src/HOL/Polynomial.thy
 author huffman Thu Jan 15 09:17:15 2009 -0800 (2009-01-15) changeset 29537 50345a0f9df8 parent 29480 4e08ee896e81 child 29539 abfe2af6883e permissions -rw-r--r--
rename divmod_poly to pdivmod
```     1 (*  Title:      HOL/Polynomial.thy
```
```     2     Author:     Brian Huffman
```
```     3                 Based on an earlier development by Clemens Ballarin
```
```     4 *)
```
```     5
```
```     6 header {* Univariate Polynomials *}
```
```     7
```
```     8 theory Polynomial
```
```     9 imports Plain SetInterval
```
```    10 begin
```
```    11
```
```    12 subsection {* Definition of type @{text poly} *}
```
```    13
```
```    14 typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
```
```    15   morphisms coeff Abs_poly
```
```    16   by auto
```
```    17
```
```    18 lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
```
```    19 by (simp add: coeff_inject [symmetric] expand_fun_eq)
```
```    20
```
```    21 lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
```
```    22 by (simp add: expand_poly_eq)
```
```    23
```
```    24
```
```    25 subsection {* Degree of a polynomial *}
```
```    26
```
```    27 definition
```
```    28   degree :: "'a::zero poly \<Rightarrow> nat" where
```
```    29   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
```
```    30
```
```    31 lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
```
```    32 proof -
```
```    33   have "coeff p \<in> Poly"
```
```    34     by (rule coeff)
```
```    35   hence "\<exists>n. \<forall>i>n. coeff p i = 0"
```
```    36     unfolding Poly_def by simp
```
```    37   hence "\<forall>i>degree p. coeff p i = 0"
```
```    38     unfolding degree_def by (rule LeastI_ex)
```
```    39   moreover assume "degree p < n"
```
```    40   ultimately show ?thesis by simp
```
```    41 qed
```
```    42
```
```    43 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
```
```    44   by (erule contrapos_np, rule coeff_eq_0, simp)
```
```    45
```
```    46 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
```
```    47   unfolding degree_def by (erule Least_le)
```
```    48
```
```    49 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
```
```    50   unfolding degree_def by (drule not_less_Least, simp)
```
```    51
```
```    52
```
```    53 subsection {* The zero polynomial *}
```
```    54
```
```    55 instantiation poly :: (zero) zero
```
```    56 begin
```
```    57
```
```    58 definition
```
```    59   zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
```
```    60
```
```    61 instance ..
```
```    62 end
```
```    63
```
```    64 lemma coeff_0 [simp]: "coeff 0 n = 0"
```
```    65   unfolding zero_poly_def
```
```    66   by (simp add: Abs_poly_inverse Poly_def)
```
```    67
```
```    68 lemma degree_0 [simp]: "degree 0 = 0"
```
```    69   by (rule order_antisym [OF degree_le le0]) simp
```
```    70
```
```    71 lemma leading_coeff_neq_0:
```
```    72   assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
```
```    73 proof (cases "degree p")
```
```    74   case 0
```
```    75   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
```
```    76     by (simp add: expand_poly_eq)
```
```    77   then obtain n where "coeff p n \<noteq> 0" ..
```
```    78   hence "n \<le> degree p" by (rule le_degree)
```
```    79   with `coeff p n \<noteq> 0` and `degree p = 0`
```
```    80   show "coeff p (degree p) \<noteq> 0" by simp
```
```    81 next
```
```    82   case (Suc n)
```
```    83   from `degree p = Suc n` have "n < degree p" by simp
```
```    84   hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
```
```    85   then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
```
```    86   from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
```
```    87   also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
```
```    88   finally have "degree p = i" .
```
```    89   with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
```
```    90 qed
```
```    91
```
```    92 lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
```
```    93   by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
```
```    94
```
```    95
```
```    96 subsection {* List-style constructor for polynomials *}
```
```    97
```
```    98 definition
```
```    99   pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   100 where
```
```   101   [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
```
```   102
```
```   103 syntax
```
```   104   "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
```
```   105
```
```   106 translations
```
```   107   "[:x, xs:]" == "CONST pCons x [:xs:]"
```
```   108   "[:x:]" == "CONST pCons x 0"
```
```   109
```
```   110 lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
```
```   111   unfolding Poly_def by (auto split: nat.split)
```
```   112
```
```   113 lemma coeff_pCons:
```
```   114   "coeff (pCons a p) = nat_case a (coeff p)"
```
```   115   unfolding pCons_def
```
```   116   by (simp add: Abs_poly_inverse Poly_nat_case coeff)
```
```   117
```
```   118 lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
```
```   119   by (simp add: coeff_pCons)
```
```   120
```
```   121 lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
```
```   122   by (simp add: coeff_pCons)
```
```   123
```
```   124 lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
```
```   125 by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   126
```
```   127 lemma degree_pCons_eq:
```
```   128   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
```
```   129 apply (rule order_antisym [OF degree_pCons_le])
```
```   130 apply (rule le_degree, simp)
```
```   131 done
```
```   132
```
```   133 lemma degree_pCons_0: "degree (pCons a 0) = 0"
```
```   134 apply (rule order_antisym [OF _ le0])
```
```   135 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   136 done
```
```   137
```
```   138 lemma degree_pCons_eq_if [simp]:
```
```   139   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
```
```   140 apply (cases "p = 0", simp_all)
```
```   141 apply (rule order_antisym [OF _ le0])
```
```   142 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   143 apply (rule order_antisym [OF degree_pCons_le])
