src/HOL/Library/Polynomial.thy
 author haftmann Sun Mar 18 08:57:45 2012 +0100 (2012-03-18) changeset 47002 9435d419109a parent 46031 ac6bae9fdc2f child 47108 2a1953f0d20d permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Polynomial.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Clemens Ballarin
```
```     4 *)
```
```     5
```
```     6 header {* Univariate Polynomials *}
```
```     7
```
```     8 theory Polynomial
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 subsection {* Definition of type @{text poly} *}
```
```    13
```
```    14 definition "Poly = {f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
```
```    15
```
```    16 typedef (open) 'a poly = "Poly :: (nat => 'a::zero) set"
```
```    17   morphisms coeff Abs_poly
```
```    18   unfolding Poly_def by auto
```
```    19
```
```    20 (* FIXME should be named poly_eq_iff *)
```
```    21 lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
```
```    22   by (simp add: coeff_inject [symmetric] fun_eq_iff)
```
```    23
```
```    24 (* FIXME should be named poly_eqI *)
```
```    25 lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
```
```    26   by (simp add: expand_poly_eq)
```
```    27
```
```    28
```
```    29 subsection {* Degree of a polynomial *}
```
```    30
```
```    31 definition
```
```    32   degree :: "'a::zero poly \<Rightarrow> nat" where
```
```    33   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
```
```    34
```
```    35 lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
```
```    36 proof -
```
```    37   have "coeff p \<in> Poly"
```
```    38     by (rule coeff)
```
```    39   hence "\<exists>n. \<forall>i>n. coeff p i = 0"
```
```    40     unfolding Poly_def by simp
```
```    41   hence "\<forall>i>degree p. coeff p i = 0"
```
```    42     unfolding degree_def by (rule LeastI_ex)
```
```    43   moreover assume "degree p < n"
```
```    44   ultimately show ?thesis by simp
```
```    45 qed
```
```    46
```
```    47 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
```
```    48   by (erule contrapos_np, rule coeff_eq_0, simp)
```
```    49
```
```    50 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
```
```    51   unfolding degree_def by (erule Least_le)
```
```    52
```
```    53 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
```
```    54   unfolding degree_def by (drule not_less_Least, simp)
```
```    55
```
```    56
```
```    57 subsection {* The zero polynomial *}
```
```    58
```
```    59 instantiation poly :: (zero) zero
```
```    60 begin
```
```    61
```
```    62 definition
```
```    63   zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
```
```    64
```
```    65 instance ..
```
```    66 end
```
```    67
```
```    68 lemma coeff_0 [simp]: "coeff 0 n = 0"
```
```    69   unfolding zero_poly_def
```
```    70   by (simp add: Abs_poly_inverse Poly_def)
```
```    71
```
```    72 lemma degree_0 [simp]: "degree 0 = 0"
```
```    73   by (rule order_antisym [OF degree_le le0]) simp
```
```    74
```
```    75 lemma leading_coeff_neq_0:
```
```    76   assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
```
```    77 proof (cases "degree p")
```
```    78   case 0
```
```    79   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
```
```    80     by (simp add: expand_poly_eq)
```
```    81   then obtain n where "coeff p n \<noteq> 0" ..
```
```    82   hence "n \<le> degree p" by (rule le_degree)
```
```    83   with `coeff p n \<noteq> 0` and `degree p = 0`
```
```    84   show "coeff p (degree p) \<noteq> 0" by simp
```
```    85 next
```
```    86   case (Suc n)
```
```    87   from `degree p = Suc n` have "n < degree p" by simp
```
```    88   hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
```
```    89   then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
```
```    90   from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
```
```    91   also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
```
```    92   finally have "degree p = i" .
```
```    93   with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
```
```    94 qed
```
```    95
```
```    96 lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
```
```    97   by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
```
```    98
```
```    99
```
```   100 subsection {* List-style constructor for polynomials *}
```
```   101
```
```   102 definition
```
```   103   pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   104 where
```
```   105   "pCons a p = Abs_poly (nat_case a (coeff p))"
```
```   106
```
```   107 syntax
```
```   108   "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
```
```   109
```
```   110 translations
```
```   111   "[:x, xs:]" == "CONST pCons x [:xs:]"
```
```   112   "[:x:]" == "CONST pCons x 0"
```
```   113   "[:x:]" <= "CONST pCons x (_constrain 0 t)"
```
```   114
```
```   115 lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
```
```   116   unfolding Poly_def by (auto split: nat.split)
```
```   117
```
```   118 lemma coeff_pCons:
```
```   119   "coeff (pCons a p) = nat_case a (coeff p)"
```
```   120   unfolding pCons_def
```
```   121   by (simp add: Abs_poly_inverse Poly_nat_case coeff)
```
```   122
```
```   123 lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
```
```   124   by (simp add: coeff_pCons)
```
```   125
```
```   126 lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
```
```   127   by (simp add: coeff_pCons)
```
```   128
```
```   129 lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
```
```   130 by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   131
```
```   132 lemma degree_pCons_eq:
```
```   133   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
```
```   134 apply (rule order_antisym [OF degree_pCons_le])
```
```   135 apply (rule le_degree, simp)
```
```   136 done
```
```   137
```
```   138 lemma degree_pCons_0: "degree (pCons a 0) = 0"
```
```   139 apply (rule order_antisym [OF _ le0])
```
```   140 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   141 done
```
```   142
```
```   143 lemma degree_pCons_eq_if [simp]:
```
```   144   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
```
```   145 apply (cases "p = 0", simp_all)
```
```   146 apply (rule order_antisym [OF _ le0])
```
```   147 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   148 apply (rule order_antisym [OF degree_pCons_le])
```
```   149 apply (rule le_degree, simp)
```
```   150 done
```
```   151
```
```   152 lemma pCons_0_0 [simp, code_post]: "pCons 0 0 = 0"
```
```   153 by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   154
```
```   155 lemma pCons_eq_iff [simp]:
```
```   156   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
```
```   157 proof (safe)
```
```   158   assume "pCons a p = pCons b q"
```
```   159   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
```
```   160   then show "a = b" by simp
```
```   161 next
```
```   162   assume "pCons a p = pCons b q"
```
```   163   then have "\<forall>n. coeff (pCons a p) (Suc n) =
```
```   164                  coeff (pCons b q) (Suc n)" by simp
```
```   165   then show "p = q" by (simp add: expand_poly_eq)
```
```   166 qed
```
```   167
```
```   168 lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
```
```   169   using pCons_eq_iff [of a p 0 0] by simp
```
```   170
```
```   171 lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
```
```   172   unfolding Poly_def
```
```   173   by (clarify, rule_tac x=n in exI, simp)
```
```   174
```
```   175 lemma pCons_cases [cases type: poly]:
```
```   176   obtains (pCons) a q where "p = pCons a q"
```
```   177 proof
```
```   178   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
```
```   179     by (rule poly_ext)
```
```   180        (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
```
```   181              split: nat.split)
```
```   182 qed
```
```   183
```
```   184 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
```
```   185   assumes zero: "P 0"
```
```   186   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
```
```   187   shows "P p"
```
```   188 proof (induct p rule: measure_induct_rule [where f=degree])
```
```   189   case (less p)
```
```   190   obtain a q where "p = pCons a q" by (rule pCons_cases)
```
```   191   have "P q"
```
```   192   proof (cases "q = 0")
```
```   193     case True
```
```   194     then show "P q" by (simp add: zero)
```
```   195   next
```
```   196     case False
```
```   197     then have "degree (pCons a q) = Suc (degree q)"
```
```   198       by (rule degree_pCons_eq)
```
```   199     then have "degree q < degree p"
```
```   200       using `p = pCons a q` by simp
```
```   201     then show "P q"
```
```   202       by (rule less.hyps)
```
```   203   qed
```
```   204   then have "P (pCons a q)"
```
```   205     by (rule pCons)
```
```   206   then show ?case
```
```   207     using `p = pCons a q` by simp
```
```   208 qed
```
```   209
```
```   210
```
```   211 subsection {* Recursion combinator for polynomials *}
```
```   212
```
```   213 function
```
```   214   poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
```
```   215 where
```
