src/HOL/Computational_Algebra/Polynomial_FPS.thy
 author Manuel Eberl Mon Aug 21 20:49:15 2017 +0200 (2017-08-21) changeset 66480 4b8d1df8933b parent 65486 d801126a14cb child 67399 eab6ce8368fa permissions -rw-r--r--
HOL-Analysis: Convergent FPS and infinite sums
1 (*  Title:      HOL/Computational_Algebra/Polynomial_FPS.thy
2     Author:     Manuel Eberl, TU München
3 *)
5 section \<open>Converting polynomials to formal power series\<close>
7 theory Polynomial_FPS
8   imports Polynomial Formal_Power_Series
9 begin
11 context
12   includes fps_notation
13 begin
15 definition fps_of_poly where
16   "fps_of_poly p = Abs_fps (coeff p)"
18 lemma fps_of_poly_eq_iff: "fps_of_poly p = fps_of_poly q \<longleftrightarrow> p = q"
19   by (simp add: fps_of_poly_def poly_eq_iff fps_eq_iff)
21 lemma fps_of_poly_nth [simp]: "fps_of_poly p \$ n = coeff p n"
22   by (simp add: fps_of_poly_def)
24 lemma fps_of_poly_const: "fps_of_poly [:c:] = fps_const c"
25 proof (subst fps_eq_iff, clarify)
26   fix n :: nat show "fps_of_poly [:c:] \$ n = fps_const c \$ n"
27     by (cases n) (auto simp: fps_of_poly_def)
28 qed
30 lemma fps_of_poly_0 [simp]: "fps_of_poly 0 = 0"
31   by (subst fps_const_0_eq_0 [symmetric], subst fps_of_poly_const [symmetric]) simp
33 lemma fps_of_poly_1 [simp]: "fps_of_poly 1 = 1"
34   by (simp add: fps_eq_iff)
36 lemma fps_of_poly_1' [simp]: "fps_of_poly [:1:] = 1"
37   by (subst fps_const_1_eq_1 [symmetric], subst fps_of_poly_const [symmetric])
38      (simp add: one_poly_def)
40 lemma fps_of_poly_numeral [simp]: "fps_of_poly (numeral n) = numeral n"
41   by (simp add: numeral_fps_const fps_of_poly_const [symmetric] numeral_poly)
43 lemma fps_of_poly_numeral' [simp]: "fps_of_poly [:numeral n:] = numeral n"
44   by (simp add: numeral_fps_const fps_of_poly_const [symmetric] numeral_poly)
46 lemma fps_of_poly_fps_X [simp]: "fps_of_poly [:0, 1:] = fps_X"
47   by (auto simp add: fps_of_poly_def fps_eq_iff coeff_pCons split: nat.split)
49 lemma fps_of_poly_add: "fps_of_poly (p + q) = fps_of_poly p + fps_of_poly q"
50   by (simp add: fps_of_poly_def plus_poly.rep_eq fps_plus_def)
52 lemma fps_of_poly_diff: "fps_of_poly (p - q) = fps_of_poly p - fps_of_poly q"
53   by (simp add: fps_of_poly_def minus_poly.rep_eq fps_minus_def)
55 lemma fps_of_poly_uminus: "fps_of_poly (-p) = -fps_of_poly p"
56   by (simp add: fps_of_poly_def uminus_poly.rep_eq fps_uminus_def)
58 lemma fps_of_poly_mult: "fps_of_poly (p * q) = fps_of_poly p * fps_of_poly q"
59   by (simp add: fps_of_poly_def fps_times_def fps_eq_iff coeff_mult atLeast0AtMost)
61 lemma fps_of_poly_smult:
62   "fps_of_poly (smult c p) = fps_const c * fps_of_poly p"
63   using fps_of_poly_mult[of "[:c:]" p] by (simp add: fps_of_poly_mult fps_of_poly_const)
65 lemma fps_of_poly_sum: "fps_of_poly (sum f A) = sum (\<lambda>x. fps_of_poly (f x)) A"
66   by (cases "finite A", induction rule: finite_induct) (simp_all add: fps_of_poly_add)
68 lemma fps_of_poly_sum_list: "fps_of_poly (sum_list xs) = sum_list (map fps_of_poly xs)"
69   by (induction xs) (simp_all add: fps_of_poly_add)
71 lemma fps_of_poly_prod: "fps_of_poly (prod f A) = prod (\<lambda>x. fps_of_poly (f x)) A"
72   by (cases "finite A", induction rule: finite_induct) (simp_all add: fps_of_poly_mult)
74 lemma fps_of_poly_prod_list: "fps_of_poly (prod_list xs) = prod_list (map fps_of_poly xs)"
75   by (induction xs) (simp_all add: fps_of_poly_mult)
77 lemma fps_of_poly_pCons:
78   "fps_of_poly (pCons (c :: 'a :: semiring_1) p) = fps_const c + fps_of_poly p * fps_X"
79   by (subst fps_mult_fps_X_commute [symmetric], intro fps_ext)
80      (auto simp: fps_of_poly_def coeff_pCons split: nat.split)
82 lemma fps_of_poly_pderiv: "fps_of_poly (pderiv p) = fps_deriv (fps_of_poly p)"
