src/HOL/Polynomial.thy
 author huffman Mon Jan 12 08:15:07 2009 -0800 (2009-01-12) changeset 29453 de4f26f59135 parent 29451 5f0cb3fa530d child 29454 b0f586f38dd7 permissions -rw-r--r--
 huffman@29451 ` 1` ```(* Title: HOL/Polynomial.thy ``` huffman@29451 ` 2` ``` Author: Brian Huffman ``` huffman@29451 ` 3` ``` Based on an earlier development by Clemens Ballarin ``` huffman@29451 ` 4` ```*) ``` huffman@29451 ` 5` huffman@29451 ` 6` ```header {* Univariate Polynomials *} ``` huffman@29451 ` 7` huffman@29451 ` 8` ```theory Polynomial ``` huffman@29451 ` 9` ```imports Plain SetInterval ``` huffman@29451 ` 10` ```begin ``` huffman@29451 ` 11` huffman@29451 ` 12` ```subsection {* Definition of type @{text poly} *} ``` huffman@29451 ` 13` huffman@29451 ` 14` ```typedef (Poly) 'a poly = "{f::nat \ 'a::zero. \n. \i>n. f i = 0}" ``` huffman@29451 ` 15` ``` morphisms coeff Abs_poly ``` huffman@29451 ` 16` ``` by auto ``` huffman@29451 ` 17` huffman@29451 ` 18` ```lemma expand_poly_eq: "p = q \ (\n. coeff p n = coeff q n)" ``` huffman@29451 ` 19` ```by (simp add: coeff_inject [symmetric] expand_fun_eq) ``` huffman@29451 ` 20` huffman@29451 ` 21` ```lemma poly_ext: "(\n. coeff p n = coeff q n) \ p = q" ``` huffman@29451 ` 22` ```by (simp add: expand_poly_eq) ``` huffman@29451 ` 23` huffman@29451 ` 24` huffman@29451 ` 25` ```subsection {* Degree of a polynomial *} ``` huffman@29451 ` 26` huffman@29451 ` 27` ```definition ``` huffman@29451 ` 28` ``` degree :: "'a::zero poly \ nat" where ``` huffman@29451 ` 29` ``` "degree p = (LEAST n. \i>n. coeff p i = 0)" ``` huffman@29451 ` 30` huffman@29451 ` 31` ```lemma coeff_eq_0: "degree p < n \ coeff p n = 0" ``` huffman@29451 ` 32` ```proof - ``` huffman@29451 ` 33` ``` have "coeff p \ Poly" ``` huffman@29451 ` 34` ``` by (rule coeff) ``` huffman@29451 ` 35` ``` hence "\n. \i>n. coeff p i = 0" ``` huffman@29451 ` 36` ``` unfolding Poly_def by simp ``` huffman@29451 ` 37` ``` hence "\i>degree p. coeff p i = 0" ``` huffman@29451 ` 38` ``` unfolding degree_def by (rule LeastI_ex) ``` huffman@29451 ` 39` ``` moreover assume "degree p < n" ``` huffman@29451 ` 40` ``` ultimately show ?thesis by simp ``` huffman@29451 ` 41` ```qed ``` huffman@29451 ` 42` huffman@29451 ` 43` ```lemma le_degree: "coeff p n \ 0 \ n \ degree p" ``` huffman@29451 ` 44` ``` by (erule contrapos_np, rule coeff_eq_0, simp) ``` huffman@29451 ` 45` huffman@29451 ` 46` ```lemma degree_le: "\i>n. coeff p i = 0 \ degree p \ n" ``` huffman@29451 ` 47` ``` unfolding degree_def by (erule Least_le) ``` huffman@29451 ` 48` huffman@29451 ` 49` ```lemma less_degree_imp: "n < degree p \ \i>n. coeff p i \ 0" ``` huffman@29451 ` 50` ``` unfolding degree_def by (drule not_less_Least, simp) ``` huffman@29451 ` 51` huffman@29451 ` 52` huffman@29451 ` 53` ```subsection {* The zero polynomial *} ``` huffman@29451 ` 54` huffman@29451 ` 55` ```instantiation poly :: (zero) zero ``` huffman@29451 ` 56` ```begin ``` huffman@29451 ` 57` huffman@29451 ` 58` ```definition ``` huffman@29451 ` 59` ``` zero_poly_def: "0 = Abs_poly (\n. 0)" ``` huffman@29451 ` 60` huffman@29451 ` 61` ```instance .. ``` huffman@29451 ` 62` ```end ``` huffman@29451 ` 63` huffman@29451 ` 64` ```lemma coeff_0 [simp]: "coeff 0 n = 0" ``` huffman@29451 ` 65` ``` unfolding zero_poly_def ``` huffman@29451 ` 66` ``` by (simp add: Abs_poly_inverse Poly_def) ``` huffman@29451 ` 67` huffman@29451 ` 68` ```lemma degree_0 [simp]: "degree 0 = 0" ``` huffman@29451 ` 69` ``` by (rule order_antisym [OF degree_le le0]) simp ``` huffman@29451 ` 70` huffman@29451 ` 71` ```lemma leading_coeff_neq_0: ``` huffman@29451 ` 72` ``` assumes "p \ 0" shows "coeff p (degree p) \ 0" ``` huffman@29451 ` 73` ```proof (cases "degree p") ``` huffman@29451 ` 74` ``` case 0 ``` huffman@29451 ` 75` ``` from `p \ 0` have "\n. coeff p n \ 0" ``` huffman@29451 ` 76` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 77` ``` then obtain n where "coeff p n \ 0" .. ``` huffman@29451 ` 78` ``` hence "n \ degree p" by (rule le_degree) ``` huffman@29451 ` 79` ``` with `coeff p n \ 0` and `degree p = 0` ``` huffman@29451 ` 80` ``` show "coeff p (degree p) \ 0" by simp ``` huffman@29451 ` 81` ```next ``` huffman@29451 ` 82` ``` case (Suc n) ``` huffman@29451 ` 83` ``` from `degree p = Suc n` have "n < degree p" by simp ``` huffman@29451 ` 84` ``` hence "\i>n. coeff p i \ 0" by (rule less_degree_imp) ``` huffman@29451 ` 85` ``` then obtain i where "n < i" and "coeff p i \ 0" by fast ``` huffman@29451 ` 86` ``` from `degree p = Suc n` and `n < i` have "degree p \ i" by simp ``` huffman@29451 ` 87` ``` also from `coeff p i \ 0` have "i \ degree p" by (rule le_degree) ``` huffman@29451 ` 88` ``` finally have "degree p = i" . ``` huffman@29451 ` 89` ``` with `coeff p i \ 0` show "coeff p (degree p) \ 0" by simp ``` huffman@29451 ` 90` ```qed ``` huffman@29451 ` 91` huffman@29451 ` 92` ```lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \ p = 0" ``` huffman@29451 ` 93` ``` by (cases "p = 0", simp, simp add: leading_coeff_neq_0) ``` huffman@29451 ` 94` huffman@29451 ` 95` huffman@29451 ` 96` ```subsection {* List-style constructor for polynomials *} ``` huffman@29451 ` 97` huffman@29451 ` 98` ```definition ``` huffman@29451 ` 99` ``` pCons :: "'a::zero \ 'a poly \ 'a poly" ``` huffman@29451 ` 100` ```where ``` huffman@29451 ` 101` ``` [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))" ``` huffman@29451 ` 102` huffman@29451 ` 103` ```lemma Poly_nat_case: "f \ Poly \ nat_case a f \ Poly" ``` huffman@29451 ` 104` ``` unfolding Poly_def by (auto split: nat.split) ``` huffman@29451 ` 105` huffman@29451 ` 106` ```lemma coeff_pCons: ``` huffman@29451 ` 107` ``` "coeff (pCons a p) = nat_case a (coeff p)" ``` huffman@29451 ` 108` ``` unfolding pCons_def ``` huffman@29451 ` 109` ``` by (simp add: Abs_poly_inverse Poly_nat_case coeff) ``` huffman@29451 ` 110` huffman@29451 ` 111` ```lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a" ``` huffman@29451 ` 112` ``` by (simp add: coeff_pCons) ``` huffman@29451 ` 113` huffman@29451 ` 114` ```lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" ``` huffman@29451 ` 115` ``` by (simp add: coeff_pCons) ``` huffman@29451 ` 116` huffman@29451 ` 117` ```lemma degree_pCons_le: "degree (pCons a p) \ Suc (degree p)" ``` huffman@29451 ` 118` ```by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split) ``` huffman@29451 ` 119` huffman@29451 ` 120` ```lemma degree_pCons_eq: ``` huffman@29451 ` 121` ``` "p \ 0 \ degree (pCons a p) = Suc (degree p)" ``` huffman@29451 ` 122` ```apply (rule order_antisym [OF degree_pCons_le]) ``` huffman@29451 ` 123` ```apply (rule le_degree, simp) ``` huffman@29451 ` 124` ```done ``` huffman@29451 ` 125` huffman@29451 ` 126` ```lemma degree_pCons_0: "degree (pCons a 0) = 0" ``` huffman@29451 ` 127` ```apply (rule order_antisym [OF _ le0]) ``` huffman@29451 ` 128` ```apply (rule degree_le, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 129` ```done ``` huffman@29451 ` 130` huffman@29451 ` 131` ```lemma degree_pCons_eq_if: ``` huffman@29451 ` 132` ``` "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" ``` huffman@29451 ` 133` ```apply (cases "p = 0", simp_all) ``` huffman@29451 ` 134` ```apply (rule order_antisym [OF _ le0]) ``` huffman@29451 ` 135` ```apply (rule degree_le, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 136` ```apply (rule order_antisym [OF degree_pCons_le]) ``` huffman@29451 ` 137` ```apply (rule le_degree, simp) ``` huffman@29451 ` 138` ```done ``` huffman@29451 ` 139` huffman@29451 ` 140` ```lemma pCons_0_0 [simp]: "pCons 0 0 = 0" ``` huffman@29451 ` 141` ```by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 142` huffman@29451 ` 143` ```lemma pCons_eq_iff [simp]: ``` huffman@29451 ` 144` ``` "pCons a p = pCons b q \ a = b \ p = q" ``` huffman@29451 ` 145` ```proof (safe) ``` huffman@29451 ` 146` ``` assume "pCons a p = pCons b q" ``` huffman@29451 ` 147` ``` then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp ``` huffman@29451 ` 148` ``` then show "a = b" by simp ``` huffman@29451 ` 149` ```next ``` huffman@29451 ` 150` ``` assume "pCons a p = pCons b q" ``` huffman@29451 ` 151` ``` then have "\n. coeff (pCons a p) (Suc n) = ``` huffman@29451 ` 152` ``` coeff (pCons b q) (Suc n)" by simp ``` huffman@29451 ` 153` ``` then show "p = q" by (simp add: expand_poly_eq) ``` huffman@29451 ` 154` ```qed ``` huffman@29451 ` 155` huffman@29451 ` 156` ```lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \ a = 0 \ p = 0" ``` huffman@29451 ` 157` ``` using pCons_eq_iff [of a p 0 0] by simp ``` huffman@29451 ` 158` huffman@29451 ` 159` ```lemma Poly_Suc: "f \ Poly \ (\n. f (Suc n)) \ Poly" ``` huffman@29451 ` 160` ``` unfolding Poly_def ``` huffman@29451 ` 161` ``` by (clarify, rule_tac x=n in exI, simp) ``` huffman@29451 ` 162` huffman@29451 ` 163` ```lemma pCons_cases [cases type: poly]: ``` huffman@29451 ` 164` ``` obtains (pCons) a q where "p = pCons a q" ``` huffman@29451 ` 165` ```proof ``` huffman@29451 ` 166` ``` show "p = pCons (coeff p 0) (Abs_poly (\n. coeff p (Suc n)))" ``` huffman@29451 ` 167` ``` by (rule poly_ext) ``` huffman@29451 ` 168` ``` (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons ``` huffman@29451 ` 169` ``` split: nat.split) ``` huffman@29451 ` 170` ```qed ``` huffman@29451 ` 171` huffman@29451 ` 172` ```lemma pCons_induct [case_names 0 pCons, induct type: poly]: ``` huffman@29451 ` 173` ``` assumes zero: "P 0" ``` huffman@29451 ` 174` ``` assumes pCons: "\a p. P p \ P (pCons a p)" ``` huffman@29451 ` 175` ``` shows "P p" ``` huffman@29451 ` 176` ```proof (induct p rule: measure_induct_rule [where f=degree]) ``` huffman@29451 ` 177` ``` case (less p) ``` huffman@29451 ` 178` ``` obtain a q where "p = pCons a q" by (rule pCons_cases) ``` huffman@29451 ` 179` ``` have "P q" ``` huffman@29451 ` 180` ``` proof (cases "q = 0") ``` huffman@29451 ` 181` ``` case True ``` huffman@29451 ` 182` ``` then show "P q" by (simp add: zero) ``` huffman@29451 ` 183` ``` next ``` huffman@29451 ` 184` ``` case False ``` huffman@29451 ` 185` ``` then have "degree (pCons a q) = Suc (degree q)" ``` huffman@29451 ` 186` ``` by (rule degree_pCons_eq) ``` huffman@29451 ` 187` ``` then have "degree q < degree p" ``` huffman@29451 ` 188` ``` using `p = pCons a q` by simp ``` huffman@29451 ` 189` ``` then show "P q" ``` huffman@29451 ` 190` ``` by (rule less.hyps) ``` huffman@29451 ` 191` ``` qed ``` huffman@29451 ` 192` ``` then have "P (pCons a q)" ``` huffman@29451 ` 193` ``` by (rule pCons) ``` huffman@29451 ` 194` ``` then show ?case ``` huffman@29451 ` 195` ``` using `p = pCons a q` by simp ``` huffman@29451 ` 196` ```qed ``` huffman@29451 ` 197` huffman@29451 ` 198` huffman@29451 ` 199` ```subsection {* Monomials *} ``` huffman@29451 ` 200` huffman@29451 ` 201` ```definition ``` huffman@29451 ` 202` ``` monom :: "'a \ nat \ 'a::zero poly" where ``` huffman@29451 ` 203` ``` "monom a m = Abs_poly (\n. if m = n then a else 0)" ``` huffman@29451 ` 204` huffman@29451 ` 205` ```lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)" ``` huffman@29451 ` 206` ``` unfolding monom_def ``` huffman@29451 ` 207` ``` by (subst Abs_poly_inverse, auto simp add: Poly_def) ``` huffman@29451 ` 208` huffman@29451 ` 209` ```lemma monom_0: "monom a 0 = pCons a 0" ``` huffman@29451 ` 210` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 211` huffman@29451 ` 212` ```lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" ``` huffman@29451 ` 213` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 214` huffman@29451 ` 215` ```lemma monom_eq_0 [simp]: "monom 0 n = 0" ``` huffman@29451 ` 216` ``` by (rule poly_ext) simp ``` huffman@29451 ` 217` huffman@29451 ` 218` ```lemma monom_eq_0_iff [simp]: "monom a n = 0 \ a = 0" ``` huffman@29451 ` 219` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 220` huffman@29451 ` 221` ```lemma monom_eq_iff [simp]: "monom a n = monom b n \ a = b" ``` huffman@29451 ` 222` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 223` huffman@29451 ` 224` ```lemma degree_monom_le: "degree (monom a n) \ n" ``` huffman@29451 ` 225` ``` by (rule degree_le, simp) ``` huffman@29451 ` 226` huffman@29451 ` 227` ```lemma degree_monom_eq: "a \ 0 \ degree (monom a n) = n" ``` huffman@29451 ` 228` ``` apply (rule order_antisym [OF degree_monom_le]) ``` huffman@29451 ` 229` ``` apply (rule le_degree, simp) ``` huffman@29451 ` 230` ``` done ``` huffman@29451 ` 231` huffman@29451 ` 232` huffman@29451 ` 233` ```subsection {* Addition and subtraction *} ``` huffman@29451 ` 234` huffman@29451 ` 235` ```instantiation poly :: (comm_monoid_add) comm_monoid_add ``` huffman@29451 ` 236` ```begin ``` huffman@29451 ` 237` huffman@29451 ` 238` ```definition ``` huffman@29451 ` 239` ``` plus_poly_def [code del]: ``` huffman@29451 ` 240` ``` "p + q = Abs_poly (\n. coeff p n + coeff q n)" ``` huffman@29451 ` 241` huffman@29451 ` 242` ```lemma Poly_add: ``` huffman@29451 ` 243` ``` fixes f g :: "nat \ 'a" ``` huffman@29451 ` 244` ``` shows "\f \ Poly; g \ Poly\ \ (\n. f n + g n) \ Poly" ``` huffman@29451 ` 245` ``` unfolding Poly_def ``` huffman@29451 ` 246` ``` apply (clarify, rename_tac m n) ``` huffman@29451 ` 247` ``` apply (rule_tac x="max m n" in exI, simp) ``` huffman@29451 ` 248` ``` done ``` huffman@29451 ` 249` huffman@29451 ` 250` ```lemma coeff_add [simp]: ``` huffman@29451 ` 251` ``` "coeff (p + q) n = coeff p n + coeff q n" ``` huffman@29451 ` 252` ``` unfolding plus_poly_def ``` huffman@29451 ` 253` ``` by (simp add: Abs_poly_inverse coeff Poly_add) ``` huffman@29451 ` 254` huffman@29451 ` 255` ```instance proof ``` huffman@29451 ` 256` ``` fix p q r :: "'a poly" ``` huffman@29451 ` 257` ``` show "(p + q) + r = p + (q + r)" ``` huffman@29451 ` 258` ``` by (simp add: expand_poly_eq add_assoc) ``` huffman@29451 ` 259` ``` show "p + q = q + p" ``` huffman@29451 ` 260` ``` by (simp add: expand_poly_eq add_commute) ``` huffman@29451 ` 261` ``` show "0 + p = p" ``` huffman@29451 ` 262` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 263` ```qed ``` huffman@29451 ` 264` huffman@29451 ` 265` ```end ``` huffman@29451 ` 266` huffman@29451 ` 267` ```instantiation poly :: (ab_group_add) ab_group_add ``` huffman@29451 ` 268` ```begin ``` huffman@29451 ` 269` huffman@29451 ` 270` ```definition ``` huffman@29451 ` 271` ``` uminus_poly_def [code del]: ``` huffman@29451 ` 272` ``` "- p = Abs_poly (\n. - coeff p n)" ``` huffman@29451 ` 273` huffman@29451 ` 274` ```definition ``` huffman@29451 ` 275` ``` minus_poly_def [code del]: ``` huffman@29451 ` 276` ``` "p - q = Abs_poly (\n. coeff p n - coeff q n)" ``` huffman@29451 ` 277` huffman@29451 ` 278` ```lemma Poly_minus: ``` huffman@29451 ` 279` ``` fixes f :: "nat \ 'a" ``` huffman@29451 ` 280` ``` shows "f \ Poly \ (\n. - f n) \ Poly" ``` huffman@29451 ` 281` ``` unfolding Poly_def by simp ``` huffman@29451 ` 282` huffman@29451 ` 283` ```lemma Poly_diff: ``` huffman@29451 ` 284` ``` fixes f g :: "nat \ 'a" ``` huffman@29451 ` 285` ``` shows "\f \ Poly; g \ Poly\ \ (\n. f n - g n) \ Poly" ``` huffman@29451 ` 286` ``` unfolding diff_minus by (simp add: Poly_add Poly_minus) ``` huffman@29451 ` 287` huffman@29451 ` 288` ```lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" ``` huffman@29451 ` 289` ``` unfolding uminus_poly_def ``` huffman@29451 ` 290` ``` by (simp add: Abs_poly_inverse coeff Poly_minus) ``` huffman@29451 ` 291` huffman@29451 ` 292` ```lemma coeff_diff [simp]: ``` huffman@29451 ` 293` ``` "coeff (p - q) n = coeff p n - coeff q n" ``` huffman@29451 ` 294` ``` unfolding minus_poly_def ``` huffman@29451 ` 295` ``` by (simp add: Abs_poly_inverse coeff Poly_diff) ``` huffman@29451 ` 296` huffman@29451 ` 297` ```instance proof ``` huffman@29451 ` 298` ``` fix p q :: "'a poly" ``` huffman@29451 ` 299` ``` show "- p + p = 0" ``` huffman@29451 ` 300` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 301` ``` show "p - q = p + - q" ``` huffman@29451 ` 302` ``` by (simp add: expand_poly_eq diff_minus) ``` huffman@29451 ` 303` ```qed ``` huffman@29451 ` 304` huffman@29451 ` 305` ```end ``` huffman@29451 ` 306` huffman@29451 ` 307` ```lemma add_pCons [simp]: ``` huffman@29451 ` 308` ``` "pCons a p + pCons b q = pCons (a + b) (p + q)" ``` huffman@29451 ` 309` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 310` huffman@29451 ` 311` ```lemma minus_pCons [simp]: ``` huffman@29451 ` 312` ``` "- pCons a p = pCons (- a) (- p)" ``` huffman@29451 ` 313` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 314` huffman@29451 ` 315` ```lemma diff_pCons [simp]: ``` huffman@29451 ` 316` ``` "pCons a p - pCons b q = pCons (a - b) (p - q)" ``` huffman@29451 ` 317` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 318` huffman@29451 ` 319` ```lemma degree_add_le: "degree (p + q) \ max (degree p) (degree q)" ``` huffman@29451 ` 320` ``` by (rule degree_le, auto simp add: coeff_eq_0) ``` huffman@29451 ` 321` huffman@29453 ` 322` ```lemma degree_add_less: ``` huffman@29453 ` 323` ``` "\degree p < n; degree q < n\ \ degree (p + q) < n" ``` huffman@29453 ` 324` ``` by (auto intro: le_less_trans degree_add_le) ``` huffman@29453 ` 325` huffman@29451 ` 326` ```lemma degree_add_eq_right: ``` huffman@29451 ` 327` ``` "degree p < degree q \ degree (p + q) = degree q" ``` huffman@29451 ` 328` ``` apply (cases "q = 0", simp) ``` huffman@29451 ` 329` ``` apply (rule order_antisym) ``` huffman@29451 ` 330` ``` apply (rule ord_le_eq_trans [OF degree_add_le]) ``` huffman@29451 ` 331` ``` apply simp ``` huffman@29451 ` 332` ``` apply (rule le_degree) ``` huffman@29451 ` 333` ``` apply (simp add: coeff_eq_0) ``` huffman@29451 ` 334` ``` done ``` huffman@29451 ` 335` huffman@29451 ` 336` ```lemma degree_add_eq_left: ``` huffman@29451 ` 337` ``` "degree q < degree p \ degree (p + q) = degree p" ``` huffman@29451 ` 338` ``` using degree_add_eq_right [of q p] ``` huffman@29451 ` 339` ``` by (simp add: add_commute) ``` huffman@29451 ` 340` huffman@29451 ` 341` ```lemma degree_minus [simp]: "degree (- p) = degree p" ``` huffman@29451 ` 342` ``` unfolding degree_def by simp ``` huffman@29451 ` 343` huffman@29451 ` 344` ```lemma degree_diff_le: "degree (p - q) \ max (degree p) (degree q)" ``` huffman@29451 ` 345` ``` using degree_add_le [where p=p and q="-q"] ``` huffman@29451 ` 346` ``` by (simp add: diff_minus) ``` huffman@29451 ` 347` huffman@29453 ` 348` ```lemma degree_diff_less: ``` huffman@29453 ` 349` ``` "\degree p < n; degree q < n\ \ degree (p - q) < n" ``` huffman@29453 ` 350` ``` by (auto intro: le_less_trans degree_diff_le) ``` huffman@29453 ` 351` huffman@29451 ` 352` ```lemma add_monom: "monom a n + monom b n = monom (a + b) n" ``` huffman@29451 ` 353` ``` by (rule poly_ext) simp ``` huffman@29451 ` 