src/HOL/Library/Polynomial.thy
 author haftmann Thu Dec 29 10:47:56 2011 +0100 (2011-12-29) changeset 46031 ac6bae9fdc2f parent 45928 874845660119 child 47002 9435d419109a permissions -rw-r--r--
tuned declaration
 wenzelm@41959 ` 1` ```(* Title: HOL/Library/Polynomial.thy ``` huffman@29451 ` 2` ``` Author: Brian Huffman ``` wenzelm@41959 ` 3` ``` Author: Clemens Ballarin ``` huffman@29451 ` 4` ```*) ``` huffman@29451 ` 5` huffman@29451 ` 6` ```header {* Univariate Polynomials *} ``` huffman@29451 ` 7` huffman@29451 ` 8` ```theory Polynomial ``` haftmann@30738 ` 9` ```imports Main ``` huffman@29451 ` 10` ```begin ``` huffman@29451 ` 11` huffman@29451 ` 12` ```subsection {* Definition of type @{text poly} *} ``` huffman@29451 ` 13` wenzelm@45694 ` 14` ```definition "Poly = {f::nat \ 'a::zero. \n. \i>n. f i = 0}" ``` wenzelm@45694 ` 15` wenzelm@45694 ` 16` ```typedef (open) 'a poly = "Poly :: (nat => 'a::zero) set" ``` huffman@29451 ` 17` ``` morphisms coeff Abs_poly ``` wenzelm@45694 ` 18` ``` unfolding Poly_def by auto ``` huffman@29451 ` 19` huffman@29451 ` 20` ```lemma expand_poly_eq: "p = q \ (\n. coeff p n = coeff q n)" ``` wenzelm@45694 ` 21` ``` by (simp add: coeff_inject [symmetric] fun_eq_iff) ``` huffman@29451 ` 22` huffman@29451 ` 23` ```lemma poly_ext: "(\n. coeff p n = coeff q n) \ p = q" ``` wenzelm@45694 ` 24` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 25` huffman@29451 ` 26` huffman@29451 ` 27` ```subsection {* Degree of a polynomial *} ``` huffman@29451 ` 28` huffman@29451 ` 29` ```definition ``` huffman@29451 ` 30` ``` degree :: "'a::zero poly \ nat" where ``` huffman@29451 ` 31` ``` "degree p = (LEAST n. \i>n. coeff p i = 0)" ``` huffman@29451 ` 32` huffman@29451 ` 33` ```lemma coeff_eq_0: "degree p < n \ coeff p n = 0" ``` huffman@29451 ` 34` ```proof - ``` huffman@29451 ` 35` ``` have "coeff p \ Poly" ``` huffman@29451 ` 36` ``` by (rule coeff) ``` huffman@29451 ` 37` ``` hence "\n. \i>n. coeff p i = 0" ``` huffman@29451 ` 38` ``` unfolding Poly_def by simp ``` huffman@29451 ` 39` ``` hence "\i>degree p. coeff p i = 0" ``` huffman@29451 ` 40` ``` unfolding degree_def by (rule LeastI_ex) ``` huffman@29451 ` 41` ``` moreover assume "degree p < n" ``` huffman@29451 ` 42` ``` ultimately show ?thesis by simp ``` huffman@29451 ` 43` ```qed ``` huffman@29451 ` 44` huffman@29451 ` 45` ```lemma le_degree: "coeff p n \ 0 \ n \ degree p" ``` huffman@29451 ` 46` ``` by (erule contrapos_np, rule coeff_eq_0, simp) ``` huffman@29451 ` 47` huffman@29451 ` 48` ```lemma degree_le: "\i>n. coeff p i = 0 \ degree p \ n" ``` huffman@29451 ` 49` ``` unfolding degree_def by (erule Least_le) ``` huffman@29451 ` 50` huffman@29451 ` 51` ```lemma less_degree_imp: "n < degree p \ \i>n. coeff p i \ 0" ``` huffman@29451 ` 52` ``` unfolding degree_def by (drule not_less_Least, simp) ``` huffman@29451 ` 53` huffman@29451 ` 54` huffman@29451 ` 55` ```subsection {* The zero polynomial *} ``` huffman@29451 ` 56` huffman@29451 ` 57` ```instantiation poly :: (zero) zero ``` huffman@29451 ` 58` ```begin ``` huffman@29451 ` 59` huffman@29451 ` 60` ```definition ``` huffman@29451 ` 61` ``` zero_poly_def: "0 = Abs_poly (\n. 0)" ``` huffman@29451 ` 62` huffman@29451 ` 63` ```instance .. ``` huffman@29451 ` 64` ```end ``` huffman@29451 ` 65` huffman@29451 ` 66` ```lemma coeff_0 [simp]: "coeff 0 n = 0" ``` huffman@29451 ` 67` ``` unfolding zero_poly_def ``` huffman@29451 ` 68` ``` by (simp add: Abs_poly_inverse Poly_def) ``` huffman@29451 ` 69` huffman@29451 ` 70` ```lemma degree_0 [simp]: "degree 0 = 0" ``` huffman@29451 ` 71` ``` by (rule order_antisym [OF degree_le le0]) simp ``` huffman@29451 ` 72` huffman@29451 ` 73` ```lemma leading_coeff_neq_0: ``` huffman@29451 ` 74` ``` assumes "p \ 0" shows "coeff p (degree p) \ 0" ``` huffman@29451 ` 75` ```proof (cases "degree p") ``` huffman@29451 ` 76` ``` case 0 ``` huffman@29451 ` 77` ``` from `p \ 0` have "\n. coeff p n \ 0" ``` huffman@29451 ` 78` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 79` ``` then obtain n where "coeff p n \ 0" .. ``` huffman@29451 ` 80` ``` hence "n \ degree p" by (rule le_degree) ``` huffman@29451 ` 81` ``` with `coeff p n \ 0` and `degree p = 0` ``` huffman@29451 ` 82` ``` show "coeff p (degree p) \ 0" by simp ``` huffman@29451 ` 83` ```next ``` huffman@29451 ` 84` ``` case (Suc n) ``` huffman@29451 ` 85` ``` from `degree p = Suc n` have "n < degree p" by simp ``` huffman@29451 ` 86` ``` hence "\i>n. coeff p i \ 0" by (rule less_degree_imp) ``` huffman@29451 ` 87` ``` then obtain i where "n < i" and "coeff p i \ 0" by fast ``` huffman@29451 ` 88` ``` from `degree p = Suc n` and `n < i` have "degree p \ i" by simp ``` huffman@29451 ` 89` ``` also from `coeff p i \ 0` have "i \ degree p" by (rule le_degree) ``` huffman@29451 ` 90` ``` finally have "degree p = i" . ``` huffman@29451 ` 91` ``` with `coeff p i \ 0` show "coeff p (degree p) \ 0" by simp ``` huffman@29451 ` 92` ```qed ``` huffman@29451 ` 93` huffman@29451 ` 94` ```lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \ p = 0" ``` huffman@29451 ` 95` ``` by (cases "p = 0", simp, simp add: leading_coeff_neq_0) ``` huffman@29451 ` 96` huffman@29451 ` 97` huffman@29451 ` 98` ```subsection {* List-style constructor for polynomials *} ``` huffman@29451 ` 99` huffman@29451 ` 100` ```definition ``` huffman@29451 ` 101` ``` pCons :: "'a::zero \ 'a poly \ 'a poly" ``` huffman@29451 ` 102` ```where ``` haftmann@37765 ` 103` ``` "pCons a p = Abs_poly (nat_case a (coeff p))" ``` huffman@29451 ` 104` huffman@29455 ` 105` ```syntax ``` huffman@29455 ` 106` ``` "_poly" :: "args \ 'a poly" ("[:(_):]") ``` huffman@29455 ` 107` huffman@29455 ` 108` ```translations ``` huffman@29455 ` 109` ``` "[:x, xs:]" == "CONST pCons x [:xs:]" ``` huffman@29455 ` 110` ``` "[:x:]" == "CONST pCons x 0" ``` huffman@30155 ` 111` ``` "[:x:]" <= "CONST pCons x (_constrain 0 t)" ``` huffman@29455 ` 112` huffman@29451 ` 113` ```lemma Poly_nat_case: "f \ Poly \ nat_case a f \ Poly" ``` huffman@29451 ` 114` ``` unfolding Poly_def by (auto split: nat.split) ``` huffman@29451 ` 115` huffman@29451 ` 116` ```lemma coeff_pCons: ``` huffman@29451 ` 117` ``` "coeff (pCons a p) = nat_case a (coeff p)" ``` huffman@29451 ` 118` ``` unfolding pCons_def ``` huffman@29451 ` 119` ``` by (simp add: Abs_poly_inverse Poly_nat_case coeff) ``` huffman@29451 ` 120` huffman@29451 ` 121` ```lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a" ``` huffman@29451 ` 122` ``` by (simp add: coeff_pCons) ``` huffman@29451 ` 123` huffman@29451 ` 124` ```lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" ``` huffman@29451 ` 125` ``` by (simp add: coeff_pCons) ``` huffman@29451 ` 126` huffman@29451 ` 127` ```lemma degree_pCons_le: "degree (pCons a p) \ Suc (degree p)" ``` huffman@29451 ` 128` ```by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split) ``` huffman@29451 ` 129` huffman@29451 ` 130` ```lemma degree_pCons_eq: ``` huffman@29451 ` 131` ``` "p \ 0 \ degree (pCons a p) = Suc (degree p)" ``` huffman@29451 ` 132` ```apply (rule order_antisym [OF degree_pCons_le]) ``` huffman@29451 ` 133` ```apply (rule le_degree, simp) ``` huffman@29451 ` 134` ```done ``` huffman@29451 ` 135` huffman@29451 ` 136` ```lemma degree_pCons_0: "degree (pCons a 0) = 0" ``` huffman@29451 ` 137` ```apply (rule order_antisym [OF _ le0]) ``` huffman@29451 ` 138` ```apply (rule degree_le, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 139` ```done ``` huffman@29451 ` 140` huffman@29460 ` 141` ```lemma degree_pCons_eq_if [simp]: ``` huffman@29451 ` 142` ``` "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" ``` huffman@29451 ` 143` ```apply (cases "p = 0", simp_all) ``` huffman@29451 ` 144` ```apply (rule order_antisym [OF _ le0]) ``` huffman@29451 ` 145` ```apply (rule degree_le, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 146` ```apply (rule order_antisym [OF degree_pCons_le]) ``` huffman@29451 ` 147` ```apply (rule le_degree, simp) ``` huffman@29451 ` 148` ```done ``` huffman@29451 ` 149` haftmann@46031 ` 150` ```lemma pCons_0_0 [simp, code_post]: "pCons 0 0 = 0" ``` huffman@29451 ` 151` ```by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 152` huffman@29451 ` 153` ```lemma pCons_eq_iff [simp]: ``` huffman@29451 ` 154` ``` "pCons a p = pCons b q \ a = b \ p = q" ``` huffman@29451 ` 155` ```proof (safe) ``` huffman@29451 ` 156` ``` assume "pCons a p = pCons b q" ``` huffman@29451 ` 157` ``` then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp ``` huffman@29451 ` 158` ``` then show "a = b" by simp ``` huffman@29451 ` 159` ```next ``` huffman@29451 ` 160` ``` assume "pCons a p = pCons b q" ``` huffman@29451 ` 161` ``` then have "\n. coeff (pCons a p) (Suc n) = ``` huffman@29451 ` 162` ``` coeff (pCons b q) (Suc n)" by simp ``` huffman@29451 ` 163` ``` then show "p = q" by (simp add: expand_poly_eq) ``` huffman@29451 ` 164` ```qed ``` huffman@29451 ` 165` huffman@29451 ` 166` ```lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \ a = 0 \ p = 0" ``` huffman@29451 ` 167` ``` using pCons_eq_iff [of a p 0 0] by simp ``` huffman@29451 ` 168` huffman@29451 ` 169` ```lemma Poly_Suc: "f \ Poly \ (\n. f (Suc n)) \ Poly" ``` huffman@29451 ` 170` ``` unfolding Poly_def ``` huffman@29451 ` 171` ``` by (clarify, rule_tac x=n in exI, simp) ``` huffman@29451 ` 172` huffman@29451 ` 173` ```lemma pCons_cases [cases type: poly]: ``` huffman@29451 ` 174` ``` obtains (pCons) a q where "p = pCons a q" ``` huffman@29451 ` 175` ```proof ``` huffman@29451 ` 176` ``` show "p = pCons (coeff p 0) (Abs_poly (\n. coeff p (Suc n)))" ``` huffman@29451 ` 177` ``` by (rule poly_ext) ``` huffman@29451 ` 178` ``` (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons ``` huffman@29451 ` 179` ``` split: nat.split) ``` huffman@29451 ` 180` ```qed ``` huffman@29451 ` 181` huffman@29451 ` 182` ```lemma pCons_induct [case_names 0 pCons, induct type: poly]: ``` huffman@29451 ` 183` ``` assumes zero: "P 0" ``` huffman@29451 ` 184` ``` assumes pCons: "\a p. P p \ P (pCons a p)" ``` huffman@29451 ` 185` ``` shows "P p" ``` huffman@29451 ` 186` ```proof (induct p rule: measure_induct_rule [where f=degree]) ``` huffman@29451 ` 187` ``` case (less p) ``` huffman@29451 ` 188` ``` obtain a q where "p = pCons a q" by (rule pCons_cases) ``` huffman@29451 ` 189` ``` have "P q" ``` huffman@29451 ` 190` ``` proof (cases "q = 0") ``` huffman@29451 ` 191` ``` case True ``` huffman@29451 ` 192` ``` then show "P q" by (simp add: zero) ``` huffman@29451 ` 193` ``` next ``` huffman@29451 ` 194` ``` case False ``` huffman@29451 ` 195` ``` then have "degree (pCons a q) = Suc (degree q)" ``` huffman@29451 ` 196` ``` by (rule degree_pCons_eq) ``` huffman@29451 ` 197` ``` then have "degree q < degree p" ``` huffman@29451 ` 198` ``` using `p = pCons a q` by simp ``` huffman@29451 ` 199` ``` then show "P q" ``` huffman@29451 ` 200` ``` by (rule