```   216   poly_rec_pCons_eq_if [simp del]:
```
```   217     "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
```
```   218 by (case_tac x, rename_tac q, case_tac q, auto)
```
```   219
```
```   220 termination poly_rec
```
```   221 by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
```
```   222    (simp add: degree_pCons_eq)
```
```   223
```
```   224 lemma poly_rec_0:
```
```   225   "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
```
```   226   using poly_rec_pCons_eq_if [of z f 0 0] by simp
```
```   227
```
```   228 lemma poly_rec_pCons:
```
```   229   "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
```
```   230   by (simp add: poly_rec_pCons_eq_if poly_rec_0)
```
```   231
```
```   232
```
```   233 subsection {* Monomials *}
```
```   234
```
```   235 definition
```
```   236   monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
```
```   237   "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
```
```   238
```
```   239 lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
```
```   240   unfolding monom_def
```
```   241   by (subst Abs_poly_inverse, auto simp add: Poly_def)
```
```   242
```
```   243 lemma monom_0: "monom a 0 = pCons a 0"
```
```   244   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   245
```
```   246 lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
```
```   247   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   248
```
```   249 lemma monom_eq_0 [simp]: "monom 0 n = 0"
```
```   250   by (rule poly_ext) simp
```
```   251
```
```   252 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
```
```   253   by (simp add: expand_poly_eq)
```
```   254
```
```   255 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
```
```   256   by (simp add: expand_poly_eq)
```
```   257
```
```   258 lemma degree_monom_le: "degree (monom a n) \<le> n"
```
```   259   by (rule degree_le, simp)
```
```   260
```
```   261 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
```
```   262   apply (rule order_antisym [OF degree_monom_le])
```
```   263   apply (rule le_degree, simp)
```
```   264   done
```
```   265
```
```   266
```
```   267 subsection {* Addition and subtraction *}
```
```   268
```
```   269 instantiation poly :: (comm_monoid_add) comm_monoid_add
```
```   270 begin
```
```   271
```
```   272 definition
```
```   273   plus_poly_def:
```
```   274     "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
```
```   275
```
```   276 lemma Poly_add:
```
```   277   fixes f g :: "nat \<Rightarrow> 'a"
```
```   278   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
```
```   279   unfolding Poly_def
```
```   280   apply (clarify, rename_tac m n)
```
```   281   apply (rule_tac x="max m n" in exI, simp)
```
```   282   done
```
```   283
```
```   284 lemma coeff_add [simp]:
```
```   285   "coeff (p + q) n = coeff p n + coeff q n"
```
```   286   unfolding plus_poly_def
```
```   287   by (simp add: Abs_poly_inverse coeff Poly_add)
```
```   288
```
```   289 instance proof
```
```   290   fix p q r :: "'a poly"
```
```   291   show "(p + q) + r = p + (q + r)"
```
```   292     by (simp add: expand_poly_eq add_assoc)
```
```   293   show "p + q = q + p"
```
```   294     by (simp add: expand_poly_eq add_commute)
```
```   295   show "0 + p = p"
```
```   296     by (simp add: expand_poly_eq)
```
```   297 qed
```
```   298
```
```   299 end
```
```   300
```
```   301 instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
```
```   302 proof
```
```   303   fix p q r :: "'a poly"
```
```   304   assume "p + q = p + r" thus "q = r"
```
```   305     by (simp add: expand_poly_eq)
```
```   306 qed
```
```   307
```
```   308 instantiation poly :: (ab_group_add) ab_group_add
```
```   309 begin
```
```   310
```
```   311 definition
```
```   312   uminus_poly_def:
```
```   313     "- p = Abs_poly (\<lambda>n. - coeff p n)"
```
```   314
```
```   315 definition
```
```   316   minus_poly_def:
```
```   317     "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
```
```   318
```
```   319 lemma Poly_minus:
```
```   320   fixes f :: "nat \<Rightarrow> 'a"
```
```   321   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
```
```   322   unfolding Poly_def by simp
```
```   323
```
```   324 lemma Poly_diff:
```
```   325   fixes f g :: "nat \<Rightarrow> 'a"
```
```   326   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
```
```   327   unfolding diff_minus by (simp add: Poly_add Poly_minus)
```
```   328
```
```   329 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
```
```   330   unfolding uminus_poly_def
```
```   331   by (simp add: Abs_poly_inverse coeff Poly_minus)
```
```   332
```
```   333 lemma coeff_diff [simp]:
```
```   334   "coeff (p - q) n = coeff p n - coeff q n"
```
```   335   unfolding minus_poly_def
```
```   336   by (simp add: Abs_poly_inverse coeff Poly_diff)
```
```   337
```
```   338 instance proof
```
```   339   fix p q :: "'a poly"
```
```   340   show "- p + p = 0"
```
```   341     by (simp add: expand_poly_eq)
```
```   342   show "p - q = p + - q"
```
```   343     by (simp add: expand_poly_eq diff_minus)
```
```   344 qed
```
```   345
```
```   346 end
```
```   347
```
```   348 lemma add_pCons [simp]:
```
```   349   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```   350   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   351
```
```   352 lemma minus_pCons [simp]:
```
```   353   "- pCons a p = pCons (- a) (- p)"
```
```   354   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   355
```
```   356 lemma diff_pCons [simp]:
```
```   357   "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```   358   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   359
```
```   360 lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
```
```   361   by (rule degree_le, auto simp add: coeff_eq_0)
```
```   362
```
```   363 lemma degree_add_le:
```
```   364   "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
```
```   365   by (auto intro: order_trans degree_add_le_max)
```
```   366
```
```   367 lemma degree_add_less:
```
```   368   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
```
```   369   by (auto intro: le_less_trans degree_add_le_max)
```
```   370
```
```   371 lemma degree_add_eq_right:
```
```   372   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
```
```   373   apply (cases "q = 0", simp)
```
```   374   apply (rule order_antisym)
```
```   375   apply (simp add: degree_add_le)
```
```   376   apply (rule le_degree)
```
```   377   apply (simp add: coeff_eq_0)
```
```   378   done
```
```   379
```
```   380 lemma degree_add_eq_left:
```
```   381   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
```
```   382   using degree_add_eq_right [of q p]
```
```   383   by (simp add: add_commute)
```
```   384
```
```   385 lemma degree_minus [simp]: "degree (- p) = degree p"
```
```   386   unfolding degree_def by simp
```
```   387
```
```   388 lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
```
```   389   using degree_add_le [where p=p and q="-q"]
```
```   390   by (simp add: diff_minus)
```
```   391
```
```   392 lemma degree_diff_le:
```
```   393   "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
```
```   394   by (simp add: diff_minus degree_add_le)
```
```   395
```
```   396 lemma degree_diff_less:
```
```   397   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
```
```   398   by (simp add: diff_minus degree_add_less)
```
```   399
```
```   400 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
```
```   401   by (rule poly_ext) simp
```
```   402
```
```   403 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
```
```   404   by (rule poly_ext) simp
```
```   405
```
```   406 lemma minus_monom: "- monom a n = monom (-a) n"
```
```   407   by (rule poly_ext) simp
```
```   408
```
```   409 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
```
```   410   by (cases "finite A", induct set: finite, simp_all)
```
```   411
```
```   412 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
```
```   413   by (rule poly_ext) (simp add: coeff_setsum)
```
```   414
```
```   415
```
```   416 subsection {* Multiplication by a constant *}
```
```   417
```
```   418 definition
```
```   419   smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   420   "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
```
```   421
```
```   422 lemma Poly_smult:
```
```   423   fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
```
```   424   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
```
```   425   unfolding Poly_def
```
```   426   by (clarify, rule_tac x=n in exI, simp)
```
```   427
```
```   428 lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
```
```   429   unfolding smult_def
```
```   430   by (simp add: Abs_poly_inverse Poly_smult coeff)
```
```   431
```
```   432 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
```
```   433   by (rule degree_le, simp add: coeff_eq_0)
```
```   434
```
```   435 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
```
```   436   by (rule poly_ext, simp add: mult_assoc)
```
```   437
```
```   438 lemma smult_0_right [simp]: "smult a 0 = 0"
```
```   439   by (rule poly_ext, simp)
