 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6297 |
lemma fps_hypergeo_1_0[simp]: "fps_hypergeo [1] [] c = 1/(1 - fps_const c * fps_X)" |
52902 | 6298 |
proof - |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6299 |
let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * fps_X)" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6300 |
have th0: "(fps_const c * fps_X) $ 0 = 0" by simp |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6301 |
show ?thesis unfolding gp[OF th0, symmetric] |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6302 |
by (simp add: fps_eq_iff pochhammer_fact[symmetric] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6303 |
fps_compose_nth power_mult_distrib if_distrib cong del: if_weak_cong) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6304 |
qed |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6305 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6306 |
lemma fps_hypergeo_B[simp]: "fps_hypergeo [-a] [] (- 1) = fps_binomial a" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6307 |
by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6308 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6309 |
lemma fps_hypergeo_0[simp]: "fps_hypergeo as bs c $ 0 = 1" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6310 |
apply simp |
54452 | 6311 |
apply (subgoal_tac "\<forall>as. foldl (\<lambda>(r::'a) (a::'a). r) 1 as = 1") |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6312 |
apply auto |
48757 | 6313 |
apply (induct_tac as) |
6314 |
apply auto |
|
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6315 |
done |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6316 |
|
53196 | 6317 |
lemma foldl_prod_prod: |
54452 | 6318 |
"foldl (\<lambda>(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (\<lambda>r x. r * g x) w as = |
6319 |
foldl (\<lambda>r x. r * f x * g x) (v * w) as" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6320 |
by (induct as arbitrary: v w) (simp_all add: algebra_simps) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6321 |
|
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6322 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6323 |
lemma fps_hypergeo_rec: |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6324 |
"fps_hypergeo as bs c $ Suc n = ((foldl (\<lambda>r a. r* (a + of_nat n)) c as) / |
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6325 |
(foldl (\<lambda>r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * fps_hypergeo as bs c $ n" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6326 |
apply (simp del: of_nat_Suc of_nat_add fact_Suc) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6327 |
apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc) |
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6328 |
unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63367
diff
changeset
|
6329 |
apply (simp add: algebra_simps) |
52902 | 6330 |
done |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6331 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6332 |
lemma fps_XD_nth[simp]: "fps_XD a $ n = of_nat n * a$n" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6333 |
by (simp add: fps_XD_def) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6334 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6335 |
lemma fps_XD_0th[simp]: "fps_XD a $ 0 = 0" |
60501 | 6336 |
by simp |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6337 |
lemma fps_XD_Suc[simp]:" fps_XD a $ Suc n = of_nat (Suc n) * a $ Suc n" |
60501 | 6338 |
by simp |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6339 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6340 |
definition "fps_XDp c a = fps_XD a + fps_const c * a" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6341 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6342 |
lemma fps_XDp_nth[simp]: "fps_XDp c a $ n = (c + of_nat n) * a$n" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6343 |
by (simp add: fps_XDp_def algebra_simps) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6344 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6345 |
lemma fps_XDp_commute: "fps_XDp b \<circ> fps_XDp (c::'a::comm_ring_1) = fps_XDp c \<circ> fps_XDp b" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6346 |
by (simp add: fps_XDp_def fun_eq_iff fps_eq_iff algebra_simps) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6347 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6348 |