```
```   144 apply (rule le_degree, simp)
```
```   145 done
```
```   146
```
```   147 lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
```
```   148 by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   149
```
```   150 lemma pCons_eq_iff [simp]:
```
```   151   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
```
```   152 proof (safe)
```
```   153   assume "pCons a p = pCons b q"
```
```   154   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
```
```   155   then show "a = b" by simp
```
```   156 next
```
```   157   assume "pCons a p = pCons b q"
```
```   158   then have "\<forall>n. coeff (pCons a p) (Suc n) =
```
```   159                  coeff (pCons b q) (Suc n)" by simp
```
```   160   then show "p = q" by (simp add: expand_poly_eq)
```
```   161 qed
```
```   162
```
```   163 lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
```
```   164   using pCons_eq_iff [of a p 0 0] by simp
```
```   165
```
```   166 lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
```
```   167   unfolding Poly_def
```
```   168   by (clarify, rule_tac x=n in exI, simp)
```
```   169
```
```   170 lemma pCons_cases [cases type: poly]:
```
```   171   obtains (pCons) a q where "p = pCons a q"
```
```   172 proof
```
```   173   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
```
```   174     by (rule poly_ext)
```
```   175        (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
```
```   176              split: nat.split)
```
```   177 qed
```
```   178
```
```   179 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
```
```   180   assumes zero: "P 0"
```
```   181   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
```
```   182   shows "P p"
```
```   183 proof (induct p rule: measure_induct_rule [where f=degree])
```
```   184   case (less p)
```
```   185   obtain a q where "p = pCons a q" by (rule pCons_cases)
```
```   186   have "P q"
```
```   187   proof (cases "q = 0")
```
```   188     case True
```
```   189     then show "P q" by (simp add: zero)
```
```   190   next
```
```   191     case False
```
```   192     then have "degree (pCons a q) = Suc (degree q)"
```
```   193       by (rule degree_pCons_eq)
```
```   194     then have "degree q < degree p"
```
```   195       using `p = pCons a q` by simp
```
```   196     then show "P q"
```
```   197       by (rule less.hyps)
```
```   198   qed
```
```   199   then have "P (pCons a q)"
```
```   200     by (rule pCons)
```
```   201   then show ?case
```
```   202     using `p = pCons a q` by simp
```
```   203 qed
```
```   204
```
```   205
```
```   206 subsection {* Recursion combinator for polynomials *}
```
```   207
```
```   208 function
```
```   209   poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
```
```   210 where
```
```   211   poly_rec_pCons_eq_if [simp del, code del]:
```
```   212     "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
```
```   213 by (case_tac x, rename_tac q, case_tac q, auto)
```
```   214
```
```   215 termination poly_rec
```
```   216 by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
```
```   217    (simp add: degree_pCons_eq)
```
```   218
```
```   219 lemma poly_rec_0:
```
```   220   "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
```
```   221   using poly_rec_pCons_eq_if [of z f 0 0] by simp
```
```   222
```
```   223 lemma poly_rec_pCons:
```
```   224   "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
```
```   225   by (simp add: poly_rec_pCons_eq_if poly_rec_0)
```
```   226
```
```   227
```
```   228 subsection {* Monomials *}
```
```   229
```
```   230 definition
```
```   231   monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
```
```   232   "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
```
```   233
```
```   234 lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
```
```   235   unfolding monom_def
```
```   236   by (subst Abs_poly_inverse, auto simp add: Poly_def)
```
```   237
```
```   238 lemma monom_0: "monom a 0 = pCons a 0"
```
```   239   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   240
```
```   241 lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
```
```   242   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   243
```
```   244 lemma monom_eq_0 [simp]: "monom 0 n = 0"
```
```   245   by (rule poly_ext) simp
```
```   246
```
```   247 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
```
```   248   by (simp add: expand_poly_eq)
```
```   249
```
```   250 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
```
```   251   by (simp add: expand_poly_eq)
```
```   252
```
```   253 lemma degree_monom_le: "degree (monom a n) \<le> n"
```
```   254   by (rule degree_le, simp)
```
```   255
```
```   256 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
```
```   257   apply (rule order_antisym [OF degree_monom_le])
```
```   258   apply (rule le_degree, simp)
```
```   259   done
```
```   260
```
```   261
```
```   262 subsection {* Addition and subtraction *}
```
```   263
```
```   264 instantiation poly :: (comm_monoid_add) comm_monoid_add
```
```   265 begin
```
```   266
```
```   267 definition
```
```   268   plus_poly_def [code del]:
```
```   269     "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
```
```   270
```
```   271 lemma Poly_add:
```
```   272   fixes f g :: "nat \<Rightarrow> 'a"
```
```   273   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
```
```   274   unfolding Poly_def
```
```   275   apply (clarify, rename_tac m n)
```
```   276   apply (rule_tac x="max m n" in exI, simp)
```
```   277   done
```
```   278
```
```   279 lemma coeff_add [simp]:
```
```   280   "coeff (p + q) n = coeff p n + coeff q n"
```
```   281   unfolding plus_poly_def
```
```   282   by (simp add: Abs_poly_inverse coeff Poly_add)
```
```   283
```
```   284 instance proof
```
```   285   fix p q r :: "'a poly"
```
```   286   show "(p + q) + r = p + (q + r)"
```
```   287     by (simp add: expand_poly_eq add_assoc)
```
```   288   show "p + q = q + p"
```
```   289     by (simp add: expand_poly_eq add_commute)
```
```   290   show "0 + p = p"
```
```   291     by (simp add: expand_poly_eq)
```
```   292 qed
```
```   293
```
```   294 end
```
```   295
```