less.hyps) ``` huffman@29451 ` 201` ``` qed ``` huffman@29451 ` 202` ``` then have "P (pCons a q)" ``` huffman@29451 ` 203` ``` by (rule pCons) ``` huffman@29451 ` 204` ``` then show ?case ``` huffman@29451 ` 205` ``` using `p = pCons a q` by simp ``` huffman@29451 ` 206` ```qed ``` huffman@29451 ` 207` huffman@29451 ` 208` huffman@29454 ` 209` ```subsection {* Recursion combinator for polynomials *} ``` huffman@29454 ` 210` huffman@29454 ` 211` ```function ``` huffman@29454 ` 212` ``` poly_rec :: "'b \ ('a::zero \ 'a poly \ 'b \ 'b) \ 'a poly \ 'b" ``` huffman@29454 ` 213` ```where ``` haftmann@37765 ` 214` ``` poly_rec_pCons_eq_if [simp del]: ``` huffman@29454 ` 215` ``` "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)" ``` huffman@29454 ` 216` ```by (case_tac x, rename_tac q, case_tac q, auto) ``` huffman@29454 ` 217` huffman@29454 ` 218` ```termination poly_rec ``` huffman@29454 ` 219` ```by (relation "measure (degree \ snd \ snd)", simp) ``` huffman@29454 ` 220` ``` (simp add: degree_pCons_eq) ``` huffman@29454 ` 221` huffman@29454 ` 222` ```lemma poly_rec_0: ``` huffman@29454 ` 223` ``` "f 0 0 z = z \ poly_rec z f 0 = z" ``` huffman@29454 ` 224` ``` using poly_rec_pCons_eq_if [of z f 0 0] by simp ``` huffman@29454 ` 225` huffman@29454 ` 226` ```lemma poly_rec_pCons: ``` huffman@29454 ` 227` ``` "f 0 0 z = z \ poly_rec z f (pCons a p) = f a p (poly_rec z f p)" ``` huffman@29454 ` 228` ``` by (simp add: poly_rec_pCons_eq_if poly_rec_0) ``` huffman@29454 ` 229` huffman@29454 ` 230` huffman@29451 ` 231` ```subsection {* Monomials *} ``` huffman@29451 ` 232` huffman@29451 ` 233` ```definition ``` huffman@29451 ` 234` ``` monom :: "'a \ nat \ 'a::zero poly" where ``` huffman@29451 ` 235` ``` "monom a m = Abs_poly (\n. if m = n then a else 0)" ``` huffman@29451 ` 236` huffman@29451 ` 237` ```lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)" ``` huffman@29451 ` 238` ``` unfolding monom_def ``` huffman@29451 ` 239` ``` by (subst Abs_poly_inverse, auto simp add: Poly_def) ``` huffman@29451 ` 240` huffman@29451 ` 241` ```lemma monom_0: "monom a 0 = pCons a 0" ``` huffman@29451 ` 242` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 243` huffman@29451 ` 244` ```lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" ``` huffman@29451 ` 245` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 246` huffman@29451 ` 247` ```lemma monom_eq_0 [simp]: "monom 0 n = 0" ``` huffman@29451 ` 248` ``` by (rule poly_ext) simp ``` huffman@29451 ` 249` huffman@29451 ` 250` ```lemma monom_eq_0_iff [simp]: "monom a n = 0 \ a = 0" ``` huffman@29451 ` 251` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 252` huffman@29451 ` 253` ```lemma monom_eq_iff [simp]: "monom a n = monom b n \ a = b" ``` huffman@29451 ` 254` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 255` huffman@29451 ` 256` ```lemma degree_monom_le: "degree (monom a n) \ n" ``` huffman@29451 ` 257` ``` by (rule degree_le, simp) ``` huffman@29451 ` 258` huffman@29451 ` 259` ```lemma degree_monom_eq: "a \ 0 \ degree (monom a n) = n" ``` huffman@29451 ` 260` ``` apply (rule order_antisym [OF degree_monom_le]) ``` huffman@29451 ` 261` ``` apply (rule le_degree, simp) ``` huffman@29451 ` 262` ``` done ``` huffman@29451 ` 263` huffman@29451 ` 264` huffman@29451 ` 265` ```subsection {* Addition and subtraction *} ``` huffman@29451 ` 266` huffman@29451 ` 267` ```instantiation poly :: (comm_monoid_add) comm_monoid_add ``` huffman@29451 ` 268` ```begin ``` huffman@29451 ` 269` huffman@29451 ` 270` ```definition ``` haftmann@37765 ` 271` ``` plus_poly_def: ``` huffman@29451 ` 272` ``` "p + q = Abs_poly (\n. coeff p n + coeff q n)" ``` huffman@29451 ` 273` huffman@29451 ` 274` ```lemma Poly_add: ``` huffman@29451 ` 275` ``` fixes f g :: "nat \ 'a" ``` huffman@29451 ` 276` ``` shows "\f \ Poly; g \ Poly\ \ (\n. f n + g n) \ Poly" ``` huffman@29451 ` 277` ``` unfolding Poly_def ``` huffman@29451 ` 278` ``` apply (clarify, rename_tac m n) ``` huffman@29451 ` 279` ``` apply (rule_tac x="max m n" in exI, simp) ``` huffman@29451 ` 280` ``` done ``` huffman@29451 ` 281` huffman@29451 ` 282` ```lemma coeff_add [simp]: ``` huffman@29451 ` 283` ``` "coeff (p + q) n = coeff p n + coeff q n" ``` huffman@29451 ` 284` ``` unfolding plus_poly_def ``` huffman@29451 ` 285` ``` by (simp add: Abs_poly_inverse coeff Poly_add) ``` huffman@29451 ` 286` huffman@29451 ` 287` ```instance proof ``` huffman@29451 ` 288` ``` fix p q r :: "'a poly" ``` huffman@29451 ` 289` ``` show "(p + q) + r = p + (q + r)" ``` huffman@29451 ` 290` ``` by (simp add: expand_poly_eq add_assoc) ``` huffman@29451 ` 291` ``` show "p + q = q + p" ``` huffman@29451 ` 292` ``` by (simp add: expand_poly_eq add_commute) ``` huffman@29451 ` 293` ``` show "0 + p = p" ``` huffman@29451 ` 294` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 295` ```qed ``` huffman@29451 ` 296` huffman@29451 ` 297` ```end ``` huffman@29451 ` 298` huffman@29904 ` 299` ```instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add ``` huffman@29540 ` 300` ```proof ``` huffman@29540 ` 301` ``` fix p q r :: "'a poly" ``` huffman@29540 ` 302` ``` assume "p + q = p + r" thus "q = r" ``` huffman@29540 ` 303` ``` by (simp add: expand_poly_eq) ``` huffman@29540 ` 304` ```qed ``` huffman@29540 ` 305` huffman@29451 ` 306` ```instantiation poly :: (ab_group_add) ab_group_add ``` huffman@29451 ` 307` ```begin ``` huffman@29451 ` 308` huffman@29451 ` 309` ```definition ``` haftmann@37765 ` 310` ``` uminus_poly_def: ``` huffman@29451 ` 311` ``` "- p = Abs_poly (\n. - coeff p n)" ``` huffman@29451 ` 312` huffman@29451 ` 313` ```definition ``` haftmann@37765 ` 314` ``` minus_poly_def: ``` huffman@29451 ` 315` ``` "p - q = Abs_poly (\n. coeff p n - coeff q n)" ``` huffman@29451 ` 316` huffman@29451 ` 317` ```lemma Poly_minus: ``` huffman@29451 ` 318` ``` fixes f :: "nat \ 'a" ``` huffman@29451 ` 319` ``` shows "f \ Poly \ (\n. - f n) \ Poly" ``` huffman@29451 ` 320` ``` unfolding Poly_def by simp ``` huffman@29451 ` 321` huffman@29451 ` 322` ```lemma Poly_diff: ``` huffman@29451 ` 323` ``` fixes f g :: "nat \ 'a" ``` huffman@29451 ` 324` ``` shows "\f \ Poly; g \ Poly\ \ (\n. f n - g n) \ Poly" ``` huffman@29451 ` 325` ``` unfolding diff_minus by (simp add: Poly_add Poly_minus) ``` huffman@29451 ` 326` huffman@29451 ` 327` ```lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" ``` huffman@29451 ` 328` ``` unfolding uminus_poly_def ``` huffman@29451 ` 329` ``` by (simp add: Abs_poly_inverse coeff Poly_minus) ``` huffman@29451 ` 330` huffman@29451 ` 331` ```lemma coeff_diff [simp]: ``` huffman@29451 ` 332` ``` "coeff (p - q) n = coeff p n - coeff q n" ``` huffman@29451 ` 333` ``` unfolding minus_poly_def ``` huffman@29451 ` 334` ``` by (simp add: Abs_poly_inverse coeff Poly_diff) ``` huffman@29451 ` 335` huffman@29451 ` 336` ```instance proof ``` huffman@29451 ` 337` ``` fix p q :: "'a poly" ``` huffman@29451 ` 338` ``` show "- p + p = 0" ``` huffman@29451 ` 339` ``` by (simp add: expand_poly_eq) ``` huffman@29451 ` 340` ``` show "p - q = p + - q" ``` huffman@29451 ` 341` ``` by (simp add: expand_poly_eq diff_minus) ``` huffman@29451 ` 342` ```qed ``` huffman@29451 ` 343` huffman@29451 ` 344` ```end ``` huffman@29451 ` 345` huffman@29451 ` 346` ```lemma add_pCons [simp]: ``` huffman@29451 ` 347` ``` "pCons a p + pCons b q = pCons (a + b) (p + q)" ``` huffman@29451 ` 348` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 349` huffman@29451 ` 350` ```lemma minus_pCons [simp]: ``` huffman@29451 ` 351` ``` "- pCons a p = pCons (- a) (- p)" ``` huffman@29451 ` 352` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 353` huffman@29451 ` 354` ```lemma diff_pCons [simp]: ``` huffman@29451 ` 355` ``` "pCons a p - pCons b q = pCons (a - b) (p - q)" ``` huffman@29451 ` 356` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 357` huffman@29539 ` 358` ```lemma degree_add_le_max: "degree (p + q) \ max (degree p) (degree q)" ``` huffman@29451 ` 359` ``` by (rule degree_le, auto simp add: coeff_eq_0) ``` huffman@29451 ` 360` huffman@29539 ` 361` ```lemma degree_add_le: ``` huffman@29539 ` 362` ``` "\degree p \ n; degree q \ n\ \ degree (p + q) \ n" ``` huffman@29539 ` 363` ``` by (auto intro: order_trans degree_add_le_max) ``` huffman@29539 ` 364` huffman@29453 ` 365` ```lemma degree_add_less: ``` huffman@29453 ` 366` ``` "\degree p < n; degree q < n\ \ degree (p + q) < n" ``` huffman@29539 ` 367` ``` by (auto intro: le_less_trans degree_add_le_max) ``` huffman@29453 ` 368` huffman@29451 ` 369` ```lemma degree_add_eq_right: ``` huffman@29451 ` 370` ``` "degree p < degree q \ degree (p + q) = degree q" ``` huffman@29451 ` 371` ``` apply (cases "q = 0", simp) ``` huffman@29451 ` 372` ``` apply (rule order_antisym) ``` huffman@29539 ` 373` ``` apply (simp add: degree_add_le) ``` huffman@29451 ` 374` ``` apply (rule le_degree) ``` huffman@29451 ` 375` ``` apply (simp add: coeff_eq_0) ``` huffman@29451 ` 376` ``` done ``` huffman@29451 ` 377` huffman@29451 ` 378` ```lemma degree_add_eq_left: ``` huffman@29451 ` 379` ``` "degree q < degree p \ degree (p + q) = degree p" ``` huffman@29451 ` 380` ``` using degree_add_eq_right [of q p] ``` huffman@29451 ` 381` ``` by (simp add: add_commute) ``` huffman@29451 ` 382` huffman@29451 ` 383` ```lemma degree_minus [simp]: "degree (- p) = degree p" ``` huffman@29451 ` 384` ``` unfolding degree_def by simp ``` huffman@29451 ` 385` huffman@29539 ` 386` ```lemma degree_diff_le_max: "degree (p - q) \ max (degree p) (degree q)" ``` huffman@29451 ` 387` ``` using degree_add_le [where p=p and q="-q"] ``` huffman@29451 ` 388` ``` by (simp add: diff_minus) ``` huffman@29451 ` 389` huffman@29539 ` 390` ```lemma degree_diff_le: ``` huffman@29539 ` 391` ``` "\degree p \ n; degree q \ n\ \ degree (p - q) \ n" ``` huffman@29539 ` 392` ``` by (simp add: diff_minus degree_add_le) ``` huffman@29539 ` 393` huffman@29453 ` 394` ```lemma degree_diff_less: ``` huffman@29453 ` 395` ``` "\degree p < n; degree q < n\ \ degree (p - q) < n" ``` huffman@29539 ` 396` ``` by (simp add: diff_minus degree_add_less) ``` huffman@29453 ` 397` huffman@29451 ` 398` ```lemma add_monom: "monom a n + monom b n = monom (a + b) n" ``` huffman@29451 ` 399` ``` by (rule poly_ext) simp ``` huffman@29451 ` 400` huffman@29451 ` 401` ```lemma diff_monom: "monom a n - monom b n = monom (a - b) n" ``` huffman@29451 ` 402` ``` by (rule poly_ext) simp ``` huffman@29451 ` 403` huffman@29451 ` 404` ```lemma minus_monom: "- monom a n = monom (-a) n" ``` huffman@29451 ` 405` ``` by (rule poly_ext) simp ``` huffman@29451 ` 406` huffman@29451 ` 407` ```lemma coeff_setsum: "coeff (\x\A. p x) i = (\x\A. coeff (p x) i)" ``` huffman@29451 ` 408` ``` by (cases "finite A", induct set: finite, simp_all) ``` huffman@29451 ` 409` huffman@29451 ` 410` ```lemma monom_setsum: "monom (\x\A. a x) n = (\x\A. monom (a x) n)" ``` huffman@29451 ` 411` ``` by (rule