354` huffman@29451 ` 355` ```lemma diff_monom: "monom a n - monom b n = monom (a - b) n" ``` huffman@29451 ` 356` ``` by (rule poly_ext) simp ``` huffman@29451 ` 357` huffman@29451 ` 358` ```lemma minus_monom: "- monom a n = monom (-a) n" ``` huffman@29451 ` 359` ``` by (rule poly_ext) simp ``` huffman@29451 ` 360` huffman@29451 ` 361` ```lemma coeff_setsum: "coeff (\x\A. p x) i = (\x\A. coeff (p x) i)" ``` huffman@29451 ` 362` ``` by (cases "finite A", induct set: finite, simp_all) ``` huffman@29451 ` 363` huffman@29451 ` 364` ```lemma monom_setsum: "monom (\x\A. a x) n = (\x\A. monom (a x) n)" ``` huffman@29451 ` 365` ``` by (rule poly_ext) (simp add: coeff_setsum) ``` huffman@29451 ` 366` huffman@29451 ` 367` huffman@29451 ` 368` ```subsection {* Multiplication by a constant *} ``` huffman@29451 ` 369` huffman@29451 ` 370` ```definition ``` huffman@29451 ` 371` ``` smult :: "'a::comm_semiring_0 \ 'a poly \ 'a poly" where ``` huffman@29451 ` 372` ``` "smult a p = Abs_poly (\n. a * coeff p n)" ``` huffman@29451 ` 373` huffman@29451 ` 374` ```lemma Poly_smult: ``` huffman@29451 ` 375` ``` fixes f :: "nat \ 'a::comm_semiring_0" ``` huffman@29451 ` 376` ``` shows "f \ Poly \ (\n. a * f n) \ Poly" ``` huffman@29451 ` 377` ``` unfolding Poly_def ``` huffman@29451 ` 378` ``` by (clarify, rule_tac x=n in exI, simp) ``` huffman@29451 ` 379` huffman@29451 ` 380` ```lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" ``` huffman@29451 ` 381` ``` unfolding smult_def ``` huffman@29451 ` 382` ``` by (simp add: Abs_poly_inverse Poly_smult coeff) ``` huffman@29451 ` 383` huffman@29451 ` 384` ```lemma degree_smult_le: "degree (smult a p) \ degree p" ``` huffman@29451 ` 385` ``` by (rule degree_le, simp add: coeff_eq_0) ``` huffman@29451 ` 386` huffman@29451 ` 387` ```lemma smult_smult: "smult a (smult b p) = smult (a * b) p" ``` huffman@29451 ` 388` ``` by (rule poly_ext, simp add: mult_assoc) ``` huffman@29451 ` 389` huffman@29451 ` 390` ```lemma smult_0_right [simp]: "smult a 0 = 0" ``` huffman@29451 ` 391` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 392` huffman@29451 ` 393` ```lemma smult_0_left [simp]: "smult 0 p = 0" ``` huffman@29451 ` 394` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 395` huffman@29451 ` 396` ```lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" ``` huffman@29451 ` 397` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 398` huffman@29451 ` 399` ```lemma smult_add_right: ``` huffman@29451 ` 400` ``` "smult a (p + q) = smult a p + smult a q" ``` huffman@29451 ` 401` ``` by (rule poly_ext, simp add: ring_simps) ``` huffman@29451 ` 402` huffman@29451 ` 403` ```lemma smult_add_left: ``` huffman@29451 ` 404` ``` "smult (a + b) p = smult a p + smult b p" ``` huffman@29451 ` 405` ``` by (rule poly_ext, simp add: ring_simps) ``` huffman@29451 ` 406` huffman@29451 ` 407` ```lemma smult_minus_right: ``` huffman@29451 ` 408` ``` "smult (a::'a::comm_ring) (- p) = - smult a p" ``` huffman@29451 ` 409` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 410` huffman@29451 ` 411` ```lemma smult_minus_left: ``` huffman@29451 ` 412` ``` "smult (- a::'a::comm_ring) p = - smult a p" ``` huffman@29451 ` 413` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 414` huffman@29451 ` 415` ```lemma smult_diff_right: ``` huffman@29451 ` 416` ``` "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" ``` huffman@29451 ` 417` ``` by (rule poly_ext, simp add: ring_simps) ``` huffman@29451 ` 418` huffman@29451 ` 419` ```lemma smult_diff_left: ``` huffman@29451 ` 420` ``` "smult (a - b::'a::comm_ring) p = smult a p - smult b p" ``` huffman@29451 ` 421` ``` by (rule poly_ext, simp add: ring_simps) ``` huffman@29451 ` 422` huffman@29451 ` 423` ```lemma smult_pCons [simp]: ``` huffman@29451 ` 424` ``` "smult a (pCons b p) = pCons (a * b) (smult a p)" ``` huffman@29451 ` 425` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 426` huffman@29451 ` 427` ```lemma smult_monom: "smult a (monom b n) = monom (a * b) n" ``` huffman@29451 ` 428` ``` by (induct n, simp add: monom_0, simp add: monom_Suc) ``` huffman@29451 ` 429` huffman@29451 ` 430` huffman@29451 ` 431` ```subsection {* Multiplication of polynomials *} ``` huffman@29451 ` 432` huffman@29451 ` 433` ```lemma Poly_mult_lemma: ``` huffman@29451 ` 434` ``` fixes f g :: "nat \ 'a::comm_semiring_0" and m n :: nat ``` huffman@29451 ` 435` ``` assumes "\i>m. f i = 0" ``` huffman@29451 ` 436` ``` assumes "\j>n. g j = 0" ``` huffman@29451 ` 437` ``` shows "\k>m+n. (\i\k. f i * g (k-i)) = 0" ``` huffman@29451 ` 438` ```proof (clarify) ``` huffman@29451 ` 439` ``` fix k :: nat ``` huffman@29451 ` 440` ``` assume "m + n < k" ``` huffman@29451 ` 441` ``` show "(\i\k. f i * g (k - i)) = 0" ``` huffman@29451 ` 442` ``` proof (rule setsum_0' [rule_format]) ``` huffman@29451 ` 443` ``` fix i :: nat ``` huffman@29451 ` 444` ``` assume "i \ {..k}" hence "i \ k" by simp ``` huffman@29451 ` 445` ``` with `m + n < k` have "m < i \ n < k - i" by arith ``` huffman@29451 ` 446` ``` thus "f i * g (k - i) = 0" ``` huffman@29451 ` 447` ``` using prems by auto ``` huffman@29451 ` 448` ``` qed ``` huffman@29451 ` 449` ```qed ``` huffman@29451 ` 450` huffman@29451 ` 451` ```lemma Poly_mult: ``` huffman@29451 ` 452` ``` fixes f g :: "nat \ 'a::comm_semiring_0" ``` huffman@29451 ` 453` ``` shows "\f \ Poly; g \ Poly\ \ (\n. \i\n. f i * g (n-i)) \ Poly" ``` huffman@29451 ` 454` ``` unfolding Poly_def ``` huffman@29451 ` 455` ``` by (safe, rule exI, rule Poly_mult_lemma) ``` huffman@29451 ` 456` huffman@29451 ` 457` ```lemma poly_mult_assoc_lemma: ``` huffman@29451 ` 458` ``` fixes k :: nat and f :: "nat \ nat \ nat \ 'a::comm_monoid_add" ``` huffman@29451 ` 459` ``` shows "(\j\k. \i\j. f i (j - i) (n - j)) = ``` huffman@29451 ` 460` ``` (\j\k. \i\k - j. f j i (n - j - i))" ``` huffman@29451 ` 461` ```proof (induct k) ``` huffman@29451 ` 462` ``` case 0 show ?case by simp ``` huffman@29451 ` 463` ```next ``` huffman@29451 ` 464` ``` case (Suc k) thus ?case ``` huffman@29451 ` 465` ``` by (simp add: Suc_diff_le setsum_addf add_assoc ``` huffman@29451 ` 466` ``` cong: strong_setsum_cong) ``` huffman@29451 ` 467` ```qed ``` huffman@29451 ` 468` huffman@29451 ` 469` ```lemma poly_mult_commute_lemma: ``` huffman@29451 ` 470` ``` fixes n :: nat and f :: "nat \ nat \ 'a::comm_monoid_add" ``` huffman@29451 ` 471` ``` shows "(\i\n. f i (n - i)) = (\i\n. f (n - i) i)" ``` huffman@29451 ` 472` ```proof (rule setsum_reindex_cong) ``` huffman@29451 ` 473` ``` show "inj_on (\i. n - i) {..n}" ``` huffman@29451 ` 474` ``` by (rule inj_onI) simp ``` huffman@29451 ` 475` ``` show "{..n} = (\i. n - i) ` {..n}" ``` huffman@29451 ` 476` ``` by (auto, rule_tac x="n - x" in image_eqI, simp_all) ``` huffman@29451 ` 477` ```next ``` huffman@29451 ` 478` ``` fix i assume "i \ {..n}" ``` huffman@29451 ` 479` ``` hence "n - (n - i) = i" by simp ``` huffman@29451 ` 480` ``` thus "f (n - i) i = f (n - i) (n - (n - i))" by simp ``` huffman@29451 ` 481` ```qed ``` huffman@29451 ` 482` huffman@29451 ` 483` ```text {* TODO: move to appropriate theory *} ``` huffman@29451 ` 484` ```lemma setsum_atMost_Suc_shift: ``` huffman@29451 ` 485` ``` fixes f :: "nat \ 'a::comm_monoid_add" ``` huffman@29451 ` 486` ``` shows "(\i\Suc n. f i) = f 0 + (\i\n. f (Suc i))" ``` huffman@29451 ` 487` ```proof (induct n) ``` huffman@29451 ` 488` ``` case 0 show ?case by simp ``` huffman@29451 ` 489` ```next ``` huffman@29451 ` 490` ``` case (Suc n) note IH = this ``` huffman@29451 ` 491` ``` have "(\i\Suc (Suc n). f i) = (\i\Suc n. f i) + f (Suc (Suc n))" ``` huffman@29451 ` 492` ``` by (rule setsum_atMost_Suc) ``` huffman@29451 ` 493` ``` also have "(\i\Suc n. f i) = f 0 + (\i\n. f (Suc i))" ``` huffman@29451 ` 494` ``` by (rule IH) ``` huffman@29451 ` 495` ``` also have "f 0 + (\i\n. f (Suc i)) + f (Suc (Suc n)) = ``` huffman@29451 ` 496` ``` f 0 + ((\i\n. f (Suc i)) + f (Suc (Suc n)))" ``` huffman@29451 ` 497` ``` by (rule add_assoc) ``` huffman@29451 ` 498` ``` also have "(\i\n. f (Suc i)) + f (Suc (Suc n)) = (\i\Suc n. f (Suc i))" ``` huffman@29451 ` 499` ``` by (rule setsum_atMost_Suc [symmetric]) ``` huffman@29451 ` 500` ``` finally show ?case . ``` huffman@29451 ` 501` ```qed ``` huffman@29451 ` 502` huffman@29451 ` 503` ```instantiation poly :: (comm_semiring_0) comm_semiring_0 ``` huffman@29451 ` 504` ```begin ``` huffman@29451 ` 505` huffman@29451 ` 506` ```definition ``` huffman@29451 ` 507` ``` times_poly_def: ``` huffman@29451 ` 508` ``` "p * q = Abs_poly (\n. \i\n. coeff p i * coeff q (n-i))" ``` huffman@29451 ` 509` huffman@29451 ` 510` ```lemma coeff_mult: ``` huffman@29451 ` 511` ``` "coeff (p * q) n = (\i\n. coeff p i * coeff q (n-i))" ``` huffman@29451 ` 512` ``` unfolding times_poly_def ``` huffman@29451 ` 513` ``` by (simp add: Abs_poly_inverse coeff Poly_mult) ``` huffman@29451 ` 514` huffman@29451 ` 515` ```instance proof ``` huffman@29451 ` 516` ``` fix p q r :: "'a poly" ``` huffman@29451 ` 517` ``` show 0: "0 * p = 0" ``` huffman@29451 ` 518` ``` by (simp add: expand_poly_eq coeff_mult) ``` huffman@29451 ` 519` ``` show "p * 0 = 0" ``` huffman@29451 ` 520` ``` by (simp add: expand_poly_eq coeff_mult) ``` huffman@29451 ` 521` ``` show "(p + q) * r = p * r + q * r" ``` huffman@29451 ` 522` ``` by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf) ``` huffman@29451 ` 523` ``` show "(p * q) * r = p * (q * r)" ``` huffman@29451 ` 524` ``` proof (rule poly_ext) ``` huffman@29451 ` 525` ``` fix n :: nat ``` huffman@29451 ` 526` ``` have "(\j\n. \i\j. coeff p i * coeff q (j - i) * coeff r (n - j)) = ``` huffman@29451 ` 527` ``` (\j\n. \i\n - j. coeff p j * coeff q i * coeff r (n - j - i))" ``` huffman@29451 ` 528` ``` by (rule poly_mult_assoc_lemma) ``` huffman@29451 ` 529` ``` thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n" ``` huffman@29451 ` 530` ``` by (simp add: coeff_mult setsum_right_distrib ``` huffman@29451 ` 531` ``` setsum_left_distrib mult_assoc) ``` huffman@29451 ` 532` ``` qed ``` huffman@29451 ` 533` ``` show "p * q = q * p" ``` huffman@29451 ` 534` ``` proof (rule poly_ext) ``` huffman@29451 ` 535` ``` fix n :: nat ``` huffman@29451 ` 536` ``` have "(\i\n. coeff p i * coeff q (n - i)) = ``` huffman@29451 ` 537` ``` (\i\n. coeff p (n - i) * coeff q i)" ``` huffman@29451 ` 538` ``` by (rule poly_mult_commute_lemma) ``` huffman@29451 ` 539` ``` thus "coeff (p * q) n = coeff (q * p) n" ``` huffman@29451 ` 540` ``` by (simp add: coeff_mult mult_commute) ``` huffman@29451 ` 541` ``` qed ``` huffman@29451 ` 542` ```qed ``` huffman@29451 ` 543` huffman@29451 ` 544` ```end ``` huffman@29451 ` 545` huffman@29451 ` 546` ```lemma degree_mult_le: "degree (p * q) \ degree p + degree q" ``` huffman@29451 ` 547` ```apply (rule degree_le, simp add: coeff_mult) ``` huffman@29451 ` 548` ```apply (rule Poly_mult_lemma) ``` huffman@29451 ` 549` ```apply (simp_all add: coeff_eq_0) ``` huffman@29451 ` 550` ```done ``` huffman@29451 ` 551` huffman@29451 ` 552` ```lemma mult_pCons_left [simp]: ``` huffman@29451 ` 553` ``` "pCons a p * q = smult a q + pCons 0 (p * q)" ``` huffman@29451 ` 554` ```apply (rule poly_ext) ``` huffman@29451 ` 555` ```apply (case_tac n) ``` huffman@29451 ` 556` ```apply (simp add: coeff_mult) ``` huffman@29451 ` 557` ```apply (simp add: coeff_mult setsum_atMost_Suc_shift ``` huffman@29451 ` 558` ``` del: setsum_atMost_Suc) ``` huffman@29451 ` 559` ```done ``` huffman@29451 ` 560` huffman@29451 ` 561` ```lemma mult_pCons_right [simp]: ``` huffman@29451 ` 562` ``` "p * pCons a q = smult a p + pCons 0 (p * q)" ``` huffman@29451 ` 563` ``` using mult_pCons_left [of a q p] by (simp add: mult_commute) ``` huffman@29451 ` 564` huffman@29451 ` 565` ```lemma mult_smult_left: "smult a p * q = smult a (p * q)" ``` huffman@29451 ` 566` ``` by (induct p, simp, simp add: smult_add_right smult_smult) ``` huffman@29451 ` 567` huffman@29451 ` 568` ```lemma mult_smult_right: "p * smult a q = smult a (p * q)" ``` huffman@29451 ` 569` ``` using mult_smult_left [of a q p] by (simp add: mult_commute) ``` huffman@29451 ` 570` huffman@29451 ` 571` ```lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" ``` huffman@29451 ` 572` ``` by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) ``` huffman@29451 ` 573` huffman@29451 ` 574` huffman@29451 ` 575` ```subsection {* The unit polynomial and exponentiation *} ``` huffman@29451 ` 576` huffman@29451 ` 577` ```instantiation poly :: (comm_semiring_1) comm_semiring_1 ``` huffman@29451 ` 