```   296 instantiation poly :: (ab_group_add) ab_group_add
```
```   297 begin
```
```   298
```
```   299 definition
```
```   300   uminus_poly_def [code del]:
```
```   301     "- p = Abs_poly (\<lambda>n. - coeff p n)"
```
```   302
```
```   303 definition
```
```   304   minus_poly_def [code del]:
```
```   305     "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
```
```   306
```
```   307 lemma Poly_minus:
```
```   308   fixes f :: "nat \<Rightarrow> 'a"
```
```   309   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
```
```   310   unfolding Poly_def by simp
```
```   311
```
```   312 lemma Poly_diff:
```
```   313   fixes f g :: "nat \<Rightarrow> 'a"
```
```   314   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
```
```   315   unfolding diff_minus by (simp add: Poly_add Poly_minus)
```
```   316
```
```   317 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
```
```   318   unfolding uminus_poly_def
```
```   319   by (simp add: Abs_poly_inverse coeff Poly_minus)
```
```   320
```
```   321 lemma coeff_diff [simp]:
```
```   322   "coeff (p - q) n = coeff p n - coeff q n"
```
```   323   unfolding minus_poly_def
```
```   324   by (simp add: Abs_poly_inverse coeff Poly_diff)
```
```   325
```
```   326 instance proof
```
```   327   fix p q :: "'a poly"
```
```   328   show "- p + p = 0"
```
```   329     by (simp add: expand_poly_eq)
```
```   330   show "p - q = p + - q"
```
```   331     by (simp add: expand_poly_eq diff_minus)
```
```   332 qed
```
```   333
```
```   334 end
```
```   335
```
```   336 lemma add_pCons [simp]:
```
```   337   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```   338   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   339
```
```   340 lemma minus_pCons [simp]:
```
```   341   "- pCons a p = pCons (- a) (- p)"
```
```   342   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   343
```
```   344 lemma diff_pCons [simp]:
```
```   345   "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```   346   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   347
```
```   348 lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
```
```   349   by (rule degree_le, auto simp add: coeff_eq_0)
```
```   350
```
```   351 lemma degree_add_less:
```
```   352   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
```
```   353   by (auto intro: le_less_trans degree_add_le)
```
```   354
```
```   355 lemma degree_add_eq_right:
```
```   356   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
```
```   357   apply (cases "q = 0", simp)
```
```   358   apply (rule order_antisym)
```
```   359   apply (rule ord_le_eq_trans [OF degree_add_le])
```
```   360   apply simp
```
```   361   apply (rule le_degree)
```
```   362   apply (simp add: coeff_eq_0)
```
```   363   done
```
```   364
```
```   365 lemma degree_add_eq_left:
```
```   366   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
```
```   367   using degree_add_eq_right [of q p]
```
```   368   by (simp add: add_commute)
```
```   369
```
```   370 lemma degree_minus [simp]: "degree (- p) = degree p"
```
```   371   unfolding degree_def by simp
```
```   372
```
```   373 lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
```
```   374   using degree_add_le [where p=p and q="-q"]
```
```   375   by (simp add: diff_minus)
```
```   376
```
```   377 lemma degree_diff_less:
```
```   378   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
```
```   379   by (auto intro: le_less_trans degree_diff_le)
```
```   380
```
```   381 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
```
```   382   by (rule poly_ext) simp
```
```   383
```
```   384 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
```
```   385   by (rule poly_ext) simp
```
```   386
```
```   387 lemma minus_monom: "- monom a n = monom (-a) n"
```
```   388   by (rule poly_ext) simp
```
```   389
```
```   390 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
```
```   391   by (cases "finite A", induct set: finite, simp_all)
```
```   392
```
```   393 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
```
```   394   by (rule poly_ext) (simp add: coeff_setsum)
```
```   395
```
```   396
```
```   397 subsection {* Multiplication by a constant *}
```
```   398
```
```   399 definition
```
```   400   smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   401   "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
```
```   402
```
```   403 lemma Poly_smult:
```
```   404   fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
```
```   405   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
```
```   406   unfolding Poly_def
```
```   407   by (clarify, rule_tac x=n in exI, simp)
```
```   408
```
```   409 lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
```
```   410   unfolding smult_def
```
```   411   by (simp add: Abs_poly_inverse Poly_smult coeff)
```
```   412
```
```   413 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
```
```   414   by (rule degree_le, simp add: coeff_eq_0)
```
```   415
```
```   416 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
```
```   417   by (rule poly_ext, simp add: mult_assoc)
```
```   418
```
```   419 lemma smult_0_right [simp]: "smult a 0 = 0"
```
```   420   by (rule poly_ext, simp)
```
```   421
```
```   422 lemma smult_0_left [simp]: "smult 0 p = 0"
```
```   423   by (rule poly_ext, simp)
```
```   424
```
```   425 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
```
```   426   by (rule poly_ext, simp)
```
```   427
```
```   428 lemma smult_add_right:
```
```   429   "smult a (p + q) = smult a p + smult a q"
```
```   430   by (rule poly_ext, simp add: ring_simps)
```
```   431
```
```   432 lemma smult_add_left:
```
```   433   "smult (a + b) p = smult a p + smult b p"
```
```   434   by (rule poly_ext, simp add: ring_simps)
```
```   435
```
```   436 lemma smult_minus_right [simp]:
```
```   437   "smult (a::'a::comm_ring) (- p) = - smult a p"
```
```   438   by (rule poly_ext, simp)
```
```   439
```
```   440 lemma smult_minus_left [simp]:
```
```   441   "smult (- a::'a::comm_ring) p = - smult a p"
```
```   442   by (rule poly_ext, simp)
```
```   443
```
```   444 lemma smult_diff_right:
```