```
```   440
```
```   441 lemma smult_0_left [simp]: "smult 0 p = 0"
```
```   442   by (rule poly_ext, simp)
```
```   443
```
```   444 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
```
```   445   by (rule poly_ext, simp)
```
```   446
```
```   447 lemma smult_add_right:
```
```   448   "smult a (p + q) = smult a p + smult a q"
```
```   449   by (rule poly_ext, simp add: algebra_simps)
```
```   450
```
```   451 lemma smult_add_left:
```
```   452   "smult (a + b) p = smult a p + smult b p"
```
```   453   by (rule poly_ext, simp add: algebra_simps)
```
```   454
```
```   455 lemma smult_minus_right [simp]:
```
```   456   "smult (a::'a::comm_ring) (- p) = - smult a p"
```
```   457   by (rule poly_ext, simp)
```
```   458
```
```   459 lemma smult_minus_left [simp]:
```
```   460   "smult (- a::'a::comm_ring) p = - smult a p"
```
```   461   by (rule poly_ext, simp)
```
```   462
```
```   463 lemma smult_diff_right:
```
```   464   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
```
```   465   by (rule poly_ext, simp add: algebra_simps)
```
```   466
```
```   467 lemma smult_diff_left:
```
```   468   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
```
```   469   by (rule poly_ext, simp add: algebra_simps)
```
```   470
```
```   471 lemmas smult_distribs =
```
```   472   smult_add_left smult_add_right
```
```   473   smult_diff_left smult_diff_right
```
```   474
```
```   475 lemma smult_pCons [simp]:
```
```   476   "smult a (pCons b p) = pCons (a * b) (smult a p)"
```
```   477   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   478
```
```   479 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
```
```   480   by (induct n, simp add: monom_0, simp add: monom_Suc)
```
```   481
```
```   482 lemma degree_smult_eq [simp]:
```
```   483   fixes a :: "'a::idom"
```
```   484   shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
```
```   485   by (cases "a = 0", simp, simp add: degree_def)
```
```   486
```
```   487 lemma smult_eq_0_iff [simp]:
```
```   488   fixes a :: "'a::idom"
```
```   489   shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
```
```   490   by (simp add: expand_poly_eq)
```
```   491
```
```   492
```
```   493 subsection {* Multiplication of polynomials *}
```
```   494
```
```   495 text {* TODO: move to SetInterval.thy *}
```
```   496 lemma setsum_atMost_Suc_shift:
```
```   497   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
```
```   498   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   499 proof (induct n)
```
```   500   case 0 show ?case by simp
```
```   501 next
```
```   502   case (Suc n) note IH = this
```
```   503   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
```
```   504     by (rule setsum_atMost_Suc)
```
```   505   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   506     by (rule IH)
```
```   507   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
```
```   508              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
```
```   509     by (rule add_assoc)
```
```   510   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
```
```   511     by (rule setsum_atMost_Suc [symmetric])
```
```   512   finally show ?case .
```
```   513 qed
```
```   514
```
```   515 instantiation poly :: (comm_semiring_0) comm_semiring_0
```
```   516 begin
```
```   517
```
```   518 definition
```
```   519   times_poly_def:
```
```   520     "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
```
```   521
```
```   522 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
```
```   523   unfolding times_poly_def by (simp add: poly_rec_0)
```
```   524
```
```   525 lemma mult_pCons_left [simp]:
```
```   526   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```   527   unfolding times_poly_def by (simp add: poly_rec_pCons)
```
```   528
```
```   529 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
```
```   530   by (induct p, simp add: mult_poly_0_left, simp)
```
```   531
```
```   532 lemma mult_pCons_right [simp]:
```
```   533   "p * pCons a q = smult a p + pCons 0 (p * q)"
```
```   534   by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)
```
```   535
```
```   536 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
```
```   537
```
```   538 lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
```
```   539   by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
```
```   540
```
```   541 lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
```
```   542   by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
```
```   543
```
```   544 lemma mult_poly_add_left:
```
```   545   fixes p q r :: "'a poly"
```
```   546   shows "(p + q) * r = p * r + q * r"
```
```   547   by (induct r, simp add: mult_poly_0,
```
```   548                 simp add: smult_distribs algebra_simps)
```
```   549
```
```   550 instance proof
```
```   551   fix p q r :: "'a poly"
```
```   552   show 0: "0 * p = 0"
```
```   553     by (rule mult_poly_0_left)
```
```   554   show "p * 0 = 0"
```
```   555     by (rule mult_poly_0_right)
```
```   556   show "(p + q) * r = p * r + q * r"
```
```   557     by (rule mult_poly_add_left)
```
```   558   show "(p * q) * r = p * (q * r)"
```
```   559     by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
```
```   560   show "p * q = q * p"
```
```   561     by (induct p, simp add: mult_poly_0, simp)
```
```   562 qed
```
```   563
```
```   564 end
```
```   565
```
```   566 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
```
```   567
```
```   568 lemma coeff_mult:
```
```   569   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   570 proof (induct p arbitrary: n)
```
```   571   case 0 show ?case by simp
```
```   572 next
```
```   573   case (pCons a p n) thus ?case
```
```   574     by (cases n, simp, simp add: setsum_atMost_Suc_shift
```
```   575                             del: setsum_atMost_Suc)
```
```   576 qed
```
```   577
```
```   578 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
```
```   579 apply (rule degree_le)
```
```   580 apply (induct p)
```
```   581 apply simp
```
```   582 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   583 done
```
```   584
```
```   585 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
```
```   586   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
```
```   587
```
```   588
```
```   589 subsection {* The unit polynomial and exponentiation *}
```
```   590
```
```   591 instantiation poly :: (comm_semiring_1) comm_semiring_1
```
```   592 begin
```
```   593
```
```   594 definition
```
```   595   one_poly_def:
```
```   596     "1 = pCons 1 0"
```
```   597
```
```   598 instance proof
```
```   599   fix p :: "'a poly" show "1 * p = p"
```
```   600     unfolding one_poly_def
```
```   601     by simp
```
```   602 next
```
```   603   show "0 \<noteq> (1::'a poly)"
```
```   604     unfolding one_poly_def by simp
```
```   605 qed
```
```   606
```
```   607 end
```
```   608
```
```   609 instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
```
```   610
```
```   611 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
```
```   612   unfolding one_poly_def
```
```   613   by (simp add: coeff_pCons split: nat.split)
```
```   614
```
```   615 lemma degree_1 [simp]: "degree 1 = 0"
```
```   616   unfolding one_poly_def
```
```   617   by (rule degree_pCons_0)
```
```   618
```
```   619 text {* Lemmas about divisibility *}
```
```   620
```
```   621 lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
```
```   622 proof -
```
```   623   assume "p dvd q"
```
```   624   then obtain k where "q = p * k" ..
```
```   625   then have "smult a q = p * smult a k" by simp
```
```   626   then show "p dvd smult a q" ..
```
```   627 qed
```
```   628
```
```   629 lemma dvd_smult_cancel:
```
```   630   fixes a :: "'a::field"
```
```   631   shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
```
```   632   by (drule dvd_smult [where a="inverse a"]) simp
```
```   633
```
```   634 lemma dvd_smult_iff:
```
```   635   fixes a :: "'a::field"
```
```   636   shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
```
```   637   by (safe elim!: dvd_smult dvd_smult_cancel)
```
```   638
```
```   639 lemma smult_dvd_cancel:
```
```   640   "smult a p dvd q \<Longrightarrow> p dvd q"
```
```   641 proof -
```
```   642   assume "smult a p dvd q"
```
```   643   then obtain k where "q = smult a p * k" ..
```
```   644   then have "q = p * smult a k" by simp
```
```   645   then show "p dvd q" ..
```
```   646 qed
```
```   647
```
```   648 lemma smult_dvd:
```
```   649   fixes a :: "'a::field"
```
```   650   shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
```
```   651   by (rule smult_dvd_cancel [where a="inverse a"]) simp
```
```   652
```
```   653 lemma smult_dvd_iff:
```
```   654   fixes a :: "'a::field"
```
```   655   shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
```
```   656   by (auto elim: smult_dvd smult_dvd_cancel)
```
```   657
```
```   658 lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
```
```   659 by (induct n, simp, auto intro: order_trans degree_mult_le)
```
```   660
```
```   661 instance poly :: (comm_ring) comm_ring ..
