src/HOL/Computational_Algebra/Polynomial.thy
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```     1 (*  Title:      HOL/Computational_Algebra/Polynomial.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Clemens Ballarin
```
```     4     Author:     Amine Chaieb
```
```     5     Author:     Florian Haftmann
```
```     6 *)
```
```     7
```
```     8 section \<open>Polynomials as type over a ring structure\<close>
```
```     9
```
```    10 theory Polynomial
```
```    11 imports
```
```    12   HOL.Deriv
```
```    13   "HOL-Library.More_List"
```
```    14   "HOL-Library.Infinite_Set"
```
```    15   Factorial_Ring
```
```    16 begin
```
```    17
```
```    18 subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close>
```
```    19
```
```    20 definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
```
```    21   where "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
```
```    22
```
```    23 lemma cCons_0_Nil_eq [simp]: "0 ## [] = []"
```
```    24   by (simp add: cCons_def)
```
```    25
```
```    26 lemma cCons_Cons_eq [simp]: "x ## y # ys = x # y # ys"
```
```    27   by (simp add: cCons_def)
```
```    28
```
```    29 lemma cCons_append_Cons_eq [simp]: "x ## xs @ y # ys = x # xs @ y # ys"
```
```    30   by (simp add: cCons_def)
```
```    31
```
```    32 lemma cCons_not_0_eq [simp]: "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
```
```    33   by (simp add: cCons_def)
```
```    34
```
```    35 lemma strip_while_not_0_Cons_eq [simp]:
```
```    36   "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
```
```    37 proof (cases "x = 0")
```
```    38   case False
```
```    39   then show ?thesis by simp
```
```    40 next
```
```    41   case True
```
```    42   show ?thesis
```
```    43   proof (induct xs rule: rev_induct)
```
```    44     case Nil
```
```    45     with True show ?case by simp
```
```    46   next
```
```    47     case (snoc y ys)
```
```    48     then show ?case
```
```    49       by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
```
```    50   qed
```
```    51 qed
```
```    52
```
```    53 lemma tl_cCons [simp]: "tl (x ## xs) = xs"
```
```    54   by (simp add: cCons_def)
```
```    55
```
```    56
```
```    57 subsection \<open>Definition of type \<open>poly\<close>\<close>
```
```    58
```
```    59 typedef (overloaded) 'a poly = "{f :: nat \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n. f n = 0}"
```
```    60   morphisms coeff Abs_poly
```
```    61   by (auto intro!: ALL_MOST)
```
```    62
```
```    63 setup_lifting type_definition_poly
```
```    64
```
```    65 lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
```
```    66   by (simp add: coeff_inject [symmetric] fun_eq_iff)
```
```    67
```
```    68 lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
```
```    69   by (simp add: poly_eq_iff)
```
```    70
```
```    71 lemma MOST_coeff_eq_0: "\<forall>\<^sub>\<infinity> n. coeff p n = 0"
```
```    72   using coeff [of p] by simp
```
```    73
```
```    74
```
```    75 subsection \<open>Degree of a polynomial\<close>
```
```    76
```
```    77 definition degree :: "'a::zero poly \<Rightarrow> nat"
```
```    78   where "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
```
```    79
```
```    80 lemma coeff_eq_0:
```
```    81   assumes "degree p < n"
```
```    82   shows "coeff p n = 0"
```
```    83 proof -
```
```    84   have "\<exists>n. \<forall>i>n. coeff p i = 0"
```
```    85     using MOST_coeff_eq_0 by (simp add: MOST_nat)
```
```    86   then have "\<forall>i>degree p. coeff p i = 0"
```
```    87     unfolding degree_def by (rule LeastI_ex)
```
```    88   with assms show ?thesis by simp
```
```    89 qed
```
```    90
```
```    91 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
```
```    92   by (erule contrapos_np, rule coeff_eq_0, simp)
```
```    93
```
```    94 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
```
```    95   unfolding degree_def by (erule Least_le)
```
```    96
```
```    97 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
```
```    98   unfolding degree_def by (drule not_less_Least, simp)
```
```    99
```
```   100
```
```   101 subsection \<open>The zero polynomial\<close>
```
```   102
```
```   103 instantiation poly :: (zero) zero
```
```   104 begin
```
```   105
```
```   106 lift_definition zero_poly :: "'a poly"
```
```   107   is "\<lambda>_. 0"
```
```   108   by (rule MOST_I) simp
```
```   109
```
```   110 instance ..
```
```   111
```
```   112 end
```
```   113
```
```   114 lemma coeff_0 [simp]: "coeff 0 n = 0"
```
```   115   by transfer rule
```
```   116
```
```   117 lemma degree_0 [simp]: "degree 0 = 0"
```
```   118   by (rule order_antisym [OF degree_le le0]) simp
```
```   119
```
```   120 lemma leading_coeff_neq_0:
```
```   121   assumes "p \<noteq> 0"
```
```   122   shows "coeff p (degree p) \<noteq> 0"
```
```   123 proof (cases "degree p")
```
```   124   case 0
```
```   125   from \<open>p \<noteq> 0\<close> obtain n where "coeff p n \<noteq> 0"
```
```   126     by (auto simp add: poly_eq_iff)
```
```   127   then have "n \<le> degree p"
```
```   128     by (rule le_degree)
```
```   129   with \<open>coeff p n \<noteq> 0\<close> and \<open>degree p = 0\<close> show "coeff p (degree p) \<noteq> 0"
```
```   130     by simp
```
```   131 next
```
```   132   case (Suc n)
```
```   133   from \<open>degree p = Suc n\<close> have "n < degree p"
```
```   134     by simp
```
```   135   then have "\<exists>i>n. coeff p i \<noteq> 0"
```
```   136     by (rule less_degree_imp)
```
```   137   then obtain i where "n < i" and "coeff p i \<noteq> 0"
```
```   138     by blast
```
```   139   from \<open>degree p = Suc n\<close> and \<open>n < i\<close> have "degree p \<le> i"
```
```   140     by simp
```
```   141   also from \<open>coeff p i \<noteq> 0\<close> have "i \<le> degree p"
```
```   142     by (rule le_degree)
```
```   143   finally have "degree p = i" .
```
```   144   with \<open>coeff p i \<noteq> 0\<close> show "coeff p (degree p) \<noteq> 0" by simp
```
```   145 qed
```
```   146
```
```   147 lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
```
```   148   by (cases "p = 0") (simp_all add: leading_coeff_neq_0)
```
```   149
```
```   150 lemma eq_zero_or_degree_less:
```
```   151   assumes "degree p \<le> n" and "coeff p n = 0"
```
```   152   shows "p = 0 \<or> degree p < n"
```
```   153 proof (cases n)
```
```   154   case 0
```
```   155   with \<open>degree p \<le> n\<close> and \<open>coeff p n = 0\<close> have "coeff p (degree p) = 0"
```
```   156     by simp
```
```   157   then have "p = 0" by simp
```
```   158   then show ?thesis ..
```
```   159 next
```
```   160   case (Suc m)
```
```   161   from \<open>degree p \<le> n\<close> have "\<forall>i>n. coeff p i = 0"
```
```   162     by (simp add: coeff_eq_0)
```
```   163   with \<open>coeff p n = 0\<close> have "\<forall>i\<ge>n. coeff p i = 0"
```
```   164     by (simp add: le_less)
```
```   165   with \<open>n = Suc m\<close> have "\<forall>i>m. coeff p i = 0"
```
```   166     by (simp add: less_eq_Suc_le)
```
```   167   then have "degree p \<le> m"
```
```   168     by (rule degree_le)
```
```   169   with \<open>n = Suc m\<close> have "degree p < n"
```
```   170     by (simp add: less_Suc_eq_le)
```
```   171   then show ?thesis ..
```
```   172 qed
```
```   173
```
```   174 lemma coeff_0_degree_minus_1: "coeff rrr dr = 0 \<Longrightarrow> degree rrr \<le> dr \<Longrightarrow> degree rrr \<le> dr - 1"
```
```   175   using eq_zero_or_degree_less by fastforce
```
```   176
```
```   177
```
```   178 subsection \<open>List-style constructor for polynomials\<close>
```
```   179
```
```   180 lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   181   is "\<lambda>a p. case_nat a (coeff p)"
```
```   182   by (rule MOST_SucD) (simp add: MOST_coeff_eq_0)
```
```   183
```
```   184 lemmas coeff_pCons = pCons.rep_eq
```
```   185
```
```   186 lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
```
```   187   by transfer simp
```
```   188
```
```   189 lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
```
```   190   by (simp add: coeff_pCons)
```
```   191
```
```   192 lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
```
```   193   by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   194
```
```   195 lemma degree_pCons_eq: "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
```
```   196   apply (rule order_antisym [OF degree_pCons_le])
```
```   197   apply (rule le_degree, simp)
```
```   198   done
```
```   199
```
```   200 lemma degree_pCons_0: "degree (pCons a 0) = 0"
```
```   201   apply (rule order_antisym [OF _ le0])
```
```   202   apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   203   done
```
```   204
```
```   205 lemma degree_pCons_eq_if [simp]: "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
```
```   206   apply (cases "p = 0", simp_all)
```
```   207   apply (rule order_antisym [OF _ le0])
```
```   208   apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   209   apply (rule order_antisym [OF degree_pCons_le])
```
```   210   apply (rule le_degree, simp)
```
```   211   done
```
```   212
```
```   213 lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
```
```   214   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   215
```
```   216 lemma pCons_eq_iff [simp]: "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
```
```   217 proof safe
```
```   218   assume "pCons a p = pCons b q"
```
```   219   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0"
```
```   220     by simp
```
```   221   then show "a = b"
```
```   222     by simp
```
```   223 next
```
```   224   assume "pCons a p = pCons b q"
```
```   225   then have "coeff (pCons a p) (Suc n) = coeff (pCons b q) (Suc n)" for n
```
```   226     by simp
```
```   227   then show "p = q"
```
```   228     by (simp add: poly_eq_iff)
```
```   229 qed
```
```   230
```
```   231 lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
```
```   232   using pCons_eq_iff [of a p 0 0] by simp
```
```   233
```
```   234 lemma pCons_cases [cases type: poly]:
```
```   235   obtains (pCons) a q where "p = pCons a q"
```
```   236 proof
```
```   237   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
```
```   238     by transfer
```
```   239       (simp_all add: MOST_inj[where f=Suc and P="\<lambda>n. p n = 0" for p] fun_eq_iff Abs_poly_inverse
```
```   240         split: nat.split)
```
```   241 qed
```
```   242
```
```   243 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
```
```   244   assumes zero: "P 0"
```
```   245   assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"
```
```   246   shows "P p"
```
```   247 proof (induct p rule: measure_induct_rule [where f=degree])
```
```   248   case (less p)
```
```   249   obtain a q where "p = pCons a q" by (rule pCons_cases)
```
```   250   have "P q"
```
```   251   proof (cases "q = 0")
```
```   252     case True
```
```   253     then show "P q" by (simp add: zero)
```
```   254   next
```
```   255     case False
```
```   256     then have "degree (pCons a q) = Suc (degree q)"
```
```   257       by (rule degree_pCons_eq)
```
```   258     with \<open>p = pCons a q\<close> have "degree q < degree p"
```
```   259       by simp
```
```   260     then show "P q"
```
```   261       by (rule less.hyps)
```
```   262   qed
```
```   263   have "P (pCons a q)"
```
```   264   proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")
```
```   265     case True
```
```   266     with \<open>P q\<close> show ?thesis by (auto intro: pCons)
```
```   267   next
```
```   268     case False
```
```   269     with zero show ?thesis by simp
```
```   270   qed
```
```   271   with \<open>p = pCons a q\<close> show ?case
```
```   272     by simp
```
```   273 qed
```
```   274
```
```   275 lemma degree_eq_zeroE:
```
```   276   fixes p :: "'a::zero poly"
```
```   277   assumes "degree p = 0"
```
```   278   obtains a where "p = pCons a 0"
```
```   279 proof -
```
```   280   obtain a q where p: "p = pCons a q"
```
```   281     by (cases p)
```
```   282   with assms have "q = 0"
```
```   283     by (cases "q = 0") simp_all
```
```   284   with p have "p = pCons a 0"
```
```   285     by simp
```
```   286   then show thesis ..
```
```   287 qed
```
```   288
```
```   289
```
```   290 subsection \<open>Quickcheck generator for polynomials\<close>
```
```   291
```
```   292 quickcheck_generator poly constructors: "0 :: _ poly", pCons
```
```   293
```
```   294
```
```   295 subsection \<open>List-style syntax for polynomials\<close>
```
```   296
```
```   297 syntax "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
```
```   298 translations
```
```   299   "[:x, xs:]" \<rightleftharpoons> "CONST pCons x [:xs:]"
```
```   300   "[:x:]" \<rightleftharpoons> "CONST pCons x 0"
```
```   301   "[:x:]" \<leftharpoondown> "CONST pCons x (_constrain 0 t)"
```
```   302
```
```   303
```
```   304 subsection \<open>Representation of polynomials by lists of coefficients\<close>
```
```   305
```
```   306 primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
```
```   307   where
```
```   308     [code_post]: "Poly [] = 0"
```
```   309   | [code_post]: "Poly (a # as) = pCons a (Poly as)"
```
```   310
```
```   311 lemma Poly_replicate_0 [simp]: "Poly (replicate n 0) = 0"
```
```   312   by (induct n) simp_all
```
```   313
```
```   314 lemma Poly_eq_0: "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
```
```   315   by (induct as) (auto simp add: Cons_replicate_eq)
```
```   316
```
```   317 lemma Poly_append_replicate_zero [simp]: "Poly (as @ replicate n 0) = Poly as"
```
```   318   by (induct as) simp_all
```
```   319
```
```   320 lemma Poly_snoc_zero [simp]: "Poly (as @ [0]) = Poly as"
```
```   321   using Poly_append_replicate_zero [of as 1] by simp
```
```   322
```
```   323 lemma Poly_cCons_eq_pCons_Poly [simp]: "Poly (a ## p) = pCons a (Poly p)"
```
```   324   by (simp add: cCons_def)
```
```   325
```
```   326 lemma Poly_on_rev_starting_with_0 [simp]: "hd as = 0 \<Longrightarrow> Poly (rev (tl as)) = Poly (rev as)"
```
```   327   by (cases as) simp_all
```
```   328
```
```   329 lemma degree_Poly: "degree (Poly xs) \<le> length xs"
```
```   330   by (induct xs) simp_all
```
```   331
```
```   332 lemma coeff_Poly_eq [simp]: "coeff (Poly xs) = nth_default 0 xs"
```
```   333   by (induct xs) (simp_all add: fun_eq_iff coeff_pCons split: nat.splits)
```
```   334
```
```   335 definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
```
```   336   where "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
```
```   337
```
```   338 lemma coeffs_eq_Nil [simp]: "coeffs p = [] \<longleftrightarrow> p = 0"
```
```   339   by (simp add: coeffs_def)
```
```   340
```
```   341 lemma not_0_coeffs_not_Nil: "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
```
```   342   by simp
```
```   343
```
```   344 lemma coeffs_0_eq_Nil [simp]: "coeffs 0 = []"
```
```   345   by simp
```
```   346
```
```   347 lemma coeffs_pCons_eq_cCons [simp]: "coeffs (pCons a p) = a ## coeffs p"
```
```   348 proof -
```
```   349   have *: "\<forall>m\<in>set ms. m > 0 \<Longrightarrow> map (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)"
```
```   350     for ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
```
```   351     by (induct ms) (auto split: nat.split)
```
```   352   show ?thesis
```
```   353     by (simp add: * coeffs_def upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)
```
```   354 qed
```
```   355
```
```   356 lemma length_coeffs: "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = degree p + 1"
```
```   357   by (simp add: coeffs_def)
```
```   358
```
```   359 lemma coeffs_nth: "p \<noteq> 0 \<Longrightarrow> n \<le> degree p \<Longrightarrow> coeffs p ! n = coeff p n"
```
```   360   by (auto simp: coeffs_def simp del: upt_Suc)
```
```   361
```
```   362 lemma coeff_in_coeffs: "p \<noteq> 0 \<Longrightarrow> n \<le> degree p \<Longrightarrow> coeff p n \<in> set (coeffs p)"
```
```   363   using coeffs_nth [of p n, symmetric] by (simp add: length_coeffs)
```
```   364
```
```   365 lemma not_0_cCons_eq [simp]: "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
```
```   366   by (simp add: cCons_def)
```
```   367
```
```   368 lemma Poly_coeffs [simp, code abstype]: "Poly (coeffs p) = p"
```
```   369   by (induct p) auto
```
```   370
```
```   371 lemma coeffs_Poly [simp]: "coeffs (Poly as) = strip_while (HOL.eq 0) as"
```
```   372 proof (induct as)
```
```   373   case Nil
```
```   374   then show ?case by simp
```
```   375 next
```
```   376   case (Cons a as)
```
```   377   from replicate_length_same [of as 0] have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
```
```   378     by (auto dest: sym [of _ as])
```
```   379   with Cons show ?case by auto
```
```   380 qed
```
```   381
```
```   382 lemma no_trailing_coeffs [simp]:
```
```   383   "no_trailing (HOL.eq 0) (coeffs p)"
```
```   384   by (induct p)  auto
```
```   385
```
```   386 lemma strip_while_coeffs [simp]:
```
```   387   "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
```
```   388   by simp
```
```   389
```
```   390 lemma coeffs_eq_iff: "p = q \<longleftrightarrow> coeffs p = coeffs q"
```
```   391   (is "?P \<longleftrightarrow> ?Q")
```
```   392 proof
```
```   393   assume ?P
```
```   394   then show ?Q by simp
```
```   395 next
```
```   396   assume ?Q
```
```   397   then have "Poly (coeffs p) = Poly (coeffs q)" by simp
```
```   398   then show ?P by simp
```
```   399 qed
```
```   400
```
```   401 lemma nth_default_coeffs_eq: "nth_default 0 (coeffs p) = coeff p"
```
```   402   by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
```
```   403
```
```   404 lemma [code]: "coeff p = nth_default 0 (coeffs p)"
```
```   405   by (simp add: nth_default_coeffs_eq)
```
```   406
```
```   407 lemma coeffs_eqI:
```
```   408   assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
```
```   409   assumes zero: "no_trailing (HOL.eq 0) xs"
```
```   410   shows "coeffs p = xs"
```
```   411 proof -
```
```   412   from coeff have "p = Poly xs"
```
```   413     by (simp add: poly_eq_iff)
```
```   414   with zero show ?thesis by simp
```
```   415 qed
```
```   416
```
```   417 lemma degree_eq_length_coeffs [code]: "degree p = length (coeffs p) - 1"
```
```   418   by (simp add: coeffs_def)
```
```   419
```
```   420 lemma length_coeffs_degree: "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
```
```   421   by (induct p) (auto simp: cCons_def)
```
```   422
```
```   423 lemma [code abstract]: "coeffs 0 = []"
```
```   424   by (fact coeffs_0_eq_Nil)
```
```   425
```
```   426 lemma [code abstract]: "coeffs (pCons a p) = a ## coeffs p"
```
```   427   by (fact coeffs_pCons_eq_cCons)
```
```   428
```
```   429 lemma set_coeffs_subset_singleton_0_iff [simp]:
```
```   430   "set (coeffs p) \<subseteq> {0} \<longleftrightarrow> p = 0"
```
```   431   by (auto simp add: coeffs_def intro: classical)
```
```   432
```
```   433 lemma set_coeffs_not_only_0 [simp]:
```
```   434   "set (coeffs p) \<noteq> {0}"
```
```   435   by (auto simp add: set_eq_subset)
```
```   436
```
```   437 lemma forall_coeffs_conv:
```
```   438   "(\<forall>n. P (coeff p n)) \<longleftrightarrow> (\<forall>c \<in> set (coeffs p). P c)" if "P 0"
```
```   439   using that by (auto simp add: coeffs_def)
```
```   440     (metis atLeastLessThan_iff coeff_eq_0 not_less_iff_gr_or_eq zero_le)
```
```   441
```
```   442 instantiation poly :: ("{zero, equal}") equal
```
```   443 begin
```
```   444
```
```   445 definition [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
```
```   446
```
```   447 instance
```
```   448   by standard (simp add: equal equal_poly_def coeffs_eq_iff)
```
```   449
```
```   450 end
```
```   451
```
```   452 lemma [code nbe]: "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
```
```   453   by (fact equal_refl)
```
```   454
```
```   455 definition is_zero :: "'a::zero poly \<Rightarrow> bool"
```
```   456   where [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
```
```   457
```
```   458 lemma is_zero_null [code_abbrev]: "is_zero p \<longleftrightarrow> p = 0"
```
```   459   by (simp add: is_zero_def null_def)
```
```   460
```
```   461
```
```   462 subsubsection \<open>Reconstructing the polynomial from the list\<close>
```
```   463   \<comment> \<open>contributed by Sebastiaan J.C. Joosten and RenÃ© Thiemann\<close>
```
```   464
```
```   465 definition poly_of_list :: "'a::comm_monoid_add list \<Rightarrow> 'a poly"
```
```   466   where [simp]: "poly_of_list = Poly"
```
```   467
```
```   468 lemma poly_of_list_impl [code abstract]: "coeffs (poly_of_list as) = strip_while (HOL.eq 0) as"
```
```   469   by simp
```
```   470
```
```   471
```
```   472 subsection \<open>Fold combinator for polynomials\<close>
```
```   473
```
```   474 definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   475   where "fold_coeffs f p = foldr f (coeffs p)"
```
```   476
```
```   477 lemma fold_coeffs_0_eq [simp]: "fold_coeffs f 0 = id"
```
```   478   by (simp add: fold_coeffs_def)
```
```   479
```
```   480 lemma fold_coeffs_pCons_eq [simp]: "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   481   by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
```
```   482
```
```   483 lemma fold_coeffs_pCons_0_0_eq [simp]: "fold_coeffs f (pCons 0 0) = id"
```
```   484   by (simp add: fold_coeffs_def)
```
```   485
```
```   486 lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
```
```   487   "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   488   by (simp add: fold_coeffs_def)
```
```   489
```
```   490 lemma fold_coeffs_pCons_not_0_0_eq [simp]:
```
```   491   "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   492   by (simp add: fold_coeffs_def)
```
```   493
```
```   494
```
```   495 subsection \<open>Canonical morphism on polynomials -- evaluation\<close>
```
```   496
```
```   497 definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   498   where "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" \<comment> \<open>The Horner Schema\<close>
```
```   499
```
```   500 lemma poly_0 [simp]: "poly 0 x = 0"
```
```   501   by (simp add: poly_def)
```
```   502
```
```   503 lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
```
```   504   by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
```
```   505
```
```   506 lemma poly_altdef: "poly p x = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"
```
```   507   for x :: "'a::{comm_semiring_0,semiring_1}"
```
```   508 proof (induction p rule: pCons_induct)
```
```   509   case 0
```
```   510   then show ?case
```
```   511     by simp
```
```   512 next
```
```   513   case (pCons a p)
```
```   514   show ?case
```
```   515   proof (cases "p = 0")
```
```   516     case True
```
```   517     then show ?thesis by simp
```
```   518   next
```
```   519     case False
```
```   520     let ?p' = "pCons a p"
```
```   521     note poly_pCons[of a p x]
```
```   522     also note pCons.IH
```
```   523     also have "a + x * (\<Sum>i\<le>degree p. coeff p i * x ^ i) =
```
```   524         coeff ?p' 0 * x^0 + (\<Sum>i\<le>degree p. coeff ?p' (Suc i) * x^Suc i)"
```
```   525       by (simp add: field_simps sum_distrib_left coeff_pCons)
```
```   526     also note sum_atMost_Suc_shift[symmetric]
```
```   527     also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric]
```
```   528     finally show ?thesis .