```   445   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
```
```   446   by (rule poly_ext, simp add: ring_simps)
```
```   447
```
```   448 lemma smult_diff_left:
```
```   449   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
```
```   450   by (rule poly_ext, simp add: ring_simps)
```
```   451
```
```   452 lemmas smult_distribs =
```
```   453   smult_add_left smult_add_right
```
```   454   smult_diff_left smult_diff_right
```
```   455
```
```   456 lemma smult_pCons [simp]:
```
```   457   "smult a (pCons b p) = pCons (a * b) (smult a p)"
```
```   458   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   459
```
```   460 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
```
```   461   by (induct n, simp add: monom_0, simp add: monom_Suc)
```
```   462
```
```   463
```
```   464 subsection {* Multiplication of polynomials *}
```
```   465
```
```   466 text {* TODO: move to SetInterval.thy *}
```
```   467 lemma setsum_atMost_Suc_shift:
```
```   468   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
```
```   469   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   470 proof (induct n)
```
```   471   case 0 show ?case by simp
```
```   472 next
```
```   473   case (Suc n) note IH = this
```
```   474   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
```
```   475     by (rule setsum_atMost_Suc)
```
```   476   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   477     by (rule IH)
```
```   478   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
```
```   479              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
```
```   480     by (rule add_assoc)
```
```   481   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
```
```   482     by (rule setsum_atMost_Suc [symmetric])
```
```   483   finally show ?case .
```
```   484 qed
```
```   485
```
```   486 instantiation poly :: (comm_semiring_0) comm_semiring_0
```
```   487 begin
```
```   488
```
```   489 definition
```
```   490   times_poly_def [code del]:
```
```   491     "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
```
```   492
```
```   493 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
```
```   494   unfolding times_poly_def by (simp add: poly_rec_0)
```
```   495
```
```   496 lemma mult_pCons_left [simp]:
```
```   497   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```   498   unfolding times_poly_def by (simp add: poly_rec_pCons)
```
```   499
```
```   500 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
```
```   501   by (induct p, simp add: mult_poly_0_left, simp)
```
```   502
```
```   503 lemma mult_pCons_right [simp]:
```
```   504   "p * pCons a q = smult a p + pCons 0 (p * q)"
```
```   505   by (induct p, simp add: mult_poly_0_left, simp add: ring_simps)
```
```   506
```
```   507 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
```
```   508
```
```   509 lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
```
```   510   by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
```
```   511
```
```   512 lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
```
```   513   by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
```
```   514
```
```   515 lemma mult_poly_add_left:
```
```   516   fixes p q r :: "'a poly"
```
```   517   shows "(p + q) * r = p * r + q * r"
```
```   518   by (induct r, simp add: mult_poly_0,
```
```   519                 simp add: smult_distribs group_simps)
```
```   520
```
```   521 instance proof
```
```   522   fix p q r :: "'a poly"
```
```   523   show 0: "0 * p = 0"
```
```   524     by (rule mult_poly_0_left)
```
```   525   show "p * 0 = 0"
```
```   526     by (rule mult_poly_0_right)
```
```   527   show "(p + q) * r = p * r + q * r"
```
```   528     by (rule mult_poly_add_left)
```
```   529   show "(p * q) * r = p * (q * r)"
```
```   530     by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
```
```   531   show "p * q = q * p"
```
```   532     by (induct p, simp add: mult_poly_0, simp)
```
```   533 qed
```
```   534
```
```   535 end
```
```   536
```
```   537 lemma coeff_mult:
```
```   538   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   539 proof (induct p arbitrary: n)
```
```   540   case 0 show ?case by simp
```
```   541 next
```
```   542   case (pCons a p n) thus ?case
```
```   543     by (cases n, simp, simp add: setsum_atMost_Suc_shift
```
```   544                             del: setsum_atMost_Suc)
```
```   545 qed
```
```   546
```
```   547 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
```
```   548 apply (rule degree_le)
```
```   549 apply (induct p)
```
```   550 apply simp
```
```   551 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   552 done
```
```   553
```
```   554 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
```
```   555   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
```
```   556
```
```   557
```
```   558 subsection {* The unit polynomial and exponentiation *}
```
```   559
```
```   560 instantiation poly :: (comm_semiring_1) comm_semiring_1
```
```   561 begin
```
```   562
```
```   563 definition
```
```   564   one_poly_def:
```
```   565     "1 = pCons 1 0"
```
```   566
```
```   567 instance proof
```
```   568   fix p :: "'a poly" show "1 * p = p"
```
```   569     unfolding one_poly_def
```
```   570     by simp
```
```   571 next
```
```   572   show "0 \<noteq> (1::'a poly)"
```
```   573     unfolding one_poly_def by simp
```
```   574 qed
```
```   575
```
```   576 end
```
```   577
```
```   578 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
```
```   579   unfolding one_poly_def
```
```   580   by (simp add: coeff_pCons split: nat.split)
```
```   581
```
```   582 lemma degree_1 [simp]: "degree 1 = 0"
```
```   583   unfolding one_poly_def
```
```   584   by (rule degree_pCons_0)
```
```   585
```
```   586 instantiation poly :: (comm_semiring_1) recpower
```
```   587 begin
```
```   588
```
```   589 primrec power_poly where
```
```   590   power_poly_0: "(p::'a poly) ^ 0 = 1"
```
```   591 | power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
```
```   592
```
```   593 instance
```
```   594   by default simp_all
```
```   595
```
```   596 end
```
```   597
```
```   598 instance poly :: (comm_ring) comm_ring ..