```
```   662
```
```   663 instance poly :: (comm_ring_1) comm_ring_1 ..
```
```   664
```
```   665 instantiation poly :: (comm_ring_1) number_ring
```
```   666 begin
```
```   667
```
```   668 definition
```
```   669   "number_of k = (of_int k :: 'a poly)"
```
```   670
```
```   671 instance
```
```   672   by default (rule number_of_poly_def)
```
```   673
```
```   674 end
```
```   675
```
```   676
```
```   677 subsection {* Polynomials form an integral domain *}
```
```   678
```
```   679 lemma coeff_mult_degree_sum:
```
```   680   "coeff (p * q) (degree p + degree q) =
```
```   681    coeff p (degree p) * coeff q (degree q)"
```
```   682   by (induct p, simp, simp add: coeff_eq_0)
```
```   683
```
```   684 instance poly :: (idom) idom
```
```   685 proof
```
```   686   fix p q :: "'a poly"
```
```   687   assume "p \<noteq> 0" and "q \<noteq> 0"
```
```   688   have "coeff (p * q) (degree p + degree q) =
```
```   689         coeff p (degree p) * coeff q (degree q)"
```
```   690     by (rule coeff_mult_degree_sum)
```
```   691   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
```
```   692     using `p \<noteq> 0` and `q \<noteq> 0` by simp
```
```   693   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
```
```   694   thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
```
```   695 qed
```
```   696
```
```   697 lemma degree_mult_eq:
```
```   698   fixes p q :: "'a::idom poly"
```
```   699   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
```
```   700 apply (rule order_antisym [OF degree_mult_le le_degree])
```
```   701 apply (simp add: coeff_mult_degree_sum)
```
```   702 done
```
```   703
```
```   704 lemma dvd_imp_degree_le:
```
```   705   fixes p q :: "'a::idom poly"
```
```   706   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
```
```   707   by (erule dvdE, simp add: degree_mult_eq)
```
```   708
```
```   709
```
```   710 subsection {* Polynomials form an ordered integral domain *}
```
```   711
```
```   712 definition
```
```   713   pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
```
```   714 where
```
```   715   "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
```
```   716
```
```   717 lemma pos_poly_pCons:
```
```   718   "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
```
```   719   unfolding pos_poly_def by simp
```
```   720
```
```   721 lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
```
```   722   unfolding pos_poly_def by simp
```
```   723
```
```   724 lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
```
```   725   apply (induct p arbitrary: q, simp)
```
```   726   apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
```
```   727   done
```
```   728
```
```   729 lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
```
```   730   unfolding pos_poly_def
```
```   731   apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
```
```   732   apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)
```
```   733   apply auto
```
```   734   done
```
```   735
```
```   736 lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
```
```   737 by (induct p) (auto simp add: pos_poly_pCons)
```
```   738
```
```   739 instantiation poly :: (linordered_idom) linordered_idom
```
```   740 begin
```
```   741
```
```   742 definition
```
```   743   "x < y \<longleftrightarrow> pos_poly (y - x)"
```
```   744
```
```   745 definition
```
```   746   "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
```
```   747
```
```   748 definition
```
```   749   "abs (x::'a poly) = (if x < 0 then - x else x)"
```
```   750
```
```   751 definition
```
```   752   "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```   753
```
```   754 instance proof
```
```   755   fix x y :: "'a poly"
```
```   756   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```   757     unfolding less_eq_poly_def less_poly_def
```
```   758     apply safe
```
```   759     apply simp
```
```   760     apply (drule (1) pos_poly_add)
```
```   761     apply simp
```
```   762     done
```
```   763 next
```
```   764   fix x :: "'a poly" show "x \<le> x"
```
```   765     unfolding less_eq_poly_def by simp
```
```   766 next
```
```   767   fix x y z :: "'a poly"
```
```   768   assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
```
```   769     unfolding less_eq_poly_def
```
```   770     apply safe
```
```   771     apply (drule (1) pos_poly_add)
```
```   772     apply (simp add: algebra_simps)
```
```   773     done
```
```   774 next
```
```   775   fix x y :: "'a poly"
```
```   776   assume "x \<le> y" and "y \<le> x" thus "x = y"
```
```   777     unfolding less_eq_poly_def
```
```   778     apply safe
```
```   779     apply (drule (1) pos_poly_add)
```
```   780     apply simp
```
```   781     done
```
```   782 next
```
```   783   fix x y z :: "'a poly"
```
```   784   assume "x \<le> y" thus "z + x \<le> z + y"
```
```   785     unfolding less_eq_poly_def
```
```   786     apply safe
```
```   787     apply (simp add: algebra_simps)
```
```   788     done
```
```   789 next
```
```   790   fix x y :: "'a poly"
```
```   791   show "x \<le> y \<or> y \<le> x"
```
```   792     unfolding less_eq_poly_def
```
```   793     using pos_poly_total [of "x - y"]
```
```   794     by auto
```
```   795 next
```
```   796   fix x y z :: "'a poly"
```
```   797   assume "x < y" and "0 < z"
```
```   798   thus "z * x < z * y"
```
```   799     unfolding less_poly_def
```
```   800     by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
```
```   801 next
```
```   802   fix x :: "'a poly"
```
```   803   show "\<bar>x\<bar> = (if x < 0 then - x else x)"
```
```   804     by (rule abs_poly_def)
```
```   805 next
```
```   806   fix x :: "'a poly"
```
```   807   show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```   808     by (rule sgn_poly_def)
```
```   809 qed
```
```   810
```
```   811 end
```
```   812
```
```   813 text {* TODO: Simplification rules for comparisons *}
```
```   814
```
```   815
```
```   816 subsection {* Long division of polynomials *}
```
```   817
```
```   818 definition
```
```   819   pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
```
```   820 where
```
```   821   "pdivmod_rel x y q r \<longleftrightarrow>
```
```   822     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
```
```   823
```
```   824 lemma pdivmod_rel_0:
```
```   825   "pdivmod_rel 0 y 0 0"
```
```   826   unfolding pdivmod_rel_def by simp
```
```   827
```
```   828 lemma pdivmod_rel_by_0:
```
```   829   "pdivmod_rel x 0 0 x"
```
```   830   unfolding pdivmod_rel_def by simp
```
```   831
```
```   832 lemma eq_zero_or_degree_less:
```
```   833   assumes "degree p \<le> n" and "coeff p n = 0"
```
```   834   shows "p = 0 \<or> degree p < n"
```
```   835 proof (cases n)
```
```   836   case 0
```
```   837   with `degree p \<le> n` and `coeff p n = 0`
```
```   838   have "coeff p (degree p) = 0" by simp
```
```   839   then have "p = 0" by simp
```
```   840   then show ?thesis ..
```
```   841 next
```
```   842   case (Suc m)
```
```   843   have "\<forall>i>n. coeff p i = 0"
```
```   844     using `degree p \<le> n` by (simp add: coeff_eq_0)
```
```   845   then have "\<forall>i\<ge>n. coeff p i = 0"
```
```   846     using `coeff p n = 0` by (simp add: le_less)
```
```   847   then have "\<forall>i>m. coeff p i = 0"
```
```   848     using `n = Suc m` by (simp add: less_eq_Suc_le)
```
```   849   then have "degree p \<le> m"
```
```   850     by (rule degree_le)
```
```   851   then have "degree p < n"
```
```   852     using `n = Suc m` by (simp add: less_Suc_eq_le)
```
```   853   then show ?thesis ..