```
```   529   qed
```
```   530 qed
```
```   531
```
```   532 lemma poly_0_coeff_0: "poly p 0 = coeff p 0"
```
```   533   by (cases p) (auto simp: poly_altdef)
```
```   534
```
```   535
```
```   536 subsection \<open>Monomials\<close>
```
```   537
```
```   538 lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
```
```   539   is "\<lambda>a m n. if m = n then a else 0"
```
```   540   by (simp add: MOST_iff_cofinite)
```
```   541
```
```   542 lemma coeff_monom [simp]: "coeff (monom a m) n = (if m = n then a else 0)"
```
```   543   by transfer rule
```
```   544
```
```   545 lemma monom_0: "monom a 0 = pCons a 0"
```
```   546   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   547
```
```   548 lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
```
```   549   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   550
```
```   551 lemma monom_eq_0 [simp]: "monom 0 n = 0"
```
```   552   by (rule poly_eqI) simp
```
```   553
```
```   554 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
```
```   555   by (simp add: poly_eq_iff)
```
```   556
```
```   557 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
```
```   558   by (simp add: poly_eq_iff)
```
```   559
```
```   560 lemma degree_monom_le: "degree (monom a n) \<le> n"
```
```   561   by (rule degree_le, simp)
```
```   562
```
```   563 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
```
```   564   apply (rule order_antisym [OF degree_monom_le])
```
```   565   apply (rule le_degree)
```
```   566   apply simp
```
```   567   done
```
```   568
```
```   569 lemma coeffs_monom [code abstract]:
```
```   570   "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
```
```   571   by (induct n) (simp_all add: monom_0 monom_Suc)
```
```   572
```
```   573 lemma fold_coeffs_monom [simp]: "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
```
```   574   by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
```
```   575
```
```   576 lemma poly_monom: "poly (monom a n) x = a * x ^ n"
```
```   577   for a x :: "'a::comm_semiring_1"
```
```   578   by (cases "a = 0", simp_all) (induct n, simp_all add: mult.left_commute poly_def)
```
```   579
```
```   580 lemma monom_eq_iff': "monom c n = monom d m \<longleftrightarrow>  c = d \<and> (c = 0 \<or> n = m)"
```
```   581   by (auto simp: poly_eq_iff)
```
```   582
```
```   583 lemma monom_eq_const_iff: "monom c n = [:d:] \<longleftrightarrow> c = d \<and> (c = 0 \<or> n = 0)"
```
```   584   using monom_eq_iff'[of c n d 0] by (simp add: monom_0)
```
```   585
```
```   586
```
```   587 subsection \<open>Leading coefficient\<close>
```
```   588
```
```   589 abbreviation lead_coeff:: "'a::zero poly \<Rightarrow> 'a"
```
```   590   where "lead_coeff p \<equiv> coeff p (degree p)"
```
```   591
```
```   592 lemma lead_coeff_pCons[simp]:
```
```   593   "p \<noteq> 0 \<Longrightarrow> lead_coeff (pCons a p) = lead_coeff p"
```
```   594   "p = 0 \<Longrightarrow> lead_coeff (pCons a p) = a"
```
```   595   by auto
```
```   596
```
```   597 lemma lead_coeff_monom [simp]: "lead_coeff (monom c n) = c"
```
```   598   by (cases "c = 0") (simp_all add: degree_monom_eq)
```
```   599
```
```   600 lemma last_coeffs_eq_coeff_degree:
```
```   601   "last (coeffs p) = lead_coeff p" if "p \<noteq> 0"
```
```   602   using that by (simp add: coeffs_def)
```
```   603
```
```   604
```
```   605 subsection \<open>Addition and subtraction\<close>
```
```   606
```
```   607 instantiation poly :: (comm_monoid_add) comm_monoid_add
```
```   608 begin
```
```   609
```
```   610 lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   611   is "\<lambda>p q n. coeff p n + coeff q n"
```
```   612 proof -
```
```   613   fix q p :: "'a poly"
```
```   614   show "\<forall>\<^sub>\<infinity>n. coeff p n + coeff q n = 0"
```
```   615     using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
```
```   616 qed
```
```   617
```
```   618 lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n"
```
```   619   by (simp add: plus_poly.rep_eq)
```
```   620
```
```   621 instance
```
```   622 proof
```
```   623   fix p q r :: "'a poly"
```
```   624   show "(p + q) + r = p + (q + r)"
```
```   625     by (simp add: poly_eq_iff add.assoc)
```
```   626   show "p + q = q + p"
```
```   627     by (simp add: poly_eq_iff add.commute)
```
```   628   show "0 + p = p"
```
```   629     by (simp add: poly_eq_iff)
```
```   630 qed
```
```   631
```
```   632 end
```
```   633
```
```   634 instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
```
```   635 begin
```
```   636
```
```   637 lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   638   is "\<lambda>p q n. coeff p n - coeff q n"
```
```   639 proof -
```
```   640   fix q p :: "'a poly"
```
```   641   show "\<forall>\<^sub>\<infinity>n. coeff p n - coeff q n = 0"
```
```   642     using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
```
```   643 qed
```
```   644
```
```   645 lemma coeff_diff [simp]: "coeff (p - q) n = coeff p n - coeff q n"
```
```   646   by (simp add: minus_poly.rep_eq)
```
```   647
```
```   648 instance
```
```   649 proof
```
```   650   fix p q r :: "'a poly"
```
```   651   show "p + q - p = q"
```
```   652     by (simp add: poly_eq_iff)
```
```   653   show "p - q - r = p - (q + r)"
```
```   654     by (simp add: poly_eq_iff diff_diff_eq)
```
```   655 qed
```
```   656
```
```   657 end
```
```   658
```
```   659 instantiation poly :: (ab_group_add) ab_group_add
```
```   660 begin
```
```   661
```
```   662 lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
```
```   663   is "\<lambda>p n. - coeff p n"
```
```   664 proof -
```
```   665   fix p :: "'a poly"
```
```   666   show "\<forall>\<^sub>\<infinity>n. - coeff p n = 0"
```
```   667     using MOST_coeff_eq_0 by simp
```
```   668 qed
```
```   669
```
```   670 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
```
```   671   by (simp add: uminus_poly.rep_eq)
```
```   672
```
```   673 instance
```
```   674 proof
```
```   675   fix p q :: "'a poly"
```
```   676   show "- p + p = 0"
```
```   677     by (simp add: poly_eq_iff)
```
```   678   show "p - q = p + - q"
```
```   679     by (simp add: poly_eq_iff)
```
```   680 qed
```
```   681
```
```   682 end
```
```   683
```
```   684 lemma add_pCons [simp]: "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```   685   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   686
```
```   687 lemma minus_pCons [simp]: "- pCons a p = pCons (- a) (- p)"
```
```   688   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   689
```
```   690 lemma diff_pCons [simp]: "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```   691   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   692
```
```   693 lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
```
```   694   by (rule degree_le) (auto simp add: coeff_eq_0)
```
```   695
```
```   696 lemma degree_add_le: "degree p \<le> n \<Longrightarrow> degree q \<le> n \<Longrightarrow> degree (p + q) \<le> n"
```
```   697   by (auto intro: order_trans degree_add_le_max)
```
```   698
```
```   699 lemma degree_add_less: "degree p < n \<Longrightarrow> degree q < n \<Longrightarrow> degree (p + q) < n"
```
```   700   by (auto intro: le_less_trans degree_add_le_max)
```
```   701
```
```   702 lemma degree_add_eq_right: "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
```
```   703   apply (cases "q = 0")
```
```   704    apply simp
```
```   705   apply (rule order_antisym)
```
```   706    apply (simp add: degree_add_le)
```
```   707   apply (rule le_degree)
```
```   708   apply (simp add: coeff_eq_0)
```
```   709   done
```
```   710
```
```   711 lemma degree_add_eq_left: "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
```
```   712   using degree_add_eq_right [of q p] by (simp add: add.commute)
```
```   713
```
```   714 lemma degree_minus [simp]: "degree (- p) = degree p"
```
```   715   by (simp add: degree_def)
```
```   716
```
```   717 lemma lead_coeff_add_le: "degree p < degree q \<Longrightarrow> lead_coeff (p + q) = lead_coeff q"
```
```   718   by (metis coeff_add coeff_eq_0 monoid_add_class.add.left_neutral degree_add_eq_right)
```
```   719
```
```   720 lemma lead_coeff_minus: "lead_coeff (- p) = - lead_coeff p"
```
```   721   by (metis coeff_minus degree_minus)
```
```   722
```
```   723 lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
```
```   724   for p q :: "'a::ab_group_add poly"
```
```   725   using degree_add_le [where p=p and q="-q"] by simp
```
```   726
```
```   727 lemma degree_diff_le: "degree p \<le> n \<Longrightarrow> degree q \<le> n \<Longrightarrow> degree (p - q) \<le> n"
```
```   728   for p q :: "'a::ab_group_add poly"
```
```   729   using degree_add_le [of p n "- q"] by simp
```
```   730
```
```   731 lemma degree_diff_less: "degree p < n \<Longrightarrow> degree q < n \<Longrightarrow> degree (p - q) < n"
```
```   732   for p q :: "'a::ab_group_add poly"
```
```   733   using degree_add_less [of p n "- q"] by simp
```
```   734
```
```   735 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
```
```   736   by (rule poly_eqI) simp
```
```   737
```
```   738 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
```
```   739   by (rule poly_eqI) simp
```
```   740
```
```   741 lemma minus_monom: "- monom a n = monom (- a) n"
```
```   742   by (rule poly_eqI) simp
```
```   743
```
```   744 lemma coeff_sum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
```
```   745   by (induct A rule: infinite_finite_induct) simp_all
```
```   746
```
```   747 lemma monom_sum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
```
```   748   by (rule poly_eqI) (simp add: coeff_sum)
```
```   749
```
```   750 fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   751   where
```
```   752     "plus_coeffs xs [] = xs"
```
```   753   | "plus_coeffs [] ys = ys"
```
```   754   | "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
```
```   755
```
```   756 lemma coeffs_plus_eq_plus_coeffs [code abstract]:
```
```   757   "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
```
```   758 proof -
```
```   759   have *: "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
```
```   760     for xs ys :: "'a list" and n
```
```   761   proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
```
```   762     case (3 x xs y ys n)
```
```   763     then show ?case
```
```   764       by (cases n) (auto simp add: cCons_def)
```
```   765   qed simp_all
```
```   766   have **: "no_trailing (HOL.eq 0) (plus_coeffs xs ys)"
```
```   767     if "no_trailing (HOL.eq 0) xs" and "no_trailing (HOL.eq 0) ys"
```
```   768     for xs ys :: "'a list"
```
```   769     using that by (induct xs ys rule: plus_coeffs.induct) (simp_all add: cCons_def)
```
```   770   show ?thesis
```
```   771     by (rule coeffs_eqI) (auto simp add: * nth_default_coeffs_eq intro: **)
```
```   772 qed
```
```   773
```
```   774 lemma coeffs_uminus [code abstract]:
```
```   775   "coeffs (- p) = map uminus (coeffs p)"
```
```   776 proof -
```
```   777   have eq_0: "HOL.eq 0 \<circ> uminus = HOL.eq (0::'a)"
```
```   778     by (simp add: fun_eq_iff)
```
```   779   show ?thesis
```
```   780     by (rule coeffs_eqI) (simp_all add: nth_default_map_eq nth_default_coeffs_eq no_trailing_map eq_0)
```
```   781 qed
```
```   782
```
```   783 lemma [code]: "p - q = p + - q"
```
```   784   for p q :: "'a::ab_group_add poly"
```
```   785   by (fact diff_conv_add_uminus)
```
```   786
```
```   787 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
```
```   788   apply (induct p arbitrary: q)
```
```   789    apply simp
```
```   790   apply (case_tac q, simp, simp add: algebra_simps)
```
```   791   done
```
```   792
```
```   793 lemma poly_minus [simp]: "poly (- p) x = - poly p x"
```
```   794   for x :: "'a::comm_ring"
```
```   795   by (induct p) simp_all
```
```   796
```
```   797 lemma poly_diff [simp]: "poly (p - q) x = poly p x - poly q x"
```
```   798   for x :: "'a::comm_ring"
```
```   799   using poly_add [of p "- q" x] by simp
```
```   800
```
```   801 lemma poly_sum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
```
```   802   by (induct A rule: infinite_finite_induct) simp_all
```
```   803
```
```   804 lemma degree_sum_le: "finite S \<Longrightarrow> (\<And>p. p \<in> S \<Longrightarrow> degree (f p) \<le> n) \<Longrightarrow> degree (sum f S) \<le> n"
```
```   805 proof (induct S rule: finite_induct)
```
```   806   case empty
```
```   807   then show ?case by simp
```
```   808 next
```
```   809   case (insert p S)
```
```   810   then have "degree (sum f S) \<le> n" "degree (f p) \<le> n"
```
```   811     by auto
```
```   812   then show ?case
```
```   813     unfolding sum.insert[OF insert(1-2)] by (metis degree_add_le)
```
```   814 qed
```
```   815
```
```   816 lemma poly_as_sum_of_monoms':
```
```   817   assumes "degree p \<le> n"
```
```   818   shows "(\<Sum>i\<le>n. monom (coeff p i) i) = p"
```
```   819 proof -
```
```   820   have eq: "\<And>i. {..n} \<inter> {i} = (if i \<le> n then {i} else {})"
```
```   821     by auto
```
```   822   from assms show ?thesis
```
```   823     by (simp add: poly_eq_iff coeff_sum coeff_eq_0 sum.If_cases eq
```
```   824         if_distrib[where f="\<lambda>x. x * a" for a])
```
```   825 qed
```
```   826
```
```   827 lemma poly_as_sum_of_monoms: "(\<Sum>i\<le>degree p. monom (coeff p i) i) = p"
```
```   828   by (intro poly_as_sum_of_monoms' order_refl)
```
```   829
```
```   830 lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)"
```
```   831   by (induct xs) (simp_all add: monom_0 monom_Suc)
```
```   832
```
```   833
```
```   834 subsection \<open>Multiplication by a constant, polynomial multiplication and the unit polynomial\<close>
```
```   835
```
```   836 lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   837   is "\<lambda>a p n. a * coeff p n"
```
```   838 proof -
```
```   839   fix a :: 'a and p :: "'a poly"
```
```   840   show "\<forall>\<^sub>\<infinity> i. a * coeff p i = 0"
```
```   841     using MOST_coeff_eq_0[of p] by eventually_elim simp
```
```   842 qed
```
```   843
```
```   844 lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
```
```   845   by (simp add: smult.rep_eq)
```
```   846
```
```   847 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
```
```   848   by (rule degree_le) (simp add: coeff_eq_0)
```
```   849
```
```   850 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
```
```   851   by (rule poly_eqI) (simp add: mult.assoc)
```
```   852
```
```   853 lemma smult_0_right [simp]: "smult a 0 = 0"
```
```   854   by (rule poly_eqI) simp
```
```   855
```
```   856 lemma smult_0_left [simp]: "smult 0 p = 0"
```
```   857   by (rule poly_eqI) simp
```
```   858
```
```   859 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
```
```   860   by (rule poly_eqI) simp
```
```   861
```
```   862 lemma smult_add_right: "smult a (p + q) = smult a p + smult a q"
```
```   863   by (rule poly_eqI) (simp add: algebra_simps)
```
```   864
```
```   865 lemma smult_add_left: "smult (a + b) p = smult a p + smult b p"
```
```   866   by (rule poly_eqI) (simp add: algebra_simps)
```
```   867
```
```   868 lemma smult_minus_right [simp]: "smult a (- p) = - smult a p"
```
```   869   for a :: "'a::comm_ring"
```
```   870   by (rule poly_eqI) simp
```
```   871
```
```   872 lemma smult_minus_left [simp]: "smult (- a) p = - smult a p"
```
```   873   for a :: "'a::comm_ring"
```
```   874   by (rule poly_eqI) simp
```
```   875
```
```   876 lemma smult_diff_right: "smult a (p - q) = smult a p - smult a q"
```
```   877   for a :: "'a::comm_ring"
```
```   878   by (rule poly_eqI) (simp add: algebra_simps)
```
```   879
```
```   880 lemma smult_diff_left: "smult (a - b) p = smult a p - smult b p"
```
```   881   for a b :: "'a::comm_ring"
```
```   882   by (rule poly_eqI) (simp add: algebra_simps)
```
```   883
```
```   884 lemmas smult_distribs =
```
```   885   smult_add_left smult_add_right
```
```   886   smult_diff_left smult_diff_right
```
```   887
```
```   888 lemma smult_pCons [simp]: "smult a (pCons b p) = pCons (a * b) (smult a p)"
```
```   889   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   890
```
```   891 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
```
```   892   by (induct n) (simp_all add: monom_0 monom_Suc)
```
```   893
```
```   894 lemma smult_Poly: "smult c (Poly xs) = Poly (map (op * c) xs)"
```
```   895   by (auto simp: poly_eq_iff nth_default_def)
```
```   896
```
```   897 lemma degree_smult_eq [simp]: "degree (smult a p) = (if a = 0 then 0 else degree p)"
```
```   898   for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}"
```
```   899   by (cases "a = 0") (simp_all add: degree_def)
```
```   900
```
```   901 lemma smult_eq_0_iff [simp]: "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
```
```   902   for a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}"
```
```   903   by (simp add: poly_eq_iff)
```
```   904
```
```   905 lemma coeffs_smult [code abstract]:
```
```   906   "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
```
```   907   for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```   908 proof -
```
```   909   have eq_0: "HOL.eq 0 \<circ> times a = HOL.eq (0::'a)" if "a \<noteq> 0"
```
```   910     using that by (simp add: fun_eq_iff)
```
```   911   show ?thesis
```
```   912     by (rule coeffs_eqI) (auto simp add: no_trailing_map nth_default_map_eq nth_default_coeffs_eq eq_0)
```
```   913 qed
```
```   914
```
```   915 lemma smult_eq_iff:
```
```   916   fixes b :: "'a :: field"
```
```   917   assumes "b \<noteq> 0"
```