```
```   599
```
```   600 instance poly :: (comm_ring_1) comm_ring_1 ..
```
```   601
```
```   602 instantiation poly :: (comm_ring_1) number_ring
```
```   603 begin
```
```   604
```
```   605 definition
```
```   606   "number_of k = (of_int k :: 'a poly)"
```
```   607
```
```   608 instance
```
```   609   by default (rule number_of_poly_def)
```
```   610
```
```   611 end
```
```   612
```
```   613
```
```   614 subsection {* Polynomials form an integral domain *}
```
```   615
```
```   616 lemma coeff_mult_degree_sum:
```
```   617   "coeff (p * q) (degree p + degree q) =
```
```   618    coeff p (degree p) * coeff q (degree q)"
```
```   619   by (induct p, simp, simp add: coeff_eq_0)
```
```   620
```
```   621 instance poly :: (idom) idom
```
```   622 proof
```
```   623   fix p q :: "'a poly"
```
```   624   assume "p \<noteq> 0" and "q \<noteq> 0"
```
```   625   have "coeff (p * q) (degree p + degree q) =
```
```   626         coeff p (degree p) * coeff q (degree q)"
```
```   627     by (rule coeff_mult_degree_sum)
```
```   628   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
```
```   629     using `p \<noteq> 0` and `q \<noteq> 0` by simp
```
```   630   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
```
```   631   thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
```
```   632 qed
```
```   633
```
```   634 lemma degree_mult_eq:
```
```   635   fixes p q :: "'a::idom poly"
```
```   636   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
```
```   637 apply (rule order_antisym [OF degree_mult_le le_degree])
```
```   638 apply (simp add: coeff_mult_degree_sum)
```
```   639 done
```
```   640
```
```   641 lemma dvd_imp_degree_le:
```
```   642   fixes p q :: "'a::idom poly"
```
```   643   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
```
```   644   by (erule dvdE, simp add: degree_mult_eq)
```
```   645
```
```   646
```
```   647 subsection {* Long division of polynomials *}
```
```   648
```
```   649 definition
```
```   650   pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
```
```   651 where
```
```   652   [code del]:
```
```   653   "pdivmod_rel x y q r \<longleftrightarrow>
```
```   654     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
```
```   655
```
```   656 lemma pdivmod_rel_0:
```
```   657   "pdivmod_rel 0 y 0 0"
```
```   658   unfolding pdivmod_rel_def by simp
```
```   659
```
```   660 lemma pdivmod_rel_by_0:
```
```   661   "pdivmod_rel x 0 0 x"
```
```   662   unfolding pdivmod_rel_def by simp
```
```   663
```
```   664 lemma eq_zero_or_degree_less:
```
```   665   assumes "degree p \<le> n" and "coeff p n = 0"
```
```   666   shows "p = 0 \<or> degree p < n"
```
```   667 proof (cases n)
```
```   668   case 0
```
```   669   with `degree p \<le> n` and `coeff p n = 0`
```
```   670   have "coeff p (degree p) = 0" by simp
```
```   671   then have "p = 0" by simp
```
```   672   then show ?thesis ..
```
```   673 next
```
```   674   case (Suc m)
```
```   675   have "\<forall>i>n. coeff p i = 0"
```
```   676     using `degree p \<le> n` by (simp add: coeff_eq_0)
```
```   677   then have "\<forall>i\<ge>n. coeff p i = 0"
```
```   678     using `coeff p n = 0` by (simp add: le_less)
```
```   679   then have "\<forall>i>m. coeff p i = 0"
```
```   680     using `n = Suc m` by (simp add: less_eq_Suc_le)
```
```   681   then have "degree p \<le> m"
```
```   682     by (rule degree_le)
```
```   683   then have "degree p < n"
```
```   684     using `n = Suc m` by (simp add: less_Suc_eq_le)
```
```   685   then show ?thesis ..
```
```   686 qed
```
```   687
```
```   688 lemma pdivmod_rel_pCons:
```
```   689   assumes rel: "pdivmod_rel x y q r"
```
```   690   assumes y: "y \<noteq> 0"
```
```   691   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
```
```   692   shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
```
```   693     (is "pdivmod_rel ?x y ?q ?r")
```
```   694 proof -
```
```   695   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
```
```   696     using assms unfolding pdivmod_rel_def by simp_all
```
```   697
```
```   698   have 1: "?x = ?q * y + ?r"
```
```   699     using b x by simp
```
```   700
```
```   701   have 2: "?r = 0 \<or> degree ?r < degree y"
```
```   702   proof (rule eq_zero_or_degree_less)
```
```   703     have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
```
```   704       by (rule degree_diff_le)
```
```   705     also have "\<dots> \<le> degree y"
```
```   706     proof (rule min_max.le_supI)
```
```   707       show "degree (pCons a r) \<le> degree y"
```
```   708         using r by auto
```
```   709       show "degree (smult b y) \<le> degree y"
```
```   710         by (rule degree_smult_le)
```
```   711     qed
```
```   712     finally show "degree ?r \<le> degree y" .