```
```   854 qed
```
```   855
```
```   856 lemma pdivmod_rel_pCons:
```
```   857   assumes rel: "pdivmod_rel x y q r"
```
```   858   assumes y: "y \<noteq> 0"
```
```   859   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
```
```   860   shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
```
```   861     (is "pdivmod_rel ?x y ?q ?r")
```
```   862 proof -
```
```   863   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
```
```   864     using assms unfolding pdivmod_rel_def by simp_all
```
```   865
```
```   866   have 1: "?x = ?q * y + ?r"
```
```   867     using b x by simp
```
```   868
```
```   869   have 2: "?r = 0 \<or> degree ?r < degree y"
```
```   870   proof (rule eq_zero_or_degree_less)
```
```   871     show "degree ?r \<le> degree y"
```
```   872     proof (rule degree_diff_le)
```
```   873       show "degree (pCons a r) \<le> degree y"
```
```   874         using r by auto
```
```   875       show "degree (smult b y) \<le> degree y"
```
```   876         by (rule degree_smult_le)
```
```   877     qed
```
```   878   next
```
```   879     show "coeff ?r (degree y) = 0"
```
```   880       using `y \<noteq> 0` unfolding b by simp
```
```   881   qed
```
```   882
```
```   883   from 1 2 show ?thesis
```
```   884     unfolding pdivmod_rel_def
```
```   885     using `y \<noteq> 0` by simp
```
```   886 qed
```
```   887
```
```   888 lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
```
```   889 apply (cases "y = 0")
```
```   890 apply (fast intro!: pdivmod_rel_by_0)
```
```   891 apply (induct x)
```
```   892 apply (fast intro!: pdivmod_rel_0)
```
```   893 apply (fast intro!: pdivmod_rel_pCons)
```
```   894 done
```
```   895
```
```   896 lemma pdivmod_rel_unique:
```
```   897   assumes 1: "pdivmod_rel x y q1 r1"
```
```   898   assumes 2: "pdivmod_rel x y q2 r2"
```
```   899   shows "q1 = q2 \<and> r1 = r2"
```
```   900 proof (cases "y = 0")
```
```   901   assume "y = 0" with assms show ?thesis
```
```   902     by (simp add: pdivmod_rel_def)
```
```   903 next
```
```   904   assume [simp]: "y \<noteq> 0"
```
```   905   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
```
```   906     unfolding pdivmod_rel_def by simp_all
```
```   907   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
```
```   908     unfolding pdivmod_rel_def by simp_all
```
```   909   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
```
```   910     by (simp add: algebra_simps)
```
```   911   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
```
```   912     by (auto intro: degree_diff_less)
```
```   913
```
```   914   show "q1 = q2 \<and> r1 = r2"
```
```   915   proof (rule ccontr)
```
```   916     assume "\<not> (q1 = q2 \<and> r1 = r2)"
```
```   917     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
```
```   918     with r3 have "degree (r2 - r1) < degree y" by simp
```
```   919     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
```
```   920     also have "\<dots> = degree ((q1 - q2) * y)"
```
```   921       using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
```
```   922     also have "\<dots> = degree (r2 - r1)"
```
```   923       using q3 by simp
```
```   924     finally have "degree (r2 - r1) < degree (r2 - r1)" .
```
```   925     then show "False" by simp
```
```   926   qed
```
```   927 qed
```
```   928
```
```   929 lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
```
```   930 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
```
```   931
```
```   932 lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
```
```   933 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
```
```   934
```
```   935 lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]
```
```   936
```
```   937 lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]
```
```   938
```
```   939 instantiation poly :: (field) ring_div
```
```   940 begin
```
```   941
```
```   942 definition div_poly where
```
```   943   "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
```
```   944
```
```   945 definition mod_poly where
```
```   946   "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
```
```   947
```
```   948 lemma div_poly_eq:
```
```   949   "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
```
```   950 unfolding div_poly_def
```
```   951 by (fast elim: pdivmod_rel_unique_div)
```
```   952
```
```   953 lemma mod_poly_eq:
```
```   954   "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
```
```   955 unfolding mod_poly_def
```
```   956 by (fast elim: pdivmod_rel_unique_mod)
```
```   957
```
```   958 lemma pdivmod_rel:
```
```   959   "pdivmod_rel x y (x div y) (x mod y)"
```
```   960 proof -
```
```   961   from pdivmod_rel_exists
```
```   962     obtain q r where "pdivmod_rel x y q r" by fast
```
```   963   thus ?thesis
```
```   964     by (simp add: div_poly_eq mod_poly_eq)
```
```   965 qed
```
```   966
```
```   967 instance proof
```
```   968   fix x y :: "'a poly"
```
```   969   show "x div y * y + x mod y = x"
```
```   970     using pdivmod_rel [of x y]
```
```   971     by (simp add: pdivmod_rel_def)
```
```   972 next
```
```   973   fix x :: "'a poly"
```
```   974   have "pdivmod_rel x 0 0 x"
```
```   975     by (rule pdivmod_rel_by_0)
```
```   976   thus "x div 0 = 0"
```
```   977     by (rule div_poly_eq)
```
```   978 next
```
```   979   fix y :: "'a poly"
```
```   980   have "pdivmod_rel 0 y 0 0"
```
```   981     by (rule pdivmod_rel_0)
```
```   982   thus "0 div y = 0"
```
```   983     by (rule div_poly_eq)
```
```   984 next
```
```   985   fix x y z :: "'a poly"
```
```   986   assume "y \<noteq> 0"
```
```   987   hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
```
```   988     using pdivmod_rel [of x y]
```
```   989     by (simp add: pdivmod_rel_def left_distrib)
```
```   990   thus "(x + z * y) div y = z + x div y"
```
```   991     by (rule div_poly_eq)
```
```   992 next
```
```   993   fix x y z :: "'a poly"
```
```   994   assume "x \<noteq> 0"
```
```   995   show "(x * y) div (x * z) = y div z"
```
```   996   proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
```
```   997     have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
```
```   998       by (rule pdivmod_rel_by_0)
```
```   999     then have [simp]: "\<And>x::'a poly. x div 0 = 0"
```
```  1000       by (rule div_poly_eq)
```
```  1001     have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
```
```  1002       by (rule pdivmod_rel_0)
```
```  1003     then have [simp]: "\<And>x::'a poly. 0 div x = 0"
```
```  1004       by (rule div_poly_eq)
```
```  1005     case False then show ?thesis by auto
```
```  1006   next
```
```  1007     case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
```
```  1008     with `x \<noteq> 0`
```
```  1009     have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
```
```  1010       by (auto simp add: pdivmod_rel_def algebra_simps)
```
```  1011         (rule classical, simp add: degree_mult_eq)
```
```  1012     moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
```
```  1013     ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
```
```  1014     then show ?thesis by (simp add: div_poly_eq)
```
```  1015   qed
```
```  1016 qed
```
```  1017
```
```  1018 end
```
```  1019
```
```  1020 lemma degree_mod_less:
```
```  1021   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
```
```  1022   using pdivmod_rel [of x y]
```
```  1023   unfolding pdivmod_rel_def by simp
```
```  1024
```
```  1025 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
```
```  1026 proof -
```
```  1027   assume "degree x < degree y"
```
```  1028   hence "pdivmod_rel x y 0 x"
```
```  1029     by (simp add: pdivmod_rel_def)
```
```  1030   thus "x div y = 0" by (rule div_poly_eq)
```
```  1031 qed
```
```  1032
```
```  1033 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
```
```  1034 proof -
```
```  1035   assume "degree x < degree y"
```
```  1036   hence "pdivmod_rel x y 0 x"
```
```  1037     by (simp add: pdivmod_rel_def)
```
```  1038   thus "x mod y = x" by (rule mod_poly_eq)
```
```  1039 qed
```
```  1040
```
```  1041 lemma pdivmod_rel_smult_left:
```
```  1042   "pdivmod_rel x y q r
```
```  1043     \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
```
```  1044   unfolding pdivmod_rel_def by (simp add: smult_add_right)
```
```  1045
```
```  1046 lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
```
```  1047   by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
```
```  1048
```
```  1049 lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
```
```  1050   by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
```
```  1051
```
```  1052 lemma poly_div_minus_left [simp]:
```
```  1053   fixes x y :: "'a::field poly"
```
```  1054   shows "(- x) div y = - (x div y)"
```
```  1055   using div_smult_left [of "- 1::'a"] by simp
```
```  1056
```
```  1057 lemma poly_mod_minus_left [simp]:
```
```  1058   fixes x y :: "'a::field poly"
```
```  1059   shows "(- x) mod y = - (x mod y)"
```
```  1060   using mod_smult_left [of "- 1::'a"] by simp
```
```  1061
```
```  1062 lemma pdivmod_rel_smult_right:
```
```  1063   "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
```
```  1064     \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
```
```  1065   unfolding pdivmod_rel_def by simp
```
```  1066
```
```  1067 lemma div_smult_right:
```
```  1068   "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
```
```  1069   by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
```
```  1070
```
```  1071 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
```