lemma fps_XDp0 [simp]: "fps_XDp 0 = fps_XD" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
6349 |
by (simp add: fun_eq_iff fps_eq_iff) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6350 |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6351 |
lemma fps_XDp_fps_integral [simp]: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6352 |
fixes a :: "'a::{division_ring,ring_char_0} fps" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6353 |
shows "fps_XDp 0 (fps_integral a c) = fps_X * a" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6354 |
using fps_deriv_fps_integral[of a c] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6355 |
by (simp add: fps_XD_def) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6356 |
|
65396
b42167902f57
moved AFP material to Formal_Power_Series; renamed E/L/F in Formal_Power_Series
eberlm <eberlm@in.tum.de>
parents:
64786
diff
changeset
|
6357 |
lemma fps_hypergeo_minus_nat: |
68442 | 6358 |
"fps_hypergeo [- of_nat n] [- of_nat (n + m)] (c::'a::field_char_0) $ k = |
54452 | 6359 |
(if k \<le> n then |
52902 | 6360 |
pochhammer (- of_nat n) k * c ^ k / (pochhammer (- of_nat (n + m)) k * of_nat (fact k)) |
6361 |
else 0)" |
|
68442 | 6362 |
"fps_hypergeo [- of_nat m] [- of_nat (m + n)] (c::'a::field_char_0) $ k = |
54452 | 6363 |
(if k \<le> m then |
52902 | 6364 |
pochhammer (- of_nat m) k * c ^ k / (pochhammer (- of_nat (m + n)) k * of_nat (fact k)) |
6365 |
else 0)" |
|
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6366 |
by (simp_all add: pochhammer_eq_0_iff) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6367 |
|
64267 | 6368 |
lemma sum_eq_if: "sum f {(n::nat) .. m} = (if m < n then 0 else f n + sum f {n+1 .. m})" |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6369 |
apply simp |
64267 | 6370 |
apply (subst sum.insert[symmetric]) |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69791
diff
changeset
|
6371 |
apply (auto simp add: not_less sum.atLeast_Suc_atMost) |
52902 | 6372 |
done |
6373 |
||
6374 |
lemma pochhammer_rec_if: "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))" |
|
6375 |
by (cases n) (simp_all add: pochhammer_rec) |
|
6376 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6377 |
lemma fps_XDp_foldr_nth [simp]: "foldr (\<lambda>c r. fps_XDp c \<circ> r) cs (\<lambda>c. fps_XDp c a) c0 $ n = |
54452 | 6378 |
foldr (\<lambda>c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6379 |
by (induct cs arbitrary: c0) (simp_all add: algebra_simps) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6380 |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6381 |
lemma genric_fps_XDp_foldr_nth: |
54452 | 6382 |
assumes f: "\<forall>n c a. f c a $ n = (of_nat n + k c) * a$n" |
54681 | 6383 |
shows "foldr (\<lambda>c r. f c \<circ> r) cs (\<lambda>c. g c a) c0 $ n = |
54452 | 6384 |
foldr (\<lambda>c r. (k c + of_nat n) * r) cs (g c0 a $ n)" |
69791
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69597
diff
changeset
|
6385 |
by (induct cs arbitrary: c0) (simp_all add: algebra_simps f) |
32160
63686057cbe8
Vandermonde vs Pochhammer; Hypergeometric series - very basic facts
chaieb
parents:
32157
diff
changeset
|
6386 |
|
51107 | 6387 |
lemma dist_less_imp_nth_equal: |
6388 |
assumes "dist f g < inverse (2 ^ i)" |
|
52902 | 6389 |
and"j \<le> i" |
51107 | 6390 |
shows "f $ j = g $ j" |
54263
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
6391 |
proof (rule ccontr) |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
6392 |
assume "f $ j \<noteq> g $ j" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6393 |
hence "f \<noteq> g" by auto |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6394 |
with assms have "i < subdegree (f - g)" |
62390 | 6395 |
by (simp add: if_split_asm dist_fps_def) |
54263
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
6396 |
also have "\<dots> \<le> j" |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6397 |
using \<open>f $ j \<noteq> g $ j\<close> by (intro subdegree_leI) simp_all |
60500 | 6398 |
finally show False using \<open>j \<le> i\<close> by simp |
52902 | 6399 |
qed |
51107 | 6400 |
|
6401 |
lemma nth_equal_imp_dist_less: |
|
6402 |
assumes "\<And>j. j \<le> i \<Longrightarrow> f $ j = g $ j" |
|
6403 |
shows "dist f g < inverse (2 ^ i)" |
|
52902 | 6404 |
proof (cases "f = g") |
60501 | 6405 |
case True |
6406 |
then show ?thesis by simp |
|
6407 |
next |
|
52902 | 6408 |
case False |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6409 |