poly_ext) (simp add: coeff_setsum) ``` huffman@29451 ` 412` huffman@29451 ` 413` huffman@29451 ` 414` ```subsection {* Multiplication by a constant *} ``` huffman@29451 ` 415` huffman@29451 ` 416` ```definition ``` huffman@29451 ` 417` ``` smult :: "'a::comm_semiring_0 \ 'a poly \ 'a poly" where ``` huffman@29451 ` 418` ``` "smult a p = Abs_poly (\n. a * coeff p n)" ``` huffman@29451 ` 419` huffman@29451 ` 420` ```lemma Poly_smult: ``` huffman@29451 ` 421` ``` fixes f :: "nat \ 'a::comm_semiring_0" ``` huffman@29451 ` 422` ``` shows "f \ Poly \ (\n. a * f n) \ Poly" ``` huffman@29451 ` 423` ``` unfolding Poly_def ``` huffman@29451 ` 424` ``` by (clarify, rule_tac x=n in exI, simp) ``` huffman@29451 ` 425` huffman@29451 ` 426` ```lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" ``` huffman@29451 ` 427` ``` unfolding smult_def ``` huffman@29451 ` 428` ``` by (simp add: Abs_poly_inverse Poly_smult coeff) ``` huffman@29451 ` 429` huffman@29451 ` 430` ```lemma degree_smult_le: "degree (smult a p) \ degree p" ``` huffman@29451 ` 431` ``` by (rule degree_le, simp add: coeff_eq_0) ``` huffman@29451 ` 432` huffman@29472 ` 433` ```lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" ``` huffman@29451 ` 434` ``` by (rule poly_ext, simp add: mult_assoc) ``` huffman@29451 ` 435` huffman@29451 ` 436` ```lemma smult_0_right [simp]: "smult a 0 = 0" ``` huffman@29451 ` 437` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 438` huffman@29451 ` 439` ```lemma smult_0_left [simp]: "smult 0 p = 0" ``` huffman@29451 ` 440` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 441` huffman@29451 ` 442` ```lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" ``` huffman@29451 ` 443` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 444` huffman@29451 ` 445` ```lemma smult_add_right: ``` huffman@29451 ` 446` ``` "smult a (p + q) = smult a p + smult a q" ``` nipkow@29667 ` 447` ``` by (rule poly_ext, simp add: algebra_simps) ``` huffman@29451 ` 448` huffman@29451 ` 449` ```lemma smult_add_left: ``` huffman@29451 ` 450` ``` "smult (a + b) p = smult a p + smult b p" ``` nipkow@29667 ` 451` ``` by (rule poly_ext, simp add: algebra_simps) ``` huffman@29451 ` 452` huffman@29457 ` 453` ```lemma smult_minus_right [simp]: ``` huffman@29451 ` 454` ``` "smult (a::'a::comm_ring) (- p) = - smult a p" ``` huffman@29451 ` 455` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 456` huffman@29457 ` 457` ```lemma smult_minus_left [simp]: ``` huffman@29451 ` 458` ``` "smult (- a::'a::comm_ring) p = - smult a p" ``` huffman@29451 ` 459` ``` by (rule poly_ext, simp) ``` huffman@29451 ` 460` huffman@29451 ` 461` ```lemma smult_diff_right: ``` huffman@29451 ` 462` ``` "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" ``` nipkow@29667 ` 463` ``` by (rule poly_ext, simp add: algebra_simps) ``` huffman@29451 ` 464` huffman@29451 ` 465` ```lemma smult_diff_left: ``` huffman@29451 ` 466` ``` "smult (a - b::'a::comm_ring) p = smult a p - smult b p" ``` nipkow@29667 ` 467` ``` by (rule poly_ext, simp add: algebra_simps) ``` huffman@29451 ` 468` huffman@29472 ` 469` ```lemmas smult_distribs = ``` huffman@29472 ` 470` ``` smult_add_left smult_add_right ``` huffman@29472 ` 471` ``` smult_diff_left smult_diff_right ``` huffman@29472 ` 472` huffman@29451 ` 473` ```lemma smult_pCons [simp]: ``` huffman@29451 ` 474` ``` "smult a (pCons b p) = pCons (a * b) (smult a p)" ``` huffman@29451 ` 475` ``` by (rule poly_ext, simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 476` huffman@29451 ` 477` ```lemma smult_monom: "smult a (monom b n) = monom (a * b) n" ``` huffman@29451 ` 478` ``` by (induct n, simp add: monom_0, simp add: monom_Suc) ``` huffman@29451 ` 479` huffman@29659 ` 480` ```lemma degree_smult_eq [simp]: ``` huffman@29659 ` 481` ``` fixes a :: "'a::idom" ``` huffman@29659 ` 482` ``` shows "degree (smult a p) = (if a = 0 then 0 else degree p)" ``` huffman@29659 ` 483` ``` by (cases "a = 0", simp, simp add: degree_def) ``` huffman@29659 ` 484` huffman@29659 ` 485` ```lemma smult_eq_0_iff [simp]: ``` huffman@29659 ` 486` ``` fixes a :: "'a::idom" ``` huffman@29659 ` 487` ``` shows "smult a p = 0 \ a = 0 \ p = 0" ``` huffman@29659 ` 488` ``` by (simp add: expand_poly_eq) ``` huffman@29659 ` 489` huffman@29451 ` 490` huffman@29451 ` 491` ```subsection {* Multiplication of polynomials *} ``` huffman@29451 ` 492` huffman@29474 ` 493` ```text {* TODO: move to SetInterval.thy *} ``` huffman@29451 ` 494` ```lemma setsum_atMost_Suc_shift: ``` huffman@29451 ` 495` ``` fixes f :: "nat \ 'a::comm_monoid_add" ``` huffman@29451 ` 496` ``` shows "(\i\Suc n. f i) = f 0 + (\i\n. f (Suc i))" ``` huffman@29451 ` 497` ```proof (induct n) ``` huffman@29451 ` 498` ``` case 0 show ?case by simp ``` huffman@29451 ` 499` ```next ``` huffman@29451 ` 500` ``` case (Suc n) note IH = this ``` huffman@29451 ` 501` ``` have "(\i\Suc (Suc n). f i) = (\i\Suc n. f i) + f (Suc (Suc n))" ``` huffman@29451 ` 502` ``` by (rule setsum_atMost_Suc) ``` huffman@29451 ` 503` ``` also have "(\i\Suc n. f i) = f 0 + (\i\n. f (Suc i))" ``` huffman@29451 ` 504` ``` by (rule IH) ``` huffman@29451 ` 505` ``` also have "f 0 + (\i\n. f (Suc i)) + f (Suc (Suc n)) = ``` huffman@29451 ` 506` ``` f 0 + ((\i\n. f (Suc i)) + f (Suc (Suc n)))" ``` huffman@29451 ` 507` ``` by (rule add_assoc) ``` huffman@29451 ` 508` ``` also have "(\i\n. f (Suc i)) + f (Suc (Suc n)) = (\i\Suc n. f (Suc i))" ``` huffman@29451 ` 509` ``` by (rule setsum_atMost_Suc [symmetric]) ``` huffman@29451 ` 510` ``` finally show ?case . ``` huffman@29451 ` 511` ```qed ``` huffman@29451 ` 512` huffman@29451 ` 513` ```instantiation poly :: (comm_semiring_0) comm_semiring_0 ``` huffman@29451 ` 514` ```begin ``` huffman@29451 ` 515` huffman@29451 ` 516` ```definition ``` haftmann@37765 ` 517` ``` times_poly_def: ``` huffman@29474 ` 518` ``` "p * q = poly_rec 0 (\a p pq. smult a q + pCons 0 pq) p" ``` huffman@29474 ` 519` huffman@29474 ` 520` ```lemma mult_poly_0_left: "(0::'a poly) * q = 0" ``` huffman@29474 ` 521` ``` unfolding times_poly_def by (simp add: poly_rec_0) ``` huffman@29474 ` 522` huffman@29474 ` 523` ```lemma mult_pCons_left [simp]: ``` huffman@29474 ` 524` ``` "pCons a p * q = smult a q + pCons 0 (p * q)" ``` huffman@29474 ` 525` ``` unfolding times_poly_def by (simp add: poly_rec_pCons) ``` huffman@29474 ` 526` huffman@29474 ` 527` ```lemma mult_poly_0_right: "p * (0::'a poly) = 0" ``` huffman@29474 ` 528` ``` by (induct p, simp add: mult_poly_0_left, simp) ``` huffman@29451 ` 529` huffman@29474 ` 530` ```lemma mult_pCons_right [simp]: ``` huffman@29474 ` 531` ``` "p * pCons a q = smult a p + pCons 0 (p * q)" ``` nipkow@29667 ` 532` ``` by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps) ``` huffman@29474 ` 533` huffman@29474 ` 534` ```lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right ``` huffman@29474 ` 535` huffman@29474 ` 536` ```lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" ``` huffman@29474 ` 537` ``` by (induct p, simp add: mult_poly_0, simp add: smult_add_right) ``` huffman@29474 ` 538` huffman@29474 ` 539` ```lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" ``` huffman@29474 ` 540` ``` by (induct q, simp add: mult_poly_0, simp add: smult_add_right) ``` huffman@29474 ` 541` huffman@29474 ` 542` ```lemma mult_poly_add_left: ``` huffman@29474 ` 543` ``` fixes p q r :: "'a poly" ``` huffman@29474 ` 544` ``` shows "(p + q) * r = p * r + q * r" ``` huffman@29474 ` 545` ``` by (induct r, simp add: mult_poly_0, ``` nipkow@29667 ` 546` ``` simp add: smult_distribs algebra_simps) ``` huffman@29451 ` 547` huffman@29451 ` 548` ```instance proof ``` huffman@29451 ` 549` ``` fix p q r :: "'a poly" ``` huffman@29451 ` 550` ``` show 0: "0 * p = 0" ``` huffman@29474 ` 551` ``` by (rule mult_poly_0_left) ``` huffman@29451 ` 552` ``` show "p * 0 = 0" ``` huffman@29474 ` 553` ``` by (rule mult_poly_0_right) ``` huffman@29451 ` 554` ``` show "(p + q) * r = p * r + q * r" ``` huffman@29474 ` 555` ``` by (rule mult_poly_add_left) ``` huffman@29451 ` 556` ``` show "(p * q) * r = p * (q * r)" ``` huffman@29474 ` 557` ``` by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) ``` huffman@29451 ` 558` ``` show "p * q = q * p" ``` huffman@29474 ` 559` ``` by (induct p, simp add: mult_poly_0, simp) ``` huffman@29451 ` 560` ```qed ``` huffman@29451 ` 561` huffman@29451 ` 562` ```end ``` huffman@29451 ` 563` huffman@29540 ` 564` ```instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. ``` huffman@29540 ` 565` huffman@29474 ` 566` ```lemma coeff_mult: ``` huffman@29474 ` 567` ``` "coeff (p * q) n = (\i\n. coeff p i * coeff q (n-i))" ``` huffman@29474 ` 568` ```proof (induct p arbitrary: n) ``` huffman@29474 ` 569` ``` case 0 show ?case by simp ``` huffman@29474 ` 570` ```next ``` huffman@29474 ` 571` ``` case (pCons a p n) thus ?case ``` huffman@29474 ` 572` ``` by (cases n, simp, simp add: setsum_atMost_Suc_shift ``` huffman@29474 ` 573` ``` del: setsum_atMost_Suc) ``` huffman@29474 ` 574` ```qed ``` huffman@29451 ` 575` huffman@29474 ` 576` ```lemma degree_mult_le: "degree (p * q) \ degree p + degree q" ``` huffman@29474 ` 577` ```apply (rule degree_le) ``` huffman@29474 ` 578` ```apply (induct p) ``` huffman@29474 ` 579` ```apply simp ``` huffman@29474 ` 580` ```apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) ``` huffman@29451 ` 581` ```done ``` huffman@29451 ` 582` huffman@29451 ` 583` ```lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" ``` huffman@29451 ` 584` ``` by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) ``` huffman@29451 ` 585` huffman@29451 ` 586` huffman@29451 ` 587` ```subsection {* The unit polynomial and exponentiation *} ``` huffman@29451 ` 588` huffman@29451 ` 589` ```instantiation poly :: (comm_semiring_1) comm_semiring_1 ``` huffman@29451 ` 590` ```begin ``` huffman@29451 ` 591` huffman@29451 ` 592` ```definition ``` huffman@29451 ` 593` ``` one_poly_def: ``` huffman@29451 ` 594` ``` "1 = pCons 1 0" ``` huffman@29451 ` 595` huffman@29451 ` 596` ```instance proof ``` huffman@29451 ` 597` ``` fix p :: "'a poly" show "1 * p = p" ``` huffman@29451 ` 598` ``` unfolding one_poly_def ``` huffman@29451 ` 599` ``` by simp ``` huffman@29451 ` 600` ```next ``` huffman@29451 ` 601` ``` show "0 \ (1::'a poly)" ``` huffman@29451 ` 602` ``` unfolding one_poly_def by simp ``` huffman@29451 ` 603` ```qed ``` huffman@29451 ` 604` huffman@29451 ` 605` ```end ``` huffman@29451 ` 606` huffman@29540 ` 607` ```instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. ``` huffman@29540 ` 608` huffman@29451 ` 609` ```lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" ``` huffman@29451 ` 610` ``` unfolding one_poly_def ``` huffman@29451 ` 611` ``` by (simp add: coeff_pCons split: nat.split) ``` huffman@29451 ` 612` huffman@29451 ` 613` ```lemma degree_1 [simp]: "degree 1 = 0" ``` huffman@29451 ` 614` ``` unfolding one_poly_def ``` huffman@29451 ` 615` ``` by (rule degree_pCons_0) ``` huffman@29451 ` 616` huffman@29979 ` 617` ```text {* Lemmas about divisibility *} ``` huffman@29979 ` 618` huffman@29979 ` 619` ```lemma dvd_smult: "p dvd q \ p dvd smult a q" ``` huffman@29979 ` 620` ```proof - ``` huffman@29979 ` 621` ``` assume "p dvd q" ``` huffman@29979 ` 622` ``` then obtain k where "q = p * k" .. ``` huffman@29979 ` 623` ``` then have "smult a q = p * smult a k" by simp ``` huffman@29979 ` 624` ``` then show "p dvd smult a q" .. ``` huffman@29979 ` 625` ```qed ``` huffman@29979 ` 626` huffman@29979 ` 627` ```lemma dvd_smult_cancel: ``` huffman@29979 ` 628` ``` fixes a :: "'a::field" ``` huffman@29979 ` 629` ``` shows "p dvd smult a q \ a \ 0 \ p dvd q" ``` huffman@29979 ` 630` ``` by (drule dvd_smult [where a="inverse a"]) simp ``` huffman@29979 ` 631` huffman@29979 ` 632` ```lemma dvd_smult_iff: ``` huffman@29979 ` 633` ``` fixes a :: "'a::field" ``` huffman@29979 ` 634` ``` shows "a \ 0 \ p dvd smult a q \ p dvd q" ``` huffman@29979 ` 635` ``` by (safe elim!: dvd_smult dvd_smult_cancel) ``` huffman@29979 ` 636` huffman@31663 ` 637` ```lemma smult_dvd_cancel: ``` huffman@31663 ` 638` ``` "smult a p dvd q \ p dvd q" ``` huffman@31663 ` 639` ```proof - ``` huffman@31663 ` 640` ``` assume "smult a p dvd q" ``` huffman@31663 ` 641` ``` then obtain k where "q = smult a p * k" .. ``` huffman@31663 ` 642` ``` then have "q = p * smult a k" by simp ``` huffman@31663 ` 643` ``` then show "p dvd q" .. ``` huffman@31663 ` 644` ```qed ``` huffman@31663 ` 645` huffman@31663 ` 646` ```lemma smult_dvd: ``` huffman@31663 ` 647` ``` fixes a :: "'a::field" ``` huffman@31663 ` 648` ``` shows "p dvd q \ a \ 0 \ smult a p dvd q" ``` huffman@31663 ` 649` ``` by (rule smult_dvd_cancel [where a="inverse a"]) simp ``` huffman@31663 ` 650` huffman@31663 ` 651` ```lemma smult_dvd_iff: ``` huffman@31663 ` 652` ``` fixes a :: "'a::field" ``` huffman@31663 ` 653` ``` shows "smult a p dvd q \ (if a = 0 then q = 0 else p dvd q)" ``` huffman@31663 ` 654` ``` by (auto elim: smult_dvd smult_dvd_cancel) ``` huffman@31663 ` 655` huffman@29979 ` 656` ```lemma degree_power_le: "degree (p ^ n) \ degree p * n" ``` huffman@29979 ` 657` ```by (induct n, simp, auto intro: order_trans degree_mult_le) ``` huffman@29979 ` 658` huffman@29451 ` 659` ```instance poly :: (comm_ring) comm_ring .. ``` huffman@29451 ` 660` huffman@29451 ` 661` ```instance poly :: (comm_ring_1) comm_ring_1 .. ``` huffman@29451 ` 662` huffman@29451 ` 663` ```instantiation poly :: (comm_ring_1) number_ring ``` huffman@29451 ` 664` ```begin ``` huffman@29451 ` 665` huffman@29451 ` 666` ```definition ``` huffman@29451 ` 667` ``` "number_of k = (of_int k :: 'a poly)" ``` huffman@29451 ` 668` huffman@29451 ` 669` ```instance ``` huffman@29451 ` 670` ``` by default (rule number_of_poly_def) ``` huffman@29451 ` 671` huffman@29451 ` 672` ```end ``` huffman@29451 ` 673` huffman@29451 ` 674` huffman@29451 ` 675` ```subsection {* Polynomials form an integral domain *} ``` huffman@29451 ` 676` huffman@29451 ` 677` ```lemma coeff_mult_degree_sum: ``` huffman@29451 ` 678` ``` "coeff (p * q) (degree p + degree q) = ``` huffman@29451 ` 679` ``` coeff p (degree p) * coeff q (degree q)" ``` huffman@29471 ` 680` ``` by (induct p, simp, simp add: coeff_eq_0) ``` huffman@29451 ` 681` huffman@29451 ` 682` ```instance poly :: (idom) idom ``` huffman@29451 ` 683` ```proof ``` huffman@29451 ` 684` ``` fix p q :: "'a poly" ``` huffman@29451 ` 685` ``` assume "p \ 0" and "q \ 0" ``` huffman@29451 ` 686` ``` have "coeff (p * q) (degree p + degree q) = ``` huffman@29451 ` 687` ``` coeff p (degree p) * coeff q (degree q)" ``` huffman@29451 ` 688` ``` by (rule coeff_mult_degree_sum) ``` huffman@29451 ` 689` ``` also have "coeff p (degree p) * coeff q (degree q) \ 0" ``` huffman@29451 ` 690` ``` using `p \ 0` and `q \ 0` by simp ``` huffman@29451 ` 691` ``` finally have "\n. coeff (p * q) n \ 0" .. ``` huffman@29451 ` 692` ``` thus "p * q \ 0" by (simp add: expand_poly_eq) ``` huffman@29451 ` 693` ```qed ``` huffman@29451 ` 694` huffman@29451 ` 695` ```lemma degree_mult_eq: ``` huffman@29451 ` 696` ``` fixes p q :: "'a::idom poly" ``` huffman@29451 ` 697` ``` shows "\p \ 0; q \ 0\ \ degree (p * q) = degree p + degree q" ``` huffman@29451 ` 698` ```apply (rule order_antisym [OF degree_mult_le le_degree]) ``` huffman@29451 ` 699` ```apply (simp add: coeff_mult_degree_sum) ``` huffman@29451 ` 700` ```done ``` huffman@29451 ` 701` huffman@29451 ` 702` ```lemma dvd_imp_degree_le: ``` huffman@29451 ` 703` ``` fixes p q :: "'a::idom poly" ``` huffman@29451 ` 704` ``` shows "\p dvd q; q \ 0\ \ degree p \ degree q" ``` huffman@29451 ` 705` ``` by (erule dvdE, simp add: degree_mult_eq) ``` huffman@29451 ` 706` huffman@29451 ` 707` huffman@29878 ` 708` ```subsection {* Polynomials form an ordered integral domain *} ``` huffman@29878 ` 709` huffman@29878 ` 710` ```definition ``` haftmann@35028 ` 711` ``` pos_poly :: "'a::linordered_idom poly \ bool" ``` huffman@29878 ` 712` ```where ``` huffman@29878 ` 713` ``` "pos_poly p \ 0 < coeff p (degree p)" ``` huffman@29878 ` 714` huffman@29878 ` 715` ```lemma pos_poly_pCons: ``` huffman@29878 ` 716` ``` "pos_poly (pCons a p) \ pos_poly p \ (p = 0 \ 0 < a)" ``` huffman@29878 ` 717` ``` unfolding pos_poly_def by simp ``` huffman@29878 ` 718` huffman@29878 ` 719` ```lemma not_pos_poly_0 [simp]: "\ pos_poly 0" ``` huffman@29878 ` 720` ``` unfolding pos_poly_def by simp ``` huffman@29878 ` 721` huffman@29878 ` 722` ```lemma pos_poly_add: "\pos_poly p; pos_poly q\ \ pos_poly (p + q)" ``` huffman@29878 ` 723` ``` apply (induct p arbitrary: q, simp) ``` huffman@29878 ` 724` ``` apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos) ``` huffman@29878 ` 725` ``` done ``` huffman@29878 ` 726` huffman@29878 ` 727` ```lemma pos_poly_mult: "\pos_poly p; pos_poly q\ \ pos_poly (p * q)" ``` huffman@29878 ` 728` ``` unfolding pos_poly_def ``` huffman@29878 ` 729` ``` apply (subgoal_tac "p \ 0 \ q \ 0") ``` huffman@29878 ` 730` ``` apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos) ``` huffman@29878 ` 731` ``` apply auto ``` huffman@29878 ` 732` ``` done ``` huffman@29878 ` 733` huffman@29878 ` 734` ```lemma pos_poly_total: "p = 0 \ pos_poly p \ pos_poly (- p)" ``` huffman@29878 ` 735` ```by (induct p) (auto simp add: pos_poly_pCons) ``` huffman@29878 ` 736` haftmann@35028 ` 737` ```instantiation poly :: (linordered_idom) linordered_idom ``` huffman@29878 ` 738` ```begin ``` huffman@29878 ` 739` huffman@29878 ` 740` ```definition ``` haftmann@37765 ` 741` ``` "x < y \ pos_poly (y - x)" ``` huffman@29878 ` 742` huffman@29878 ` 743` ```definition ``` haftmann@37765 ` 744` ``` "x \ y \ x = y \ pos_poly (y - x)" ``` huffman@29878 ` 745` huffman@29878 ` 746` ```definition ``` haftmann@37765 ` 747` ``` "abs (x::'a poly) = (if x < 0 then - x else x)" ``` huffman@29878 ` 748` huffman@29878 ` 749` ```definition ``` haftmann@37765 ` 750` ``` "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" ``` huffman@29878 ` 751` huffman@29878 ` 752` ```instance proof ``` huffman@29878 ` 753` ``` fix x y :: "'a poly" ``` huffman@29878 ` 754` ``` show "x < y \ x \ y \ \ y \ x" ``` huffman@29878 ` 755` ``` unfolding less_eq_poly_def less_poly_def ``` huffman@29878 ` 756` ``` apply safe ``` huffman@29878 ` 757` ``` apply simp ``` huffman@29878 ` 758` ``` apply (drule (1) pos_poly_add) ``` huffman@29878 ` 759` ``` apply simp ``` huffman@29878 ` 760` ``` done ``` huffman@29878 ` 761` ```next ``` huffman@29878 ` 762` ``` fix x :: "'a poly" show "x \ x" ``` huffman@29878 ` 763` ``` unfolding less_eq_poly_def by simp ``` huffman@29878 ` 764` ```next ``` huffman@29878 ` 765` ``` fix x y z :: "'a poly" ``` huffman@29878 ` 766` ``` assume "x \ y" and "y \ z" thus "x \ z" ``` huffman@29878 ` 767` ``` unfolding less_eq_poly_def ``` huffman@29878 ` 768` ``` apply safe ``` huffman@29878 ` 769` ``` apply (drule (1) pos_poly_add) ``` huffman@29878 ` 770` ``` apply (simp add: algebra_simps) ``` huffman@29878 ` 771` ``` done ``` huffman@29878 ` 772` ```next ``` huffman@29878 ` 773` ``` fix x y :: "'a poly" ``` huffman@29878 ` 774` ``` assume "x \ y" and "y \ x" thus "x = y" ``` huffman@29878 ` 775` ``` unfolding less_eq_poly_def ``` huffman@29878 ` 776` ``` apply safe ``` huffman@29878 ` 777` ``` apply (drule (1) pos_poly_add) ``` huffman@29878 ` 778` ``` apply simp ``` huffman@29878 ` 779` ``` done ``` huffman@29878 ` 780` ```next ``` huffman@29878 ` 781` ``` fix x y z :: "'a poly" ``` huffman@29878 ` 782` ``` assume "x \ y" thus "z + x \ z + y" ``` huffman@29878 ` 783` ``` unfolding less_eq_poly_def ``` huffman@29878 ` 784` ``` apply safe ``` huffman@29878 ` 785` ``` apply (simp add: algebra_simps) ``` huffman@29878 ` 786` ``` done ``` huffman@29878 ` 787` ```next ``` huffman@29878 ` 788` ``` fix x y :: "'a poly" ``` huffman@29878 ` 789` ``` show "x \ y \ y \ x" ``` huffman@29878 ` 790` ``` unfolding less_eq_poly_def ``` huffman@29878 ` 791` ``` using pos_poly_total [of "x - y"] ``` huffman@29878 ` 792` ``` by auto ``` huffman@29878 ` 793` ```next ``` huffman@29878 ` 794` ``` fix x y z :: "'a poly" ``` huffman@29878 ` 795` ``` assume "x < y" and "0 < z" ``` huffman@29878 ` 796` ``` thus "z * x < z * y" ``` huffman@29878 ` 797` ``` unfolding less_poly_def ``` huffman@29878 ` 798` ``` by (simp add: right_diff_distrib [symmetric] pos_poly_mult) ``` huffman@29878 ` 799` ```next ``` huffman@29878 ` 800` ``` fix x :: "'a poly" ``` huffman@29878 ` 801` ``` show "\x\ = (if x < 0 then - x else x)" ``` huffman@29878 ` 802` ``` by (rule abs_poly_def) ``` huffman@29878 ` 803` ```next ``` huffman@29878 ` 804` ``` fix x :: "'a poly" ``` huffman@29878 ` 805` ``` show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" ``` huffman@29878 ` 806` ``` by (rule sgn_poly_def) ``` huffman@29878 ` 807` ```qed ``` huffman@29878 ` 808` huffman@29878 ` 809` ```end ``` huffman@29878 ` 810` huffman@29878 ` 811` ```text {* TODO: Simplification rules for comparisons *} ``` huffman@29878 ` 812` huffman@29878 ` 813` huffman@29451 ` 814` ```subsection {* Long division of polynomials *} ``` huffman@29451 ` 815` huffman@29451 ` 816` ```definition ``` huffman@29537 ` 817` ``` pdivmod_rel :: "'a::field poly \ 'a poly \ 'a poly \ 'a poly \ bool" ``` huffman@29451 ` 818` ```where ``` huffman@29537 ` 819` ``` "pdivmod_rel x y q r \ ``` huffman@29451 ` 820` ``` x = q * y + r \ (if y = 0 then q = 0 else r = 0 \ degree r < degree y)" ``` huffman@29451 ` 821` huffman@29537 ` 822` ```lemma pdivmod_rel_0: ``` huffman@29537 ` 823` ``` "pdivmod_rel 0 y 0 0" ``` huffman@29537 ` 824` ``` unfolding pdivmod_rel_def by simp ``` huffman@29451 ` 825` huffman@29537 ` 826` ```lemma pdivmod_rel_by_0: ``` huffman@29537 ` 827` ``` "pdivmod_rel x 0 0 x" ``` huffman@29537 ` 828` ``` unfolding pdivmod_rel_def by simp ``` huffman@29451 ` 829` huffman@29451 ` 830` ```lemma