```  1072   by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
```
```  1073
```
```  1074 lemma poly_div_minus_right [simp]:
```
```  1075   fixes x y :: "'a::field poly"
```
```  1076   shows "x div (- y) = - (x div y)"
```
```  1077   using div_smult_right [of "- 1::'a"]
```
```  1078   by (simp add: nonzero_inverse_minus_eq)
```
```  1079
```
```  1080 lemma poly_mod_minus_right [simp]:
```
```  1081   fixes x y :: "'a::field poly"
```
```  1082   shows "x mod (- y) = x mod y"
```
```  1083   using mod_smult_right [of "- 1::'a"] by simp
```
```  1084
```
```  1085 lemma pdivmod_rel_mult:
```
```  1086   "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
```
```  1087     \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
```
```  1088 apply (cases "z = 0", simp add: pdivmod_rel_def)
```
```  1089 apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
```
```  1090 apply (cases "r = 0")
```
```  1091 apply (cases "r' = 0")
```
```  1092 apply (simp add: pdivmod_rel_def)
```
```  1093 apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)
```
```  1094 apply (cases "r' = 0")
```
```  1095 apply (simp add: pdivmod_rel_def degree_mult_eq)
```
```  1096 apply (simp add: pdivmod_rel_def field_simps)
```
```  1097 apply (simp add: degree_mult_eq degree_add_less)
```
```  1098 done
```
```  1099
```
```  1100 lemma poly_div_mult_right:
```
```  1101   fixes x y z :: "'a::field poly"
```
```  1102   shows "x div (y * z) = (x div y) div z"
```
```  1103   by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
```
```  1104
```
```  1105 lemma poly_mod_mult_right:
```
```  1106   fixes x y z :: "'a::field poly"
```
```  1107   shows "x mod (y * z) = y * (x div y mod z) + x mod y"
```
```  1108   by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
```
```  1109
```
```  1110 lemma mod_pCons:
```
```  1111   fixes a and x
```
```  1112   assumes y: "y \<noteq> 0"
```
```  1113   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
```
```  1114   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
```
```  1115 unfolding b
```
```  1116 apply (rule mod_poly_eq)
```
```  1117 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
```
```  1118 done
```
```  1119
```
```  1120
```
```  1121 subsection {* GCD of polynomials *}
```
```  1122
```
```  1123 function
```
```  1124   poly_gcd :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```  1125   "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x"
```
```  1126 | "y \<noteq> 0 \<Longrightarrow> poly_gcd x y = poly_gcd y (x mod y)"
```
```  1127 by auto
```
```  1128
```
```  1129 termination poly_gcd
```
```  1130 by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
```
```  1131    (auto dest: degree_mod_less)
```
```  1132
```
```  1133 declare poly_gcd.simps [simp del]
```
```  1134
```
```  1135 lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x"
```
```  1136   and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y"
```
```  1137   apply (induct x y rule: poly_gcd.induct)
```
```  1138   apply (simp_all add: poly_gcd.simps)
```
```  1139   apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
```
```  1140   apply (blast dest: dvd_mod_imp_dvd)
```
```  1141   done
```
```  1142
```
```  1143 lemma poly_gcd_greatest: "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd poly_gcd x y"
```
```  1144   by (induct x y rule: poly_gcd.induct)
```
```  1145      (simp_all add: poly_gcd.simps dvd_mod dvd_smult)
```
```  1146
```
```  1147 lemma dvd_poly_gcd_iff [iff]:
```
```  1148   "k dvd poly_gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
```
```  1149   by (blast intro!: poly_gcd_greatest intro: dvd_trans)
```
```  1150
```
```  1151 lemma poly_gcd_monic:
```
```  1152   "coeff (poly_gcd x y) (degree (poly_gcd x y)) =
```
```  1153     (if x = 0 \<and> y = 0 then 0 else 1)"
```
```  1154   by (induct x y rule: poly_gcd.induct)
```
```  1155      (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero)
```
```  1156
```
```  1157 lemma poly_gcd_zero_iff [simp]:
```
```  1158   "poly_gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```  1159   by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
```
```  1160
```
```  1161 lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0"
```
```  1162   by simp
```
```  1163
```
```  1164 lemma poly_dvd_antisym:
```
```  1165   fixes p q :: "'a::idom poly"
```
```  1166   assumes coeff: "coeff p (degree p) = coeff q (degree q)"
```
```  1167   assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
```
```  1168 proof (cases "p = 0")
```
```  1169   case True with coeff show "p = q" by simp
```
```  1170 next
```
```  1171   case False with coeff have "q \<noteq> 0" by auto
```
```  1172   have degree: "degree p = degree q"
```
```  1173     using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
```
```  1174     by (intro order_antisym dvd_imp_degree_le)
```
```  1175
```
```  1176   from `p dvd q` obtain a where a: "q = p * a" ..
```
```  1177   with `q \<noteq> 0` have "a \<noteq> 0" by auto
```
```  1178   with degree a `p \<noteq> 0` have "degree a = 0"
```
```  1179     by (simp add: degree_mult_eq)
```
```  1180   with coeff a show "p = q"
```
```  1181     by (cases a, auto split: if_splits)
```
```  1182 qed
```
```  1183
```
```  1184 lemma poly_gcd_unique:
```
```  1185   assumes dvd1: "d dvd x" and dvd2: "d dvd y"
```
```  1186     and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
```
```  1187     and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
```
```  1188   shows "poly_gcd x y = d"
```
```  1189 proof -
```
```  1190   have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)"
```
```  1191     by (simp_all add: poly_gcd_monic monic)
```
```  1192   moreover have "poly_gcd x y dvd d"
```
```  1193     using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
```
```  1194   moreover have "d dvd poly_gcd x y"
```
```  1195     using dvd1 dvd2 by (rule poly_gcd_greatest)
```
```  1196   ultimately show ?thesis
```
```  1197     by (rule poly_dvd_antisym)
```
```  1198 qed
```
```  1199
```
```  1200 interpretation poly_gcd: abel_semigroup poly_gcd
```
```  1201 proof
```
```  1202   fix x y z :: "'a poly"
```
```  1203   show "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)"
```
```  1204     by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
```
```  1205   show "poly_gcd x y = poly_gcd y x"
```
```  1206     by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1207 qed
```
```  1208
```
```  1209 lemmas poly_gcd_assoc = poly_gcd.assoc
```
```  1210 lemmas poly_gcd_commute = poly_gcd.commute
```
```  1211 lemmas poly_gcd_left_commute = poly_gcd.left_commute
```
```  1212
```
```  1213 lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
```
```  1214
```
```  1215 lemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1"
```
```  1216 by (rule poly_gcd_unique) simp_all
```
```  1217
```
```  1218 lemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1"
```
```  1219 by (rule poly_gcd_unique) simp_all
```
```  1220
```
```  1221 lemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y"
```
```  1222 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1223
```
```  1224 lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y"
```
```  1225 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1226
```
```  1227
```
```  1228 subsection {* Evaluation of polynomials *}
```
```  1229
```
```  1230 definition
```
```  1231   poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```  1232   "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
```
```  1233
```
```  1234 lemma poly_0 [simp]: "poly 0 x = 0"
```
```  1235   unfolding poly_def by (simp add: poly_rec_0)
```
```  1236
```
```  1237 lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
```
```  1238   unfolding poly_def by (simp add: poly_rec_pCons)
```
```  1239
```
```  1240 lemma poly_1 [simp]: "poly 1 x = 1"
```
```  1241   unfolding one_poly_def by simp
```
```  1242
```
```  1243 lemma poly_monom:
```
```  1244   fixes a x :: "'a::{comm_semiring_1}"
```
```  1245   shows "poly (monom a n) x = a * x ^ n"
```
```  1246   by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
```
```  1247
```
```  1248 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
```
```  1249   apply (induct p arbitrary: q, simp)
```
```  1250   apply (case_tac q, simp, simp add: algebra_simps)
```
```  1251   done
```
```  1252
```
```  1253 lemma poly_minus [simp]:
```
```  1254   fixes x :: "'a::comm_ring"
```
```  1255   shows "poly (- p) x = - poly p x"
```
```  1256   by (induct p, simp_all)
```
```  1257
```
```  1258 lemma poly_diff [simp]:
```
```  1259   fixes x :: "'a::comm_ring"
```
```  1260   shows "poly (p - q) x = poly p x - poly q x"
```
```  1261   by (simp add: diff_minus)
```
```  1262
```
```  1263 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
```
```  1264   by (cases "finite A", induct set: finite, simp_all)
```
```  1265
```
```  1266 lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
```
```  1267   by (induct p, simp, simp add: algebra_simps)
```
```  1268
```
```  1269 lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
```
```  1270   by (induct p, simp_all, simp add: algebra_simps)
```
```  1271
```
```  1272 lemma poly_power [simp]:
```
```  1273   fixes p :: "'a::{comm_semiring_1} poly"
```
```  1274   shows "poly (p ^ n) x = poly p x ^ n"
```
```  1275   by (induct n, simp, simp add: power_Suc)
```
```  1276
```
```  1277
```
```  1278 subsection {* Synthetic division *}
```
```  1279
```
```  1280 text {*
```
```  1281   Synthetic division is simply division by the
```
```  1282   linear polynomial @{term "x - c"}.