```   918   shows "smult a p = smult b q \<longleftrightarrow> smult (a / b) p = q"
```
```   919     (is "?lhs \<longleftrightarrow> ?rhs")
```
```   920 proof
```
```   921   assume ?lhs
```
```   922   also from assms have "smult (inverse b) \<dots> = q"
```
```   923     by simp
```
```   924   finally show ?rhs
```
```   925     by (simp add: field_simps)
```
```   926 next
```
```   927   assume ?rhs
```
```   928   with assms show ?lhs by auto
```
```   929 qed
```
```   930
```
```   931 instantiation poly :: (comm_semiring_0) comm_semiring_0
```
```   932 begin
```
```   933
```
```   934 definition "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
```
```   935
```
```   936 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
```
```   937   by (simp add: times_poly_def)
```
```   938
```
```   939 lemma mult_pCons_left [simp]: "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```   940   by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
```
```   941
```
```   942 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
```
```   943   by (induct p) (simp_all add: mult_poly_0_left)
```
```   944
```
```   945 lemma mult_pCons_right [simp]: "p * pCons a q = smult a p + pCons 0 (p * q)"
```
```   946   by (induct p) (simp_all add: mult_poly_0_left algebra_simps)
```
```   947
```
```   948 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
```
```   949
```
```   950 lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
```
```   951   by (induct p) (simp_all add: mult_poly_0 smult_add_right)
```
```   952
```
```   953 lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
```
```   954   by (induct q) (simp_all add: mult_poly_0 smult_add_right)
```
```   955
```
```   956 lemma mult_poly_add_left: "(p + q) * r = p * r + q * r"
```
```   957   for p q r :: "'a poly"
```
```   958   by (induct r) (simp_all add: mult_poly_0 smult_distribs algebra_simps)
```
```   959
```
```   960 instance
```
```   961 proof
```
```   962   fix p q r :: "'a poly"
```
```   963   show 0: "0 * p = 0"
```
```   964     by (rule mult_poly_0_left)
```
```   965   show "p * 0 = 0"
```
```   966     by (rule mult_poly_0_right)
```
```   967   show "(p + q) * r = p * r + q * r"
```
```   968     by (rule mult_poly_add_left)
```
```   969   show "(p * q) * r = p * (q * r)"
```
```   970     by (induct p) (simp_all add: mult_poly_0 mult_poly_add_left)
```
```   971   show "p * q = q * p"
```
```   972     by (induct p) (simp_all add: mult_poly_0)
```
```   973 qed
```
```   974
```
```   975 end
```
```   976
```
```   977 lemma coeff_mult_degree_sum:
```
```   978   "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"
```
```   979   by (induct p) (simp_all add: coeff_eq_0)
```
```   980
```
```   981 instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors
```
```   982 proof
```
```   983   fix p q :: "'a poly"
```
```   984   assume "p \<noteq> 0" and "q \<noteq> 0"
```
```   985   have "coeff (p * q) (degree p + degree q) = coeff p (degree p) * coeff q (degree q)"
```
```   986     by (rule coeff_mult_degree_sum)
```
```   987   also from \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
```
```   988     by simp
```
```   989   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
```
```   990   then show "p * q \<noteq> 0"
```
```   991     by (simp add: poly_eq_iff)
```
```   992 qed
```
```   993
```
```   994 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
```
```   995
```
```   996 lemma coeff_mult: "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   997 proof (induct p arbitrary: n)
```
```   998   case 0
```
```   999   show ?case by simp
```
```  1000 next
```
```  1001   case (pCons a p n)
```
```  1002   then show ?case
```
```  1003     by (cases n) (simp_all add: sum_atMost_Suc_shift del: sum_atMost_Suc)
```
```  1004 qed
```
```  1005
```
```  1006 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
```
```  1007   apply (rule degree_le)
```
```  1008   apply (induct p)
```
```  1009    apply simp
```
```  1010   apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```  1011   done
```
```  1012
```
```  1013 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
```
```  1014   by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc)
```
```  1015
```
```  1016 instantiation poly :: (comm_semiring_1) comm_semiring_1
```
```  1017 begin
```
```  1018
```
```  1019 lift_definition one_poly :: "'a poly"
```
```  1020   is "\<lambda>n. of_bool (n = 0)"
```
```  1021   by (rule MOST_SucD) simp
```
```  1022
```
```  1023 lemma coeff_1 [simp]:
```
```  1024   "coeff 1 n = of_bool (n = 0)"
```
```  1025   by (simp add: one_poly.rep_eq)
```
```  1026
```
```  1027 lemma one_pCons:
```
```  1028   "1 = [:1:]"
```
```  1029   by (simp add: poly_eq_iff coeff_pCons split: nat.splits)
```
```  1030
```
```  1031 lemma pCons_one:
```
```  1032   "[:1:] = 1"
```
```  1033   by (simp add: one_pCons)
```
```  1034
```
```  1035 instance
```
```  1036   by standard (simp_all add: one_pCons)
```
```  1037
```
```  1038 end
```
```  1039
```
```  1040 lemma poly_1 [simp]:
```
```  1041   "poly 1 x = 1"
```
```  1042   by (simp add: one_pCons)
```
```  1043
```
```  1044 lemma one_poly_eq_simps [simp]:
```
```  1045   "1 = [:1:] \<longleftrightarrow> True"
```
```  1046   "[:1:] = 1 \<longleftrightarrow> True"
```
```  1047   by (simp_all add: one_pCons)
```
```  1048
```
```  1049 lemma degree_1 [simp]:
```
```  1050   "degree 1 = 0"
```
```  1051   by (simp add: one_pCons)
```
```  1052
```
```  1053 lemma coeffs_1_eq [simp, code abstract]:
```
```  1054   "coeffs 1 = [1]"
```
```  1055   by (simp add: one_pCons)
```
```  1056
```
```  1057 lemma smult_one [simp]:
```
```  1058   "smult c 1 = [:c:]"
```
```  1059   by (simp add: one_pCons)
```
```  1060
```
```  1061 lemma monom_eq_1 [simp]:
```
```  1062   "monom 1 0 = 1"
```
```  1063   by (simp add: monom_0 one_pCons)
```
```  1064
```
```  1065 lemma monom_eq_1_iff:
```
```  1066   "monom c n = 1 \<longleftrightarrow> c = 1 \<and> n = 0"
```
```  1067   using monom_eq_const_iff [of c n 1] by auto
```
```  1068
```
```  1069 lemma monom_altdef:
```
```  1070   "monom c n = smult c ([:0, 1:] ^ n)"
```
```  1071   by (induct n) (simp_all add: monom_0 monom_Suc)
```
```  1072
```
```  1073 instance poly :: ("{comm_semiring_1,semiring_1_no_zero_divisors}") semiring_1_no_zero_divisors ..
```
```  1074 instance poly :: (comm_ring) comm_ring ..
```
```  1075 instance poly :: (comm_ring_1) comm_ring_1 ..
```
```  1076 instance poly :: (comm_ring_1) comm_semiring_1_cancel ..
```
```  1077
```
```  1078 lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
```
```  1079   by (induct n) (auto intro: order_trans degree_mult_le)
```
```  1080
```
```  1081 lemma coeff_0_power: "coeff (p ^ n) 0 = coeff p 0 ^ n"
```
```  1082   by (induct n) (simp_all add: coeff_mult)
```
```  1083
```
```  1084 lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
```
```  1085   by (induct p) (simp_all add: algebra_simps)
```
```  1086
```
```  1087 lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
```
```  1088   by (induct p) (simp_all add: algebra_simps)
```
```  1089
```
```  1090 lemma poly_power [simp]: "poly (p ^ n) x = poly p x ^ n"
```
```  1091   for p :: "'a::comm_semiring_1 poly"
```
```  1092   by (induct n) simp_all
```
```  1093
```
```  1094 lemma poly_prod: "poly (\<Prod>k\<in>A. p k) x = (\<Prod>k\<in>A. poly (p k) x)"
```
```  1095   by (induct A rule: infinite_finite_induct) simp_all
```
```  1096
```
```  1097 lemma degree_prod_sum_le: "finite S \<Longrightarrow> degree (prod f S) \<le> sum (degree \<circ> f) S"
```
```  1098 proof (induct S rule: finite_induct)
```
```  1099   case empty
```
```  1100   then show ?case by simp
```
```  1101 next
```
```  1102   case (insert a S)
```
```  1103   show ?case
```
```  1104     unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)]
```
```  1105     by (rule le_trans[OF degree_mult_le]) (use insert in auto)
```
```  1106 qed
```
```  1107
```
```  1108 lemma coeff_0_prod_list: "coeff (prod_list xs) 0 = prod_list (map (\<lambda>p. coeff p 0) xs)"
```
```  1109   by (induct xs) (simp_all add: coeff_mult)
```
```  1110
```
```  1111 lemma coeff_monom_mult: "coeff (monom c n * p) k = (if k < n then 0 else c * coeff p (k - n))"
```
```  1112 proof -
```
```  1113   have "coeff (monom c n * p) k = (\<Sum>i\<le>k. (if n = i then c else 0) * coeff p (k - i))"
```
```  1114     by (simp add: coeff_mult)
```
```  1115   also have "\<dots> = (\<Sum>i\<le>k. (if n = i then c * coeff p (k - i) else 0))"
```
```  1116     by (intro sum.cong) simp_all
```
```  1117   also have "\<dots> = (if k < n then 0 else c * coeff p (k - n))"
```
```  1118     by simp
```
```  1119   finally show ?thesis .
```
```  1120 qed
```
```  1121
```
```  1122 lemma monom_1_dvd_iff': "monom 1 n dvd p \<longleftrightarrow> (\<forall>k<n. coeff p k = 0)"
```
```  1123 proof
```
```  1124   assume "monom 1 n dvd p"
```
```  1125   then obtain r where "p = monom 1 n * r"
```
```  1126     by (rule dvdE)
```
```  1127   then show "\<forall>k<n. coeff p k = 0"
```
```  1128     by (simp add: coeff_mult)
```
```  1129 next
```
```  1130   assume zero: "(\<forall>k<n. coeff p k = 0)"
```
```  1131   define r where "r = Abs_poly (\<lambda>k. coeff p (k + n))"
```
```  1132   have "\<forall>\<^sub>\<infinity>k. coeff p (k + n) = 0"
```
```  1133     by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg,
```
```  1134         subst cofinite_eq_sequentially [symmetric]) transfer
```
```  1135   then have coeff_r [simp]: "coeff r k = coeff p (k + n)" for k
```
```  1136     unfolding r_def by (subst poly.Abs_poly_inverse) simp_all
```
```  1137   have "p = monom 1 n * r"
```
```  1138     by (rule poly_eqI, subst coeff_monom_mult) (simp_all add: zero)
```
```  1139   then show "monom 1 n dvd p" by simp
```
```  1140 qed
```
```  1141
```
```  1142
```
```  1143 subsection \<open>Mapping polynomials\<close>
```
```  1144
```
```  1145 definition map_poly :: "('a :: zero \<Rightarrow> 'b :: zero) \<Rightarrow> 'a poly \<Rightarrow> 'b poly"
```
```  1146   where "map_poly f p = Poly (map f (coeffs p))"
```
```  1147
```
```  1148 lemma map_poly_0 [simp]: "map_poly f 0 = 0"
```
```  1149   by (simp add: map_poly_def)
```
```  1150
```
```  1151 lemma map_poly_1: "map_poly f 1 = [:f 1:]"
```
```  1152   by (simp add: map_poly_def)
```
```  1153
```
```  1154 lemma map_poly_1' [simp]: "f 1 = 1 \<Longrightarrow> map_poly f 1 = 1"
```
```  1155   by (simp add: map_poly_def one_pCons)
```
```  1156
```
```  1157 lemma coeff_map_poly:
```
```  1158   assumes "f 0 = 0"
```
```  1159   shows "coeff (map_poly f p) n = f (coeff p n)"
```
```  1160   by (auto simp: assms map_poly_def nth_default_def coeffs_def not_less Suc_le_eq coeff_eq_0
```
```  1161       simp del: upt_Suc)
```
```  1162
```
```  1163 lemma coeffs_map_poly [code abstract]:
```
```  1164   "coeffs (map_poly f p) = strip_while (op = 0) (map f (coeffs p))"
```
```  1165   by (simp add: map_poly_def)
```
```  1166
```
```  1167 lemma coeffs_map_poly':
```
```  1168   assumes "\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0"
```
```  1169   shows "coeffs (map_poly f p) = map f (coeffs p)"
```
```  1170   using assms
```
```  1171   by (auto simp add: coeffs_map_poly strip_while_idem_iff
```
```  1172     last_coeffs_eq_coeff_degree no_trailing_unfold last_map)
```
```  1173
```
```  1174 lemma set_coeffs_map_poly:
```
```  1175   "(\<And>x. f x = 0 \<longleftrightarrow> x = 0) \<Longrightarrow> set (coeffs (map_poly f p)) = f ` set (coeffs p)"
```
```  1176   by (simp add: coeffs_map_poly')
```
```  1177
```
```  1178 lemma degree_map_poly:
```
```  1179   assumes "\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0"
```
```  1180   shows "degree (map_poly f p) = degree p"
```
```  1181   by (simp add: degree_eq_length_coeffs coeffs_map_poly' assms)
```
```  1182
```
```  1183 lemma map_poly_eq_0_iff:
```
```  1184   assumes "f 0 = 0" "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0"
```
```  1185   shows "map_poly f p = 0 \<longleftrightarrow> p = 0"
```
```  1186 proof -
```
```  1187   have "(coeff (map_poly f p) n = 0) = (coeff p n = 0)" for n
```
```  1188   proof -
```
```  1189     have "coeff (map_poly f p) n = f (coeff p n)"
```
```  1190       by (simp add: coeff_map_poly assms)
```
```  1191     also have "\<dots> = 0 \<longleftrightarrow> coeff p n = 0"
```
```  1192     proof (cases "n < length (coeffs p)")
```
```  1193       case True
```
```  1194       then have "coeff p n \<in> set (coeffs p)"
```
```  1195         by (auto simp: coeffs_def simp del: upt_Suc)
```
```  1196       with assms show "f (coeff p n) = 0 \<longleftrightarrow> coeff p n = 0"
```
```  1197         by auto
```
```  1198     next
```
```  1199       case False
```
```  1200       then show ?thesis
```
```  1201         by (auto simp: assms length_coeffs nth_default_coeffs_eq [symmetric] nth_default_def)
```
```  1202     qed
```
```  1203     finally show ?thesis .
```
```  1204   qed
```
```  1205   then show ?thesis by (auto simp: poly_eq_iff)
```
```  1206 qed
```
```  1207
```
```  1208 lemma map_poly_smult:
```
```  1209   assumes "f 0 = 0""\<And>c x. f (c * x) = f c * f x"
```
```  1210   shows "map_poly f (smult c p) = smult (f c) (map_poly f p)"
```
```  1211   by (intro poly_eqI) (simp_all add: assms coeff_map_poly)
```
```  1212
```
```  1213 lemma map_poly_pCons:
```
```  1214   assumes "f 0 = 0"
```
```  1215   shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
```
```  1216   by (intro poly_eqI) (simp_all add: assms coeff_map_poly coeff_pCons split: nat.splits)
```
```  1217
```
```  1218 lemma map_poly_map_poly:
```
```  1219   assumes "f 0 = 0" "g 0 = 0"
```
```  1220   shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
```
```  1221   by (intro poly_eqI) (simp add: coeff_map_poly assms)
```
```  1222
```
```  1223 lemma map_poly_id [simp]: "map_poly id p = p"
```
```  1224   by (simp add: map_poly_def)
```
```  1225
```
```  1226 lemma map_poly_id' [simp]: "map_poly (\<lambda>x. x) p = p"
```
```  1227   by (simp add: map_poly_def)
```
```  1228
```
```  1229 lemma map_poly_cong:
```
```  1230   assumes "(\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = g x)"
```
```  1231   shows "map_poly f p = map_poly g p"
```
```  1232 proof -
```
```  1233   from assms have "map f (coeffs p) = map g (coeffs p)"
```
```  1234     by (intro map_cong) simp_all
```
```  1235   then show ?thesis
```
```  1236     by (simp only: coeffs_eq_iff coeffs_map_poly)
```
```  1237 qed
```
```  1238
```
```  1239 lemma map_poly_monom: "f 0 = 0 \<Longrightarrow> map_poly f (monom c n) = monom (f c) n"
```
```  1240   by (intro poly_eqI) (simp_all add: coeff_map_poly)
```
```  1241
```
```  1242 lemma map_poly_idI:
```
```  1243   assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x"
```
```  1244   shows "map_poly f p = p"
```
```  1245   using map_poly_cong[OF assms, of _ id] by simp
```
```  1246
```
```  1247 lemma map_poly_idI':
```
```  1248   assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x"
```
```  1249   shows "p = map_poly f p"
```
```  1250   using map_poly_cong[OF assms, of _ id] by simp
```
```  1251
```
```  1252 lemma smult_conv_map_poly: "smult c p = map_poly (\<lambda>x. c * x) p"
```
```  1253   by (intro poly_eqI) (simp_all add: coeff_map_poly)
```
```  1254
```
```  1255
```
```  1256 subsection \<open>Conversions\<close>
```
```  1257
```
```  1258 lemma of_nat_poly:
```
```  1259   "of_nat n = [:of_nat n:]"
```
```  1260   by (induct n) (simp_all add: one_pCons)
```
```  1261
```
```  1262 lemma of_nat_monom:
```
```  1263   "of_nat n = monom (of_nat n) 0"
```
```  1264   by (simp add: of_nat_poly monom_0)
```
```  1265
```
```  1266 lemma degree_of_nat [simp]:
```
```  1267   "degree (of_nat n) = 0"
```
```  1268   by (simp add: of_nat_poly)
```
```  1269
```
```  1270 lemma lead_coeff_of_nat [simp]:
```
```  1271   "lead_coeff (of_nat n) = of_nat n"
```
```  1272   by (simp add: of_nat_poly)
```
```  1273
```
```  1274 lemma of_int_poly:
```
```  1275   "of_int k = [:of_int k:]"
```
```  1276   by (simp only: of_int_of_nat of_nat_poly) simp
```
```  1277
```
```  1278 lemma of_int_monom:
```
```  1279   "of_int k = monom (of_int k) 0"
```
```  1280   by (simp add: of_int_poly monom_0)
```
```  1281
```
```  1282 lemma degree_of_int [simp]:
```
```  1283   "degree (of_int k) = 0"
```
```  1284   by (simp add: of_int_poly)
```
```  1285
```
```  1286 lemma lead_coeff_of_int [simp]:
```
```  1287   "lead_coeff (of_int k) = of_int k"
```
```  1288   by (simp add: of_int_poly)
```
```  1289
```
```  1290 lemma numeral_poly: "numeral n = [:numeral n:]"
```
```  1291 proof -
```
```  1292   have "numeral n = of_nat (numeral n)"
```
```  1293     by simp
```
```  1294   also have "\<dots> = [:of_nat (numeral n):]"
```
```  1295     by (simp add: of_nat_poly)
```
```  1296   finally show ?thesis
```
```  1297     by simp
```
```  1298 qed
```
```  1299
```
```  1300 lemma numeral_monom:
```
```  1301   "numeral n = monom (numeral n) 0"
```
```  1302   by (simp add: numeral_poly monom_0)
```
```  1303
```
```  1304 lemma degree_numeral [simp]:
```
```  1305   "degree (numeral n) = 0"
```
```  1306   by (simp add: numeral_poly)
```
```  1307
```
```  1308 lemma lead_coeff_numeral [simp]:
```
```  1309   "lead_coeff (numeral n) = numeral n"
```
```  1310   by (simp add: numeral_poly)
```
```  1311
```
```  1312
```
```  1313 subsection \<open>Lemmas about divisibility\<close>
```
```  1314
```
```  1315 lemma dvd_smult:
```
```  1316   assumes "p dvd q"
```
```  1317   shows "p dvd smult a q"
```
```  1318 proof -
```
```  1319   from assms obtain k where "q = p * k" ..