with assms have "dist f g = inverse (2 ^ subdegree (f - g))" |
62390 | 6410 |
by (simp add: if_split_asm dist_fps_def) |
51107 | 6411 |
moreover |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6412 |
from assms and False have "i < subdegree (f - g)" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6413 |
by (intro subdegree_greaterI) simp_all |
51107 | 6414 |
ultimately show ?thesis by simp |
60501 | 6415 |
qed |
52902 | 6416 |
|
6417 |
lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) \<longleftrightarrow> (\<forall>j \<le> i. f $ j = g $ j)" |
|
51107 | 6418 |
using dist_less_imp_nth_equal nth_equal_imp_dist_less by blast |
6419 |
||
6420 |
instance fps :: (comm_ring_1) complete_space |
|
6421 |
proof |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6422 |
fix fps_X :: "nat \<Rightarrow> 'a fps" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6423 |
assume "Cauchy fps_X" |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6424 |
obtain M where M: "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. fps_X (M i) $ j = fps_X m $ j" |
60501 | 6425 |
proof - |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6426 |
have "\<exists>M. \<forall>m \<ge> M. \<forall>j\<le>i. fps_X M $ j = fps_X m $ j" for i |
60501 | 6427 |
proof - |
6428 |
have "0 < inverse ((2::real)^i)" by simp |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6429 |
from metric_CauchyD[OF \<open>Cauchy fps_X\<close> this] dist_less_imp_nth_equal |
60501 | 6430 |
show ?thesis by blast |
6431 |
qed |
|
6432 |
then show ?thesis using that by metis |
|
6433 |
qed |
|
6434 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6435 |
show "convergent fps_X" |
51107 | 6436 |
proof (rule convergentI) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6437 |
show "fps_X \<longlonglongrightarrow> Abs_fps (\<lambda>i. fps_X (M i) $ i)" |
51107 | 6438 |
unfolding tendsto_iff |
6439 |
proof safe |
|
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6440 |
fix e::real assume e: "0 < e" |
61969 | 6441 |
have "(\<lambda>n. inverse (2 ^ n) :: real) \<longlonglongrightarrow> 0" by (rule LIMSEQ_inverse_realpow_zero) simp_all |
61608
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6442 |
from this and e have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially" |
a0487caabb4a
subdegree/shift/cutoff and Euclidean ring instance for formal power series
eberlm
parents:
61585
diff
changeset
|
6443 |
by (rule order_tendstoD) |
60501 | 6444 |
then obtain i where "inverse (2 ^ i) < e" |
6445 |
by (auto simp: eventually_sequentially) |
|
6446 |
have "eventually (\<lambda>x. M i \<le> x) sequentially" |
|
6447 |
by (auto simp: eventually_sequentially) |
|
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6448 |
then show "eventually (\<lambda>x. dist (fps_X x) (Abs_fps (\<lambda>i. fps_X (M i) $ i)) < e) sequentially" |
51107 | 6449 |
proof eventually_elim |
52902 | 6450 |
fix x |
60501 | 6451 |
assume x: "M i \<le> x" |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6452 |
have "fps_X (M i) $ j = fps_X (M j) $ j" if "j \<le> i" for j |
60501 | 6453 |
using M that by (metis nat_le_linear) |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6454 |
with x have "dist (fps_X x) (Abs_fps (\<lambda>j. fps_X (M j) $ j)) < inverse (2 ^ i)" |
51107 | 6455 |
using M by (force simp: dist_less_eq_nth_equal) |
60500 | 6456 |
also note \<open>inverse (2 ^ i) < e\<close> |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6457 |
finally show "dist (fps_X x) (Abs_fps (\<lambda>j. fps_X (M j) $ j)) < e" . |
51107 | 6458 |
qed |
6459 |
qed |
|
6460 |
qed |
|
6461 |
qed |
|
6462 |
||
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6463 |
(* TODO: Figure out better notation for this thing *) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6464 |
no_notation fps_nth (infixl "$" 75) |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6465 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6466 |
bundle fps_notation |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6467 |
begin |
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6468 |
notation fps_nth (infixl "$" 75) |
29911
c790a70a3d19
declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents:
29906
diff
changeset
|
6469 |
end |
66480
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6470 |
|
4b8d1df8933b
HOL-Analysis: Convergent FPS and infinite sums
Manuel Eberl <eberlm@in.tum.de>
parents:
66466
diff
changeset
|
6471 |
end |