```
```  1283 *}
```
```  1284
```
```  1285 definition
```
```  1286   synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
```
```  1287 where
```
```  1288   "synthetic_divmod p c =
```
```  1289     poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
```
```  1290
```
```  1291 definition
```
```  1292   synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```  1293 where
```
```  1294   "synthetic_div p c = fst (synthetic_divmod p c)"
```
```  1295
```
```  1296 lemma synthetic_divmod_0 [simp]:
```
```  1297   "synthetic_divmod 0 c = (0, 0)"
```
```  1298   unfolding synthetic_divmod_def
```
```  1299   by (simp add: poly_rec_0)
```
```  1300
```
```  1301 lemma synthetic_divmod_pCons [simp]:
```
```  1302   "synthetic_divmod (pCons a p) c =
```
```  1303     (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
```
```  1304   unfolding synthetic_divmod_def
```
```  1305   by (simp add: poly_rec_pCons)
```
```  1306
```
```  1307 lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
```
```  1308   by (induct p, simp, simp add: split_def)
```
```  1309
```
```  1310 lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
```
```  1311   unfolding synthetic_div_def by simp
```
```  1312
```
```  1313 lemma synthetic_div_pCons [simp]:
```
```  1314   "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
```
```  1315   unfolding synthetic_div_def
```
```  1316   by (simp add: split_def snd_synthetic_divmod)
```
```  1317
```
```  1318 lemma synthetic_div_eq_0_iff:
```
```  1319   "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
```
```  1320   by (induct p, simp, case_tac p, simp)
```
```  1321
```
```  1322 lemma degree_synthetic_div:
```
```  1323   "degree (synthetic_div p c) = degree p - 1"
```
```  1324   by (induct p, simp, simp add: synthetic_div_eq_0_iff)
```
```  1325
```
```  1326 lemma synthetic_div_correct:
```
```  1327   "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
```
```  1328   by (induct p) simp_all
```
```  1329
```
```  1330 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
```
```  1331 by (induct p arbitrary: a) simp_all
```
```  1332
```
```  1333 lemma synthetic_div_unique:
```
```  1334   "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
```
```  1335 apply (induct p arbitrary: q r)
```
```  1336 apply (simp, frule synthetic_div_unique_lemma, simp)
```
```  1337 apply (case_tac q, force)
```
```  1338 done
```
```  1339
```
```  1340 lemma synthetic_div_correct':
```
```  1341   fixes c :: "'a::comm_ring_1"
```
```  1342   shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
```
```  1343   using synthetic_div_correct [of p c]
```
```  1344   by (simp add: algebra_simps)
```
```  1345
```
```  1346 lemma poly_eq_0_iff_dvd:
```
```  1347   fixes c :: "'a::idom"
```
```  1348   shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
```
```  1349 proof
```
```  1350   assume "poly p c = 0"
```
```  1351   with synthetic_div_correct' [of c p]
```
```  1352   have "p = [:-c, 1:] * synthetic_div p c" by simp
```
```  1353   then show "[:-c, 1:] dvd p" ..
```
```  1354 next
```
```  1355   assume "[:-c, 1:] dvd p"
```
```  1356   then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
```
```  1357   then show "poly p c = 0" by simp
```
```  1358 qed
```
```  1359
```
```  1360 lemma dvd_iff_poly_eq_0:
```
```  1361   fixes c :: "'a::idom"
```
```  1362   shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
```
```  1363   by (simp add: poly_eq_0_iff_dvd)
```
```  1364
```
```  1365 lemma poly_roots_finite:
```
```  1366   fixes p :: "'a::idom poly"
```
```  1367   shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
```
```  1368 proof (induct n \<equiv> "degree p" arbitrary: p)
```
```  1369   case (0 p)
```
```  1370   then obtain a where "a \<noteq> 0" and "p = [:a:]"
```
```  1371     by (cases p, simp split: if_splits)
```
```  1372   then show "finite {x. poly p x = 0}" by simp
```
```  1373 next
```
```  1374   case (Suc n p)
```
```  1375   show "finite {x. poly p x = 0}"
```
```  1376   proof (cases "\<exists>x. poly p x = 0")
```
```  1377     case False
```
```  1378     then show "finite {x. poly p x = 0}" by simp
```
```  1379   next
```
```  1380     case True
```
```  1381     then obtain a where "poly p a = 0" ..
```
```  1382     then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
```
```  1383     then obtain k where k: "p = [:-a, 1:] * k" ..