```
```  1320   then have "smult a q = p * smult a k" by simp
```
```  1321   then show "p dvd smult a q" ..
```
```  1322 qed
```
```  1323
```
```  1324 lemma dvd_smult_cancel: "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
```
```  1325   for a :: "'a::field"
```
```  1326   by (drule dvd_smult [where a="inverse a"]) simp
```
```  1327
```
```  1328 lemma dvd_smult_iff: "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
```
```  1329   for a :: "'a::field"
```
```  1330   by (safe elim!: dvd_smult dvd_smult_cancel)
```
```  1331
```
```  1332 lemma smult_dvd_cancel:
```
```  1333   assumes "smult a p dvd q"
```
```  1334   shows "p dvd q"
```
```  1335 proof -
```
```  1336   from assms obtain k where "q = smult a p * k" ..
```
```  1337   then have "q = p * smult a k" by simp
```
```  1338   then show "p dvd q" ..
```
```  1339 qed
```
```  1340
```
```  1341 lemma smult_dvd: "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
```
```  1342   for a :: "'a::field"
```
```  1343   by (rule smult_dvd_cancel [where a="inverse a"]) simp
```
```  1344
```
```  1345 lemma smult_dvd_iff: "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
```
```  1346   for a :: "'a::field"
```
```  1347   by (auto elim: smult_dvd smult_dvd_cancel)
```
```  1348
```
```  1349 lemma is_unit_smult_iff: "smult c p dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1"
```
```  1350 proof -
```
```  1351   have "smult c p = [:c:] * p" by simp
```
```  1352   also have "\<dots> dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1"
```
```  1353   proof safe
```
```  1354     assume *: "[:c:] * p dvd 1"
```
```  1355     then show "p dvd 1"
```
```  1356       by (rule dvd_mult_right)
```
```  1357     from * obtain q where q: "1 = [:c:] * p * q"
```
```  1358       by (rule dvdE)
```
```  1359     have "c dvd c * (coeff p 0 * coeff q 0)"
```
```  1360       by simp
```
```  1361     also have "\<dots> = coeff ([:c:] * p * q) 0"
```
```  1362       by (simp add: mult.assoc coeff_mult)
```
```  1363     also note q [symmetric]
```
```  1364     finally have "c dvd coeff 1 0" .
```
```  1365     then show "c dvd 1" by simp
```
```  1366   next
```
```  1367     assume "c dvd 1" "p dvd 1"
```
```  1368     from this(1) obtain d where "1 = c * d"
```
```  1369       by (rule dvdE)
```
```  1370     then have "1 = [:c:] * [:d:]"
```
```  1371       by (simp add: one_pCons ac_simps)
```
```  1372     then have "[:c:] dvd 1"
```
```  1373       by (rule dvdI)
```
```  1374     from mult_dvd_mono[OF this \<open>p dvd 1\<close>] show "[:c:] * p dvd 1"
```
```  1375       by simp
```
```  1376   qed
```
```  1377   finally show ?thesis .
```
```  1378 qed
```
```  1379
```
```  1380
```
```  1381 subsection \<open>Polynomials form an integral domain\<close>
```
```  1382
```
```  1383 instance poly :: (idom) idom ..
```
```  1384
```
```  1385 instance poly :: ("{ring_char_0, comm_ring_1}") ring_char_0
```
```  1386   by standard (auto simp add: of_nat_poly intro: injI)
```
```  1387
```
```  1388 lemma degree_mult_eq: "p \<noteq> 0 \<Longrightarrow> q \<noteq> 0 \<Longrightarrow> degree (p * q) = degree p + degree q"
```
```  1389   for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  1390   by (rule order_antisym [OF degree_mult_le le_degree]) (simp add: coeff_mult_degree_sum)
```
```  1391
```
```  1392 lemma degree_mult_eq_0:
```
```  1393   "degree (p * q) = 0 \<longleftrightarrow> p = 0 \<or> q = 0 \<or> (p \<noteq> 0 \<and> q \<noteq> 0 \<and> degree p = 0 \<and> degree q = 0)"
```
```  1394   for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  1395   by (auto simp: degree_mult_eq)
```
```  1396
```
```  1397 lemma degree_power_eq: "p \<noteq> 0 \<Longrightarrow> degree ((p :: 'a :: idom poly) ^ n) = n * degree p"
```
```  1398   by (induction n) (simp_all add: degree_mult_eq)
```
```  1399
```
```  1400 lemma degree_mult_right_le:
```
```  1401   fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  1402   assumes "q \<noteq> 0"
```
```  1403   shows "degree p \<le> degree (p * q)"
```
```  1404   using assms by (cases "p = 0") (simp_all add: degree_mult_eq)
```
```  1405
```
```  1406 lemma coeff_degree_mult: "coeff (p * q) (degree (p * q)) = coeff q (degree q) * coeff p (degree p)"
```
```  1407   for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  1408   by (cases "p = 0 \<or> q = 0") (auto simp: degree_mult_eq coeff_mult_degree_sum mult_ac)
```
```  1409
```
```  1410 lemma dvd_imp_degree_le: "p dvd q \<Longrightarrow> q \<noteq> 0 \<Longrightarrow> degree p \<le> degree q"
```
```  1411   for p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  1412   by (erule dvdE, hypsubst, subst degree_mult_eq) auto
```
```  1413
```
```  1414 lemma divides_degree:
```
```  1415   fixes p q :: "'a ::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  1416   assumes "p dvd q"
```
```  1417   shows "degree p \<le> degree q \<or> q = 0"
```
```  1418   by (metis dvd_imp_degree_le assms)
```
```  1419
```
```  1420 lemma const_poly_dvd_iff:
```
```  1421   fixes c :: "'a::{comm_semiring_1,semiring_no_zero_divisors}"
```
```  1422   shows "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)"
```
```  1423 proof (cases "c = 0 \<or> p = 0")
```
```  1424   case True
```
```  1425   then show ?thesis
```
```  1426     by (auto intro!: poly_eqI)
```
```  1427 next
```
```  1428   case False
```
```  1429   show ?thesis
```
```  1430   proof
```
```  1431     assume "[:c:] dvd p"
```
```  1432     then show "\<forall>n. c dvd coeff p n"
```
```  1433       by (auto elim!: dvdE simp: coeffs_def)
```
```  1434   next
```
```  1435     assume *: "\<forall>n. c dvd coeff p n"
```
```  1436     define mydiv where "mydiv x y = (SOME z. x = y * z)" for x y :: 'a
```
```  1437     have mydiv: "x = y * mydiv x y" if "y dvd x" for x y
```
```  1438       using that unfolding mydiv_def dvd_def by (rule someI_ex)
```
```  1439     define q where "q = Poly (map (\<lambda>a. mydiv a c) (coeffs p))"
```
```  1440     from False * have "p = q * [:c:]"
```
```  1441       by (intro poly_eqI)
```
```  1442         (auto simp: q_def nth_default_def not_less length_coeffs_degree coeffs_nth
```
```  1443           intro!: coeff_eq_0 mydiv)
```
```  1444     then show "[:c:] dvd p"
```
```  1445       by (simp only: dvd_triv_right)
```
```  1446   qed
```
```  1447 qed
```
```  1448
```
```  1449 lemma const_poly_dvd_const_poly_iff [simp]: "[:a:] dvd [:b:] \<longleftrightarrow> a dvd b"
```
```  1450   for a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors}"
```
```  1451   by (subst const_poly_dvd_iff) (auto simp: coeff_pCons split: nat.splits)
```
```  1452
```
```  1453 lemma lead_coeff_mult: "lead_coeff (p * q) = lead_coeff p * lead_coeff q"
```
```  1454   for p q :: "'a::{comm_semiring_0, semiring_no_zero_divisors} poly"
```
```  1455   by (cases "p = 0 \<or> q = 0") (auto simp: coeff_mult_degree_sum degree_mult_eq)
```
```  1456
```
```  1457 lemma lead_coeff_smult: "lead_coeff (smult c p) = c * lead_coeff p"
```
```  1458   for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  1459 proof -
```
```  1460   have "smult c p = [:c:] * p" by simp
```
```  1461   also have "lead_coeff \<dots> = c * lead_coeff p"
```
```  1462     by (subst lead_coeff_mult) simp_all
```
```  1463   finally show ?thesis .
```
```  1464 qed
```
```  1465
```
```  1466 lemma lead_coeff_1 [simp]: "lead_coeff 1 = 1"
```
```  1467   by simp
```
```  1468
```
```  1469 lemma lead_coeff_power: "lead_coeff (p ^ n) = lead_coeff p ^ n"
```
```  1470   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  1471   by (induct n) (simp_all add: lead_coeff_mult)
```
```  1472
```
```  1473
```
```  1474 subsection \<open>Polynomials form an ordered integral domain\<close>
```
```  1475
```
```  1476 definition pos_poly :: "'a::linordered_semidom poly \<Rightarrow> bool"
```
```  1477   where "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
```
```  1478
```
```  1479 lemma pos_poly_pCons: "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
```
```  1480   by (simp add: pos_poly_def)
```
```  1481
```
```  1482 lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
```
```  1483   by (simp add: pos_poly_def)
```
```  1484
```
```  1485 lemma pos_poly_add: "pos_poly p \<Longrightarrow> pos_poly q \<Longrightarrow> pos_poly (p + q)"
```
```  1486   apply (induct p arbitrary: q)
```
```  1487    apply simp
```
```  1488   apply (case_tac q)
```
```  1489   apply (force simp add: pos_poly_pCons add_pos_pos)
```
```  1490   done
```
```  1491
```
```  1492 lemma pos_poly_mult: "pos_poly p \<Longrightarrow> pos_poly q \<Longrightarrow> pos_poly (p * q)"
```
```  1493   unfolding pos_poly_def
```
```  1494   apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
```
```  1495    apply (simp add: degree_mult_eq coeff_mult_degree_sum)
```
```  1496   apply auto
```
```  1497   done
```
```  1498
```
```  1499 lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
```
```  1500   for p :: "'a::linordered_idom poly"
```
```  1501   by (induct p) (auto simp: pos_poly_pCons)
```
```  1502
```
```  1503 lemma pos_poly_coeffs [code]: "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)"
```
```  1504   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1505 proof
```
```  1506   assume ?rhs
```
```  1507   then show ?lhs
```
```  1508     by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
```
```  1509 next
```
```  1510   assume ?lhs
```
```  1511   then have *: "0 < coeff p (degree p)"
```
```  1512     by (simp add: pos_poly_def)
```
```  1513   then have "p \<noteq> 0"
```
```  1514     by auto
```
```  1515   with * show ?rhs
```
```  1516     by (simp add: last_coeffs_eq_coeff_degree)
```
```  1517 qed
```
```  1518
```
```  1519 instantiation poly :: (linordered_idom) linordered_idom
```
```  1520 begin
```
```  1521
```
```  1522 definition "x < y \<longleftrightarrow> pos_poly (y - x)"
```
```  1523
```
```  1524 definition "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
```
```  1525
```
```  1526 definition "\<bar>x::'a poly\<bar> = (if x < 0 then - x else x)"
```
```  1527
```
```  1528 definition "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```  1529
```
```  1530 instance
```
```  1531 proof
```
```  1532   fix x y z :: "'a poly"
```
```  1533   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```  1534     unfolding less_eq_poly_def less_poly_def
```
```  1535     apply safe
```
```  1536      apply simp
```
```  1537     apply (drule (1) pos_poly_add)
```
```  1538     apply simp
```
```  1539     done
```
```  1540   show "x \<le> x"
```
```  1541     by (simp add: less_eq_poly_def)
```
```  1542   show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
```
```  1543     unfolding less_eq_poly_def
```
```  1544     apply safe
```
```  1545     apply (drule (1) pos_poly_add)
```
```  1546     apply (simp add: algebra_simps)
```
```  1547     done
```
```  1548   show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
```
```  1549     unfolding less_eq_poly_def
```
```  1550     apply safe
```
```  1551     apply (drule (1) pos_poly_add)
```
```  1552     apply simp
```
```  1553     done
```
```  1554   show "x \<le> y \<Longrightarrow> z + x \<le> z + y"
```
```  1555     unfolding less_eq_poly_def
```
```  1556     apply safe
```
```  1557     apply (simp add: algebra_simps)
```
```  1558     done
```
```  1559   show "x \<le> y \<or> y \<le> x"
```
```  1560     unfolding less_eq_poly_def
```
```  1561     using pos_poly_total [of "x - y"]
```
```  1562     by auto
```
```  1563   show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y"
```
```  1564     by (simp add: less_poly_def right_diff_distrib [symmetric] pos_poly_mult)
```
```  1565   show "\<bar>x\<bar> = (if x < 0 then - x else x)"
```
```  1566     by (rule abs_poly_def)
```
```  1567   show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```  1568     by (rule sgn_poly_def)
```
```  1569 qed
```
```  1570
```
```  1571 end
```
```  1572
```
```  1573 text \<open>TODO: Simplification rules for comparisons\<close>
```
```  1574
```
```  1575
```
```  1576 subsection \<open>Synthetic division and polynomial roots\<close>
```
```  1577
```
```  1578 subsubsection \<open>Synthetic division\<close>
```
```  1579
```
```  1580 text \<open>Synthetic division is simply division by the linear polynomial @{term "x - c"}.\<close>
```
```  1581
```
```  1582 definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
```
```  1583   where "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
```
```  1584
```
```  1585 definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```  1586   where "synthetic_div p c = fst (synthetic_divmod p c)"
```
```  1587
```
```  1588 lemma synthetic_divmod_0 [simp]: "synthetic_divmod 0 c = (0, 0)"
```
```  1589   by (simp add: synthetic_divmod_def)
```
```  1590
```
```  1591 lemma synthetic_divmod_pCons [simp]:
```
```  1592   "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
```
```  1593   by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
```
```  1594
```
```  1595 lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
```
```  1596   by (simp add: synthetic_div_def)
```
```  1597
```
```  1598 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
```
```  1599   by (induct p arbitrary: a) simp_all
```
```  1600
```
```  1601 lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
```
```  1602   by (induct p) (simp_all add: split_def)
```
```  1603
```
```  1604 lemma synthetic_div_pCons [simp]:
```
```  1605   "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
```
```  1606   by (simp add: synthetic_div_def split_def snd_synthetic_divmod)
```
```  1607
```
```  1608 lemma synthetic_div_eq_0_iff: "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
```
```  1609 proof (induct p)
```
```  1610   case 0
```
```  1611   then show ?case by simp
```
```  1612 next
```
```  1613   case (pCons a p)
```
```  1614   then show ?case by (cases p) simp
```
```  1615 qed
```
```  1616
```
```  1617 lemma degree_synthetic_div: "degree (synthetic_div p c) = degree p - 1"
```
```  1618   by (induct p) (simp_all add: synthetic_div_eq_0_iff)
```
```  1619
```
```  1620 lemma synthetic_div_correct:
```
```  1621   "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
```
```  1622   by (induct p) simp_all
```
```  1623
```
```  1624 lemma synthetic_div_unique: "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
```
```  1625   apply (induct p arbitrary: q r)
```
```  1626    apply simp
```
```  1627    apply (frule synthetic_div_unique_lemma)
```
```  1628    apply simp
```
```  1629   apply (case_tac q, force)
```
```  1630   done
```
```  1631
```
```  1632 lemma synthetic_div_correct': "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
```
```  1633   for c :: "'a::comm_ring_1"
```
```  1634   using synthetic_div_correct [of p c] by (simp add: algebra_simps)
```
```  1635
```
```  1636
```
```  1637 subsubsection \<open>Polynomial roots\<close>
```
```  1638
```
```  1639 lemma poly_eq_0_iff_dvd: "poly p c = 0 \<longleftrightarrow> [:- c, 1:] dvd p"
```
```  1640   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1641   for c :: "'a::comm_ring_1"
```
```  1642 proof
```
```  1643   assume ?lhs
```
```  1644   with synthetic_div_correct' [of c p] have "p = [:-c, 1:] * synthetic_div p c" by simp
```
```  1645   then show ?rhs ..
```
```  1646 next
```
```  1647   assume ?rhs
```
```  1648   then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
```
```  1649   then show ?lhs by simp
```
```  1650 qed
```
```  1651
```
```  1652 lemma dvd_iff_poly_eq_0: "[:c, 1:] dvd p \<longleftrightarrow> poly p (- c) = 0"
```
```  1653   for c :: "'a::comm_ring_1"
```
```  1654   by (simp add: poly_eq_0_iff_dvd)
```
```  1655
```
```  1656 lemma poly_roots_finite: "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
```
```  1657   for p :: "'a::{comm_ring_1,ring_no_zero_divisors} poly"
```
```  1658 proof (induct n \<equiv> "degree p" arbitrary: p)
```
```  1659   case 0
```
```  1660   then obtain a where "a \<noteq> 0" and "p = [:a:]"
```
```  1661     by (cases p) (simp split: if_splits)
```
```  1662   then show "finite {x. poly p x = 0}"
```
```  1663     by simp
```
```  1664 next
```
```  1665   case (Suc n)
```
```  1666   show "finite {x. poly p x = 0}"
```
```  1667   proof (cases "\<exists>x. poly p x = 0")
```
```  1668     case False
```
```  1669     then show "finite {x. poly p x = 0}" by simp
```
```  1670   next
```
```  1671     case True
```
```  1672     then obtain a where "poly p a = 0" ..
```
```  1673     then have "[:-a, 1:] dvd p"
```
```  1674       by (simp only: poly_eq_0_iff_dvd)
```
```  1675     then obtain k where k: "p = [:-a, 1:] * k" ..
```
```  1676     with \<open>p \<noteq> 0\<close> have "k \<noteq> 0"
```
```  1677       by auto
```
```  1678     with k have "degree p = Suc (degree k)"
```
```  1679       by (simp add: degree_mult_eq del: mult_pCons_left)
```
```  1680     with \<open>Suc n = degree p\<close> have "n = degree k"
```
```  1681       by simp
```
```  1682     from this \<open>k \<noteq> 0\<close> have "finite {x. poly k x = 0}"
```
```  1683       by (rule Suc.hyps)
```
```  1684     then have "finite (insert a {x. poly k x = 0})"
```
```  1685       by simp
```
```  1686     then show "finite {x. poly p x = 0}"
```
```  1687       by (simp add: k Collect_disj_eq del: mult_pCons_left)
```
```  1688   qed
```
```  1689 qed
```
```  1690
```
```  1691 lemma poly_eq_poly_eq_iff: "poly p = poly q \<longleftrightarrow> p = q"
```
```  1692   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1693   for p q :: "'a::{comm_ring_1,ring_no_zero_divisors,ring_char_0} poly"
```
```  1694 proof
```
```  1695   assume ?rhs
```
```  1696   then show ?lhs by simp
```
```  1697 next
```
```  1698   assume ?lhs
```
```  1699   have "poly p = poly 0 \<longleftrightarrow> p = 0" for p :: "'a poly"
```
```  1700     apply (cases "p = 0")
```
```  1701      apply simp_all
```
```  1702     apply (drule poly_roots_finite)
```
```  1703     apply (auto simp add: infinite_UNIV_char_0)
```
```  1704     done
```
```  1705   from \<open>?lhs\<close> and this [of "p - q"] show ?rhs
```
```  1706     by auto
```
```  1707 qed
```
```  1708
```
```  1709 lemma poly_all_0_iff_0: "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
```
```  1710   for p :: "'a::{ring_char_0,comm_ring_1,ring_no_zero_divisors} poly"
```
```  1711   by (auto simp add: poly_eq_poly_eq_iff [symmetric])
```
```  1712
```
```  1713
```
```  1714 subsubsection \<open>Order of polynomial roots\<close>
```
```  1715
```
```  1716 definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
```
```  1717   where "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
```
```  1718
```
```  1719 lemma coeff_linear_power: "coeff ([:a, 1:] ^ n) n = 1"
```
```  1720   for a :: "'a::comm_semiring_1"
```
```  1721   apply (induct n)
```
```  1722    apply simp_all
```
```  1723   apply (subst coeff_eq_0)
```
```  1724    apply (auto intro: le_less_trans degree_power_le)
```
```  1725   done
```
```  1726
```
```  1727 lemma degree_linear_power: "degree ([:a, 1:] ^ n) = n"
```
```  1728   for a :: "'a::comm_semiring_1"
```
```  1729   apply (rule order_antisym)
```
```  1730    apply (rule ord_le_eq_trans [OF degree_power_le])
```
```  1731    apply simp
```
```  1732   apply (rule le_degree)
```
```  1733   apply (simp add: coeff_linear_power)
```
```  1734   done
```
```  1735
```
```  1736 lemma order_1: "[:-a, 1:] ^ order a p dvd p"
```
```  1737   apply (cases "p = 0")
```
```  1738    apply simp
```
```  1739   apply (cases "order a p")
```
```  1740    apply simp
```
```  1741   apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
```
```  1742    apply (drule not_less_Least)
```
```  1743    apply simp
```
```  1744   apply (fold order_def)
```
```  1745   apply simp
```
```  1746   done
```
```  1747
```
```  1748 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1749   unfolding order_def
```
```  1750   apply (rule LeastI_ex)
```
```  1751   apply (rule_tac x="degree p" in exI)
```
```  1752   apply (rule notI)
```
```  1753   apply (drule (1) dvd_imp_degree_le)
```
```  1754   apply (simp only: degree_linear_power)
```
```  1755   done
```
```  1756
```
```  1757 lemma order: "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1758   by (rule conjI [OF order_1 order_2])
```
```  1759
```
```  1760 lemma order_degree:
```
```  1761   assumes p: "p \<noteq> 0"
```
```  1762   shows "order a p \<le> degree p"
```
```  1763 proof -
```
```  1764   have "order a p = degree ([:-a, 1:] ^ order a p)"
```
```  1765     by (simp only: degree_linear_power)
```
```  1766   also from order_1 p have "\<dots> \<le> degree p"
```
```  1767     by (rule dvd_imp_degree_le)
```
```  1768   finally show ?thesis .