83   by (intro fps_ext) (simp add: fps_of_poly_nth coeff_pderiv)
85 lemma fps_of_poly_power: "fps_of_poly (p ^ n) = fps_of_poly p ^ n"
86   by (induction n) (simp_all add: fps_of_poly_mult)
88 lemma fps_of_poly_monom: "fps_of_poly (monom (c :: 'a :: comm_ring_1) n) = fps_const c * fps_X ^ n"
89   by (intro fps_ext) simp_all
91 lemma fps_of_poly_monom': "fps_of_poly (monom (1 :: 'a :: comm_ring_1) n) = fps_X ^ n"
92   by (simp add: fps_of_poly_monom)
94 lemma fps_of_poly_div:
95   assumes "(q :: 'a :: field poly) dvd p"
96   shows   "fps_of_poly (p div q) = fps_of_poly p / fps_of_poly q"
97 proof (cases "q = 0")
98   case False
99   from False fps_of_poly_eq_iff[of q 0] have nz: "fps_of_poly q \<noteq> 0" by simp
100   from assms have "p = (p div q) * q" by simp
101   also have "fps_of_poly \<dots> = fps_of_poly (p div q) * fps_of_poly q"
102     by (simp add: fps_of_poly_mult)
103   also from nz have "\<dots> / fps_of_poly q = fps_of_poly (p div q)"
104     by (intro nonzero_mult_div_cancel_right) (auto simp: fps_of_poly_0)
105   finally show ?thesis ..
106 qed simp
108 lemma fps_of_poly_divide_numeral:
109   "fps_of_poly (smult (inverse (numeral c :: 'a :: field)) p) = fps_of_poly p / numeral c"
110 proof -
111   have "smult (inverse (numeral c)) p = [:inverse (numeral c):] * p" by simp
112   also have "fps_of_poly \<dots> = fps_of_poly p / numeral c"
113     by (subst fps_of_poly_mult) (simp add: numeral_fps_const fps_of_poly_pCons)
114   finally show ?thesis by simp
115 qed
118 lemma subdegree_fps_of_poly:
119   assumes "p \<noteq> 0"
120   defines "n \<equiv> Polynomial.order 0 p"
121   shows   "subdegree (fps_of_poly p) = n"
122 proof (rule subdegreeI)
123   from assms have "monom 1 n dvd p" by (simp add: monom_1_dvd_iff)
124   thus zero: "fps_of_poly p \$ i = 0" if "i < n" for i
125     using that by (simp add: monom_1_dvd_iff')
127   from assms have "\<not>monom 1 (Suc n) dvd p"
128     by (auto simp: monom_1_dvd_iff simp del: power_Suc)
129   then obtain k where k: "k \<le> n" "fps_of_poly p \$ k \<noteq> 0"
130     by (auto simp: monom_1_dvd_iff' less_Suc_eq_le)
131   with zero[of k] have "k = n" by linarith
132   with k show "fps_of_poly p \$ n \<noteq> 0" by simp
133 qed
135 lemma fps_of_poly_dvd:
136   assumes "p dvd q"
137   shows   "fps_of_poly (p :: 'a :: field poly) dvd fps_of_poly q"
138 proof (cases "p = 0 \<or> q = 0")
139   case False
140   with assms fps_of_poly_eq_iff[of p 0] fps_of_poly_eq_iff[of q 0] show ?thesis
141     by (auto simp: fps_dvd_iff subdegree_fps_of_poly dvd_imp_order_le)
142 qed (insert assms, auto)
145 lemmas fps_of_poly_simps =
146   fps_of_poly_0 fps_of_poly_1 fps_of_poly_numeral fps_of_poly_const fps_of_poly_fps_X
147   fps_of_poly_add fps_of_poly_diff fps_of_poly_uminus fps_of_poly_mult fps_of_poly_smult
148   fps_of_poly_sum fps_of_poly_sum_list fps_of_poly_prod fps_of_poly_prod_list
149   fps_of_poly_pCons fps_of_poly_pderiv fps_of_poly_power fps_of_poly_monom
150   fps_of_poly_divide_numeral
152 lemma fps_of_poly_pcompose:
153   assumes "coeff q 0 = (0 :: 'a :: idom)"
154   shows   "fps_of_poly (pcompose p q) = fps_compose (fps_of_poly p) (fps_of_poly q)"
155   using assms by (induction p rule: pCons_induct)
156                  (auto simp: pcompose_pCons fps_of_poly_simps fps_of_poly_pCons
159 lemmas reify_fps_atom =
160   fps_of_poly_0 fps_of_poly_1' fps_of_poly_numeral' fps_of_poly_const fps_of_poly_fps_X
163 text \<open>
164   The following simproc can reduce the equality of two polynomial FPSs two equality of the
165   respective polynomials. A polynomial FPS is one that only has finitely many non-zero
166   coefficients and can therefore be written as @{term "fps_of_poly p"} for some
167   polynomial \<open>p\<close>.