```
```   713   next
```
```   714     show "coeff ?r (degree y) = 0"
```
```   715       using `y \<noteq> 0` unfolding b by simp
```
```   716   qed
```
```   717
```
```   718   from 1 2 show ?thesis
```
```   719     unfolding pdivmod_rel_def
```
```   720     using `y \<noteq> 0` by simp
```
```   721 qed
```
```   722
```
```   723 lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
```
```   724 apply (cases "y = 0")
```
```   725 apply (fast intro!: pdivmod_rel_by_0)
```
```   726 apply (induct x)
```
```   727 apply (fast intro!: pdivmod_rel_0)
```
```   728 apply (fast intro!: pdivmod_rel_pCons)
```
```   729 done
```
```   730
```
```   731 lemma pdivmod_rel_unique:
```
```   732   assumes 1: "pdivmod_rel x y q1 r1"
```
```   733   assumes 2: "pdivmod_rel x y q2 r2"
```
```   734   shows "q1 = q2 \<and> r1 = r2"
```
```   735 proof (cases "y = 0")
```
```   736   assume "y = 0" with assms show ?thesis
```
```   737     by (simp add: pdivmod_rel_def)
```
```   738 next
```
```   739   assume [simp]: "y \<noteq> 0"
```
```   740   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
```
```   741     unfolding pdivmod_rel_def by simp_all
```
```   742   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
```
```   743     unfolding pdivmod_rel_def by simp_all
```
```   744   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
```
```   745     by (simp add: ring_simps)
```
```   746   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
```
```   747     by (auto intro: degree_diff_less)
```
```   748
```
```   749   show "q1 = q2 \<and> r1 = r2"
```
```   750   proof (rule ccontr)
```
```   751     assume "\<not> (q1 = q2 \<and> r1 = r2)"
```
```   752     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
```
```   753     with r3 have "degree (r2 - r1) < degree y" by simp
```
```   754     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
```
```   755     also have "\<dots> = degree ((q1 - q2) * y)"
```
```   756       using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
```
```   757     also have "\<dots> = degree (r2 - r1)"
```
```   758       using q3 by simp
```
```   759     finally have "degree (r2 - r1) < degree (r2 - r1)" .
```
```   760     then show "False" by simp
```
```   761   qed
```
```   762 qed
```
```   763
```
```   764 lemmas pdivmod_rel_unique_div =
```
```   765   pdivmod_rel_unique [THEN conjunct1, standard]
```
```   766
```
```   767 lemmas pdivmod_rel_unique_mod =
```
```   768   pdivmod_rel_unique [THEN conjunct2, standard]
```
```   769
```
```   770 instantiation poly :: (field) ring_div
```
```   771 begin
```
```   772
```
```   773 definition div_poly where
```
```   774   [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
```
```   775
```
```   776 definition mod_poly where
```
```   777   [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
```
```   778
```
```   779 lemma div_poly_eq:
```
```   780   "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
```
```   781 unfolding div_poly_def
```
```   782 by (fast elim: pdivmod_rel_unique_div)
```
```   783
```
```   784 lemma mod_poly_eq:
```
```   785   "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
```
```   786 unfolding mod_poly_def
```
```   787 by (fast elim: pdivmod_rel_unique_mod)
```
```   788
```
```   789 lemma pdivmod_rel:
```
```   790   "pdivmod_rel x y (x div y) (x mod y)"
```
```   791 proof -
```
```   792   from pdivmod_rel_exists
```
```   793     obtain q r where "pdivmod_rel x y q r" by fast
```
```   794   thus ?thesis
```
```   795     by (simp add: div_poly_eq mod_poly_eq)
```
```   796 qed
```
```   797
```
```   798 instance proof
```
```   799   fix x y :: "'a poly"
```
```   800   show "x div y * y + x mod y = x"
```
```   801     using pdivmod_rel [of x y]
```
```   802     by (simp add: pdivmod_rel_def)
```
```   803 next
```
```   804   fix x :: "'a poly"
```
```   805   have "pdivmod_rel x 0 0 x"
```
```   806     by (rule pdivmod_rel_by_0)
```
```   807   thus "x div 0 = 0"
```
```   808     by (rule div_poly_eq)
```
```   809 next
```
```   810   fix y :: "'a poly"
```
```   811   have "pdivmod_rel 0 y 0 0"
```
```   812     by (rule pdivmod_rel_0)
```
```   813   thus "0 div y = 0"
```
```   814     by (rule div_poly_eq)
```
```   815 next
```
```   816   fix x y z :: "'a poly"
```
```   817   assume "y \<noteq> 0"
```
```   818   hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
```
```   819     using pdivmod_rel [of x y]
```
```   820     by (simp add: pdivmod_rel_def left_distrib)
```
```   821   thus "(x + z * y) div y = z + x div y"
```
```   822     by (rule div_poly_eq)
```
```   823 qed
```
```   824
```
```   825 end
```
```   826
```
```   827 lemma degree_mod_less:
```
```   828   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
```
```   829   using pdivmod_rel [of x y]
```
```   830   unfolding pdivmod_rel_def by simp
```
```   831
```
```   832 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
```
```   833 proof -
```
```   834   assume "degree x < degree y"
```
```   835   hence "pdivmod_rel x y 0 x"
```
```   836     by (simp add: pdivmod_rel_def)
```
```   837   thus "x div y = 0" by (rule div_poly_eq)
```
```   838 qed
```
```   839
```
```   840 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
```
```   841 proof -
```
```   842   assume "degree x < degree y"
```
```   843   hence "pdivmod_rel x y 0 x"
```
```   844     by (simp add: pdivmod_rel_def)
```
```   845   thus "x mod y = x" by (rule mod_poly_eq)
```
```   846 qed
```
```   847
```
```   848 lemma mod_pCons:
```
```   849   fixes a and x
```