```
```  1769 qed
```
```  1770
```
```  1771 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
```
```  1772   apply (cases "p = 0")
```
```  1773    apply simp_all
```
```  1774   apply (rule iffI)
```
```  1775    apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)
```
```  1776   unfolding poly_eq_0_iff_dvd
```
```  1777   apply (metis dvd_power dvd_trans order_1)
```
```  1778   done
```
```  1779
```
```  1780 lemma order_0I: "poly p a \<noteq> 0 \<Longrightarrow> order a p = 0"
```
```  1781   by (subst (asm) order_root) auto
```
```  1782
```
```  1783 lemma order_unique_lemma:
```
```  1784   fixes p :: "'a::idom poly"
```
```  1785   assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
```
```  1786   shows "n = order a p"
```
```  1787   unfolding Polynomial.order_def
```
```  1788   apply (rule Least_equality [symmetric])
```
```  1789    apply (fact assms)
```
```  1790   apply (rule classical)
```
```  1791   apply (erule notE)
```
```  1792   unfolding not_less_eq_eq
```
```  1793   using assms(1)
```
```  1794   apply (rule power_le_dvd)
```
```  1795   apply assumption
```
```  1796   done
```
```  1797
```
```  1798 lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
```
```  1799 proof -
```
```  1800   define i where "i = order a p"
```
```  1801   define j where "j = order a q"
```
```  1802   define t where "t = [:-a, 1:]"
```
```  1803   have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
```
```  1804     by (simp add: t_def dvd_iff_poly_eq_0)
```
```  1805   assume "p * q \<noteq> 0"
```
```  1806   then show "order a (p * q) = i + j"
```
```  1807     apply clarsimp
```
```  1808     apply (drule order [where a=a and p=p, folded i_def t_def])
```
```  1809     apply (drule order [where a=a and p=q, folded j_def t_def])
```
```  1810     apply clarify
```
```  1811     apply (erule dvdE)+
```
```  1812     apply (rule order_unique_lemma [symmetric], fold t_def)
```
```  1813      apply (simp_all add: power_add t_dvd_iff)
```
```  1814     done
```
```  1815 qed
```
```  1816
```
```  1817 lemma order_smult:
```
```  1818   assumes "c \<noteq> 0"
```
```  1819   shows "order x (smult c p) = order x p"
```
```  1820 proof (cases "p = 0")
```
```  1821   case True
```
```  1822   then show ?thesis
```
```  1823     by simp
```
```  1824 next
```
```  1825   case False
```
```  1826   have "smult c p = [:c:] * p" by simp
```
```  1827   also from assms False have "order x \<dots> = order x [:c:] + order x p"
```
```  1828     by (subst order_mult) simp_all
```
```  1829   also have "order x [:c:] = 0"
```
```  1830     by (rule order_0I) (use assms in auto)
```
```  1831   finally show ?thesis
```
```  1832     by simp
```
```  1833 qed
```
```  1834
```
```  1835 (* Next two lemmas contributed by Wenda Li *)
```
```  1836 lemma order_1_eq_0 [simp]:"order x 1 = 0"
```
```  1837   by (metis order_root poly_1 zero_neq_one)
```
```  1838
```
```  1839 lemma order_power_n_n: "order a ([:-a,1:]^n)=n"
```
```  1840 proof (induct n) (*might be proved more concisely using nat_less_induct*)
```
```  1841   case 0
```
```  1842   then show ?case
```
```  1843     by (metis order_root poly_1 power_0 zero_neq_one)
```
```  1844 next
```
```  1845   case (Suc n)
```
```  1846   have "order a ([:- a, 1:] ^ Suc n) = order a ([:- a, 1:] ^ n) + order a [:-a,1:]"
```
```  1847     by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral
```
```  1848       one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
```
```  1849   moreover have "order a [:-a,1:] = 1"
```
```  1850     unfolding order_def
```
```  1851   proof (rule Least_equality, rule notI)
```
```  1852     assume "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
```
```  1853     then have "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:])"
```
```  1854       by (rule dvd_imp_degree_le) auto
```
```  1855     then show False
```
```  1856       by auto
```
```  1857   next
```
```  1858     fix y
```
```  1859     assume *: "\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
```
```  1860     show "1 \<le> y"
```
```  1861     proof (rule ccontr)
```
```  1862       assume "\<not> 1 \<le> y"
```
```  1863       then have "y = 0" by auto
```
```  1864       then have "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
```
```  1865       with * show False by auto
```
```  1866     qed
```
```  1867   qed
```
```  1868   ultimately show ?case
```
```  1869     using Suc by auto
```
```  1870 qed
```
```  1871
```
```  1872 lemma order_0_monom [simp]: "c \<noteq> 0 \<Longrightarrow> order 0 (monom c n) = n"
```
```  1873   using order_power_n_n[of 0 n] by (simp add: monom_altdef order_smult)
```
```  1874
```
```  1875 lemma dvd_imp_order_le: "q \<noteq> 0 \<Longrightarrow> p dvd q \<Longrightarrow> Polynomial.order a p \<le> Polynomial.order a q"
```
```  1876   by (auto simp: order_mult elim: dvdE)
```
```  1877
```
```  1878 text \<open>Now justify the standard squarefree decomposition, i.e. \<open>f / gcd f f'\<close>.\<close>
```
```  1879
```
```  1880 lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
```
```  1881   apply (cases "p = 0")
```
```  1882   apply auto
```
```  1883    apply (drule order_2 [where a=a and p=p])
```
```  1884    apply (metis not_less_eq_eq power_le_dvd)
```
```  1885   apply (erule power_le_dvd [OF order_1])
```
```  1886   done
```
```  1887
```
```  1888 lemma order_decomp:
```
```  1889   assumes "p \<noteq> 0"
```
```  1890   shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
```
```  1891 proof -
```
```  1892   from assms have *: "[:- a, 1:] ^ order a p dvd p"
```
```  1893     and **: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p"
```
```  1894     by (auto dest: order)
```
```  1895   from * obtain q where q: "p = [:- a, 1:] ^ order a p * q" ..
```
```  1896   with ** have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
```
```  1897     by simp
```
```  1898   then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
```
```  1899     by simp
```
```  1900   with idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
```
```  1901   have "\<not> [:- a, 1:] dvd q" by auto
```
```  1902   with q show ?thesis by blast
```
```  1903 qed
```
```  1904
```
```  1905 lemma monom_1_dvd_iff: "p \<noteq> 0 \<Longrightarrow> monom 1 n dvd p \<longleftrightarrow> n \<le> order 0 p"
```
```  1906   using order_divides[of 0 n p] by (simp add: monom_altdef)
```
```  1907
```
```  1908
```
```  1909 subsection \<open>Additional induction rules on polynomials\<close>
```
```  1910
```
```  1911 text \<open>
```
```  1912   An induction rule for induction over the roots of a polynomial with a certain property.
```
```  1913   (e.g. all positive roots)
```
```  1914 \<close>
```
```  1915 lemma poly_root_induct [case_names 0 no_roots root]:
```
```  1916   fixes p :: "'a :: idom poly"
```
```  1917   assumes "Q 0"
```
```  1918     and "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> 0) \<Longrightarrow> Q p"
```
```  1919     and "\<And>a p. P a \<Longrightarrow> Q p \<Longrightarrow> Q ([:a, -1:] * p)"
```
```  1920   shows "Q p"
```
```  1921 proof (induction "degree p" arbitrary: p rule: less_induct)
```
```  1922   case (less p)
```
```  1923   show ?case
```
```  1924   proof (cases "p = 0")
```
```  1925     case True
```
```  1926     with assms(1) show ?thesis by simp
```
```  1927   next
```
```  1928     case False
```
```  1929     show ?thesis
```
```  1930     proof (cases "\<exists>a. P a \<and> poly p a = 0")
```
```  1931       case False
```
```  1932       then show ?thesis by (intro assms(2)) blast
```
```  1933     next
```
```  1934       case True
```
```  1935       then obtain a where a: "P a" "poly p a = 0"
```
```  1936         by blast
```
```  1937       then have "-[:-a, 1:] dvd p"
```
```  1938         by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd)
```
```  1939       then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp
```
```  1940       with False have "q \<noteq> 0" by auto
```
```  1941       have "degree p = Suc (degree q)"
```
```  1942         by (subst q, subst degree_mult_eq) (simp_all add: \<open>q \<noteq> 0\<close>)
```
```  1943       then have "Q q" by (intro less) simp
```
```  1944       with a(1) have "Q ([:a, -1:] * q)"
```
```  1945         by (rule assms(3))
```
```  1946       with q show ?thesis by simp
```
```  1947     qed
```
```  1948   qed
```
```  1949 qed
```
```  1950
```
```  1951 lemma dropWhile_replicate_append:
```
```  1952   "dropWhile (op = a) (replicate n a @ ys) = dropWhile (op = a) ys"
```
```  1953   by (induct n) simp_all
```
```  1954
```
```  1955 lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs"
```
```  1956   by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append)
```
```  1957
```
```  1958 text \<open>
```
```  1959   An induction rule for simultaneous induction over two polynomials,
```
```  1960   prepending one coefficient in each step.
```
```  1961 \<close>
```
```  1962 lemma poly_induct2 [case_names 0 pCons]:
```
```  1963   assumes "P 0 0" "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)"
```
```  1964   shows "P p q"
```
```  1965 proof -
```
```  1966   define n where "n = max (length (coeffs p)) (length (coeffs q))"
```
```  1967   define xs where "xs = coeffs p @ (replicate (n - length (coeffs p)) 0)"
```
```  1968   define ys where "ys = coeffs q @ (replicate (n - length (coeffs q)) 0)"
```
```  1969   have "length xs = length ys"
```
```  1970     by (simp add: xs_def ys_def n_def)
```
```  1971   then have "P (Poly xs) (Poly ys)"
```
```  1972     by (induct rule: list_induct2) (simp_all add: assms)
```
```  1973   also have "Poly xs = p"
```
```  1974     by (simp add: xs_def Poly_append_replicate_0)
```
```  1975   also have "Poly ys = q"
```
```  1976     by (simp add: ys_def Poly_append_replicate_0)
```
```  1977   finally show ?thesis .
```
```  1978 qed
```
```  1979
```
```  1980
```
```  1981 subsection \<open>Composition of polynomials\<close>
```
```  1982
```
```  1983 (* Several lemmas contributed by RenÃ© Thiemann and Akihisa Yamada *)
```
```  1984
```
```  1985 definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  1986   where "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
```
```  1987
```
```  1988 notation pcompose (infixl "\<circ>\<^sub>p" 71)
```
```  1989
```
```  1990 lemma pcompose_0 [simp]: "pcompose 0 q = 0"
```
```  1991   by (simp add: pcompose_def)
```
```  1992
```
```  1993 lemma pcompose_pCons: "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
```
```  1994   by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
```
```  1995
```
```  1996 lemma pcompose_1: "pcompose 1 p = 1"
```
```  1997   for p :: "'a::comm_semiring_1 poly"
```
```  1998   by (auto simp: one_pCons pcompose_pCons)
```
```  1999
```
```  2000 lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)"
```
```  2001   by (induct p) (simp_all add: pcompose_pCons)
```
```  2002
```
```  2003 lemma degree_pcompose_le: "degree (pcompose p q) \<le> degree p * degree q"
```
```  2004   apply (induct p)
```
```  2005    apply simp
```
```  2006   apply (simp add: pcompose_pCons)
```
```  2007   apply clarify
```
```  2008   apply (rule degree_add_le)
```
```  2009    apply simp
```
```  2010   apply (rule order_trans [OF degree_mult_le])
```
```  2011   apply simp
```
```  2012   done
```
```  2013
```
```  2014 lemma pcompose_add: "pcompose (p + q) r = pcompose p r + pcompose q r"
```
```  2015   for p q r :: "'a::{comm_semiring_0, ab_semigroup_add} poly"
```
```  2016 proof (induction p q rule: poly_induct2)
```
```  2017   case 0
```
```  2018   then show ?case by simp
```
```  2019 next
```
```  2020   case (pCons a p b q)
```
```  2021   have "pcompose (pCons a p + pCons b q) r = [:a + b:] + r * pcompose p r + r * pcompose q r"
```
```  2022     by (simp_all add: pcompose_pCons pCons.IH algebra_simps)
```
```  2023   also have "[:a + b:] = [:a:] + [:b:]" by simp
```
```  2024   also have "\<dots> + r * pcompose p r + r * pcompose q r =
```
```  2025     pcompose (pCons a p) r + pcompose (pCons b q) r"
```
```  2026     by (simp only: pcompose_pCons add_ac)
```
```  2027   finally show ?case .
```
```  2028 qed
```
```  2029
```
```  2030 lemma pcompose_uminus: "pcompose (-p) r = -pcompose p r"
```
```  2031   for p r :: "'a::comm_ring poly"
```
```  2032   by (induct p) (simp_all add: pcompose_pCons)
```
```  2033
```
```  2034 lemma pcompose_diff: "pcompose (p - q) r = pcompose p r - pcompose q r"
```
```  2035   for p q r :: "'a::comm_ring poly"
```
```  2036   using pcompose_add[of p "-q"] by (simp add: pcompose_uminus)
```
```  2037
```
```  2038 lemma pcompose_smult: "pcompose (smult a p) r = smult a (pcompose p r)"
```
```  2039   for p r :: "'a::comm_semiring_0 poly"
```
```  2040   by (induct p) (simp_all add: pcompose_pCons pcompose_add smult_add_right)
```
```  2041
```
```  2042 lemma pcompose_mult: "pcompose (p * q) r = pcompose p r * pcompose q r"
```
```  2043   for p q r :: "'a::comm_semiring_0 poly"
```
```  2044   by (induct p arbitrary: q) (simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps)
```
```  2045
```
```  2046 lemma pcompose_assoc: "pcompose p (pcompose q r) = pcompose (pcompose p q) r"
```
```  2047   for p q r :: "'a::comm_semiring_0 poly"
```
```  2048   by (induct p arbitrary: q) (simp_all add: pcompose_pCons pcompose_add pcompose_mult)
```
```  2049
```
```  2050 lemma pcompose_idR[simp]: "pcompose p [: 0, 1 :] = p"
```
```  2051   for p :: "'a::comm_semiring_1 poly"
```
```  2052   by (induct p) (simp_all add: pcompose_pCons)
```
```  2053
```
```  2054 lemma pcompose_sum: "pcompose (sum f A) p = sum (\<lambda>i. pcompose (f i) p) A"
```
```  2055   by (induct A rule: infinite_finite_induct) (simp_all add: pcompose_1 pcompose_add)
```
```  2056
```
```  2057 lemma pcompose_prod: "pcompose (prod f A) p = prod (\<lambda>i. pcompose (f i) p) A"
```
```  2058   by (induct A rule: infinite_finite_induct) (simp_all add: pcompose_1 pcompose_mult)
```
```  2059
```
```  2060 lemma pcompose_const [simp]: "pcompose [:a:] q = [:a:]"
```
```  2061   by (subst pcompose_pCons) simp
```
```  2062
```
```  2063 lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]"
```
```  2064   by (induct p) (auto simp add: pcompose_pCons)
```
```  2065
```
```  2066 lemma degree_pcompose: "degree (pcompose p q) = degree p * degree q"
```
```  2067   for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  2068 proof (induct p)
```
```  2069   case 0
```
```  2070   then show ?case by auto
```
```  2071 next
```
```  2072   case (pCons a p)
```
```  2073   consider "degree (q * pcompose p q) = 0" | "degree (q * pcompose p q) > 0"
```
```  2074     by blast
```
```  2075   then show ?case
```
```  2076   proof cases
```
```  2077     case prems: 1
```
```  2078     show ?thesis
```
```  2079     proof (cases "p = 0")
```
```  2080       case True
```
```  2081       then show ?thesis by auto
```
```  2082     next
```
```  2083       case False
```
```  2084       from prems have "degree q = 0 \<or> pcompose p q = 0"
```
```  2085         by (auto simp add: degree_mult_eq_0)
```
```  2086       moreover have False if "pcompose p q = 0" "degree q \<noteq> 0"
```
```  2087       proof -
```
```  2088         from pCons.hyps(2) that have "degree p = 0"
```
```  2089           by auto
```
```  2090         then obtain a1 where "p = [:a1:]"
```
```  2091           by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)
```
```  2092         with \<open>pcompose p q = 0\<close> \<open>p \<noteq> 0\<close> show False
```
```  2093           by auto
```
```  2094       qed
```
```  2095       ultimately have "degree (pCons a p) * degree q = 0"
```
```  2096         by auto
```
```  2097       moreover have "degree (pcompose (pCons a p) q) = 0"
```
```  2098       proof -
```
```  2099         from prems have "0 = max (degree [:a:]) (degree (q * pcompose p q))"
```
```  2100           by simp
```
```  2101         also have "\<dots> \<ge> degree ([:a:] + q * pcompose p q)"
```
```  2102           by (rule degree_add_le_max)
```
```  2103         finally show ?thesis
```
```  2104           by (auto simp add: pcompose_pCons)
```
```  2105       qed
```
```  2106       ultimately show ?thesis by simp
```
```  2107     qed
```
```  2108   next
```
```  2109     case prems: 2
```
```  2110     then have "p \<noteq> 0" "q \<noteq> 0" "pcompose p q \<noteq> 0"
```
```  2111       by auto
```
```  2112     from prems degree_add_eq_right [of "[:a:]"]
```
```  2113     have "degree (pcompose (pCons a p) q) = degree (q * pcompose p q)"
```
```  2114       by (auto simp: pcompose_pCons)
```
```  2115     with pCons.hyps(2) degree_mult_eq[OF \<open>q\<noteq>0\<close> \<open>pcompose p q\<noteq>0\<close>] show ?thesis
```
```  2116       by auto
```
```  2117   qed
```
```  2118 qed
```
```  2119
```
```  2120 lemma pcompose_eq_0:
```
```  2121   fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  2122   assumes "pcompose p q = 0" "degree q > 0"
```
```  2123   shows "p = 0"
```
```  2124 proof -
```
```  2125   from assms degree_pcompose [of p q] have "degree p = 0"
```
```  2126     by auto
```
```  2127   then obtain a where "p = [:a:]"
```
```  2128     by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases)
```
```  2129   with assms(1) have "a = 0"
```
```  2130     by auto
```
```  2131   with \<open>p = [:a:]\<close> show ?thesis
```
```  2132     by simp
```
```  2133 qed
```
```  2134
```
```  2135 lemma lead_coeff_comp:
```
```  2136   fixes p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  2137   assumes "degree q > 0"
```
```  2138   shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)"
```
```  2139 proof (induct p)
```
```  2140   case 0
```
```  2141   then show ?case by auto
```
```  2142 next
```
```  2143   case (pCons a p)
```
```  2144   consider "degree (q * pcompose p q) = 0" | "degree (q * pcompose p q) > 0"
```
```  2145     by blast
```
```  2146   then show ?case
```
```  2147   proof cases
```
```  2148     case prems: 1
```
```  2149     then have "pcompose p q = 0"
```
```  2150       by (metis assms degree_0 degree_mult_eq_0 neq0_conv)
```
```  2151     with pcompose_eq_0[OF _ \<open>degree q > 0\<close>] have "p = 0"
```
```  2152       by simp
```
```  2153     then show ?thesis
```
```  2154       by auto
```
```  2155   next
```
```  2156     case prems: 2
```
```  2157     then have "degree [:a:] < degree (q * pcompose p q)"
```
```  2158       by simp
```
```  2159     then have "lead_coeff ([:a:] + q * p \<circ>\<^sub>p q) = lead_coeff (q * p \<circ>\<^sub>p q)"
```
```  2160       by (rule lead_coeff_add_le)
```
```  2161     then have "lead_coeff (pcompose (pCons a p) q) = lead_coeff (q * pcompose p q)"
```
```  2162       by (simp add: pcompose_pCons)
```
```  2163     also have "\<dots> = lead_coeff q * (lead_coeff p * lead_coeff q ^ degree p)"
```
```  2164       using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp
```
```  2165     also have "\<dots> = lead_coeff p * lead_coeff q ^ (degree p + 1)"
```
```  2166       by (auto simp: mult_ac)
```
```  2167     finally show ?thesis by auto
```
```  2168   qed
```
```  2169 qed
```
```  2170
```
```  2171
```
```  2172 subsection \<open>Shifting polynomials\<close>
```
```  2173
```
```  2174 definition poly_shift :: "nat \<Rightarrow> 'a::zero poly \<Rightarrow> 'a poly"
```
```  2175   where "poly_shift n p = Abs_poly (\<lambda>i. coeff p (i + n))"
```
```  2176
```
```  2177 lemma nth_default_drop: "nth_default x (drop n xs) m = nth_default x xs (m + n)"
```
```  2178   by (auto simp add: nth_default_def add_ac)
```
```  2179
```
```  2180 lemma nth_default_take: "nth_default x (take n xs) m = (if m < n then nth_default x xs m else x)"
```
```  2181   by (auto simp add: nth_default_def add_ac)
```
```  2182
```
```  2183 lemma coeff_poly_shift: "coeff (poly_shift n p) i = coeff p (i + n)"
```
```  2184 proof -
```
```  2185   from MOST_coeff_eq_0[of p] obtain m where "\<forall>k>m. coeff p k = 0"
```
```  2186     by (auto simp: MOST_nat)
```
```  2187   then have "\<forall>k>m. coeff p (k + n) = 0"
```
```  2188     by auto
```
```  2189   then have "\<forall>\<^sub>\<infinity>k. coeff p (k + n) = 0"
```
```  2190     by (auto simp: MOST_nat)
```
```  2191   then show ?thesis
```
```  2192     by (simp add: poly_shift_def poly.Abs_poly_inverse)
```
```  2193 qed
```
```  2194
```
```  2195 lemma poly_shift_id [simp]: "poly_shift 0 = (\<lambda>x. x)"
```
```  2196   by (simp add: poly_eq_iff fun_eq_iff coeff_poly_shift)
```
```  2197
```
```  2198 lemma poly_shift_0 [simp]: "poly_shift n 0 = 0"
```
```  2199   by (simp add: poly_eq_iff coeff_poly_shift)
```
```  2200
```