578` ```begin ``` huffman@29451 ` 579` huffman@29451 ` 580` ```definition ``` huffman@29451 ` 581` ``` one_poly_def: ``` huffman@29451 ` 582` ``` "1 = pCons 1 0" ``` huffman@29451 ` 583` huffman@29451 ` 584` ```instance proof ``` huffman@29451 ` 585` ``` fix p :: "'a poly" show "1 * p = p" ``` huffman@29451 ` 586` ``` unfolding one_poly_def ``` huffman@29451 ` 587` ``` by simp ``` huffman@29451 ` 588` ```next ``` huffman@29451 ` 589` ``` show "0 \ (1::'a poly)" ``` huffman@29451 ` 590` ``` unfolding one_poly_def by simp ``` huffman@29451 ` 591` ```qed ``` huffman@29451 ` 592` huffman@29451 ` 593` ```end ``` huffman@29451 ` 594` huffman@29451 ` 595` ```lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" ``` huffman@29451 ` 596` ``` unfolding one_poly_def ``` huffman@29451 ` 597` ``` by (simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 598` huffman@29451 ` 599` ```lemma degree_1 [simp]: "degree 1 = 0" ``` huffman@29451 ` 600` ``` unfolding one_poly_def ``` huffman@29451 ` 601` ``` by (rule degree_pCons_0) ``` huffman@29451 ` 602` huffman@29451 ` 603` ```instantiation poly :: (comm_semiring_1) recpower ``` huffman@29451 ` 604` ```begin ``` huffman@29451 ` 605` huffman@29451 ` 606` ```primrec power_poly where ``` huffman@29451 ` 607` ``` power_poly_0: "(p::'a poly) ^ 0 = 1" ``` huffman@29451 ` 608` ```| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" ``` huffman@29451 ` 609` huffman@29451 ` 610` ```instance ``` huffman@29451 ` 611` ``` by default simp_all ``` huffman@29451 ` 612` huffman@29451 ` 613` ```end ``` huffman@29451 ` 614` huffman@29451 ` 615` ```instance poly :: (comm_ring) comm_ring .. ``` huffman@29451 ` 616` huffman@29451 ` 617` ```instance poly :: (comm_ring_1) comm_ring_1 .. ``` huffman@29451 ` 618` huffman@29451 ` 619` ```instantiation poly :: (comm_ring_1) number_ring ``` huffman@29451 ` 620` ```begin ``` huffman@29451 ` 621` huffman@29451 ` 622` ```definition ``` huffman@29451 ` 623` ``` "number_of k = (of_int k :: 'a poly)" ``` huffman@29451 ` 624` huffman@29451 ` 625` ```instance ``` huffman@29451 ` 626` ``` by default (rule number_of_poly_def) ``` huffman@29451 ` 627` huffman@29451 ` 628` ```end ``` huffman@29451 ` 629` huffman@29451 ` 630` huffman@29451 ` 631` ```subsection {* Polynomials form an integral domain *} ``` huffman@29451 ` 632` huffman@29451 ` 633` ```lemma coeff_mult_degree_sum: ``` huffman@29451 ` 634` ``` "coeff (p * q) (degree p + degree q) = ``` huffman@29451 ` 635` ``` coeff p (degree p) * coeff q (degree q)" ``` huffman@29451 ` 636` ``` apply (simp add: coeff_mult) ``` huffman@29451 ` 637` ``` apply (subst setsum_diff1' [where a="degree p"]) ``` huffman@29451 ` 638` ``` apply simp ``` huffman@29451 ` 639` ``` apply simp ``` huffman@29451 ` 640` ``` apply (subst setsum_0' [rule_format]) ``` huffman@29451 ` 641` ``` apply clarsimp ``` huffman@29451 ` 642` ``` apply (subgoal_tac "degree p < a \ degree q < degree p + degree q - a") ``` huffman@29451 ` 643` ``` apply (force simp add: coeff_eq_0) ``` huffman@29451 ` 644` ``` apply arith ``` huffman@29451 ` 645` ``` apply simp ``` huffman@29451 ` 646` ```done ``` huffman@29451 ` 647` huffman@29451 ` 648` ```instance poly :: (idom) idom ``` huffman@29451 ` 649` ```proof ``` huffman@29451 ` 650` ``` fix p q :: "'a poly" ``` huffman@29451 ` 651` ``` assume "p \ 0" and "q \ 0" ``` huffman@29451 ` 652` ``` have "coeff (p * q) (degree p + degree q) = ``` huffman@29451 ` 653` ``` coeff p (degree p) * coeff q (degree q)" ``` huffman@29451 ` 654` ``` by (rule coeff_mult_degree_sum) ``` huffman@29451 ` 655` ``` also have "coeff p (degree p) * coeff q (degree q) \ 0" ``` huffman@29451 ` 656` ``` using `p \ 0` and `q \ 0` by simp ``` huffman@29451 ` 657` ``` finally have "\n. coeff (p * q) n \ 0" .. ``` huffman@29451 ` 658` ``` thus "p * q \ 0" by (simp add: expand_poly_eq) ``` huffman@29451 ` 659` ```qed ``` huffman@29451 ` 660` huffman@29451 ` 661` ```lemma degree_mult_eq: ``` huffman@29451 ` 662` ``` fixes p q :: "'a::idom poly" ``` huffman@29451 ` 663` ``` shows "\p \ 0; q \ 0\ \ degree (p * q) = degree p + degree q" ``` huffman@29451 ` 664` ```apply (rule order_antisym [OF degree_mult_le le_degree]) ``` huffman@29451 ` 665` ```apply (simp add: coeff_mult_degree_sum) ``` huffman@29451 ` 666` ```done ``` huffman@29451 ` 667` huffman@29451 ` 668` ```lemma dvd_imp_degree_le: ``` huffman@29451 ` 669` ``` fixes p q :: "'a::idom poly" ``` huffman@29451 ` 670` ``` shows "\p dvd q; q \ 0\ \ degree p \ degree q" ``` huffman@29451 ` 671` ``` by (erule dvdE, simp add: degree_mult_eq) ``` huffman@29451 ` 672` huffman@29451 ` 673` huffman@29451 ` 674` ```subsection {* Long division of polynomials *} ``` huffman@29451 ` 675` huffman@29451 ` 676` ```definition ``` huffman@29451 ` 677` ``` divmod_poly_rel :: "'a::field poly \ 'a poly \ 'a poly \ 'a poly \ bool" ``` huffman@29451 ` 678` ```where ``` huffman@29451 ` 679` ``` "divmod_poly_rel x y q r \ ``` huffman@29451 ` 680` ``` x = q * y + r \ (if y = 0 then q = 0 else r = 0 \ degree r < degree y)" ``` huffman@29451 ` 681` huffman@29451 ` 682` ```lemma divmod_poly_rel_0: ``` huffman@29451 ` 683` ``` "divmod_poly_rel 0 y 0 0" ``` huffman@29451 ` 684` ``` unfolding divmod_poly_rel_def by simp ``` huffman@29451 ` 685` huffman@29451 ` 686` ```lemma divmod_poly_rel_by_0: ``` huffman@29451 ` 687` ``` "divmod_poly_rel x 0 0 x" ``` huffman@29451 ` 688` ``` unfolding divmod_poly_rel_def by simp ``` huffman@29451 ` 689` huffman@29451 ` 690` ```lemma eq_zero_or_degree_less: ``` huffman@29451 ` 691` ``` assumes "degree p \ n" and "coeff p n = 0" ``` huffman@29451 ` 692` ``` shows "p = 0 \ degree p < n" ``` huffman@29451 ` 693` ```proof (cases n) ``` huffman@29451 ` 694` ``` case 0 ``` huffman@29451 ` 695` ``` with `degree p \ n` and `coeff p n = 0` ``` huffman@29451 ` 696` ``` have "coeff p (degree p) = 0" by simp ``` huffman@29451 ` 697` ``` then have "p = 0" by simp ``` huffman@29451 ` 698` ``` then show ?thesis .. ``` huffman@29451 ` 699` ```next ``` huffman@29451 ` 700` ``` case (Suc m) ``` huffman@29451 ` 701` ``` have "\i>n. coeff p i = 0" ``` huffman@29451 ` 702` ``` using `degree p \ n` by (simp add: coeff_eq_0) ``` huffman@29451 ` 703` ``` then have "\i\n. coeff p i = 0" ``` huffman@29451 ` 704` ``` using `coeff p n = 0` by (simp add: le_less) ``` huffman@29451 ` 705` ``` then have "\i>m. coeff p i = 0" ``` huffman@29451 ` 706` ``` using `n = Suc m` by (simp add: less_eq_Suc_le) ``` huffman@29451 ` 707` ``` then have "degree p \ m" ``` huffman@29451 ` 708` ``` by (rule degree_le) ``` huffman@29451 ` 709` ``` then have "degree p < n" ``` huffman@29451 ` 710` ``` using `n = Suc m` by (simp add: less_Suc_eq_le) ``` huffman@29451 ` 711` ``` then show ?thesis .. ``` huffman@29451 ` 712` ```qed ``` huffman@29451 ` 713` huffman@29451 ` 714` ```lemma divmod_poly_rel_pCons: ``` huffman@29451 ` 715` ``` assumes rel: "divmod_poly_rel x y q r" ``` huffman@29451 ` 716` ``` assumes y: "y \ 0" ``` huffman@29451 ` 717` ``` assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" ``` huffman@29451 ` 718` ``` shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)" ``` huffman@29451 ` 719` ``` (is "divmod_poly_rel ?x y ?q ?r") ``` huffman@29451 ` 720` ```proof - ``` huffman@29451 ` 721` ``` have x: "x = q * y + r" and r: "r = 0 \ degree r < degree y" ``` huffman@29451 ` 722` ``` using assms unfolding divmod_poly_rel_def by simp_all ``` huffman@29451 ` 723` huffman@29451 ` 724` ``` have 1: "?x = ?q * y + ?r" ``` huffman@29451 ` 725` ``` using b x by simp ``` huffman@29451 ` 726` huffman@29451 ` 727` ``` have 2: "?r = 0 \ degree ?r < degree y" ``` huffman@29451 ` 728` ``` proof (rule eq_zero_or_degree_less) ``` huffman@29451 ` 729` ``` have "degree ?r \ max (degree (pCons a r)) (degree (smult b y))" ``` huffman@29451 ` 730` ``` by (rule degree_diff_le) ``` huffman@29451 ` 731` ``` also have "\ \ degree y" ``` huffman@29451 ` 732` ``` proof (rule min_max.le_supI) ``` huffman@29451 ` 733` ``` show "degree (pCons a r) \ degree y" ``` huffman@29451 ` 734` ``` using r by (auto simp add: degree_pCons_eq_if) ``` huffman@29451 ` 735` ``` show "degree (smult b y) \ degree y" ``` huffman@29451 ` 736` ``` by (rule degree_smult_le) ``` huffman@29451 ` 737` ``` qed ``` huffman@29451 ` 738` ``` finally show "degree ?r \ degree y" . ``` huffman@29451 ` 739` ``` next ``` huffman@29451 ` 740` ``` show "coeff ?r (degree y) = 0" ``` huffman@29451 ` 741` ``` using `y \ 0` unfolding b by simp ``` huffman@29451 ` 742` ``` qed ``` huffman@29451 ` 743` huffman@29451 ` 744` ``` from 1 2 show ?thesis ``` huffman@29451 ` 745` ``` unfolding divmod_poly_rel_def ``` huffman@29451 ` 746` ``` using `y \ 0` by simp ``` huffman@29451 ` 747` ```qed ``` huffman@29451 ` 748` huffman@29451 ` 749` ```lemma divmod_poly_rel_exists: "\q r. divmod_poly_rel x y q r" ``` huffman@29451 ` 750` ```apply (cases "y = 0") ``` huffman@29451 ` 751` ```apply (fast intro!: divmod_poly_rel_by_0) ``` huffman@29451 ` 752` ```apply (induct x) ``` huffman@29451 ` 753` ```apply (fast intro!: divmod_poly_rel_0) ``` huffman@29451 ` 754` ```apply (fast intro!: divmod_poly_rel_pCons) ``` huffman@29451 ` 755` ```done ``` huffman@29451 ` 756` huffman@29451 ` 757` ```lemma divmod_poly_rel_unique: ``` huffman@29451 ` 758` ``` assumes 1: "divmod_poly_rel x y q1 r1" ``` huffman@29451 ` 759` ``` assumes 2: "divmod_poly_rel x y q2 r2" ``` huffman@29451 ` 760` ``` shows "q1 = q2 \ r1 = r2" ``` huffman@29451 ` 761` ```proof (cases "y = 0") ``` huffman@29451 ` 762` ``` assume "y = 0" with assms show ?thesis ``` huffman@29451 ` 763` ``` by (simp add: divmod_poly_rel_def) ``` huffman@29451 ` 764` ```next ``` huffman@29451 ` 765` ``` assume [simp]: "y \ 0" ``` huffman@29451 ` 766` ``` from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \ degree r1 < degree y" ``` huffman@29451 ` 767` ``` unfolding divmod_poly_rel_def by simp_all ``` huffman@29451 ` 768` ``` from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \ degree r2 < degree y" ``` huffman@29451 ` 769` ``` unfolding divmod_poly_rel_def by simp_all ``` huffman@29451 ` 770` ``` from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" ``` huffman@29451 ` 771` ``` by (simp add: ring_simps) ``` huffman@29451 ` 772` ``` from r1 r2 have r3: "(r2 - r1) = 0 \ degree (r2 - r1) < degree y" ``` huffman@29453 ` 773` ``` by (auto intro: degree_diff_less) ``` huffman@29451 ` 774` huffman@29451 ` 775` ``` show "q1 = q2 \ r1 = r2" ``` huffman@29451 ` 776` ``` proof (rule ccontr) ``` huffman@29451 ` 777` ``` assume "\ (q1 = q2 \ r1 = r2)" ``` huffman@29451 ` 778` ``` with q3 have "q1 \ q2" and "r1 \ r2" by auto ``` huffman@29451 ` 779` ``` with r3 have "degree (r2 - r1) < degree y" by simp ``` huffman@29451 ` 780` ``` also have "degree y \ degree (q1 - q2) + degree y" by simp ``` huffman@29451 ` 781` ``` also have "\ = degree ((q1 - q2) * y)" ``` huffman@29451 ` 782` ``` using `q1 \ q2` by (simp add: degree_mult_eq) ``` huffman@29451 ` 783` ``` also have "\ = degree (r2 - r1)" ``` huffman@29451 ` 784` ``` using q3 by simp ``` huffman@29451 ` 785` ``` finally have "degree (r2 - r1) < degree (r2 - r1)" . ``` huffman@29451 ` 786` ``` then show "False" by simp ``` huffman@29451 ` 787` ``` qed ``` huffman@29451 ` 788` ```qed ``` huffman@29451 ` 789` huffman@29451 ` 790` ```lemmas divmod_poly_rel_unique_div = ``` huffman@29451 ` 791` ``` divmod_poly_rel_unique [THEN conjunct1, standard] ``` huffman@29451 ` 792` huffman@29451 ` 793` ```lemmas divmod_poly_rel_unique_mod = ``` huffman@29451 ` 794` ``` divmod_poly_rel_unique [THEN conjunct2, standard] ``` huffman@29451 ` 795` huffman@29451 ` 796` ```instantiation poly :: (field) ring_div ``` huffman@29451 ` 797` ```begin ``` huffman@29451 ` 798` huffman@29451 ` 799` ```definition div_poly where ``` huffman@29451 ` 800` ``` [code del]: "x div y = (THE q. \r. divmod_poly_rel x y q r)" ``` huffman@29451 ` 801` huffman@29451 ` 802` ```definition mod_poly where ``` huffman@29451 ` 803` ``` [code del]: "x mod y = (THE r. \q. divmod_poly_rel x y q r)" ``` huffman@29451 ` 804` huffman@29451 ` 805` ```lemma div_poly_eq: ``` huffman@29451 ` 806` ``` "divmod_poly_rel x y q r \ x div y = q" ``` huffman@29451 ` 807` ```unfolding div_poly_def ``` huffman@29451 ` 808` ```by (fast elim: divmod_poly_rel_unique_div) ``` huffman@29451 ` 809` huffman@29451 ` 810` ```lemma mod_poly_eq: ``` huffman@29451 ` 811` ``` "divmod_poly_rel x y q r \ x mod y = r" ``` huffman@29451 ` 812` ```unfolding mod_poly_def ``` huffman@29451 ` 813` ```by (fast elim: divmod_poly_rel_unique_mod) ``` huffman@29451 ` 814` huffman@29451 ` 815` ```lemma divmod_poly_rel: ``` huffman@29451 ` 816` ``` "divmod_poly_rel x y (x div y) (x mod y)" ``` huffman@29451 ` 817` ```proof - ``` huffman@29451 ` 818` ``` from divmod_poly_rel_exists ``` huffman@29451 ` 819` ``` obtain q r where "divmod_poly_rel x y q r" by fast ``` huffman@29451 ` 820` ``` thus ?thesis ``` huffman@29451 ` 821` ``` by (simp add: div_poly_eq mod_poly_eq) ``` huffman@29451 ` 822` ```qed ``` huffman@29451 ` 823` huffman@29451 ` 824` ```instance proof ``` huffman@29451 ` 825` ``` fix x y :: "'a poly" ``` huffman@29451 ` 826` ``` show "x div y * y + x mod y = x" ``` huffman@29451 ` 827` ``` using divmod_poly_rel [of x y] ``` huffman@29451 ` 828` ``` by (simp add: divmod_poly_rel_def) ``` huffman@29451 ` 829` ```next ``` huffman@29451 ` 830` ``` fix x :: "'a poly" ``` huffman@29451 ` 831` ``` have "divmod_poly_rel x 0 0 x" ``` huffman@29451 ` 832` ``` by (rule divmod_poly_rel_by_0) ``` huffman@29451 ` 833` ``` thus "x div 0 = 0" ``` huffman@29451 ` 834` ``` by (rule div_poly_eq) ``` huffman@29451 ` 835` ```next ``` huffman@29451 ` 836` ``` fix y :: "'a poly" ``` huffman@29451 ` 837` ``` have "divmod_poly_rel 0 y 0 0" ``` huffman@29451 ` 838` ``` by (rule divmod_poly_rel_0) ``` huffman@29451 ` 839` ``` thus "0 div y = 0" ``` huffman@29451 ` 840` ``` by (rule div_poly_eq) ``` huffman@29451 ` 841` ```next ``` huffman@29451 ` 842` ``` fix x y z :: "'a poly" ``` huffman@29451 ` 843` ``` assume "y \ 0" ``` huffman@29451 ` 844` ``` hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)" ``` huffman@29451 ` 845` ``` using divmod_poly_rel [of x y] ``` huffman@29451 ` 846` ``` by (simp add: divmod_poly_rel_def left_distrib) ``` huffman@29451 ` 847` ``` thus "(x + z * y) div y = z + x div y" ``` huffman@29451 ` 848` ``` by (rule div_poly_eq) ``` huffman@29451 ` 849` ```qed ``` huffman@29451 ` 850` huffman@29451 ` 851` ```end ``` huffman@29451 ` 852` huffman@29451 ` 853` ```lemma degree_mod_less: ``` huffman@29451 ` 854` ``` "y \ 0 \ x mod y = 0 \ degree (x mod y) < degree y" ``` huffman@29451 ` 855` ``` using divmod_poly_rel [of x y] ``` huffman@29451 ` 856` ``` unfolding divmod_poly_rel_def by simp ``` huffman@29451 ` 857` huffman@29451 ` 858` ```lemma div_poly_less: "degree x < degree y \ x div y = 0" ``` huffman@29451 ` 859` ```proof - ``` huffman@29451 ` 860` ``` assume "degree x < degree y" ``` huffman@29451 ` 861` ``` hence "divmod_poly_rel x y 0 x" ``` huffman@29451 ` 862` ``` by (simp add: divmod_poly_rel_def) ``` huffman@29451 ` 863` ``` thus "x div y = 0" by (rule div_poly_eq) ``` huffman@29451 ` 864` ```qed ``` huffman@29451 ` 865` huffman@29451 ` 866` ```lemma mod_poly_less: "degree x < degree y \ x mod y = x" ``` huffman@29451 ` 867` ```proof - ``` huffman@29451 ` 868` ``` assume "degree x < degree y" ``` huffman@29451 ` 869` ``` hence "divmod_poly_rel x y 0 x" ``` huffman@29451 ` 870` ``` by (simp add: divmod_poly_rel_def) ``` huffman@29451 ` 871` ``` thus "x mod y = x" by (rule mod_poly_eq) ``` huffman@29451 ` 872` ```qed ``` huffman@29451 ` 873` huffman@29451 ` 874` ```lemma mod_pCons: ``` huffman@29451 ` 875` ``` fixes a and x ``` huffman@29451 ` 876` ``` assumes y: "y \ 0" ``` huffman@29451 ` 877` ``` defines b: "b \ coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" ``` huffman@29451 ` 878` ``` shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" ``` huffman@29451 ` 879` ```unfolding b ``` huffman@29451 ` 880` ```apply (rule mod_poly_eq) ``` huffman@29451 ` 881` ```apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl]) ``` huffman@29451 ` 882` ```done ``` huffman@29451 ` 883` huffman@29451 ` 884` huffman@29451 ` 885` ```subsection {* Evaluation of polynomials *} ``` huffman@29451 ` 886` huffman@29451 ` 887` ```definition ``` huffman@29451 ` 888` ``` poly :: "'a::{comm_semiring_1,recpower} poly \ 'a \ 'a" where ``` huffman@29451 ` 889` ``` "poly p = (\x. \n\degree p. coeff p n * x ^ n)" ``` huffman@29451 ` 890` huffman@29451 ` 891` ```lemma poly_0 [simp]: "poly 0 x = 0" ``` huffman@29451 ` 892` ``` unfolding poly_def by simp ``` huffman@29451 ` 893` huffman@29451 ` 894` ```lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" ``` huffman@29451 ` 895` ``` unfolding poly_def ``` huffman@29451 ` 896` ``` by (simp add: degree_pCons_eq_if setsum_atMost_Suc_shift power_Suc ``` huffman@29451 ` 897` ``` setsum_left_distrib setsum_right_distrib mult_ac ``` huffman@29451 ` 898` ``` del: setsum_atMost_Suc) ``` huffman@29451 ` 899` huffman@29451 ` 900` ```lemma poly_1 [simp]: "poly 1 x = 1" ``` huffman@29451 ` 901` ``` unfolding one_poly_def by simp ``` huffman@29451 ` 902` huffman@29451 ` 903` ```lemma poly_monom: "poly (monom a n) x = a * x ^ n" ``` huffman@29451 ` 904` ``` by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) ``` huffman@29451 ` 905` huffman@29451 ` 906` ```lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" ``` huffman@29451 ` 907` ``` apply (induct p arbitrary: q, simp) ``` huffman@29451 ` 908` ``` apply (case_tac q, simp, simp add: ring_simps) ``` huffman@29451 ` 909` ``` done ``` huffman@29451 ` 910` huffman@29451 ` 911` ```lemma poly_minus [simp]: ``` huffman@29451 ` 912` ``` fixes x :: "'a::{comm_ring_1,recpower}" ``` huffman@29451 ` 913` ``` shows "poly (- p) x = - poly p x" ``` huffman@29451 ` 914` ``` by (induct p, simp_all) ``` huffman@29451 ` 915` huffman@29451 ` 916` ```lemma poly_diff [simp]: ``` huffman@29451 ` 917` ``` fixes x :: "'a::{comm_ring_1,recpower}" ``` huffman@29451 ` 918` ``` shows "poly (p - q) x = poly p x - poly q x" ``` huffman@29451 ` 919` ``` by (simp add: diff_minus) ``` huffman@29451 ` 920` huffman@29451 ` 921` ```lemma poly_setsum: "poly (\k\A. p k) x = (\k\A. poly (p k) x)" ``` huffman@29451 ` 922` ``` by (cases "finite A", induct set: finite, simp_all) ``` huffman@29451 ` 923` huffman@29451 ` 924` ```lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" ``` huffman@29451 ` 925` ``` by (induct p, simp, simp add: ring_simps) ``` huffman@29451 ` 926` huffman@29451 ` 927` ```lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" ``` huffman@29451 ` 928` ``` by (induct p, simp_all, simp add: ring_simps) ``` huffman@29451 ` 929` huffman@29451 ` 930` ```end ```