```   850   assumes y: "y \<noteq> 0"
```
```   851   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
```
```   852   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
```
```   853 unfolding b
```
```   854 apply (rule mod_poly_eq)
```
```   855 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
```
```   856 done
```
```   857
```
```   858
```
```   859 subsection {* Evaluation of polynomials *}
```
```   860
```
```   861 definition
```
```   862   poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   863   "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
```
```   864
```
```   865 lemma poly_0 [simp]: "poly 0 x = 0"
```
```   866   unfolding poly_def by (simp add: poly_rec_0)
```
```   867
```
```   868 lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
```
```   869   unfolding poly_def by (simp add: poly_rec_pCons)
```
```   870
```
```   871 lemma poly_1 [simp]: "poly 1 x = 1"
```
```   872   unfolding one_poly_def by simp
```
```   873
```
```   874 lemma poly_monom:
```
```   875   fixes a x :: "'a::{comm_semiring_1,recpower}"
```
```   876   shows "poly (monom a n) x = a * x ^ n"
```
```   877   by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
```
```   878
```
```   879 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
```
```   880   apply (induct p arbitrary: q, simp)
```
```   881   apply (case_tac q, simp, simp add: ring_simps)
```
```   882   done
```
```   883
```
```   884 lemma poly_minus [simp]:
```
```   885   fixes x :: "'a::comm_ring"
```
```   886   shows "poly (- p) x = - poly p x"
```
```   887   by (induct p, simp_all)
```
```   888
```
```   889 lemma poly_diff [simp]:
```
```   890   fixes x :: "'a::comm_ring"
```
```   891   shows "poly (p - q) x = poly p x - poly q x"
```
```   892   by (simp add: diff_minus)
```
```   893
```
```   894 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
```
```   895   by (cases "finite A", induct set: finite, simp_all)
```
```   896
```
```   897 lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
```
```   898   by (induct p, simp, simp add: ring_simps)
```
```   899
```
```   900 lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
```
```   901   by (induct p, simp_all, simp add: ring_simps)
```
```   902
```
```   903 lemma poly_power [simp]:
```
```   904   fixes p :: "'a::{comm_semiring_1,recpower} poly"
```
```   905   shows "poly (p ^ n) x = poly p x ^ n"
```
```   906   by (induct n, simp, simp add: power_Suc)
```
```   907
```
```   908
```
```   909 subsection {* Synthetic division *}
```
```   910
```
```   911 definition
```
```   912   synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
```
```   913 where [code del]:
```
```   914   "synthetic_divmod p c =
```
```   915     poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
```
```   916
```
```   917 definition
```
```   918   synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```   919 where
```
```   920   "synthetic_div p c = fst (synthetic_divmod p c)"
```
```   921
```
```   922 lemma synthetic_divmod_0 [simp]:
```
```   923   "synthetic_divmod 0 c = (0, 0)"
```
```   924   unfolding synthetic_divmod_def
```
```   925   by (simp add: poly_rec_0)
```
```   926
```
```   927 lemma synthetic_divmod_pCons [simp]:
```
```   928   "synthetic_divmod (pCons a p) c =
```
```   929     (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
```
```   930   unfolding synthetic_divmod_def
```
```   931   by (simp add: poly_rec_pCons)
```
```   932
```
```   933 lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
```
```   934   by (induct p, simp, simp add: split_def)
```
```   935
```
```   936 lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
```
```   937   unfolding synthetic_div_def by simp
```
```   938
```
```   939 lemma synthetic_div_pCons [simp]:
```
```   940   "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
```
```   941   unfolding synthetic_div_def
```
```   942   by (simp add: split_def snd_synthetic_divmod)
```
```   943
```
```   944 lemma synthetic_div_eq_0_iff:
```
```   945   "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
```
```   946   by (induct p, simp, case_tac p, simp)
```
```   947
```
```   948 lemma degree_synthetic_div:
```
```   949   "degree (synthetic_div p c) = degree p - 1"
```
```   950   by (induct p, simp, simp add: synthetic_div_eq_0_iff)
```
```   951
```
```   952 lemma synthetic_div_correct:
```
```   953   "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
```
```   954   by (induct p) simp_all
```
```   955
```
```   956 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
```
```   957 by (induct p arbitrary: a) simp_all
```
```   958
```
```   959 lemma synthetic_div_unique:
```
```   960   "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
```
```   961 apply (induct p arbitrary: q r)
```
```   962 apply (simp, frule synthetic_div_unique_lemma, simp)
```
```   963 apply (case_tac q, force)
```
```   964 done
```
```   965
```
```   966 lemma synthetic_div_correct':
```
```   967   fixes c :: "'a::comm_ring_1"
```
```   968   shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
```
```   969   using synthetic_div_correct [of p c]
```
```   970   by (simp add: group_simps)
```
```   971
```
```   972 lemma poly_eq_0_iff_dvd:
```
```   973   fixes c :: "'a::idom"
```
```   974   shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
```
```   975 proof
```
```   976   assume "poly p c = 0"
```
```   977   with synthetic_div_correct' [of c p]
```
```   978   have "p = [:-c, 1:] * synthetic_div p c" by simp
```
```   979   then show "[:-c, 1:] dvd p" ..