```  2201 lemma poly_shift_1: "poly_shift n 1 = (if n = 0 then 1 else 0)"
```
```  2202   by (simp add: poly_eq_iff coeff_poly_shift)
```
```  2203
```
```  2204 lemma poly_shift_monom: "poly_shift n (monom c m) = (if m \<ge> n then monom c (m - n) else 0)"
```
```  2205   by (auto simp add: poly_eq_iff coeff_poly_shift)
```
```  2206
```
```  2207 lemma coeffs_shift_poly [code abstract]:
```
```  2208   "coeffs (poly_shift n p) = drop n (coeffs p)"
```
```  2209 proof (cases "p = 0")
```
```  2210   case True
```
```  2211   then show ?thesis by simp
```
```  2212 next
```
```  2213   case False
```
```  2214   then show ?thesis
```
```  2215     by (intro coeffs_eqI)
```
```  2216       (simp_all add: coeff_poly_shift nth_default_drop nth_default_coeffs_eq)
```
```  2217 qed
```
```  2218
```
```  2219
```
```  2220 subsection \<open>Truncating polynomials\<close>
```
```  2221
```
```  2222 definition poly_cutoff
```
```  2223   where "poly_cutoff n p = Abs_poly (\<lambda>k. if k < n then coeff p k else 0)"
```
```  2224
```
```  2225 lemma coeff_poly_cutoff: "coeff (poly_cutoff n p) k = (if k < n then coeff p k else 0)"
```
```  2226   unfolding poly_cutoff_def
```
```  2227   by (subst poly.Abs_poly_inverse) (auto simp: MOST_nat intro: exI[of _ n])
```
```  2228
```
```  2229 lemma poly_cutoff_0 [simp]: "poly_cutoff n 0 = 0"
```
```  2230   by (simp add: poly_eq_iff coeff_poly_cutoff)
```
```  2231
```
```  2232 lemma poly_cutoff_1 [simp]: "poly_cutoff n 1 = (if n = 0 then 0 else 1)"
```
```  2233   by (simp add: poly_eq_iff coeff_poly_cutoff)
```
```  2234
```
```  2235 lemma coeffs_poly_cutoff [code abstract]:
```
```  2236   "coeffs (poly_cutoff n p) = strip_while (op = 0) (take n (coeffs p))"
```
```  2237 proof (cases "strip_while (op = 0) (take n (coeffs p)) = []")
```
```  2238   case True
```
```  2239   then have "coeff (poly_cutoff n p) k = 0" for k
```
```  2240     unfolding coeff_poly_cutoff
```
```  2241     by (auto simp: nth_default_coeffs_eq [symmetric] nth_default_def set_conv_nth)
```
```  2242   then have "poly_cutoff n p = 0"
```
```  2243     by (simp add: poly_eq_iff)
```
```  2244   then show ?thesis
```
```  2245     by (subst True) simp_all
```
```  2246 next
```
```  2247   case False
```
```  2248   have "no_trailing (op = 0) (strip_while (op = 0) (take n (coeffs p)))"
```
```  2249     by simp
```
```  2250   with False have "last (strip_while (op = 0) (take n (coeffs p))) \<noteq> 0"
```
```  2251     unfolding no_trailing_unfold by auto
```
```  2252   then show ?thesis
```
```  2253     by (intro coeffs_eqI)
```
```  2254       (simp_all add: coeff_poly_cutoff nth_default_take nth_default_coeffs_eq)
```
```  2255 qed
```
```  2256
```
```  2257
```
```  2258 subsection \<open>Reflecting polynomials\<close>
```
```  2259
```
```  2260 definition reflect_poly :: "'a::zero poly \<Rightarrow> 'a poly"
```
```  2261   where "reflect_poly p = Poly (rev (coeffs p))"
```
```  2262
```
```  2263 lemma coeffs_reflect_poly [code abstract]:
```
```  2264   "coeffs (reflect_poly p) = rev (dropWhile (op = 0) (coeffs p))"
```
```  2265   by (simp add: reflect_poly_def)
```
```  2266
```
```  2267 lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0"
```
```  2268   by (simp add: reflect_poly_def)
```
```  2269
```
```  2270 lemma reflect_poly_1 [simp]: "reflect_poly 1 = 1"
```
```  2271   by (simp add: reflect_poly_def one_pCons)
```
```  2272
```
```  2273 lemma coeff_reflect_poly:
```
```  2274   "coeff (reflect_poly p) n = (if n > degree p then 0 else coeff p (degree p - n))"
```
```  2275   by (cases "p = 0")
```
```  2276     (auto simp add: reflect_poly_def nth_default_def
```
```  2277       rev_nth degree_eq_length_coeffs coeffs_nth not_less
```
```  2278       dest: le_imp_less_Suc)
```
```  2279
```
```  2280 lemma coeff_0_reflect_poly_0_iff [simp]: "coeff (reflect_poly p) 0 = 0 \<longleftrightarrow> p = 0"
```
```  2281   by (simp add: coeff_reflect_poly)
```
```  2282
```
```  2283 lemma reflect_poly_at_0_eq_0_iff [simp]: "poly (reflect_poly p) 0 = 0 \<longleftrightarrow> p = 0"
```
```  2284   by (simp add: coeff_reflect_poly poly_0_coeff_0)
```
```  2285
```
```  2286 lemma reflect_poly_pCons':
```
```  2287   "p \<noteq> 0 \<Longrightarrow> reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))"
```
```  2288   by (intro poly_eqI)
```
```  2289     (auto simp: coeff_reflect_poly coeff_pCons not_less Suc_diff_le split: nat.split)
```
```  2290
```
```  2291 lemma reflect_poly_const [simp]: "reflect_poly [:a:] = [:a:]"
```
```  2292   by (cases "a = 0") (simp_all add: reflect_poly_def)
```
```  2293
```
```  2294 lemma poly_reflect_poly_nz:
```
```  2295   "x \<noteq> 0 \<Longrightarrow> poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)"
```
```  2296   for x :: "'a::field"
```
```  2297   by (induct rule: pCons_induct) (simp_all add: field_simps reflect_poly_pCons' poly_monom)
```
```  2298
```
```  2299 lemma coeff_0_reflect_poly [simp]: "coeff (reflect_poly p) 0 = lead_coeff p"
```
```  2300   by (simp add: coeff_reflect_poly)
```
```  2301
```
```  2302 lemma poly_reflect_poly_0 [simp]: "poly (reflect_poly p) 0 = lead_coeff p"
```
```  2303   by (simp add: poly_0_coeff_0)
```
```  2304
```
```  2305 lemma reflect_poly_reflect_poly [simp]: "coeff p 0 \<noteq> 0 \<Longrightarrow> reflect_poly (reflect_poly p) = p"
```
```  2306   by (cases p rule: pCons_cases) (simp add: reflect_poly_def )
```
```  2307
```
```  2308 lemma degree_reflect_poly_le: "degree (reflect_poly p) \<le> degree p"
```
```  2309   by (simp add: degree_eq_length_coeffs coeffs_reflect_poly length_dropWhile_le diff_le_mono)
```
```  2310
```
```  2311 lemma reflect_poly_pCons: "a \<noteq> 0 \<Longrightarrow> reflect_poly (pCons a p) = Poly (rev (a # coeffs p))"
```
```  2312   by (subst coeffs_eq_iff) (simp add: coeffs_reflect_poly)
```
```  2313
```
```  2314 lemma degree_reflect_poly_eq [simp]: "coeff p 0 \<noteq> 0 \<Longrightarrow> degree (reflect_poly p) = degree p"
```
```  2315   by (cases p rule: pCons_cases) (simp add: reflect_poly_pCons degree_eq_length_coeffs)
```
```  2316
```
```  2317 (* TODO: does this work with zero divisors as well? Probably not. *)
```
```  2318 lemma reflect_poly_mult: "reflect_poly (p * q) = reflect_poly p * reflect_poly q"
```
```  2319   for p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  2320 proof (cases "p = 0 \<or> q = 0")
```
```  2321   case False
```
```  2322   then have [simp]: "p \<noteq> 0" "q \<noteq> 0" by auto
```
```  2323   show ?thesis
```
```  2324   proof (rule poly_eqI)
```
```  2325     show "coeff (reflect_poly (p * q)) i = coeff (reflect_poly p * reflect_poly q) i" for i
```
```  2326     proof (cases "i \<le> degree (p * q)")
```
```  2327       case True
```
```  2328       define A where "A = {..i} \<inter> {i - degree q..degree p}"
```
```  2329       define B where "B = {..degree p} \<inter> {degree p - i..degree (p*q) - i}"
```
```  2330       let ?f = "\<lambda>j. degree p - j"
```
```  2331
```
```  2332       from True have "coeff (reflect_poly (p * q)) i = coeff (p * q) (degree (p * q) - i)"
```
```  2333         by (simp add: coeff_reflect_poly)
```
```  2334       also have "\<dots> = (\<Sum>j\<le>degree (p * q) - i. coeff p j * coeff q (degree (p * q) - i - j))"
```
```  2335         by (simp add: coeff_mult)
```
```  2336       also have "\<dots> = (\<Sum>j\<in>B. coeff p j * coeff q (degree (p * q) - i - j))"
```
```  2337         by (intro sum.mono_neutral_right) (auto simp: B_def degree_mult_eq not_le coeff_eq_0)
```
```  2338       also from True have "\<dots> = (\<Sum>j\<in>A. coeff p (degree p - j) * coeff q (degree q - (i - j)))"
```
```  2339         by (intro sum.reindex_bij_witness[of _ ?f ?f])
```
```  2340           (auto simp: A_def B_def degree_mult_eq add_ac)
```
```  2341       also have "\<dots> =
```
```  2342         (\<Sum>j\<le>i.
```
```  2343           if j \<in> {i - degree q..degree p}
```
```  2344           then coeff p (degree p - j) * coeff q (degree q - (i - j))
```
```  2345           else 0)"
```
```  2346         by (subst sum.inter_restrict [symmetric]) (simp_all add: A_def)
```
```  2347       also have "\<dots> = coeff (reflect_poly p * reflect_poly q) i"
```
```  2348         by (fastforce simp: coeff_mult coeff_reflect_poly intro!: sum.cong)
```
```  2349       finally show ?thesis .
```
```  2350     qed (auto simp: coeff_mult coeff_reflect_poly coeff_eq_0 degree_mult_eq intro!: sum.neutral)
```
```  2351   qed
```
```  2352 qed auto
```
```  2353
```
```  2354 lemma reflect_poly_smult: "reflect_poly (smult c p) = smult c (reflect_poly p)"
```
```  2355   for p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  2356   using reflect_poly_mult[of "[:c:]" p] by simp
```
```  2357
```
```  2358 lemma reflect_poly_power: "reflect_poly (p ^ n) = reflect_poly p ^ n"
```
```  2359   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  2360   by (induct n) (simp_all add: reflect_poly_mult)
```
```  2361
```
```  2362 lemma reflect_poly_prod: "reflect_poly (prod f A) = prod (\<lambda>x. reflect_poly (f x)) A"
```
```  2363   for f :: "_ \<Rightarrow> _::{comm_semiring_0,semiring_no_zero_divisors} poly"
```
```  2364   by (induct A rule: infinite_finite_induct) (simp_all add: reflect_poly_mult)
```
```  2365
```
```  2366 lemma reflect_poly_prod_list: "reflect_poly (prod_list xs) = prod_list (map reflect_poly xs)"
```
```  2367   for xs :: "_::{comm_semiring_0,semiring_no_zero_divisors} poly list"
```
```  2368   by (induct xs) (simp_all add: reflect_poly_mult)
```
```  2369
```
```  2370 lemma reflect_poly_Poly_nz:
```
```  2371   "no_trailing (HOL.eq 0) xs \<Longrightarrow> reflect_poly (Poly xs) = Poly (rev xs)"
```
```  2372   by (simp add: reflect_poly_def)
```
```  2373
```
```  2374 lemmas reflect_poly_simps =
```
```  2375   reflect_poly_0 reflect_poly_1 reflect_poly_const reflect_poly_smult reflect_poly_mult
```
```  2376   reflect_poly_power reflect_poly_prod reflect_poly_prod_list
```
```  2377
```
```  2378
```
```  2379 subsection \<open>Derivatives\<close>
```
```  2380
```
```  2381 function pderiv :: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) poly \<Rightarrow> 'a poly"
```
```  2382   where "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
```
```  2383   by (auto intro: pCons_cases)
```
```  2384
```
```  2385 termination pderiv
```
```  2386   by (relation "measure degree") simp_all
```
```  2387
```
```  2388 declare pderiv.simps[simp del]
```
```  2389
```
```  2390 lemma pderiv_0 [simp]: "pderiv 0 = 0"
```
```  2391   using pderiv.simps [of 0 0] by simp
```
```  2392
```
```  2393 lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
```
```  2394   by (simp add: pderiv.simps)
```
```  2395
```
```  2396 lemma pderiv_1 [simp]: "pderiv 1 = 0"
```
```  2397   by (simp add: one_pCons pderiv_pCons)
```
```  2398
```
```  2399 lemma pderiv_of_nat [simp]: "pderiv (of_nat n) = 0"
```
```  2400   and pderiv_numeral [simp]: "pderiv (numeral m) = 0"
```
```  2401   by (simp_all add: of_nat_poly numeral_poly pderiv_pCons)
```
```  2402
```
```  2403 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
```
```  2404   by (induct p arbitrary: n)
```
```  2405     (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
```
```  2406
```
```  2407 fun pderiv_coeffs_code :: "'a::{comm_semiring_1,semiring_no_zero_divisors} \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```  2408   where
```
```  2409     "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)"
```
```  2410   | "pderiv_coeffs_code f [] = []"
```
```  2411
```
```  2412 definition pderiv_coeffs :: "'a::{comm_semiring_1,semiring_no_zero_divisors} list \<Rightarrow> 'a list"
```
```  2413   where "pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)"
```
```  2414
```
```  2415 (* Efficient code for pderiv contributed by RenÃ© Thiemann and Akihisa Yamada *)
```
```  2416 lemma pderiv_coeffs_code:
```
```  2417   "nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * nth_default 0 xs n"
```
```  2418 proof (induct xs arbitrary: f n)
```
```  2419   case Nil
```
```  2420   then show ?case by simp
```
```  2421 next
```
```  2422   case (Cons x xs)
```
```  2423   show ?case
```
```  2424   proof (cases n)
```
```  2425     case 0
```
```  2426     then show ?thesis
```
```  2427       by (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0") (auto simp: cCons_def)
```
```  2428   next
```
```  2429     case n: (Suc m)
```
```  2430     show ?thesis
```
```  2431     proof (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0")
```
```  2432       case False
```
```  2433       then have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n =
```
```  2434           nth_default 0 (pderiv_coeffs_code (f + 1) xs) m"
```
```  2435         by (auto simp: cCons_def n)
```
```  2436       also have "\<dots> = (f + of_nat n) * nth_default 0 xs m"
```
```  2437         by (simp add: Cons n add_ac)
```
```  2438       finally show ?thesis
```
```  2439         by (simp add: n)
```
```  2440     next
```
```  2441       case True
```
```  2442       have empty: "pderiv_coeffs_code g xs = [] \<Longrightarrow> g + of_nat m = 0 \<or> nth_default 0 xs m = 0" for g
```
```  2443       proof (induct xs arbitrary: g m)
```
```  2444         case Nil
```
```  2445         then show ?case by simp
```
```  2446       next
```
```  2447         case (Cons x xs)
```
```  2448         from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []" and g: "g = 0 \<or> x = 0"
```
```  2449           by (auto simp: cCons_def split: if_splits)
```
```  2450         note IH = Cons(1)[OF empty]
```
```  2451         from IH[of m] IH[of "m - 1"] g show ?case
```
```  2452           by (cases m) (auto simp: field_simps)
```
```  2453       qed
```
```  2454       from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0"
```
```  2455         by (auto simp: cCons_def n)
```
```  2456       moreover from True have "(f + of_nat n) * nth_default 0 (x # xs) n = 0"
```
```  2457         by (simp add: n) (use empty[of "f+1"] in \<open>auto simp: field_simps\<close>)
```
```  2458       ultimately show ?thesis by simp
```
```  2459     qed
```
```  2460   qed
```
```  2461 qed
```
```  2462
```
```  2463 lemma coeffs_pderiv_code [code abstract]: "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
```
```  2464   unfolding pderiv_coeffs_def
```
```  2465 proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases)
```
```  2466   case (1 n)
```
```  2467   have id: "coeff p (Suc n) = nth_default 0 (map (\<lambda>i. coeff p (Suc i)) [0..<degree p]) n"
```
```  2468     by (cases "n < degree p") (auto simp: nth_default_def coeff_eq_0)
```
```  2469   show ?case
```
```  2470     unfolding coeffs_def map_upt_Suc by (auto simp: id)
```
```  2471 next
```
```  2472   case 2
```
```  2473   obtain n :: 'a and xs where defs: "tl (coeffs p) = xs" "1 = n"
```
```  2474     by simp
```
```  2475   from 2 show ?case
```
```  2476     unfolding defs by (induct xs arbitrary: n) (auto simp: cCons_def)
```
```  2477 qed
```
```  2478
```
```  2479 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
```
```  2480   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly"
```
```  2481   apply (rule iffI)
```
```  2482    apply (cases p)
```
```  2483    apply simp
```
```  2484    apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
```
```  2485   apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
```
```  2486   done
```
```  2487
```
```  2488 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
```
```  2489   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly"
```
```  2490   apply (rule order_antisym [OF degree_le])
```
```  2491    apply (simp add: coeff_pderiv coeff_eq_0)
```
```  2492   apply (cases "degree p")
```
```  2493    apply simp
```
```  2494   apply (rule le_degree)
```
```  2495   apply (simp add: coeff_pderiv del: of_nat_Suc)
```
```  2496   apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
```
```  2497   done
```
```  2498
```
```  2499 lemma not_dvd_pderiv:
```
```  2500   fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly"
```
```  2501   assumes "degree p \<noteq> 0"
```
```  2502   shows "\<not> p dvd pderiv p"
```
```  2503 proof
```
```  2504   assume dvd: "p dvd pderiv p"
```
```  2505   then obtain q where p: "pderiv p = p * q"
```
```  2506     unfolding dvd_def by auto
```
```  2507   from dvd have le: "degree p \<le> degree (pderiv p)"
```
```  2508     by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff)
```
```  2509   from assms and this [unfolded degree_pderiv]
```
```  2510     show False by auto
```
```  2511 qed
```
```  2512
```
```  2513 lemma dvd_pderiv_iff [simp]: "p dvd pderiv p \<longleftrightarrow> degree p = 0"
```
```  2514   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors,semiring_char_0} poly"
```
```  2515   using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric])
```
```  2516
```
```  2517 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
```
```  2518   by (simp add: pderiv_pCons)
```
```  2519
```
```  2520 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
```
```  2521   by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps)
```
```  2522
```
```  2523 lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p"
```
```  2524   by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps)
```
```  2525
```
```  2526 lemma pderiv_diff: "pderiv ((p :: _ :: idom poly) - q) = pderiv p - pderiv q"
```
```  2527   by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps)
```
```  2528
```
```  2529 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
```
```  2530   by (rule poly_eqI) (simp add: coeff_pderiv algebra_simps)
```
```  2531
```
```  2532 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
```
```  2533   by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
```
```  2534
```
```  2535 lemma pderiv_power_Suc: "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
```
```  2536   apply (induct n)
```
```  2537    apply simp
```
```  2538   apply (subst power_Suc)
```
```  2539   apply (subst pderiv_mult)
```
```  2540   apply (erule ssubst)
```
```  2541   apply (simp only: of_nat_Suc smult_add_left smult_1_left)
```
```  2542   apply (simp add: algebra_simps)
```
```  2543   done
```
```  2544
```
```  2545 lemma pderiv_pcompose: "pderiv (pcompose p q) = pcompose (pderiv p) q * pderiv q"
```
```  2546   by (induction p rule: pCons_induct)
```
```  2547      (auto simp: pcompose_pCons pderiv_add pderiv_mult pderiv_pCons pcompose_add algebra_simps)
```
```  2548
```
```  2549 lemma pderiv_prod: "pderiv (prod f (as)) = (\<Sum>a\<in>as. prod f (as - {a}) * pderiv (f a))"
```
```  2550 proof (induct as rule: infinite_finite_induct)
```
```  2551   case (insert a as)
```
```  2552   then have id: "prod f (insert a as) = f a * prod f as"
```
```  2553     "\<And>g. sum g (insert a as) = g a + sum g as"
```
```  2554     "insert a as - {a} = as"
```
```  2555     by auto
```
```  2556   have "prod f (insert a as - {b}) = f a * prod f (as - {b})" if "b \<in> as" for b
```
```  2557   proof -
```
```  2558     from \<open>a \<notin> as\<close> that have *: "insert a as - {b} = insert a (as - {b})"
```
```  2559       by auto
```
```  2560     show ?thesis
```
```  2561       unfolding * by (subst prod.insert) (use insert in auto)
```
```  2562   qed
```
```  2563   then show ?case
```
```  2564     unfolding id pderiv_mult insert(3) sum_distrib_left
```
```  2565     by (auto simp add: ac_simps intro!: sum.cong)
```
```  2566 qed auto
```
```  2567
```
```  2568 lemma DERIV_pow2: "DERIV (\<lambda>x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
```
```  2569   by (rule DERIV_cong, rule DERIV_pow) simp
```
```  2570 declare DERIV_pow2 [simp] DERIV_pow [simp]
```
```  2571
```
```  2572 lemma DERIV_add_const: "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. a + f x :: 'a::real_normed_field) x :> D"
```
```  2573   by (rule DERIV_cong, rule DERIV_add) auto
```
```  2574
```
```  2575 lemma poly_DERIV [simp]: "DERIV (\<lambda>x. poly p x) x :> poly (pderiv p) x"
```
```  2576   by (induct p) (auto intro!: derivative_eq_intros simp add: pderiv_pCons)
```
```  2577
```
```  2578 lemma continuous_on_poly [continuous_intros]:
```
```  2579   fixes p :: "'a :: {real_normed_field} poly"
```
```  2580   assumes "continuous_on A f"
```
```  2581   shows "continuous_on A (\<lambda>x. poly p (f x))"
```
```  2582 proof -
```
```  2583   have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))"
```
```  2584     by (intro continuous_intros assms)
```
```  2585   also have "\<dots> = (\<lambda>x. poly p (f x))"
```
```  2586     by (rule ext) (simp add: poly_altdef mult_ac)
```
```  2587   finally show ?thesis .