eq_zero_or_degree_less: ``` huffman@29451 ` 831` ``` assumes "degree p \ n" and "coeff p n = 0" ``` huffman@29451 ` 832` ``` shows "p = 0 \ degree p < n" ``` huffman@29451 ` 833` ```proof (cases n) ``` huffman@29451 ` 834` ``` case 0 ``` huffman@29451 ` 835` ``` with `degree p \ n` and `coeff p n = 0` ``` huffman@29451 ` 836` ``` have "coeff p (degree p) = 0" by simp ``` huffman@29451 ` 837` ``` then have "p = 0" by simp ``` huffman@29451 ` 838` ``` then show ?thesis .. ``` huffman@29451 ` 839` ```next ``` huffman@29451 ` 840` ``` case (Suc m) ``` huffman@29451 ` 841` ``` have "\i>n. coeff p i = 0" ``` huffman@29451 ` 842` ``` using `degree p \ n` by (simp add: coeff_eq_0) ``` huffman@29451 ` 843` ``` then have "\i\n. coeff p i = 0" ``` huffman@29451 ` 844` ``` using `coeff p n = 0` by (simp add: le_less) ``` huffman@29451 ` 845` ``` then have "\i>m. coeff p i = 0" ``` huffman@29451 ` 846` ``` using `n = Suc m` by (simp add: less_eq_Suc_le) ``` huffman@29451 ` 847` ``` then have "degree p \ m" ``` huffman@29451 ` 848` ``` by (rule degree_le) ``` huffman@29451 ` 849` ``` then have "degree p < n" ``` huffman@29451 ` 850` ``` using `n = Suc m` by (simp add: less_Suc_eq_le) ``` huffman@29451 ` 851` ``` then show ?thesis .. ``` huffman@29451 ` 852` ```qed ``` huffman@29451 ` 853` huffman@29537 ` 854` ```lemma pdivmod_rel_pCons: ``` huffman@29537 ` 855` ``` assumes rel: "pdivmod_rel x y q r" ``` huffman@29451 ` 856` ``` assumes y: "y \ 0" ``` huffman@29451 ` 857` ``` assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" ``` huffman@29537 ` 858` ``` shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)" ``` huffman@29537 ` 859` ``` (is "pdivmod_rel ?x y ?q ?r") ``` huffman@29451 ` 860` ```proof - ``` huffman@29451 ` 861` ``` have x: "x = q * y + r" and r: "r = 0 \ degree r < degree y" ``` huffman@29537 ` 862` ``` using assms unfolding pdivmod_rel_def by simp_all ``` huffman@29451 ` 863` huffman@29451 ` 864` ``` have 1: "?x = ?q * y + ?r" ``` huffman@29451 ` 865` ``` using b x by simp ``` huffman@29451 ` 866` huffman@29451 ` 867` ``` have 2: "?r = 0 \ degree ?r < degree y" ``` huffman@29451 ` 868` ``` proof (rule eq_zero_or_degree_less) ``` huffman@29539 ` 869` ``` show "degree ?r \ degree y" ``` huffman@29539 ` 870` ``` proof (rule degree_diff_le) ``` huffman@29451 ` 871` ``` show "degree (pCons a r) \ degree y" ``` huffman@29460 ` 872` ``` using r by auto ``` huffman@29451 ` 873` ``` show "degree (smult b y) \ degree y" ``` huffman@29451 ` 874` ``` by (rule degree_smult_le) ``` huffman@29451 ` 875` ``` qed ``` huffman@29451 ` 876` ``` next ``` huffman@29451 ` 877` ``` show "coeff ?r (degree y) = 0" ``` huffman@29451 ` 878` ``` using `y \ 0` unfolding b by simp ``` huffman@29451 ` 879` ``` qed ``` huffman@29451 ` 880` huffman@29451 ` 881` ``` from 1 2 show ?thesis ``` huffman@29537 ` 882` ``` unfolding pdivmod_rel_def ``` huffman@29451 ` 883` ``` using `y \ 0` by simp ``` huffman@29451 ` 884` ```qed ``` huffman@29451 ` 885` huffman@29537 ` 886` ```lemma pdivmod_rel_exists: "\q r. pdivmod_rel x y q r" ``` huffman@29451 ` 887` ```apply (cases "y = 0") ``` huffman@29537 ` 888` ```apply (fast intro!: pdivmod_rel_by_0) ``` huffman@29451 ` 889` ```apply (induct x) ``` huffman@29537 ` 890` ```apply (fast intro!: pdivmod_rel_0) ``` huffman@29537 ` 891` ```apply (fast intro!: pdivmod_rel_pCons) ``` huffman@29451 ` 892` ```done ``` huffman@29451 ` 893` huffman@29537 ` 894` ```lemma pdivmod_rel_unique: ``` huffman@29537 ` 895` ``` assumes 1: "pdivmod_rel x y q1 r1" ``` huffman@29537 ` 896` ``` assumes 2: "pdivmod_rel x y q2 r2" ``` huffman@29451 ` 897` ``` shows "q1 = q2 \ r1 = r2" ``` huffman@29451 ` 898` ```proof (cases "y = 0") ``` huffman@29451 ` 899` ``` assume "y = 0" with assms show ?thesis ``` huffman@29537 ` 900` ``` by (simp add: pdivmod_rel_def) ``` huffman@29451 ` 901` ```next ``` huffman@29451 ` 902` ``` assume [simp]: "y \ 0" ``` huffman@29451 ` 903` ``` from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \ degree r1 < degree y" ``` huffman@29537 ` 904` ``` unfolding pdivmod_rel_def by simp_all ``` huffman@29451 ` 905` ``` from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \ degree r2 < degree y" ``` huffman@29537 ` 906` ``` unfolding pdivmod_rel_def by simp_all ``` huffman@29451 ` 907` ``` from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" ``` nipkow@29667 ` 908` ``` by (simp add: algebra_simps) ``` huffman@29451 ` 909` ``` from r1 r2 have r3: "(r2 - r1) = 0 \ degree (r2 - r1) < degree y" ``` huffman@29453 ` 910` ``` by (auto intro: degree_diff_less) ``` huffman@29451 ` 911` huffman@29451 ` 912` ``` show "q1 = q2 \ r1 = r2" ``` huffman@29451 ` 913` ``` proof (rule ccontr) ``` huffman@29451 ` 914` ``` assume "\ (q1 = q2 \ r1 = r2)" ``` huffman@29451 ` 915` ``` with q3 have "q1 \ q2" and "r1 \ r2" by auto ``` huffman@29451 ` 916` ``` with r3 have "degree (r2 - r1) < degree y" by simp ``` huffman@29451 ` 917` ``` also have "degree y \ degree (q1 - q2) + degree y" by simp ``` huffman@29451 ` 918` ``` also have "\ = degree ((q1 - q2) * y)" ``` huffman@29451 ` 919` ``` using `q1 \ q2` by (simp add: degree_mult_eq) ``` huffman@29451 ` 920` ``` also have "\ = degree (r2 - r1)" ``` huffman@29451 ` 921` ``` using q3 by simp ``` huffman@29451 ` 922` ``` finally have "degree (r2 - r1) < degree (r2 - r1)" . ``` huffman@29451 ` 923` ``` then show "False" by simp ``` huffman@29451 ` 924` ``` qed ``` huffman@29451 ` 925` ```qed ``` huffman@29451 ` 926` huffman@29660 ` 927` ```lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \ q = 0 \ r = 0" ``` huffman@29660 ` 928` ```by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0) ``` huffman@29660 ` 929` huffman@29660 ` 930` ```lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \ q = 0 \ r = x" ``` huffman@29660 ` 931` ```by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0) ``` huffman@29660 ` 932` wenzelm@45605 ` 933` ```lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1] ``` huffman@29451 ` 934` wenzelm@45605 ` 935` ```lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2] ``` huffman@29451 ` 936` huffman@29451 ` 937` ```instantiation poly :: (field) ring_div ``` huffman@29451 ` 938` ```begin ``` huffman@29451 ` 939` huffman@29451 ` 940` ```definition div_poly where ``` haftmann@37765 ` 941` ``` "x div y = (THE q. \r. pdivmod_rel x y q r)" ``` huffman@29451 ` 942` huffman@29451 ` 943` ```definition mod_poly where ``` haftmann@37765 ` 944` ``` "x mod y = (THE r. \q. pdivmod_rel x y q r)" ``` huffman@29451 ` 945` huffman@29451 ` 946` ```lemma div_poly_eq: ``` huffman@29537 ` 947` ``` "pdivmod_rel x y q r \ x div y = q" ``` huffman@29451 ` 948` ```unfolding div_poly_def ``` huffman@29537 ` 949` ```by (fast elim: pdivmod_rel_unique_div) ``` huffman@29451 ` 950` huffman@29451 ` 951` ```lemma mod_poly_eq: ``` huffman@29537 ` 952` ``` "pdivmod_rel x y q r \ x mod y = r" ``` huffman@29451 ` 953` ```unfolding mod_poly_def ``` huffman@29537 ` 954` ```by (fast elim: pdivmod_rel_unique_mod) ``` huffman@29451 ` 955` huffman@29537 ` 956` ```lemma pdivmod_rel: ``` huffman@29537 ` 957` ``` "pdivmod_rel x y (x div y) (x mod y)" ``` huffman@29451 ` 958` ```proof - ``` huffman@29537 ` 959` ``` from pdivmod_rel_exists ``` huffman@29537 ` 960` ``` obtain q r where "pdivmod_rel x y q r" by fast ``` huffman@29451 ` 961` ``` thus ?thesis ``` huffman@29451 ` 962` ``` by (simp add: div_poly_eq mod_poly_eq) ``` huffman@29451 ` 963` ```qed ``` huffman@29451 ` 964` huffman@29451 ` 965` ```instance proof ``` huffman@29451 ` 966` ``` fix x y :: "'a poly" ``` huffman@29451 ` 967` ``` show "x div y * y + x mod y = x" ``` huffman@29537 ` 968` ``` using pdivmod_rel [of x y] ``` huffman@29537 ` 969` ``` by (simp add: pdivmod_rel_def) ``` huffman@29451 ` 970` ```next ``` huffman@29451 ` 971` ``` fix x :: "'a poly" ``` huffman@29537 ` 972` ``` have "pdivmod_rel x 0 0 x" ``` huffman@29537 ` 973` ``` by (rule pdivmod_rel_by_0) ``` huffman@29451 ` 974` ``` thus "x div 0 = 0" ``` huffman@29451 ` 975` ``` by (rule div_poly_eq) ``` huffman@29451 ` 976` ```next ``` huffman@29451 ` 977` ``` fix y :: "'a poly" ``` huffman@29537 ` 978` ``` have "pdivmod_rel 0 y 0 0" ``` huffman@29537 ` 979` ``` by (rule pdivmod_rel_0) ``` huffman@29451 ` 980` ``` thus "0 div y = 0" ``` huffman@29451 ` 981` ``` by (rule div_poly_eq) ``` huffman@29451 ` 982` ```next ``` huffman@29451 ` 983` ``` fix x y z :: "'a poly" ``` huffman@29451 ` 984` ``` assume "y \ 0" ``` huffman@29537 ` 985` ``` hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" ``` huffman@29537 ` 986` ``` using pdivmod_rel [of x y] ``` huffman@29537 ` 987` ``` by (simp add: pdivmod_rel_def left_distrib) ``` huffman@29451 ` 988` ``` thus "(x + z * y) div y = z + x div y" ``` huffman@29451 ` 989` ``` by (rule div_poly_eq) ``` haftmann@30930 ` 990` ```next ``` haftmann@30930 ` 991` ``` fix x y z :: "'a poly" ``` haftmann@30930 ` 992` ``` assume "x \ 0" ``` haftmann@30930 ` 993` ``` show "(x * y) div (x * z) = y div z" ``` haftmann@30930 ` 994` ``` proof (cases "y \ 0 \ z \ 0") ``` haftmann@30930 ` 995` ``` have "\x::'a poly. pdivmod_rel x 0 0 x" ``` haftmann@30930 ` 996` ``` by (rule pdivmod_rel_by_0) ``` haftmann@30930 ` 997` ``` then have [simp]: "\x::'a poly. x div 0 = 0" ``` haftmann@30930 ` 998` ``` by (rule div_poly_eq) ``` haftmann@30930 ` 999` ``` have "\x::'a poly. pdivmod_rel 0 x 0 0" ``` haftmann@30930 ` 1000` ``` by (rule pdivmod_rel_0) ``` haftmann@30930 ` 1001` ``` then have [simp]: "\x::'a poly. 0 div x = 0" ``` haftmann@30930 ` 1002` ``` by (rule div_poly_eq) ``` haftmann@30930 ` 1003` ``` case False then show ?thesis by auto ``` haftmann@30930 ` 1004` ``` next ``` haftmann@30930 ` 1005` ``` case True then have "y \ 0" and "z \ 0" by auto ``` haftmann@30930 ` 1006` ``` with `x \ 0` ``` haftmann@30930 ` 1007` ``` have "\q r. pdivmod_rel y z q r \ pdivmod_rel (x * y) (x * z) q (x * r)" ``` haftmann@30930 ` 1008` ``` by (auto simp add: pdivmod_rel_def algebra_simps) ``` haftmann@30930 ` 1009` ``` (rule classical, simp add: degree_mult_eq) ``` haftmann@30930 ` 1010` ``` moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" . ``` haftmann@30930 ` 1011` ``` ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" . ``` haftmann@30930 ` 1012` ``` then show ?thesis by (simp add: div_poly_eq) ``` haftmann@30930 ` 1013` ``` qed ``` huffman@29451 ` 1014` ```qed ``` huffman@29451 ` 1015` huffman@29451 ` 1016` ```end ``` huffman@29451 ` 1017` huffman@29451 ` 1018` ```lemma degree_mod_less: ``` huffman@29451 ` 1019` ``` "y \ 0 \ x mod y = 0 \ degree (x mod y) < degree y" ``` huffman@29537 ` 1020` ``` using pdivmod_rel [of x y] ``` huffman@29537 ` 1021` ``` unfolding pdivmod_rel_def by simp ``` huffman@29451 ` 1022` huffman@29451 ` 1023` ```lemma div_poly_less: "degree x < degree y \ x div y = 0" ``` huffman@29451 ` 1024` ```proof - ``` huffman@29451 ` 1025` ``` assume "degree x < degree y" ``` huffman@29537 ` 1026` ``` hence "pdivmod_rel x y 0 x" ``` huffman@29537 ` 1027` ``` by (simp add: pdivmod_rel_def) ``` huffman@29451 ` 1028` ``` thus "x div y = 0" by (rule div_poly_eq) ``` huffman@29451 ` 1029` ```qed ``` huffman@29451 ` 1030` huffman@29451 ` 1031` ```lemma mod_poly_less: "degree x < degree y \ x mod y = x" ``` huffman@29451 ` 1032` ```proof - ``` huffman@29451 ` 1033` ``` assume "degree x < degree y" ``` huffman@29537 ` 1034` ``` hence "pdivmod_rel x y 0 x" ``` huffman@29537 ` 1035` ``` by (simp add: pdivmod_rel_def) ``` huffman@29451 ` 1036` ``` thus "x mod y = x" by (rule mod_poly_eq) ``` huffman@29451 ` 1037` ```qed ``` huffman@29451 ` 1038` huffman@29659 ` 1039` ```lemma pdivmod_rel_smult_left: ``` huffman@29659 ` 1040` ``` "pdivmod_rel x y q r ``` huffman@29659 ` 1041` ``` \ pdivmod_rel (smult a x) y (smult a q) (smult a r)" ``` huffman@29659 ` 1042` ``` unfolding pdivmod_rel_def by (simp add: smult_add_right) ``` huffman@29659 ` 1043` huffman@29659 ` 1044` ```lemma div_smult_left: "(smult a x) div y = smult a (x div y)" ``` huffman@29659 ` 1045` ``` by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) ``` huffman@29659 ` 1046` huffman@29659 ` 1047` ```lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" ``` huffman@29659 ` 1048` ``` by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) ``` huffman@29659 ` 1049` huffman@30072 ` 1050` ```lemma poly_div_minus_left [simp]: ``` huffman@30072 ` 1051` ``` fixes x y :: "'a::field poly" ``` huffman@30072 ` 1052` ``` shows "(- x) div y = - (x div y)" ``` huffman@30072 ` 1053` ``` using div_smult_left [of "- 1::'a"] by simp ``` huffman@30072 ` 1054` huffman@30072 ` 1055` ```lemma poly_mod_minus_left [simp]: ``` huffman@30072 ` 1056` ``` fixes x y :: "'a::field poly" ``` huffman@30072 ` 1057` ``` shows "(- x) mod y = - (x mod y)" ``` huffman@30072 ` 1058` ``` using mod_smult_left [of "- 1::'a"] by simp ``` huffman@30072 ` 1059` huffman@29659 ` 1060` ```lemma pdivmod_rel_smult_right: ``` huffman@29659 ` 1061` ``` "\a \ 0; pdivmod_rel x y q r\ ``` huffman@29659 ` 1062` ``` \ pdivmod_rel x (smult a y) (smult (inverse a) q) r" ``` huffman@29659 ` 1063` ``` unfolding pdivmod_rel_def by simp ``` huffman@29659 ` 1064` huffman@29659 ` 1065` ```lemma div_smult_right: ``` huffman@29659 ` 1066` ``` "a \ 0 \ x div (smult a y) = smult (inverse a) (x div y)" ``` huffman@29659 ` 1067` ``` by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) ``` huffman@29659 ` 1068` huffman@29659 ` 1069` ```lemma mod_smult_right: "a \ 0 \ x mod (smult a y) = x mod y" ``` huffman@29659 ` 1070` ``` by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) ``` huffman@29659 ` 1071` huffman@30072 ` 1072` ```lemma poly_div_minus_right [simp]: ``` huffman@30072 ` 1073` ``` fixes x y :: "'a::field poly" ``` huffman@30072 ` 1074` ``` shows "x div (- y) = - (x div y)" ``` huffman@30072 ` 1075` ``` using div_smult_right [of "- 1::'a"] ``` huffman@30072 ` 1076` ``` by (simp add: nonzero_inverse_minus_eq) ``` huffman@30072 ` 1077` huffman@30072 ` 1078` ```lemma poly_mod_minus_right [simp]: ``` huffman@30072 ` 1079` ``` fixes x y :: "'a::field poly" ``` huffman@30072 ` 1080` ``` shows "x mod (- y) = x mod y" ``` huffman@30072 ` 1081` ``` using mod_smult_right [of "- 1::'a"] by simp ``` huffman@30072 ` 1082` huffman@29660 ` 1083` ```lemma pdivmod_rel_mult: ``` huffman@29660 ` 1084` ``` "\pdivmod_rel x y q r; pdivmod_rel q z q' r'\ ``` huffman@29660 ` 1085` ``` \ pdivmod_rel x (y * z) q' (y * r' + r)" ``` huffman@29660 ` 1086` ```apply (cases "z = 0", simp add: pdivmod_rel_def) ``` huffman@29660 ` 1087` ```apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff) ``` huffman@29660 ` 1088` ```apply (cases "r = 0") ``` huffman@29660 ` 1089` ```apply (cases "r' = 0") ``` huffman@29660 ` 1090` ```apply (simp add: pdivmod_rel_def) ``` haftmann@36350 ` 1091` ```apply (simp add: pdivmod_rel_def field_simps degree_mult_eq) ``` huffman@29660 ` 1092` ```apply (cases "r' = 0") ``` huffman@29660 ` 1093` ```apply (simp add: pdivmod_rel_def degree_mult_eq) ``` haftmann@36350 ` 1094` ```apply (simp add: pdivmod_rel_def field_simps) ``` huffman@29660 ` 1095` ```apply (simp add: degree_mult_eq degree_add_less) ``` huffman@29660 ` 1096` ```done ``` huffman@29660 ` 1097` huffman@29660 ` 1098` ```lemma poly_div_mult_right: ``` huffman@29660 ` 1099` ``` fixes x y z :: "'a::field poly" ``` huffman@29660 ` 1100` ``` shows "x div (y * z) = (x div y) div z" ``` huffman@29660 ` 1101` ``` by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) ``` huffman@29660 ` 1102` huffman@29660 ` 1103` ```lemma poly_mod_mult_right: ``` huffman@29660 ` 1104` ``` fixes x y z :: "'a::field poly" ``` huffman@29660 ` 1105` ``` shows "x mod (y * z) = y * (x div y mod z) + x mod y" ``` huffman@29660 ` 1106` ``` by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) ``` huffman@29660 ` 1107` huffman@29451 ` 1108` ```lemma mod_pCons: ``` huffman@29451 ` 1109` ``` fixes a and x ``` huffman@29451 ` 1110` ``` assumes y: "y \ 0" ``` huffman@29451 ` 1111` ``` defines b: "b \ coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" ``` huffman@29451 ` 1112` ``` shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" ``` huffman@29451 ` 1113` ```unfolding b ``` huffman@29451 ` 1114` ```apply (rule mod_poly_eq) ``` huffman@29537 ` 1115` ```apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) ``` huffman@29451 ` 1116` ```done ``` huffman@29451 ` 1117` huffman@29451 ` 1118` huffman@31663 ` 1119` ```subsection {* GCD of polynomials *} ``` huffman@31663 ` 1120` huffman@31663 ` 1121` ```function ``` huffman@31663 ` 1122` ``` poly_gcd :: "'a::field poly \ 'a poly \ 'a poly" where ``` huffman@31663 ` 1123` ``` "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x" ``` huffman@31663 ` 1124` ```| "y \ 0 \ poly_gcd x y = poly_gcd y (x mod y)" ``` huffman@31663 ` 1125` ```by auto ``` huffman@31663 ` 1126` huffman@31663 ` 1127` ```termination poly_gcd ``` huffman@31663 ` 1128` ```by (relation "measure (\(x, y). if y = 0 then 0 else Suc (degree y))") ``` huffman@31663 ` 1129` ``` (auto dest: degree_mod_less) ``` huffman@31663 ` 1130` haftmann@37765 ` 1131` ```declare poly_gcd.simps [simp del] ``` huffman@31663 ` 1132` huffman@31663 ` 1133` ```lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x" ``` huffman@31663 ` 1134` ``` and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y" ``` huffman@31663 ` 1135` ``` apply (induct x y rule: poly_gcd.induct) ``` huffman@31663 ` 1136` ``` apply (simp_all add: poly_gcd.simps) ``` nipkow@44890 ` 1137` ``` apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero) ``` huffman@31663 ` 1138` ``` apply (blast dest: dvd_mod_imp_dvd) ``` huffman@31663 ` 1139` ``` done ``` huffman@31663 ` 1140` huffman@31663 ` 1141` ```lemma poly_gcd_greatest: "k dvd x \ k dvd y \ k dvd poly_gcd x y" ``` huffman@31663 ` 1142` ``` by (induct x y rule: poly_gcd.induct) ``` huffman@31663 ` 1143` ``` (simp_all add: poly_gcd.simps dvd_mod dvd_smult) ``` huffman@31663 ` 1144` huffman@31663 ` 1145` ```lemma dvd_poly_gcd_iff [iff]: ``` huffman@31663 ` 1146` ``` "k dvd poly_gcd x y \ k dvd x \ k dvd y" ``` huffman@31663 ` 1147` ``` by (blast intro!: poly_gcd_greatest intro: dvd_trans) ``` huffman@31663 ` 1148` huffman@31663 ` 1149` ```lemma poly_gcd_monic: ``` huffman@31663 ` 1150` ``` "coeff (poly_gcd x y) (degree (poly_gcd x y)) = ``` huffman@31663 ` 1151` ``` (if x = 0 \ y = 0 then 0 else 1)" ``` huffman@31663 ` 1152` ``` by (induct x y rule: poly_gcd.induct) ``` huffman@31663 ` 1153` ``` (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero) ``` huffman@31663 ` 1154` huffman@31663 ` 1155` ```lemma poly_gcd_zero_iff [simp]: ``` huffman@31663 ` 1156` ``` "poly_gcd x y = 0 \ x = 0 \ y = 0" ``` huffman@31663 ` 1157` ``` by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff) ``` huffman@31663 ` 1158` huffman@31663 ` 1159` ```lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0" ``` huffman@31663 ` 1160` ``` by simp ``` huffman@31663 ` 1161` huffman@31663 ` 1162` ```lemma poly_dvd_antisym: ``` huffman@31663 ` 1163` ``` fixes p q :: "'a::idom poly" ``` huffman@31663 ` 1164` ``` assumes coeff: "coeff p (degree p) = coeff q (degree q)" ``` huffman@31663 ` 1165` ``` assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q" ``` huffman@31663 ` 1166` ```proof (cases "p = 0") ``` huffman@31663 ` 1167` ``` case True with coeff show "p = q" by simp ``` huffman@31663 ` 1168` ```next ``` huffman@31663 ` 1169` ``` case False with coeff have "q \ 0" by auto ``` huffman@31663 ` 1170` ``` have degree: "degree p = degree q" ``` huffman@31663 ` 1171` ``` using `p dvd q` `q dvd p` `p \ 0` `q \ 0` ``` huffman@31663 ` 1172` ``` by (intro order_antisym dvd_imp_degree_le) ``` huffman@31663 ` 1173` huffman@31663 ` 1174` ``` from `p dvd q` obtain a where a: "q = p * a" .. ``` huffman@31663 ` 1175` ``` with `q \ 0` have "a \ 0" by auto ``` huffman@31663 ` 1176` ``` with degree a `p \ 0` have "degree a = 0" ``` huffman@31663 ` 1177` ``` by (simp add: degree_mult_eq) ``` huffman@31663 ` 1178` ``` with coeff a show "p = q" ``` huffman@31663 ` 1179` ``` by (cases a, auto split: if_splits) ``` huffman@31663 ` 1180` ```qed ``` huffman@31663 ` 1181` huffman@31663 ` 1182` ```lemma poly_gcd_unique: ``` huffman@31663 ` 1183` ``` assumes dvd1: "d dvd x" and dvd2: "d dvd y" ``` huffman@31663 ` 1184` ``` and greatest: "\k. k dvd x \ k dvd y \ k dvd d" ``` huffman@31663 ` 1185` ``` and monic: "coeff d (degree d) = (if x = 0 \ y = 0 then 0 else 1)" ``` huffman@31663 ` 1186` ``` shows "poly_gcd x y = d" ``` huffman@31663 ` 1187` ```proof - ``` huffman@31663 ` 1188` ``` have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)" ``` huffman@31663 ` 1189` ``` by (simp_all add: poly_gcd_monic monic) ``` huffman@31663 ` 1190` ``` moreover have "poly_gcd x y dvd d" ``` huffman@31663 ` 1191` ``` using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest) ``` huffman@31663 ` 1192` ``` moreover have "d dvd poly_gcd x y" ``` huffman@31663 ` 1193` ``` using dvd1 dvd2 by (rule poly_gcd_greatest) ``` huffman@31663 ` 1194` ``` ultimately show ?thesis ``` huffman@31663 ` 1195` ``` by (rule poly_dvd_antisym) ``` huffman@31663 ` 1196` ```qed ``` huffman@31663 ` 1197` haftmann@37770 ` 1198` ```interpretation poly_gcd: abel_semigroup poly_gcd ``` haftmann@34973 ` 1199` ```proof ``` haftmann@34973 ` 1200` ``` fix x y z :: "'a poly" ``` haftmann@34973 ` 1201` ``` show "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)" ``` haftmann@34973 ` 1202` ``` by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic) ``` haftmann@34973 ` 1203` ``` show "poly_gcd x y = poly_gcd y x" ``` haftmann@34973 ` 1204` ``` by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) ``` haftmann@34973 ` 1205` ```qed ``` huffman@31663 ` 1206` haftmann@34973 ` 1207` ```lemmas poly_gcd_assoc = poly_gcd.assoc ``` haftmann@34973 ` 1208` ```lemmas poly_gcd_commute = poly_gcd.commute ``` haftmann@34973 ` 1209` ```lemmas poly_gcd_left_commute = poly_gcd.left_commute ``` huffman@31663 ` 1210` huffman@31663 ` 1211` ```lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute ``` huffman@31663 ` 1212` huffman@31663 ` 1213` ```lemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1" ``` huffman@31663 ` 1214` ```by (rule poly_gcd_unique) simp_all ``` huffman@31663 ` 1215` huffman@31663 ` 1216` ```lemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1" ``` huffman@31663 ` 1217` ```by (rule poly_gcd_unique) simp_all ``` huffman@31663 ` 1218` huffman@31663 ` 1219` ```lemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y" ``` huffman@31663 ` 1220` ```by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) ``` huffman@31663 ` 1221` huffman@31663 ` 1222` ```lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y" ``` huffman@31663 ` 1223` ```by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic) ``` huffman@31663 ` 1224` huffman@31663 ` 1225` huffman@29451 ` 1226` ```subsection {* Evaluation of polynomials *} ``` huffman@29451 ` 1227` huffman@29451 ` 1228` ```definition ``` huffman@29454 ` 1229` ``` poly :: "'a::comm_semiring_0 poly \ 'a \ 'a" where ``` huffman@29454 ` 1230` ``` "poly = poly_rec (\x. 0) (\a p f x. a + x * f x)" ``` huffman@29451 ` 1231` huffman@29451 ` 1232` ```lemma poly_0 [simp]: "poly 0 x = 0" ``` huffman@29454 ` 1233` ``` unfolding poly_def by (simp add: poly_rec_0) ``` huffman@29451 ` 1234` huffman@29451 ` 1235` ```lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" ``` huffman@29454 ` 1236` ``` unfolding poly_def by (simp add: poly_rec_pCons) ``` huffman@29451 ` 1237` huffman@29451 ` 1238` ```lemma poly_1 [simp]: "poly 1 x = 1" ``` huffman@29451 ` 1239` ``` unfolding one_poly_def by simp ``` huffman@29451 ` 1240` huffman@29454 ` 1241` ```lemma poly_monom: ``` haftmann@31021 ` 1242` ``` fixes a x :: "'a::{comm_semiring_1}" ``` huffman@29454 ` 1243` ``` shows "poly (monom a n) x = a * x ^ n" ``` huffman@29451 ` 1244` ``` by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) ``` huffman@29451 ` 1245` huffman@29451 ` 1246` ```lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" ``` huffman@29451 ` 1247` ``` apply (induct p arbitrary: q, simp) ``` nipkow@29667 ` 1248` ``` apply (case_tac q, simp, simp add: algebra_simps) ``` huffman@29451 ` 1249` ``` done ``` huffman@29451 ` 1250` huffman@29451 ` 1251` ```lemma poly_minus [simp]: ``` huffman@29454 ` 1252` ``` fixes x :: "'a::comm_ring" ``` huffman@29451 ` 1253` ``` shows "poly (- p) x = - poly p x" ``` huffman@29451 ` 1254` ``` by (induct p, simp_all) ``` huffman@29451 ` 1255` huffman@29451 ` 1256` ```lemma poly_diff [simp]: ``` huffman@29454 ` 1257` ``` fixes x :: "'a::comm_ring" ``` huffman@29451 ` 1258` ``` shows "poly (p - q) x = poly p x - poly q x" ``` huffman@29451 ` 1259` ``` by (simp add: diff_minus) ``` huffman@29451 ` 1260` huffman@29451 ` 1261` ```lemma poly_setsum: "poly (\k\A. p k) x = (\k\A. poly (p k) x)" ``` huffman@29451 ` 1262` ``` by (cases "finite A", induct set: finite, simp_all) ``` huffman@29451 ` 1263` huffman@29451 ` 1264` ```lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" ``` nipkow@29667 ` 1265` ``` by (induct p, simp, simp add: algebra_simps) ``` huffman@29451 ` 1266` huffman@29451 ` 1267` ```lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" ``` nipkow@29667 ` 1268` ``` by (induct p, simp_all, simp add: algebra_simps) ``` huffman@29451 ` 1269` huffman@29462 ` 1270` ```lemma poly_power [simp]: ``` haftmann@31021 ` 1271` ``` fixes p :: "'a::{comm_semiring_1} poly" ``` huffman@29462 ` 1272` ``` shows "poly (p ^ n) x = poly p x ^ n" ``` huffman@29462 ` 1273` ``` by (induct n, simp, simp add: power_Suc) ``` huffman@29462 ` 1274` huffman@29456 ` 1275` huffman@29456 ` 1276` ```subsection {* Synthetic division *} ``` huffman@29456 ` 1277` huffman@29980 ` 1278` ```text {* ``` huffman@29980 ` 1279` ``` Synthetic division is simply division by the ``` huffman@29980 ` 1280` ``` linear polynomial @{term "x - c"}. ``` huffman@29980 ` 1281` ```*} ``` huffman@29980 ` 1282` huffman@29456 ` 1283` ```definition ``` huffman@29456 ` 1284` ``` synthetic_divmod :: "'a::comm_semiring_0 poly \ 'a \ 'a poly \ 'a" ``` haftmann@37765 ` 1285` ```where ``` huffman@29456 ` 1286` ``` "synthetic_divmod p c = ``` huffman@29456 ` 1287` ``` poly_rec (0, 0) (\a p (q, r). (pCons r q, a + c * r)) p" ``` huffman@29456 ` 1288` huffman@29456 ` 1289` ```definition ``` huffman@29456 ` 1290` ``` synthetic_div :: "'a::comm_semiring_0 poly \ 'a \ 'a poly" ``` huffman@29456 ` 1291` ```where ``` huffman@29456 ` 1292` ``` "synthetic_div p c = fst (synthetic_divmod p c)" ``` huffman@29456 ` 1293` huffman@29456 ` 1294` ```lemma synthetic_divmod_0 [simp]: ``` huffman@29456 ` 1295` ``` "synthetic_divmod 0 c = (0, 0)" ``` huffman@29456 ` 1296` ``` unfolding synthetic_divmod_def ``` huffman@29456 ` 1297` ``` by (simp add: poly_rec_0) ``` huffman@29456 ` 1298` huffman@29456 ` 1299` ```lemma synthetic_divmod_pCons [simp]: ``` huffman@29456 ` 1300` ``` "synthetic_divmod (pCons a p) c = ``` huffman@29456 ` 1301` ``` (\(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" ``` huffman@29456 ` 1302` ``` unfolding synthetic_divmod_def ``` huffman@29456 ` 1303` ``` by (simp add: poly_rec_pCons) ``` huffman@29456 ` 1304` huffman@29456 ` 1305` ```lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" ``` huffman@29456 ` 1306` ``` by (induct p, simp, simp add: split_def) ``` huffman@29456 ` 1307` huffman@29456 ` 1308` ```lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" ``` huffman@29456 ` 1309` ``` unfolding synthetic_div_def by simp ``` huffman@29456 ` 1310` huffman@29456 ` 1311` ```lemma synthetic_div_pCons [simp]: ``` huffman@29456 ` 1312` ``` "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" ``` huffman@29456 ` 1313` ``` unfolding synthetic_div_def ``` huffman@29456 ` 1314` ``` by (simp add: split_def snd_synthetic_divmod) ``` huffman@29456 ` 1315` huffman@29460 ` 1316` ```lemma synthetic_div_eq_0_iff: ``` huffman@29460 ` 1317` ``` "synthetic_div p c = 0 \ degree p = 0" ``` huffman@29460 ` 1318` ``` by (induct p, simp, case_tac p, simp) ``` huffman@29460 ` 1319` huffman@29460 ` 1320` ```lemma degree_synthetic_div: ``` huffman@29460 ` 1321` ``` "degree (synthetic_div p c) = degree p - 1" ``` huffman@29460 ` 1322` ``` by (induct p, simp, simp add: synthetic_div_eq_0_iff) ``` huffman@29460 ` 1323` huffman@29457 ` 1324` ```lemma synthetic_div_correct: ``` huffman@29456 ` 1325` ``` "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" ``` huffman@29456 ` 1326` ``` by (induct p) simp_all ``` huffman@29456 ` 1327` huffman@29457 ` 1328` ```lemma synthetic_div_unique_lemma: "smult c p = pCons a p \ p = 0" ``` huffman@29457 ` 1329` ```by (induct p arbitrary: a) simp_all ``` huffman@29457 ` 1330` huffman@29457 ` 1331` ```lemma synthetic_div_unique: ``` huffman@29457 ` 1332` ``` "p + smult c q = pCons r q \ r = poly p c \ q = synthetic_div p c" ``` huffman@29457 ` 1333` ```apply (induct p arbitrary: q r) ``` huffman@29457 ` 1334` ```apply (simp, frule synthetic_div_unique_lemma, simp) ``` huffman@29457 ` 1335` ```apply (case_tac q, force) ``` huffman@29457 ` 1336` ```done ``` huffman@29457 ` 1337` huffman@29457 ` 1338` ```lemma synthetic_div_correct': ``` huffman@29457 ` 1339` ``` fixes c :: "'a::comm_ring_1" ``` huffman@29457 ` 1340` ``` shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" ``` huffman@29457 ` 1341` ``` using synthetic_div_correct [of p c] ``` nipkow@29667 ` 1342` ``` by (simp add: algebra_simps) ``` huffman@29457 ` 1343` huffman@29460 ` 1344` ```lemma poly_eq_0_iff_dvd: ``` huffman@29460 ` 1345` ``` fixes c :: "'a::idom" ``` huffman@29460 ` 1346` ``` shows "poly p c = 0 \ [:-c, 1:] dvd p" ``` huffman@29460 ` 1347` ```proof ``` huffman@29460 ` 1348` ``` assume "poly p c = 0" ``` huffman@29460 ` 1349` ``` with synthetic_div_correct' [of c p] ``` huffman@29460 ` 1350` ``` have "p = [:-c, 1:] * synthetic_div p c" by simp ``` huffman@29460 ` 1351` ``` then show "[:-c, 1:] dvd p" .. ``` huffman@29460 ` 1352` ```next ``` huffman@29460 ` 1353` ``` assume "[:-c, 1:] dvd p" ``` huffman@29460 ` 1354` ``` then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) ``` huffman@29460 ` 1355` ``` then show "poly p c = 0" by simp ``` huffman@29460 ` 1356` ```qed ``` huffman@29460 ` 1357` huffman@29460 ` 1358` ```lemma dvd_iff_poly_eq_0: ``` huffman@29460 ` 1359` ``` fixes c :: "'a::idom" ``` huffman@29460 ` 1360` ``` shows "[:c, 1:] dvd p \ poly p (-c) = 0" ``` huffman@29460 ` 1361` ``` by (simp add: poly_eq_0_iff_dvd) ``` huffman@29460 ` 1362` huffman@29462 ` 1363` ```lemma poly_roots_finite: ``` huffman@29462 ` 1364` ``` fixes p :: "'a::idom poly" ``` huffman@29462 ` 1365` ``` shows "p \ 0 \ finite {x. poly p x = 0}" ``` huffman@29462 ` 1366` ```proof (induct n \ "degree p" arbitrary: p) ``` huffman@29462 ` 1367` ``` case (0 p) ``` huffman@29462 ` 1368` ``` then obtain a where "a \ 0" and "p = [:a:]" ``` huffman@29462 ` 1369` ``` by (cases p, simp split: if_splits) ``` huffman@29462 ` 1370` ``` then show "finite {x. poly p x = 0}" by simp ``` huffman@29462 ` 1371` ```next ``` huffman@29462 ` 1372` ``` case (Suc n p) ``` huffman@29462 ` 1373` ``` show "finite {x. poly p x = 0}" ``` huffman@29462 ` 1374` ``` proof (cases "\x. poly p x = 0") ``` huffman@29462 ` 1375` ``` case False ``` huffman@29462 ` 1376` ``` then show "finite {x. poly p x = 0}" by simp ``` huffman@29462 ` 1377` ``` next ``` huffman@29462 ` 1378` ``` case True ``` huffman@29462 ` 1379` ``` then obtain a where "poly p a = 0" .. ``` huffman@29462 ` 1380` ``` then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) ``` huffman@29462 ` 1381` ``` then obtain k where k: "p = [:-a, 1:] * k" .. ``` huffman@29462 ` 1382` ``` with `p \ 0` have "k \ 0" by auto ``` huffman@29462 ` 1383` ``` with k have "degree p = Suc (degree k)" ``` huffman@29462 ` 1384` ``` by (simp add: degree_mult_eq del: mult_pCons_left) ``` huffman@29462 ` 1385` ``` with `Suc n = degree p` have "n = degree k" by simp ``` berghofe@34915 ` 1386` ``` then have "finite {x. poly k x = 0}" using `k \ 0` by (rule Suc.hyps) ``` huffman@29462 ` 1387` ``` then have "finite (insert a {x. poly k x = 0})" by simp ``` huffman@29462 ` 1388` ``` then show "finite {x. poly p x = 0}" ``` huffman@29462 ` 1389` ``` by (simp add: k uminus_add_conv_diff Collect_disj_eq ``` huffman@29462 ` 1390` ``` del: mult_pCons_left) ``` huffman@29462 ` 1391` ``` qed ``` huffman@29462 ` 1392` ```qed ``` huffman@29462 ` 1393` huffman@29979 ` 1394` ```lemma poly_zero: ``` huffman@29979 ` 1395` ``` fixes p :: "'a::{idom,ring_char_0} poly" ``` huffman@29979 ` 1396` ``` shows "poly p = poly 0 \ p = 0" ``` huffman@29979 ` 1397` ```apply (cases "p = 0", simp_all) ``` huffman@29979 ` 1398` ```apply (drule poly_roots_finite) ``` huffman@29979 ` 1399` ```apply (auto simp add: infinite_UNIV_char_0) ``` huffman@29979 ` 1400` ```done ``` huffman@29979 ` 1401` huffman@29979 ` 1402` ```lemma poly_eq_iff: ``` huffman@29979 ` 1403` ``` fixes p q :: "'a::{idom,ring_char_0} poly" ``` huffman@29979 ` 1404` ``` shows "poly p = poly q \ p = q" ``` huffman@29979 ` 1405` ``` using poly_zero [of "p - q"] ``` nipkow@39302 ` 1406` ``` by (simp add: fun_eq_iff) ``` huffman@29979 ` 1407` huffman@29478 ` 1408` huffman@29980 ` 1409` ```subsection {* Composition of polynomials *} ``` huffman@29980 ` 1410` huffman@29980 ` 1411` ```definition ``` huffman@29980 ` 1412` ``` pcompose :: "'a::comm_semiring_0 poly \ 'a poly \ 'a poly" ``` huffman@29980 ` 1413` ```where ``` huffman@29980 ` 1414` ``` "pcompose p q = poly_rec 0 (\a _ c. [:a:] + q * c) p" ``` huffman@29980 ` 1415` huffman@29980 ` 1416` ```lemma pcompose_0 [simp]: "pcompose 0 q = 0" ``` huffman@29980 ` 1417` ``` unfolding pcompose_def by (simp add: poly_rec_0) ``` huffman@29980 ` 1418` huffman@29980 ` 1419` ```lemma pcompose_pCons: ``` huffman@29980 ` 1420` ``` "pcompose (pCons a p) q = [:a:] + q * pcompose p q" ``` huffman@29980 ` 1421` ``` unfolding pcompose_def by (simp add: poly_rec_pCons) ``` huffman@29980 ` 1422` huffman@29980 ` 1423` ```lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)" ``` huffman@29980 ` 1424` ``` by (induct p) (simp_all add: pcompose_pCons) ``` huffman@29980 ` 1425` huffman@29980 ` 1426` ```lemma degree_pcompose_le: ``` huffman@29980 ` 1427` ``` "degree (pcompose p q) \ degree p * degree q" ``` huffman@29980 ` 1428` ```apply (induct p, simp) ``` huffman@29980 ` 1429` ```apply (simp add: pcompose_pCons, clarify) ``` huffman@29980 ` 1430` ```apply (rule degree_add_le, simp) ``` huffman@29980 ` 1431` ```apply (rule order_trans [OF degree_mult_le], simp) ``` huffman@29980 ` 1432` ```done ``` huffman@29980 ` 1433` huffman@29980 ` 1434` huffman@29977 ` 1435` ```subsection {* Order of polynomial roots *} ``` huffman@29977 ` 1436` huffman@29977 ` 1437` ```definition ``` huffman@29979 ` 1438` ``` order :: "'a::idom \ 'a poly \ nat" ``` huffman@29977 ` 1439` ```where ``` huffman@29977 ` 1440` ``` "order a p = (LEAST n. \ [:-a, 1:] ^ Suc n dvd p)" ``` huffman@29977 ` 1441` huffman@29977 ` 1442` ```lemma coeff_linear_power: ``` huffman@29979 ` 1443` ``` fixes a :: "'a::comm_semiring_1" ``` huffman@29977 ` 1444` ``` shows "coeff ([:a, 1:] ^ n) n = 1" ``` huffman@29977 ` 1445` ```apply (induct n, simp_all) ``` huffman@29977 ` 1446` ```apply (subst coeff_eq_0) ``` huffman@29977 ` 1447` ```apply (auto intro: le_less_trans degree_power_le) ``` huffman@29977 ` 1448` ```done ``` huffman@29977 ` 1449` huffman@29977 ` 1450` ```lemma degree_linear_power: ``` huffman@29979 ` 1451` ``` fixes a :: "'a::comm_semiring_1" ``` huffman@29977 ` 1452` ``` shows "degree ([:a, 1:] ^ n) = n" ``` huffman@29977 ` 1453` ```apply (rule order_antisym) ``` huffman@29977 ` 1454` ```apply (rule ord_le_eq_trans [OF degree_power_le], simp) ``` huffman@29977 ` 1455` ```apply (rule le_degree, simp add: coeff_linear_power) ``` huffman@29977 ` 1456` ```done ``` huffman@29977 ` 1457` huffman@29977 ` 1458` ```lemma order_1: "[:-a, 1:] ^ order a p dvd p" ``` huffman@29977 ` 1459` ```apply (cases "p = 0", simp) ``` huffman@29977 ` 1460` ```apply (cases "order a p", simp) ``` huffman@29977 ` 1461` ```apply (subgoal_tac "nat < (LEAST n. \ [:-a, 1:] ^ Suc n dvd p)") ``` huffman@29977 ` 1462` ```apply (drule not_less_Least, simp) ``` huffman@29977 ` 1463` ```apply (fold order_def, simp) ``` huffman@29977 ` 1464` ```done ``` huffman@29977 ` 1465` huffman@29977 ` 1466` ```lemma order_2: "p \ 0 \ \ [:-a, 1:] ^ Suc (order a p) dvd p" ``` huffman@29977 ` 1467` ```unfolding order_def ``` huffman@29977 ` 1468` ```apply (rule LeastI_ex) ``` huffman@29977 ` 1469` ```apply (rule_tac x="degree p" in exI) ``` huffman@29977 ` 1470` ```apply (rule notI) ``` huffman@29977 ` 1471` ```apply (drule (1) dvd_imp_degree_le) ``` huffman@29977 ` 1472` ```apply (simp only: degree_linear_power) ``` huffman@29977 ` 1473` ```done ``` huffman@29977 ` 1474` huffman@29977 ` 1475` ```lemma order: ``` huffman@29977 ` 1476` ``` "p \ 0 \ [:-a, 1:] ^ order a p dvd p \ \ [:-a, 1:] ^ Suc (order a p) dvd p" ``` huffman@29977 ` 1477` ```by (rule conjI [OF order_1 order_2]) ``` huffman@29977 ` 1478` huffman@29977 ` 1479` ```lemma order_degree: ``` huffman@29977 ` 1480` ``` assumes p: "p \ 0" ``` huffman@29977 ` 1481` ``` shows "order a p \ degree p" ``` huffman@29977 ` 1482` ```proof - ``` huffman@29977 ` 1483` ``` have "order a p = degree ([:-a, 1:] ^ order a p)" ``` huffman@29977 ` 1484` ``` by (simp only: degree_linear_power) ``` huffman@29977 ` 1485` ``` also have "\ \ degree p" ``` huffman@29977 ` 1486` ``` using order_1 p by (rule dvd_imp_degree_le) ``` huffman@29977 ` 1487` ``` finally show ?thesis . ``` huffman@29977 ` 1488` ```qed ``` huffman@29977 ` 1489` huffman@29977 ` 1490` ```lemma order_root: "poly p a = 0 \ p = 0 \ order a p \ 0" ``` huffman@29977 ` 1491` ```apply (cases "p = 0", simp_all) ``` huffman@29977 ` 1492` ```apply (rule iffI) ``` huffman@29977 ` 1493` ```apply (rule ccontr, simp) ``` huffman@29977 ` 1494` ```apply (frule order_2 [where a=a], simp) ``` huffman@29977 ` 1495` ```apply (simp add: poly_eq_0_iff_dvd) ``` huffman@29977 ` 1496` ```apply (simp add: poly_eq_0_iff_dvd) ``` huffman@29977 ` 1497` ```apply (simp only: order_def) ``` huffman@29977 ` 1498` ```apply (drule not_less_Least, simp) ``` huffman@29977 ` 1499` ```done ``` huffman@29977 ` 1500` huffman@29977 ` 1501` huffman@29478 ` 1502` ```subsection {* Configuration of the code generator *} ``` huffman@29478 ` 1503` huffman@29478 ` 1504` ```code_datatype "0::'a::zero poly" pCons ``` huffman@29478 ` 1505` bulwahn@45928 ` 1506` ```quickcheck_generator poly constructors: "0::'a::zero poly", pCons ``` bulwahn@45928 ` 1507` haftmann@38857 ` 1508` ```instantiation poly :: ("{zero, equal}") equal ``` huffman@29478 ` 1509` ```begin ``` huffman@29478 ` 1510` haftmann@37765 ` 1511` ```definition ``` haftmann@38857 ` 1512` ``` "HOL.equal (p::'a poly) q \ p = q" ``` huffman@29478 ` 1513` haftmann@38857 ` 1514` ```instance proof ``` haftmann@38857 ` 1515` ```qed (rule equal_poly_def) ``` huffman@29478 ` 1516` huffman@29451 ` 1517` ```end ``` huffman@29478 ` 1518` huffman@29478 ` 1519` ```lemma eq_poly_code [code]: ``` haftmann@38857 ` 1520` ``` "HOL.equal (0::_ poly) (0::_ poly) \ True" ``` haftmann@38857 ` 1521` ``` "HOL.equal (0::_ poly) (pCons b q) \ HOL.equal 0 b \ HOL.equal 0 q" ``` haftmann@38857 ` 1522` ``` "HOL.equal (pCons a p) (0::_ poly) \ HOL.equal a 0 \ HOL.equal p 0" ``` haftmann@38857 ` 1523` ``` "HOL.equal (pCons a p) (pCons b q) \ HOL.equal a b \ HOL.equal p q" ``` haftmann@38857 ` 1524` ``` by (simp_all add: equal) ``` haftmann@38857 ` 1525` haftmann@38857 ` 1526` ```lemma [code nbe]: ``` haftmann@38857 ` 1527` ``` "HOL.equal (p :: _ poly) p \ True" ``` haftmann@38857 ` 1528` ``` by (fact equal_refl) ``` huffman@29478 ` 1529` huffman@29478 ` 1530` ```lemmas coeff_code [code] = ``` huffman@29478 ` 1531` ``` coeff_0 coeff_pCons_0 coeff_pCons_Suc ``` huffman@29478 ` 1532` huffman@29478 ` 1533` ```lemmas degree_code [code] = ``` huffman@29478 ` 1534` ``` degree_0 degree_pCons_eq_if ``` huffman@29478 ` 1535` huffman@29478 ` 1536` ```lemmas monom_poly_code [code] = ``` huffman@29478 ` 1537` ``` monom_0 monom_Suc ``` huffman@29478 ` 1538` huffman@29478 ` 1539` ```lemma add_poly_code [code]: ``` huffman@29478 ` 1540` ``` "0 + q = (q :: _ poly)" ``` huffman@29478 ` 1541` ``` "p + 0 = (p :: _ poly)" ``` huffman@29478 ` 1542` ``` "pCons a p + pCons b q = pCons (a + b) (p + q)" ``` huffman@29478 ` 1543` ```by simp_all ``` huffman@29478 ` 1544` huffman@29478 ` 1545` ```lemma minus_poly_code [code]: ``` huffman@29478 ` 1546` ``` "- 0 = (0 :: _ poly)" ``` huffman@29478 ` 1547` ``` "- pCons a p = pCons (- a) (- p)" ``` huffman@29478 ` 1548` ```by simp_all ``` huffman@29478 ` 1549` huffman@29478 ` 1550` ```lemma diff_poly_code [code]: ``` huffman@29478 ` 1551` ``` "0 - q = (- q :: _ poly)" ``` huffman@29478 ` 1552` ``` "p - 0 = (p :: _ poly)" ``` huffman@29478 ` 1553` ``` "pCons a p - pCons b q = pCons (a - b) (p - q)" ``` huffman@29478 ` 1554` ```by simp_all ``` huffman@29478 ` 1555` huffman@29478 ` 1556` ```lemmas smult_poly_code [code] = ``` huffman@29478 ` 1557` ``` smult_0_right smult_pCons ``` huffman@29478 ` 1558` huffman@29478 ` 1559` ```lemma mult_poly_code [code]: ``` huffman@29478 ` 1560` ``` "0 * q = (0 :: _ poly)" ``` huffman@29478 ` 1561` ``` "pCons a p * q = smult a q + pCons 0 (p * q)" ``` huffman@29478 ` 1562` ```by simp_all ``` huffman@29478 ` 1563` huffman@29478 ` 1564` ```lemmas poly_code [code] = ``` huffman@29478 ` 1565` ``` poly_0 poly_pCons ``` huffman@29478 ` 1566` huffman@29478 ` 1567` ```lemmas synthetic_divmod_code [code] = ``` huffman@29478 ` 1568` ``` synthetic_divmod_0 synthetic_divmod_pCons ``` huffman@29478 ` 1569` huffman@29478 ` 1570` ```text {* code generator setup for div and mod *} ``` huffman@29478 ` 1571` huffman@29478 ` 1572` ```definition ``` huffman@29537 ` 1573` ``` pdivmod :: "'a::field poly \ 'a poly \ 'a poly \ 'a poly" ``` huffman@29478 ` 1574` ```where ``` haftmann@37765 ` 1575` ``` "pdivmod x y = (x div y, x mod y)" ``` huffman@29478 ` 1576` huffman@29537 ` 1577` ```lemma div_poly_code [code]: "x div y = fst (pdivmod x y)" ``` huffman@29537 ` 1578` ``` unfolding pdivmod_def by simp ``` huffman@29478 ` 1579` huffman@29537 ` 1580` ```lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)" ``` huffman@29537 ` 1581` ``` unfolding pdivmod_def by simp ``` huffman@29478 ` 1582` huffman@29537 ` 1583` ```lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)" ``` huffman@29537 ` 1584` ``` unfolding pdivmod_def by simp ``` huffman@29478 ` 1585` huffman@29537 ` 1586` ```lemma pdivmod_pCons [code]: ``` huffman@29537 ` 1587` ``` "pdivmod (pCons a x) y = ``` huffman@29478 ` 1588` ``` (if y = 0 then (0, pCons a x) else ``` huffman@29537 ` 1589` ``` (let (q, r) = pdivmod x y; ``` huffman@29478 ` 1590` ``` b = coeff (pCons a r) (degree y) / coeff y (degree y) ``` huffman@29478 ` 1591` ``` in (pCons b q, pCons a r - smult b y)))" ``` huffman@29537 ` 1592` ```apply (simp add: pdivmod_def Let_def, safe) ``` huffman@29478 ` 1593` ```apply (rule div_poly_eq) ``` huffman@29537 ` 1594` ```apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) ``` huffman@29478 ` 1595` ```apply (rule mod_poly_eq) ``` huffman@29537 ` 1596` ```apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) ``` huffman@29478 ` 1597` ```done ``` huffman@29478 ` 1598` huffman@31663 ` 1599` ```lemma poly_gcd_code [code]: ``` huffman@31663 ` 1600` ``` "poly_gcd x y = ``` huffman@31663 ` 1601` ``` (if y = 0 then smult (inverse (coeff x (degree x))) x ``` huffman@31663 ` 1602` ``` else poly_gcd y (x mod y))" ``` huffman@31663 ` 1603` ``` by (simp add: poly_gcd.simps) ``` huffman@31663 ` 1604` huffman@29478 ` 1605` ```end ```