```
```   980 next
```
```   981   assume "[:-c, 1:] dvd p"
```
```   982   then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
```
```   983   then show "poly p c = 0" by simp
```
```   984 qed
```
```   985
```
```   986 lemma dvd_iff_poly_eq_0:
```
```   987   fixes c :: "'a::idom"
```
```   988   shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
```
```   989   by (simp add: poly_eq_0_iff_dvd)
```
```   990
```
```   991 lemma poly_roots_finite:
```
```   992   fixes p :: "'a::idom poly"
```
```   993   shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
```
```   994 proof (induct n \<equiv> "degree p" arbitrary: p)
```
```   995   case (0 p)
```
```   996   then obtain a where "a \<noteq> 0" and "p = [:a:]"
```
```   997     by (cases p, simp split: if_splits)
```
```   998   then show "finite {x. poly p x = 0}" by simp
```
```   999 next
```
```  1000   case (Suc n p)
```
```  1001   show "finite {x. poly p x = 0}"
```
```  1002   proof (cases "\<exists>x. poly p x = 0")
```
```  1003     case False
```
```  1004     then show "finite {x. poly p x = 0}" by simp
```
```  1005   next
```
```  1006     case True
```
```  1007     then obtain a where "poly p a = 0" ..
```
```  1008     then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
```
```  1009     then obtain k where k: "p = [:-a, 1:] * k" ..
```
```  1010     with `p \<noteq> 0` have "k \<noteq> 0" by auto
```
```  1011     with k have "degree p = Suc (degree k)"
```
```  1012       by (simp add: degree_mult_eq del: mult_pCons_left)
```
```  1013     with `Suc n = degree p` have "n = degree k" by simp
```
```  1014     with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps)
```
```  1015     then have "finite (insert a {x. poly k x = 0})" by simp
```
```  1016     then show "finite {x. poly p x = 0}"
```
```  1017       by (simp add: k uminus_add_conv_diff Collect_disj_eq
```
```  1018                del: mult_pCons_left)
```
```  1019   qed
```
```  1020 qed
```
```  1021
```
```  1022
```
```  1023 subsection {* Configuration of the code generator *}
```
```  1024
```
```  1025 code_datatype "0::'a::zero poly" pCons
```
```  1026
```
```  1027 declare pCons_0_0 [code post]
```
```  1028
```
```  1029 instantiation poly :: ("{zero,eq}") eq
```
```  1030 begin
```
```  1031
```
```  1032 definition [code del]:
```
```  1033   "eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q"
```
```  1034
```
```  1035 instance
```
```  1036   by default (rule eq_poly_def)
```
```  1037
```
```  1038 end
```
```  1039
```
```  1040 lemma eq_poly_code [code]:
```
```  1041   "eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
```
```  1042   "eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q"
```
```  1043   "eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0"
```
```  1044   "eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q"
```
```  1045 unfolding eq by simp_all
```
```  1046
```
```  1047 lemmas coeff_code [code] =
```
```  1048   coeff_0 coeff_pCons_0 coeff_pCons_Suc
```
```  1049
```
```  1050 lemmas degree_code [code] =
```
```  1051   degree_0 degree_pCons_eq_if
```
```  1052
```
```  1053 lemmas monom_poly_code [code] =
```
```  1054   monom_0 monom_Suc
```
```  1055
```
```  1056 lemma add_poly_code [code]:
```
```  1057   "0 + q = (q :: _ poly)"
```
```  1058   "p + 0 = (p :: _ poly)"
```
```  1059   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```  1060 by simp_all
```
```  1061
```
```  1062 lemma minus_poly_code [code]:
```
```  1063   "- 0 = (0 :: _ poly)"
```
```  1064   "- pCons a p = pCons (- a) (- p)"
```
```  1065 by simp_all
```
```  1066
```
```  1067 lemma diff_poly_code [code]:
```
```  1068   "0 - q = (- q :: _ poly)"
```
```  1069   "p - 0 = (p :: _ poly)"
```
```  1070   "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```  1071 by simp_all
```
```  1072
```
```  1073 lemmas smult_poly_code [code] =
```
```  1074   smult_0_right smult_pCons
```
```  1075
```
```  1076 lemma mult_poly_code [code]:
```
```  1077   "0 * q = (0 :: _ poly)"
```
```  1078   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```  1079 by simp_all
```
```  1080
```
```  1081 lemmas poly_code [code] =
```
```  1082   poly_0 poly_pCons
```
```  1083
```
```  1084 lemmas synthetic_divmod_code [code] =
```
```  1085   synthetic_divmod_0 synthetic_divmod_pCons
```
```  1086
```
```  1087 text {* code generator setup for div and mod *}
```
```  1088
```
```  1089 definition
```
```  1090   pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
```
```  1091 where
```
```  1092   [code del]: "pdivmod x y = (x div y, x mod y)"
```
```  1093
```
```  1094 lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
```
```  1095   unfolding pdivmod_def by simp
```
```  1096
```
```  1097 lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
```
```  1098   unfolding pdivmod_def by simp
```
```  1099
```
```  1100 lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
```
```  1101   unfolding pdivmod_def by simp
```
```  1102
```
```  1103 lemma pdivmod_pCons [code]:
```
```  1104   "pdivmod (pCons a x) y =
```
```  1105     (if y = 0 then (0, pCons a x) else
```
```  1106       (let (q, r) = pdivmod x y;
```
```  1107            b = coeff (pCons a r) (degree y) / coeff y (degree y)
```
```  1108         in (pCons b q, pCons a r - smult b y)))"
```
```  1109 apply (simp add: pdivmod_def Let_def, safe)
```
```  1110 apply (rule div_poly_eq)
```
```  1111 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1112 apply (rule mod_poly_eq)
```
```  1113 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1114 done
```
```  1115
```
```  1116 end
```