```
```  2588 qed
```
```  2589
```
```  2590 text \<open>Consequences of the derivative theorem above.\<close>
```
```  2591
```
```  2592 lemma poly_differentiable[simp]: "(\<lambda>x. poly p x) differentiable (at x)"
```
```  2593   for x :: real
```
```  2594   by (simp add: real_differentiable_def) (blast intro: poly_DERIV)
```
```  2595
```
```  2596 lemma poly_isCont[simp]: "isCont (\<lambda>x. poly p x) x"
```
```  2597   for x :: real
```
```  2598   by (rule poly_DERIV [THEN DERIV_isCont])
```
```  2599
```
```  2600 lemma poly_IVT_pos: "a < b \<Longrightarrow> poly p a < 0 \<Longrightarrow> 0 < poly p b \<Longrightarrow> \<exists>x. a < x \<and> x < b \<and> poly p x = 0"
```
```  2601   for a b :: real
```
```  2602   using IVT_objl [of "poly p" a 0 b] by (auto simp add: order_le_less)
```
```  2603
```
```  2604 lemma poly_IVT_neg: "a < b \<Longrightarrow> 0 < poly p a \<Longrightarrow> poly p b < 0 \<Longrightarrow> \<exists>x. a < x \<and> x < b \<and> poly p x = 0"
```
```  2605   for a b :: real
```
```  2606   using poly_IVT_pos [where p = "- p"] by simp
```
```  2607
```
```  2608 lemma poly_IVT: "a < b \<Longrightarrow> poly p a * poly p b < 0 \<Longrightarrow> \<exists>x>a. x < b \<and> poly p x = 0"
```
```  2609   for p :: "real poly"
```
```  2610   by (metis less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)
```
```  2611
```
```  2612 lemma poly_MVT: "a < b \<Longrightarrow> \<exists>x. a < x \<and> x < b \<and> poly p b - poly p a = (b - a) * poly (pderiv p) x"
```
```  2613   for a b :: real
```
```  2614   using MVT [of a b "poly p"]
```
```  2615   apply auto
```
```  2616   apply (rule_tac x = z in exI)
```
```  2617   apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
```
```  2618   done
```
```  2619
```
```  2620 lemma poly_MVT':
```
```  2621   fixes a b :: real
```
```  2622   assumes "{min a b..max a b} \<subseteq> A"
```
```  2623   shows "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) x"
```
```  2624 proof (cases a b rule: linorder_cases)
```
```  2625   case less
```
```  2626   from poly_MVT[OF less, of p] guess x by (elim exE conjE)
```
```  2627   then show ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])
```
```  2628 next
```
```  2629   case greater
```
```  2630   from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
```
```  2631   then show ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
```
```  2632 qed (use assms in auto)
```
```  2633
```
```  2634 lemma poly_pinfty_gt_lc:
```
```  2635   fixes p :: "real poly"
```
```  2636   assumes "lead_coeff p > 0"
```
```  2637   shows "\<exists>n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p"
```
```  2638   using assms
```
```  2639 proof (induct p)
```
```  2640   case 0
```
```  2641   then show ?case by auto
```
```  2642 next
```
```  2643   case (pCons a p)
```
```  2644   from this(1) consider "a \<noteq> 0" "p = 0" | "p \<noteq> 0" by auto
```
```  2645   then show ?case
```
```  2646   proof cases
```
```  2647     case 1
```
```  2648     then show ?thesis by auto
```
```  2649   next
```
```  2650     case 2
```
```  2651     with pCons obtain n1 where gte_lcoeff: "\<forall>x\<ge>n1. lead_coeff p \<le> poly p x"
```
```  2652       by auto
```
```  2653     from pCons(3) \<open>p \<noteq> 0\<close> have gt_0: "lead_coeff p > 0" by auto
```
```  2654     define n where "n = max n1 (1 + \<bar>a\<bar> / lead_coeff p)"
```
```  2655     have "lead_coeff (pCons a p) \<le> poly (pCons a p) x" if "n \<le> x" for x
```
```  2656     proof -
```
```  2657       from gte_lcoeff that have "lead_coeff p \<le> poly p x"
```
```  2658         by (auto simp: n_def)
```
```  2659       with gt_0 have "\<bar>a\<bar> / lead_coeff p \<ge> \<bar>a\<bar> / poly p x" and "poly p x > 0"
```
```  2660         by (auto intro: frac_le)
```
```  2661       with \<open>n \<le> x\<close>[unfolded n_def] have "x \<ge> 1 + \<bar>a\<bar> / poly p x"
```
```  2662         by auto
```
```  2663       with \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x > 0\<close> \<open>p \<noteq> 0\<close>
```
```  2664       show "lead_coeff (pCons a p) \<le> poly (pCons a p) x"
```
```  2665         by (auto simp: field_simps)
```
```  2666     qed
```
```  2667     then show ?thesis by blast
```
```  2668   qed
```
```  2669 qed
```
```  2670
```
```  2671 lemma lemma_order_pderiv1:
```
```  2672   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
```
```  2673     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
```
```  2674   by (simp only: pderiv_mult pderiv_power_Suc) (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
```
```  2675
```
```  2676 lemma lemma_order_pderiv:
```
```  2677   fixes p :: "'a :: field_char_0 poly"
```
```  2678   assumes n: "0 < n"
```
```  2679     and pd: "pderiv p \<noteq> 0"
```
```  2680     and pe: "p = [:- a, 1:] ^ n * q"
```
```  2681     and nd: "\<not> [:- a, 1:] dvd q"
```
```  2682   shows "n = Suc (order a (pderiv p))"
```
```  2683 proof -
```
```  2684   from assms have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
```
```  2685     by auto
```
```  2686   from assms obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
```
```  2687     by (cases n) auto
```
```  2688   have *: "k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l" for k l
```
```  2689     by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff)
```
```  2690   have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))"
```
```  2691   proof (rule order_unique_lemma)
```
```  2692     show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
```
```  2693       apply (subst lemma_order_pderiv1)
```
```  2694       apply (rule dvd_add)
```
```  2695        apply (metis dvdI dvd_mult2 power_Suc2)
```
```  2696       apply (metis dvd_smult dvd_triv_right)
```
```  2697       done
```
```  2698     show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
```
```  2699       apply (subst lemma_order_pderiv1)
```
```  2700       apply (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
```
```  2701       done
```
```  2702   qed
```
```  2703   then show ?thesis
```
```  2704     by (metis \<open>n = Suc n'\<close> pe)
```
```  2705 qed
```
```  2706
```
```  2707 lemma order_pderiv: "pderiv p \<noteq> 0 \<Longrightarrow> order a p \<noteq> 0 \<Longrightarrow> order a p = Suc (order a (pderiv p))"
```
```  2708   for p :: "'a::field_char_0 poly"
```
```  2709   apply (cases "p = 0")
```
```  2710    apply simp
```
```  2711   apply (drule_tac a = a and p = p in order_decomp)
```
```  2712   using neq0_conv
```
```  2713   apply (blast intro: lemma_order_pderiv)
```
```  2714   done
```
```  2715
```
```  2716 lemma poly_squarefree_decomp_order:
```
```  2717   fixes p :: "'a::field_char_0 poly"
```
```  2718   assumes "pderiv p \<noteq> 0"
```
```  2719     and p: "p = q * d"
```
```  2720     and p': "pderiv p = e * d"
```
```  2721     and d: "d = r * p + s * pderiv p"
```
```  2722   shows "order a q = (if order a p = 0 then 0 else 1)"
```
```  2723 proof (rule classical)
```
```  2724   assume 1: "\<not> ?thesis"
```
```  2725   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
```
```  2726   with p have "order a p = order a q + order a d"
```
```  2727     by (simp add: order_mult)
```
```  2728   with 1 have "order a p \<noteq> 0"
```
```  2729     by (auto split: if_splits)
```
```  2730   from \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> have "order a (pderiv p) = order a e + order a d"
```
```  2731     by (simp add: order_mult)
```
```  2732   from \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> have "order a p = Suc (order a (pderiv p))"
```
```  2733     by (rule order_pderiv)
```
```  2734   from \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> have "d \<noteq> 0"
```
```  2735     by simp
```
```  2736   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
```
```  2737     apply (simp add: d)
```
```  2738     apply (rule dvd_add)
```
```  2739      apply (rule dvd_mult)
```
```  2740      apply (simp add: order_divides \<open>p \<noteq> 0\<close> \<open>order a p = Suc (order a (pderiv p))\<close>)
```
```  2741     apply (rule dvd_mult)
```
```  2742     apply (simp add: order_divides)
```
```  2743     done
```
```  2744   with \<open>d \<noteq> 0\<close> have "order a (pderiv p) \<le> order a d"
```
```  2745     by (simp add: order_divides)
```
```  2746   show ?thesis
```
```  2747     using \<open>order a p = order a q + order a d\<close>
```
```  2748       and \<open>order a (pderiv p) = order a e + order a d\<close>
```
```  2749       and \<open>order a p = Suc (order a (pderiv p))\<close>
```
```  2750       and \<open>order a (pderiv p) \<le> order a d\<close>
```
```  2751     by auto
```
```  2752 qed
```
```  2753
```
```  2754 lemma poly_squarefree_decomp_order2:
```
```  2755   "pderiv p \<noteq> 0 \<Longrightarrow> p = q * d \<Longrightarrow> pderiv p = e * d \<Longrightarrow>
```
```  2756     d = r * p + s * pderiv p \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```  2757   for p :: "'a::field_char_0 poly"
```
```  2758   by (blast intro: poly_squarefree_decomp_order)
```
```  2759
```
```  2760 lemma order_pderiv2:
```
```  2761   "pderiv p \<noteq> 0 \<Longrightarrow> order a p \<noteq> 0 \<Longrightarrow> order a (pderiv p) = n \<longleftrightarrow> order a p = Suc n"
```
```  2762   for p :: "'a::field_char_0 poly"
```
```  2763   by (auto dest: order_pderiv)
```
```  2764
```
```  2765 definition rsquarefree :: "'a::idom poly \<Rightarrow> bool"
```
```  2766   where "rsquarefree p \<longleftrightarrow> p \<noteq> 0 \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
```
```  2767
```
```  2768 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
```
```  2769   for p :: "'a::{semidom,semiring_char_0} poly"
```
```  2770   by (cases p) (auto simp: pderiv_eq_0_iff split: if_splits)
```
```  2771
```
```  2772 lemma rsquarefree_roots: "rsquarefree p \<longleftrightarrow> (\<forall>a. \<not> (poly p a = 0 \<and> poly (pderiv p) a = 0))"
```
```  2773   for p :: "'a::field_char_0 poly"
```
```  2774   apply (simp add: rsquarefree_def)
```
```  2775   apply (case_tac "p = 0")
```
```  2776    apply simp
```
```  2777   apply simp
```
```  2778   apply (case_tac "pderiv p = 0")
```
```  2779    apply simp
```
```  2780    apply (drule pderiv_iszero, clarsimp)
```
```  2781    apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
```
```  2782   apply (force simp add: order_root order_pderiv2)
```
```  2783   done
```
```  2784
```
```  2785 lemma poly_squarefree_decomp:
```
```  2786   fixes p :: "'a::field_char_0 poly"
```
```  2787   assumes "pderiv p \<noteq> 0"
```
```  2788     and "p = q * d"
```
```  2789     and "pderiv p = e * d"
```
```  2790     and "d = r * p + s * pderiv p"
```
```  2791   shows "rsquarefree q \<and> (\<forall>a. poly q a = 0 \<longleftrightarrow> poly p a = 0)"
```
```  2792 proof -
```
```  2793   from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
```
```  2794   with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
```
```  2795   from assms have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```  2796     by (rule poly_squarefree_decomp_order2)
```
```  2797   with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
```
```  2798     by (simp add: rsquarefree_def order_root)
```
```  2799 qed
```
```  2800
```
```  2801
```
```  2802 subsection \<open>Algebraic numbers\<close>
```
```  2803
```
```  2804 text \<open>
```
```  2805   Algebraic numbers can be defined in two equivalent ways: all real numbers that are
```
```  2806   roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry
```
```  2807   uses the rational definition, but we need the integer definition.
```
```  2808
```
```  2809   The equivalence is obvious since any rational polynomial can be multiplied with the
```
```  2810   LCM of its coefficients, yielding an integer polynomial with the same roots.
```
```  2811 \<close>
```
```  2812
```
```  2813 definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool"
```
```  2814   where "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
```
```  2815
```
```  2816 lemma algebraicI: "(\<And>i. coeff p i \<in> \<int>) \<Longrightarrow> p \<noteq> 0 \<Longrightarrow> poly p x = 0 \<Longrightarrow> algebraic x"
```
```  2817   unfolding algebraic_def by blast
```
```  2818
```
```  2819 lemma algebraicE:
```
```  2820   assumes "algebraic x"
```
```  2821   obtains p where "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
```
```  2822   using assms unfolding algebraic_def by blast
```
```  2823
```
```  2824 lemma algebraic_altdef: "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
```
```  2825   for p :: "'a::field_char_0 poly"
```
```  2826 proof safe
```
```  2827   fix p
```
```  2828   assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0"
```
```  2829   define cs where "cs = coeffs p"
```
```  2830   from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'"
```
```  2831     unfolding Rats_def by blast
```
```  2832   then obtain f where f: "coeff p i = of_rat (f (coeff p i))" for i
```
```  2833     by (subst (asm) bchoice_iff) blast
```
```  2834   define cs' where "cs' = map (quotient_of \<circ> f) (coeffs p)"
```
```  2835   define d where "d = Lcm (set (map snd cs'))"
```
```  2836   define p' where "p' = smult (of_int d) p"
```
```  2837
```
```  2838   have "coeff p' n \<in> \<int>" for n
```
```  2839   proof (cases "n \<le> degree p")
```
```  2840     case True
```
```  2841     define c where "c = coeff p n"
```
```  2842     define a where "a = fst (quotient_of (f (coeff p n)))"
```
```  2843     define b where "b = snd (quotient_of (f (coeff p n)))"
```
```  2844     have b_pos: "b > 0"
```
```  2845       unfolding b_def using quotient_of_denom_pos' by simp
```
```  2846     have "coeff p' n = of_int d * coeff p n"
```
```  2847       by (simp add: p'_def)
```
```  2848     also have "coeff p n = of_rat (of_int a / of_int b)"
```
```  2849       unfolding a_def b_def
```
```  2850       by (subst quotient_of_div [of "f (coeff p n)", symmetric]) (simp_all add: f [symmetric])
```
```  2851     also have "of_int d * \<dots> = of_rat (of_int (a*d) / of_int b)"
```
```  2852       by (simp add: of_rat_mult of_rat_divide)
```
```  2853     also from nz True have "b \<in> snd ` set cs'"
```
```  2854       by (force simp: cs'_def o_def b_def coeffs_def simp del: upt_Suc)
```
```  2855     then have "b dvd (a * d)"
```
```  2856       by (simp add: d_def)
```
```  2857     then have "of_int (a * d) / of_int b \<in> (\<int> :: rat set)"
```
```  2858       by (rule of_int_divide_in_Ints)
```
```  2859     then have "of_rat (of_int (a * d) / of_int b) \<in> \<int>" by (elim Ints_cases) auto
```
```  2860     finally show ?thesis .