169   This may sound trivial, but it covers a number of annoying side conditions like
170   @{term "1 + fps_X \<noteq> 0"} that would otherwise not be solved automatically.
171 \<close>
173 ML \<open>
175 (* TODO: Support for division *)
176 signature POLY_FPS = sig
178 val reify_conv : conv
179 val eq_conv : conv
180 val eq_simproc : cterm -> thm option
182 end
185 structure Poly_Fps = struct
187 fun const_binop_conv s conv ct =
188   case Thm.term_of ct of
189     (Const (s', _) \$ _ \$ _) =>
190       if s = s' then
191         Conv.binop_conv conv ct
192       else
193         raise CTERM ("const_binop_conv", [ct])
194   | _ => raise CTERM ("const_binop_conv", [ct])
196 fun reify_conv ct =
197   let
198     val rewr = Conv.rewrs_conv o map (fn thm => thm RS @{thm eq_reflection})
199     val un = Conv.arg_conv reify_conv
200     val bin = Conv.binop_conv reify_conv
201   in
202     case Thm.term_of ct of
203       (Const (@{const_name "fps_of_poly"}, _) \$ _) => ct |> Conv.all_conv
204     | (Const (@{const_name "Groups.plus"}, _) \$ _ \$ _) => ct |> (
205         bin then_conv rewr @{thms fps_of_poly_add [symmetric]})
206     | (Const (@{const_name "Groups.uminus"}, _) \$ _) => ct |> (
207         un then_conv rewr @{thms fps_of_poly_uminus [symmetric]})
208     | (Const (@{const_name "Groups.minus"}, _) \$ _ \$ _) => ct |> (
209         bin then_conv rewr @{thms fps_of_poly_diff [symmetric]})
210     | (Const (@{const_name "Groups.times"}, _) \$ _ \$ _) => ct |> (
211         bin then_conv rewr @{thms fps_of_poly_mult [symmetric]})
212     | (Const (@{const_name "Rings.divide"}, _) \$ _ \$ (Const (@{const_name "Num.numeral"}, _) \$ _))
213         => ct |> (Conv.fun_conv (Conv.arg_conv reify_conv)
214              then_conv rewr @{thms fps_of_poly_divide_numeral [symmetric]})
215     | (Const (@{const_name "Power.power"}, _) \$ Const (@{const_name "fps_X"},_) \$ _) => ct |> (
216         rewr @{thms fps_of_poly_monom' [symmetric]})
217     | (Const (@{const_name "Power.power"}, _) \$ _ \$ _) => ct |> (
218         Conv.fun_conv (Conv.arg_conv reify_conv)
219         then_conv rewr @{thms fps_of_poly_power [symmetric]})
220     | _ => ct |> (
221         rewr @{thms reify_fps_atom [symmetric]})
222   end
225 fun eq_conv ct =
226   case Thm.term_of ct of
227     (Const (@{const_name "HOL.eq"}, _) \$ _ \$ _) => ct |> (
228       Conv.binop_conv reify_conv
229       then_conv Conv.rewr_conv @{thm fps_of_poly_eq_iff[THEN eq_reflection]})
230   | _ => raise CTERM ("poly_fps_eq_conv", [ct])
232 val eq_simproc = try eq_conv
234 end
235 \<close>
237 simproc_setup poly_fps_eq ("(f :: 'a fps) = g") = \<open>K (K Poly_Fps.eq_simproc)\<close>
239 lemma fps_of_poly_linear: "fps_of_poly [:a,1 :: 'a :: field:] = fps_X + fps_const a"
240   by simp
242 lemma fps_of_poly_linear': "fps_of_poly [:1,a :: 'a :: field:] = 1 + fps_const a * fps_X"
243   by simp
245 lemma fps_of_poly_cutoff [simp]:
246   "fps_of_poly (poly_cutoff n p) = fps_cutoff n (fps_of_poly p)"
247   by (simp add: fps_eq_iff coeff_poly_cutoff)
249 lemma fps_of_poly_shift [simp]: "fps_of_poly (poly_shift n p) = fps_shift n (fps_of_poly p)"