```
```  1384     with `p \<noteq> 0` have "k \<noteq> 0" by auto
```
```  1385     with k have "degree p = Suc (degree k)"
```
```  1386       by (simp add: degree_mult_eq del: mult_pCons_left)
```
```  1387     with `Suc n = degree p` have "n = degree k" by simp
```
```  1388     then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
```
```  1389     then have "finite (insert a {x. poly k x = 0})" by simp
```
```  1390     then show "finite {x. poly p x = 0}"
```
```  1391       by (simp add: k uminus_add_conv_diff Collect_disj_eq
```
```  1392                del: mult_pCons_left)
```
```  1393   qed
```
```  1394 qed
```
```  1395
```
```  1396 lemma poly_zero:
```
```  1397   fixes p :: "'a::{idom,ring_char_0} poly"
```
```  1398   shows "poly p = poly 0 \<longleftrightarrow> p = 0"
```
```  1399 apply (cases "p = 0", simp_all)
```
```  1400 apply (drule poly_roots_finite)
```
```  1401 apply (auto simp add: infinite_UNIV_char_0)
```
```  1402 done
```
```  1403
```
```  1404 lemma poly_eq_iff:
```
```  1405   fixes p q :: "'a::{idom,ring_char_0} poly"
```
```  1406   shows "poly p = poly q \<longleftrightarrow> p = q"
```
```  1407   using poly_zero [of "p - q"]
```
```  1408   by (simp add: fun_eq_iff)
```
```  1409
```
```  1410
```
```  1411 subsection {* Composition of polynomials *}
```
```  1412
```
```  1413 definition
```
```  1414   pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  1415 where
```
```  1416   "pcompose p q = poly_rec 0 (\<lambda>a _ c. [:a:] + q * c) p"
```
```  1417
```
```  1418 lemma pcompose_0 [simp]: "pcompose 0 q = 0"
```
```  1419   unfolding pcompose_def by (simp add: poly_rec_0)
```
```  1420
```
```  1421 lemma pcompose_pCons:
```
```  1422   "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
```
```  1423   unfolding pcompose_def by (simp add: poly_rec_pCons)
```
```  1424
```
```  1425 lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)"
```
```  1426   by (induct p) (simp_all add: pcompose_pCons)
```
```  1427
```
```  1428 lemma degree_pcompose_le:
```
```  1429   "degree (pcompose p q) \<le> degree p * degree q"
```
```  1430 apply (induct p, simp)
```
```  1431 apply (simp add: pcompose_pCons, clarify)
```
```  1432 apply (rule degree_add_le, simp)
```
```  1433 apply (rule order_trans [OF degree_mult_le], simp)
```
```  1434 done
```
```  1435
```
```  1436
```
```  1437 subsection {* Order of polynomial roots *}
```
```  1438
```
```  1439 definition
```
```  1440   order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
```
```  1441 where
```
```  1442   "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
```
```  1443
```
```  1444 lemma coeff_linear_power:
```
```  1445   fixes a :: "'a::comm_semiring_1"
```
```  1446   shows "coeff ([:a, 1:] ^ n) n = 1"
```
```  1447 apply (induct n, simp_all)
```
```  1448 apply (subst coeff_eq_0)
```
```  1449 apply (auto intro: le_less_trans degree_power_le)
```
```  1450 done
```
```  1451
```
```  1452 lemma degree_linear_power:
```
```  1453   fixes a :: "'a::comm_semiring_1"
```
```  1454   shows "degree ([:a, 1:] ^ n) = n"
```
```  1455 apply (rule order_antisym)
```
```  1456 apply (rule ord_le_eq_trans [OF degree_power_le], simp)
```
```  1457 apply (rule le_degree, simp add: coeff_linear_power)
```
```  1458 done
```
```  1459
```
```  1460 lemma order_1: "[:-a, 1:] ^ order a p dvd p"
```
```  1461 apply (cases "p = 0", simp)
```
```  1462 apply (cases "order a p", simp)
```
```  1463 apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
```
```  1464 apply (drule not_less_Least, simp)
```
```  1465 apply (fold order_def, simp)
```
```  1466 done
```
```  1467
```
```  1468 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1469 unfolding order_def
```
```  1470 apply (rule LeastI_ex)
```
```  1471 apply (rule_tac x="degree p" in exI)
```
```  1472 apply (rule notI)
```
```  1473 apply (drule (1) dvd_imp_degree_le)
```
```  1474 apply (simp only: degree_linear_power)
```
```  1475 done
```
```  1476
```
```  1477 lemma order:
```
```  1478   "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1479 by (rule conjI [OF order_1 order_2])
```
```  1480
```
```  1481 lemma order_degree:
```
```  1482   assumes p: "p \<noteq> 0"
```
```  1483   shows "order a p \<le> degree p"
```
```  1484 proof -
```
```  1485   have "order a p = degree ([:-a, 1:] ^ order a p)"
```
```  1486     by (simp only: degree_linear_power)
```
```  1487   also have "\<dots> \<le> degree p"
```
```  1488     using order_1 p by (rule dvd_imp_degree_le)
```
```  1489   finally show ?thesis .
```
```  1490 qed
```
```  1491
```
```  1492 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
```
```  1493 apply (cases "p = 0", simp_all)
```
```  1494 apply (rule iffI)
```
```  1495 apply (rule ccontr, simp)
```
```  1496 apply (frule order_2 [where a=a], simp)
```
```  1497 apply (simp add: poly_eq_0_iff_dvd)
```
```  1498 apply (simp add: poly_eq_0_iff_dvd)
```
```  1499 apply (simp only: order_def)
```
```  1500 apply (drule not_less_Least, simp)
```
```  1501 done
```
```  1502
```
```  1503
```
```  1504 subsection {* Configuration of the code generator *}
```
```  1505
```
```  1506 code_datatype "0::'a::zero poly" pCons
```
```  1507
```
```  1508 quickcheck_generator poly constructors: "0::'a::zero poly", pCons
```
```  1509
```
```  1510 instantiation poly :: ("{zero, equal}") equal
```
```  1511 begin
```
```  1512
```
```  1513 definition
```
```  1514   "HOL.equal (p::'a poly) q \<longleftrightarrow> p = q"
```
```  1515
```
```  1516 instance proof
```
```  1517 qed (rule equal_poly_def)
```
```  1518
```
```  1519 end
```
```  1520
```
```  1521 lemma eq_poly_code [code]:
```
```  1522   "HOL.equal (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
```
```  1523   "HOL.equal (0::_ poly) (pCons b q) \<longleftrightarrow> HOL.equal 0 b \<and> HOL.equal 0 q"
```
```  1524   "HOL.equal (pCons a p) (0::_ poly) \<longleftrightarrow> HOL.equal a 0 \<and> HOL.equal p 0"
```
```  1525   "HOL.equal (pCons a p) (pCons b q) \<longleftrightarrow> HOL.equal a b \<and> HOL.equal p q"
```
```  1526   by (simp_all add: equal)
```
```  1527
```
```  1528 lemma [code nbe]:
```
```  1529   "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
```
```  1530   by (fact equal_refl)
```
```  1531
```
```  1532 lemmas coeff_code [code] =
```
```  1533   coeff_0 coeff_pCons_0 coeff_pCons_Suc
```
```  1534
```
```  1535 lemmas degree_code [code] =
```
```  1536   degree_0 degree_pCons_eq_if
```
```  1537
```
```  1538 lemmas monom_poly_code [code] =
```
```  1539   monom_0 monom_Suc
```
```  1540
```
```  1541 lemma add_poly_code [code]:
```
```  1542   "0 + q = (q :: _ poly)"
```
```  1543   "p + 0 = (p :: _ poly)"
```
```  1544   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```  1545 by simp_all
```
```  1546
```
```  1547 lemma minus_poly_code [code]:
```
```  1548   "- 0 = (0 :: _ poly)"
```
```  1549   "- pCons a p = pCons (- a) (- p)"
```
```  1550 by simp_all
```
```  1551
```
```  1552 lemma diff_poly_code [code]:
```
```  1553   "0 - q = (- q :: _ poly)"
```
```  1554   "p - 0 = (p :: _ poly)"
```
```  1555   "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```  1556 by simp_all
```
```  1557
```
```  1558 lemmas smult_poly_code [code] =
```
```  1559   smult_0_right smult_pCons
```
```  1560
```
```  1561 lemma mult_poly_code [code]:
```
```  1562   "0 * q = (0 :: _ poly)"
```
```  1563   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```  1564 by simp_all
```
```  1565
```
```  1566 lemmas poly_code [code] =
```
```  1567   poly_0 poly_pCons
```
```  1568
```
```  1569 lemmas synthetic_divmod_code [code] =
```
```  1570   synthetic_divmod_0 synthetic_divmod_pCons
```
```  1571
```
```  1572 text {* code generator setup for div and mod *}
```
```  1573
```
```  1574 definition
```
```  1575   pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
```
```  1576 where
```
```  1577   "pdivmod x y = (x div y, x mod y)"
```
```  1578
```
```  1579 lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
```
```  1580   unfolding pdivmod_def by simp
```
```  1581
```
```  1582 lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
```
```  1583   unfolding pdivmod_def by simp
```
```  1584
```
```  1585 lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
```
```  1586   unfolding pdivmod_def by simp
```
```  1587
```
```  1588 lemma pdivmod_pCons [code]:
```
```  1589   "pdivmod (pCons a x) y =
```
```  1590     (if y = 0 then (0, pCons a x) else
```
```  1591       (let (q, r) = pdivmod x y;
```
```  1592            b = coeff (pCons a r) (degree y) / coeff y (degree y)
```
```  1593         in (pCons b q, pCons a r - smult b y)))"
```
```  1594 apply (simp add: pdivmod_def Let_def, safe)
```
```  1595 apply (rule div_poly_eq)
```
```  1596 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1597 apply (rule mod_poly_eq)
```
```  1598 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1599 done
```
```  1600
```
```  1601 lemma poly_gcd_code [code]:
```
```  1602   "poly_gcd x y =
```
```  1603     (if y = 0 then smult (inverse (coeff x (degree x))) x
```
```  1604               else poly_gcd y (x mod y))"
```
```  1605   by (simp add: poly_gcd.simps)
```
```  1606
```
```  1607 end
```