250   by (simp add: fps_eq_iff coeff_poly_shift)
253 definition poly_subdegree :: "'a::zero poly \<Rightarrow> nat" where
254   "poly_subdegree p = subdegree (fps_of_poly p)"
256 lemma coeff_less_poly_subdegree:
257   "k < poly_subdegree p \<Longrightarrow> coeff p k = 0"
258   unfolding poly_subdegree_def using nth_less_subdegree_zero[of k "fps_of_poly p"] by simp
260 (* TODO: Move ? *)
261 definition prefix_length :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat" where
262   "prefix_length P xs = length (takeWhile P xs)"
264 primrec prefix_length_aux :: "('a \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> nat" where
265   "prefix_length_aux P acc [] = acc"
266 | "prefix_length_aux P acc (x#xs) = (if P x then prefix_length_aux P (Suc acc) xs else acc)"
268 lemma prefix_length_aux_correct: "prefix_length_aux P acc xs = prefix_length P xs + acc"
269   by (induction xs arbitrary: acc) (simp_all add: prefix_length_def)
271 lemma prefix_length_code [code]: "prefix_length P xs = prefix_length_aux P 0 xs"
272   by (simp add: prefix_length_aux_correct)
274 lemma prefix_length_le_length: "prefix_length P xs \<le> length xs"
275   by (induction xs) (simp_all add: prefix_length_def)
277 lemma prefix_length_less_length: "(\<exists>x\<in>set xs. \<not>P x) \<Longrightarrow> prefix_length P xs < length xs"
278   by (induction xs) (simp_all add: prefix_length_def)
280 lemma nth_prefix_length:
281   "(\<exists>x\<in>set xs. \<not>P x) \<Longrightarrow> \<not>P (xs ! prefix_length P xs)"
282   by (induction xs) (simp_all add: prefix_length_def)
284 lemma nth_less_prefix_length:
285   "n < prefix_length P xs \<Longrightarrow> P (xs ! n)"
286   by (induction xs arbitrary: n)
287      (auto simp: prefix_length_def nth_Cons split: if_splits nat.splits)
288 (* END TODO *)
290 lemma poly_subdegree_code [code]: "poly_subdegree p = prefix_length (op = 0) (coeffs p)"
291 proof (cases "p = 0")
292   case False
293   note [simp] = this
294   define n where "n = prefix_length (op = 0) (coeffs p)"
295   from False have "\<exists>k. coeff p k \<noteq> 0" by (auto simp: poly_eq_iff)
296   hence ex: "\<exists>x\<in>set (coeffs p). x \<noteq> 0" by (auto simp: coeffs_def)
297   hence n_less: "n < length (coeffs p)" and nonzero: "coeffs p ! n \<noteq> 0"
298     unfolding n_def by (auto intro!: prefix_length_less_length nth_prefix_length)
299   show ?thesis unfolding poly_subdegree_def
300   proof (intro subdegreeI)
301     from n_less have "fps_of_poly p \$ n = coeffs p ! n"
302       by (subst coeffs_nth) (simp_all add: degree_eq_length_coeffs)
303     with nonzero show "fps_of_poly p \$ prefix_length (op = 0) (coeffs p) \<noteq> 0"
304       unfolding n_def by simp
305   next
306     fix k assume A: "k < prefix_length (op = 0) (coeffs p)"
307     also have "\<dots> \<le> length (coeffs p)" by (rule prefix_length_le_length)
308     finally show "fps_of_poly p \$ k = 0"
309       using nth_less_prefix_length[OF A]
310       by (simp add: coeffs_nth degree_eq_length_coeffs)
311   qed
312 qed (simp_all add: poly_subdegree_def prefix_length_def)
314 end
316 end