```
```  2861   next
```
```  2862     case False
```
```  2863     then show ?thesis
```
```  2864       by (auto simp: p'_def not_le coeff_eq_0)
```
```  2865   qed
```
```  2866   moreover have "set (map snd cs') \<subseteq> {0<..}"
```
```  2867     unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc)
```
```  2868   then have "d \<noteq> 0"
```
```  2869     unfolding d_def by (induct cs') simp_all
```
```  2870   with nz have "p' \<noteq> 0" by (simp add: p'_def)
```
```  2871   moreover from root have "poly p' x = 0"
```
```  2872     by (simp add: p'_def)
```
```  2873   ultimately show "algebraic x"
```
```  2874     unfolding algebraic_def by blast
```
```  2875 next
```
```  2876   assume "algebraic x"
```
```  2877   then obtain p where p: "coeff p i \<in> \<int>" "poly p x = 0" "p \<noteq> 0" for i
```
```  2878     by (force simp: algebraic_def)
```
```  2879   moreover have "coeff p i \<in> \<int> \<Longrightarrow> coeff p i \<in> \<rat>" for i
```
```  2880     by (elim Ints_cases) simp
```
```  2881   ultimately show "\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0" by auto
```
```  2882 qed
```
```  2883
```
```  2884
```
```  2885 subsection \<open>Division of polynomials\<close>
```
```  2886
```
```  2887 subsubsection \<open>Division in general\<close>
```
```  2888
```
```  2889 instantiation poly :: (idom_divide) idom_divide
```
```  2890 begin
```
```  2891
```
```  2892 fun divide_poly_main :: "'a \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly"
```
```  2893   where
```
```  2894     "divide_poly_main lc q r d dr (Suc n) =
```
```  2895       (let cr = coeff r dr; a = cr div lc; mon = monom a n in
```
```  2896         if False \<or> a * lc = cr then (* False \<or> is only because of problem in function-package *)
```
```  2897           divide_poly_main
```
```  2898             lc
```
```  2899             (q + mon)
```
```  2900             (r - mon * d)
```
```  2901             d (dr - 1) n else 0)"
```
```  2902   | "divide_poly_main lc q r d dr 0 = q"
```
```  2903
```
```  2904 definition divide_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  2905   where "divide_poly f g =
```
```  2906     (if g = 0 then 0
```
```  2907      else
```
```  2908       divide_poly_main (coeff g (degree g)) 0 f g (degree f)
```
```  2909         (1 + length (coeffs f) - length (coeffs g)))"
```
```  2910
```
```  2911 lemma divide_poly_main:
```
```  2912   assumes d: "d \<noteq> 0" "lc = coeff d (degree d)"
```
```  2913     and "degree (d * r) \<le> dr" "divide_poly_main lc q (d * r) d dr n = q'"
```
```  2914     and "n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> d * r = 0"
```
```  2915   shows "q' = q + r"
```
```  2916   using assms(3-)
```
```  2917 proof (induct n arbitrary: q r dr)
```
```  2918   case (Suc n)
```
```  2919   let ?rr = "d * r"
```
```  2920   let ?a = "coeff ?rr dr"
```
```  2921   let ?qq = "?a div lc"
```
```  2922   define b where [simp]: "b = monom ?qq n"
```
```  2923   let ?rrr =  "d * (r - b)"
```
```  2924   let ?qqq = "q + b"
```
```  2925   note res = Suc(3)
```
```  2926   from Suc(4) have dr: "dr = n + degree d" by auto
```
```  2927   from d have lc: "lc \<noteq> 0" by auto
```
```  2928   have "coeff (b * d) dr = coeff b n * coeff d (degree d)"
```
```  2929   proof (cases "?qq = 0")
```
```  2930     case True
```
```  2931     then show ?thesis by simp
```
```  2932   next
```
```  2933     case False
```
```  2934     then have n: "n = degree b"
```
```  2935       by (simp add: degree_monom_eq)
```
```  2936     show ?thesis
```
```  2937       unfolding n dr by (simp add: coeff_mult_degree_sum)
```
```  2938   qed
```
```  2939   also have "\<dots> = lc * coeff b n"
```
```  2940     by (simp add: d)
```
```  2941   finally have c2: "coeff (b * d) dr = lc * coeff b n" .
```
```  2942   have rrr: "?rrr = ?rr - b * d"
```
```  2943     by (simp add: field_simps)
```
```  2944   have c1: "coeff (d * r) dr = lc * coeff r n"
```
```  2945   proof (cases "degree r = n")
```
```  2946     case True
```
```  2947     with Suc(2) show ?thesis
```
```  2948       unfolding dr using coeff_mult_degree_sum[of d r] d by (auto simp: ac_simps)
```
```  2949   next
```
```  2950     case False
```
```  2951     from dr Suc(2) have "degree r \<le> n"
```
```  2952       by auto
```
```  2953         (metis add.commute add_le_cancel_left d(1) degree_0 degree_mult_eq
```
```  2954           diff_is_0_eq diff_zero le_cases)
```
```  2955     with False have r_n: "degree r < n"
```
```  2956       by auto
```
```  2957     then have right: "lc * coeff r n = 0"
```
```  2958       by (simp add: coeff_eq_0)
```
```  2959     have "coeff (d * r) dr = coeff (d * r) (degree d + n)"
```
```  2960       by (simp add: dr ac_simps)
```
```  2961     also from r_n have "\<dots> = 0"
```
```  2962       by (metis False Suc.prems(1) add.commute add_left_imp_eq coeff_degree_mult coeff_eq_0
```
```  2963         coeff_mult_degree_sum degree_mult_le dr le_eq_less_or_eq)
```
```  2964     finally show ?thesis
```
```  2965       by (simp only: right)
```
```  2966   qed
```
```  2967   have c0: "coeff ?rrr dr = 0"
```
```  2968     and id: "lc * (coeff (d * r) dr div lc) = coeff (d * r) dr"
```
```  2969     unfolding rrr coeff_diff c2
```
```  2970     unfolding b_def coeff_monom coeff_smult c1 using lc by auto
```
```  2971   from res[unfolded divide_poly_main.simps[of lc q] Let_def] id
```
```  2972   have res: "divide_poly_main lc ?qqq ?rrr d (dr - 1) n = q'"
```
```  2973     by (simp del: divide_poly_main.simps add: field_simps)
```
```  2974   note IH = Suc(1)[OF _ res]
```
```  2975   from Suc(4) have dr: "dr = n + degree d" by auto
```
```  2976   from Suc(2) have deg_rr: "degree ?rr \<le> dr" by auto
```
```  2977   have deg_bd: "degree (b * d) \<le> dr"
```
```  2978     unfolding dr b_def by (rule order.trans[OF degree_mult_le]) (auto simp: degree_monom_le)
```
```  2979   have "degree ?rrr \<le> dr"
```
```  2980     unfolding rrr by (rule degree_diff_le[OF deg_rr deg_bd])
```
```  2981   with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)"
```
```  2982     by (rule coeff_0_degree_minus_1)
```
```  2983   have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0"
```
```  2984   proof (cases dr)
```
```  2985     case 0
```
```  2986     with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0"
```
```  2987       by auto
```
```  2988     with deg_rrr have "degree ?rrr = 0"
```
```  2989       by simp
```
```  2990     from degree_eq_zeroE[OF this] obtain a where rrr: "?rrr = [:a:]"
```
```  2991       by metis
```
```  2992     show ?thesis
```
```  2993       unfolding 0 using c0 unfolding rrr 0 by simp
```
```  2994   next
```
```  2995     case _: Suc
```
```  2996     with Suc(4) show ?thesis by auto
```
```  2997   qed
```
```  2998   from IH[OF deg_rrr this] show ?case
```
```  2999     by simp
```
```  3000 next
```
```  3001   case 0
```
```  3002   show ?case
```
```  3003   proof (cases "r = 0")
```
```  3004     case True
```
```  3005     with 0 show ?thesis by auto
```
```  3006   next
```
```  3007     case False
```
```  3008     from d False have "degree (d * r) = degree d + degree r"
```
```  3009       by (subst degree_mult_eq) auto
```
```  3010     with 0 d show ?thesis by auto
```
```  3011   qed
```
```  3012 qed
```
```  3013
```
```  3014 lemma divide_poly_main_0: "divide_poly_main 0 0 r d dr n = 0"
```
```  3015 proof (induct n arbitrary: r d dr)
```
```  3016   case 0
```
```  3017   then show ?case by simp
```
```  3018 next
```
```  3019   case Suc
```
```  3020   show ?case
```
```  3021     unfolding divide_poly_main.simps[of _ _ r] Let_def
```
```  3022     by (simp add: Suc del: divide_poly_main.simps)
```
```  3023 qed
```
```  3024
```
```  3025 lemma divide_poly:
```
```  3026   assumes g: "g \<noteq> 0"
```
```  3027   shows "(f * g) div g = (f :: 'a poly)"
```
```  3028 proof -
```
```  3029   have len: "length (coeffs f) = Suc (degree f)" if "f \<noteq> 0" for f :: "'a poly"
```
```  3030     using that unfolding degree_eq_length_coeffs by auto
```
```  3031   have "divide_poly_main (coeff g (degree g)) 0 (g * f) g (degree (g * f))
```
```  3032     (1 + length (coeffs (g * f)) - length (coeffs  g)) = (f * g) div g"
```
```  3033     by (simp add: divide_poly_def Let_def ac_simps)
```
```  3034   note main = divide_poly_main[OF g refl le_refl this]
```
```  3035   have "(f * g) div g = 0 + f"
```
```  3036   proof (rule main, goal_cases)
```
```  3037     case 1
```
```  3038     show ?case
```
```  3039     proof (cases "f = 0")
```
```  3040       case True
```
```  3041       with g show ?thesis
```
```  3042         by (auto simp: degree_eq_length_coeffs)
```
```  3043     next
```
```  3044       case False
```
```  3045       with g have fg: "g * f \<noteq> 0" by auto
```
```  3046       show ?thesis
```
```  3047         unfolding len[OF fg] len[OF g] by auto
```
```  3048     qed
```
```  3049   qed
```
```  3050   then show ?thesis by simp
```
```  3051 qed
```
```  3052
```
```  3053 lemma divide_poly_0: "f div 0 = 0"
```
```  3054   for f :: "'a poly"
```
```  3055   by (simp add: divide_poly_def Let_def divide_poly_main_0)
```
```  3056
```
```  3057 instance
```
```  3058   by standard (auto simp: divide_poly divide_poly_0)
```
```  3059
```
```  3060 end
```
```  3061
```
```  3062 instance poly :: (idom_divide) algebraic_semidom ..
```
```  3063
```
```  3064 lemma div_const_poly_conv_map_poly:
```
```  3065   assumes "[:c:] dvd p"
```
```  3066   shows "p div [:c:] = map_poly (\<lambda>x. x div c) p"
```
```  3067 proof (cases "c = 0")
```
```  3068   case True
```
```  3069   then show ?thesis
```
```  3070     by (auto intro!: poly_eqI simp: coeff_map_poly)
```
```  3071 next
```
```  3072   case False
```
```  3073   from assms obtain q where p: "p = [:c:] * q" by (rule dvdE)
```
```  3074   moreover {
```
```  3075     have "smult c q = [:c:] * q"
```
```  3076       by simp
```
```  3077     also have "\<dots> div [:c:] = q"
```
```  3078       by (rule nonzero_mult_div_cancel_left) (use False in auto)
```
```  3079     finally have "smult c q div [:c:] = q" .
```
```  3080   }
```
```  3081   ultimately show ?thesis by (intro poly_eqI) (auto simp: coeff_map_poly False)
```
```  3082 qed
```
```  3083
```
```  3084 lemma is_unit_monom_0:
```
```  3085   fixes a :: "'a::field"
```
```  3086   assumes "a \<noteq> 0"
```
```  3087   shows "is_unit (monom a 0)"
```
```  3088 proof
```
```  3089   from assms show "1 = monom a 0 * monom (inverse a) 0"
```
```  3090     by (simp add: mult_monom)
```
```  3091 qed
```
```  3092
```
```  3093 lemma is_unit_triv: "a \<noteq> 0 \<Longrightarrow> is_unit [:a:]"
```
```  3094   for a :: "'a::field"
```
```  3095   by (simp add: is_unit_monom_0 monom_0 [symmetric])
```
```  3096
```
```  3097 lemma is_unit_iff_degree:
```
```  3098   fixes p :: "'a::field poly"
```
```  3099   assumes "p \<noteq> 0"
```
```  3100   shows "is_unit p \<longleftrightarrow> degree p = 0"
```
```  3101     (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3102 proof
```
```  3103   assume ?rhs
```
```  3104   then obtain a where "p = [:a:]"
```
```  3105     by (rule degree_eq_zeroE)
```
```  3106   with assms show ?lhs
```
```  3107     by (simp add: is_unit_triv)
```
```  3108 next
```
```  3109   assume ?lhs
```
```  3110   then obtain q where "q \<noteq> 0" "p * q = 1" ..
```
```  3111   then have "degree (p * q) = degree 1"
```
```  3112     by simp
```
```  3113   with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0"
```
```  3114     by (simp add: degree_mult_eq)
```
```  3115   then show ?rhs by simp
```
```  3116 qed
```
```  3117
```
```  3118 lemma is_unit_pCons_iff: "is_unit (pCons a p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0"
```
```  3119   for p :: "'a::field poly"
```
```  3120   by (cases "p = 0") (auto simp: is_unit_triv is_unit_iff_degree)
```
```  3121
```
```  3122 lemma is_unit_monom_trival: "is_unit p \<Longrightarrow> monom (coeff p (degree p)) 0 = p"
```
```  3123   for p :: "'a::field poly"
```
```  3124   by (cases p) (simp_all add: monom_0 is_unit_pCons_iff)
```
```  3125
```
```  3126 lemma is_unit_const_poly_iff: "[:c:] dvd 1 \<longleftrightarrow> c dvd 1"
```
```  3127   for c :: "'a::{comm_semiring_1,semiring_no_zero_divisors}"
```
```  3128   by (auto simp: one_pCons)
```
```  3129
```
```  3130 lemma is_unit_polyE:
```
```  3131   fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  3132   assumes "p dvd 1"
```
```  3133   obtains c where "p = [:c:]" "c dvd 1"
```
```  3134 proof -
```
```  3135   from assms obtain q where "1 = p * q"
```
```  3136     by (rule dvdE)
```
```  3137   then have "p \<noteq> 0" and "q \<noteq> 0"
```
```  3138     by auto
```
```  3139   from \<open>1 = p * q\<close> have "degree 1 = degree (p * q)"
```
```  3140     by simp
```
```  3141   also from \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> have "\<dots> = degree p + degree q"
```
```  3142     by (simp add: degree_mult_eq)
```
```  3143   finally have "degree p = 0" by simp
```
```  3144   with degree_eq_zeroE obtain c where c: "p = [:c:]" .
```
```  3145   with \<open>p dvd 1\<close> have "c dvd 1"
```
```  3146     by (simp add: is_unit_const_poly_iff)
```
```  3147   with c show thesis ..
```
```  3148 qed
```
```  3149
```
```  3150 lemma is_unit_polyE':
```
```  3151   fixes p :: "'a::field poly"
```
```  3152   assumes "is_unit p"
```
```  3153   obtains a where "p = monom a 0" and "a \<noteq> 0"
```
```  3154 proof -
```
```  3155   obtain a q where "p = pCons a q"
```
```  3156     by (cases p)
```
```  3157   with assms have "p = [:a:]" and "a \<noteq> 0"
```
```  3158     by (simp_all add: is_unit_pCons_iff)
```
```  3159   with that show thesis by (simp add: monom_0)
```
```  3160 qed
```
```  3161
```
```  3162 lemma is_unit_poly_iff: "p dvd 1 \<longleftrightarrow> (\<exists>c. p = [:c:] \<and> c dvd 1)"
```
```  3163   for p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
```
```  3164   by (auto elim: is_unit_polyE simp add: is_unit_const_poly_iff)
```
```  3165
```
```  3166
```
```  3167 subsubsection \<open>Pseudo-Division\<close>
```
```  3168
```
```  3169 text \<open>This part is by RenÃ© Thiemann and Akihisa Yamada.\<close>
```
```  3170
```
```  3171 fun pseudo_divmod_main ::
```
```  3172   "'a :: comm_ring_1  \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly \<times> 'a poly"
```
```  3173   where
```
```  3174     "pseudo_divmod_main lc q r d dr (Suc n) =
```
```  3175       (let
```
```  3176         rr = smult lc r;
```
```  3177         qq = coeff r dr;
```
```  3178         rrr = rr - monom qq n * d;
```
```  3179         qqq = smult lc q + monom qq n
```
```  3180        in pseudo_divmod_main lc qqq rrr d (dr - 1) n)"
```
```  3181   | "pseudo_divmod_main lc q r d dr 0 = (q,r)"
```
```  3182
```
```  3183 definition pseudo_divmod :: "'a :: comm_ring_1 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
```
```  3184   where "pseudo_divmod p q \<equiv>
```
```  3185     if q = 0 then (0, p)
```
```  3186     else
```
```  3187       pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p)
```
```  3188         (1 + length (coeffs p) - length (coeffs q))"
```
```  3189
```
```  3190 lemma pseudo_divmod_main:
```
```  3191   assumes d: "d \<noteq> 0" "lc = coeff d (degree d)"
```
```  3192     and "degree r \<le> dr" "pseudo_divmod_main lc q r d dr n = (q',r')"
```
```  3193     and "n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> r = 0"
```
```  3194   shows "(r' = 0 \<or> degree r' < degree d) \<and> smult (lc^n) (d * q + r) = d * q' + r'"
```
```  3195   using assms(3-)
```
```  3196 proof (induct n arbitrary: q r dr)
```
```  3197   case 0
```
```  3198   then show ?case by auto
```
```  3199 next
```
```  3200   case (Suc n)
```
```  3201   let ?rr = "smult lc r"
```
```  3202   let ?qq = "coeff r dr"
```
```  3203   define b where [simp]: "b = monom ?qq n"
```
```  3204   let ?rrr = "?rr - b * d"
```
```  3205   let ?qqq = "smult lc q + b"
```
```  3206   note res = Suc(3)
```
```  3207   from res[unfolded pseudo_divmod_main.simps[of lc q] Let_def]
```
```  3208   have res: "pseudo_divmod_main lc ?qqq ?rrr d (dr - 1) n = (q',r')"
```
```  3209     by (simp del: pseudo_divmod_main.simps)
```
```  3210   from Suc(4) have dr: "dr = n + degree d" by auto
```
```  3211   have "coeff (b * d) dr = coeff b n * coeff d (degree d)"
```
```  3212   proof (cases "?qq = 0")
```
```  3213     case True
```
```  3214     then show ?thesis by auto
```
```  3215   next
```
```  3216     case False
```
```  3217     then have n: "n = degree b"
```
```  3218       by (simp add: degree_monom_eq)
```
```  3219     show ?thesis
```
```  3220       unfolding n dr by (simp add: coeff_mult_degree_sum)
```
```  3221   qed
```
```  3222   also have "\<dots> = lc * coeff b n" by (simp add: d)
```
```  3223   finally have "coeff (b * d) dr = lc * coeff b n" .
```
```  3224   moreover have "coeff ?rr dr = lc * coeff r dr"
```
```  3225     by simp
```
```  3226   ultimately have c0: "coeff ?rrr dr = 0"
```
```  3227     by auto
```
```  3228   from Suc(4) have dr: "dr = n + degree d" by auto
```
```  3229   have deg_rr: "degree ?rr \<le> dr"
```
```  3230     using Suc(2) degree_smult_le dual_order.trans by blast
```
```  3231   have deg_bd: "degree (b * d) \<le> dr"
```
```  3232     unfolding dr by (rule order.trans[OF degree_mult_le]) (auto simp: degree_monom_le)
```
```  3233   have "degree ?rrr \<le> dr"
```
```  3234     using degree_diff_le[OF deg_rr deg_bd] by auto
```
```  3235   with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)"
```
```  3236     by (rule coeff_0_degree_minus_1)
```
```  3237   have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0"
```
```  3238   proof (cases dr)
```
```  3239     case 0
```
```  3240     with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto
```
```  3241     with deg_rrr have "degree ?rrr = 0" by simp
```
```  3242     then have "\<exists>a. ?rrr = [:a:]"
```
```  3243       by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)
```
```  3244     from this obtain a where rrr: "?rrr = [:a:]"
```
```  3245       by auto
```
```  3246     show ?thesis
```
```  3247       unfolding 0 using c0 unfolding rrr 0 by simp
```
```  3248   next
```
```  3249     case _: Suc
```
```  3250     with Suc(4) show ?thesis by auto
```
```  3251   qed
```
```  3252   note IH = Suc(1)[OF deg_rrr res this]
```
```  3253   show ?case
```
```  3254   proof (intro conjI)
```
```  3255     from IH show "r' = 0 \<or> degree r' < degree d"
```
```  3256       by blast
```
```  3257     show "smult (lc ^ Suc n) (d * q + r) = d * q' + r'"
```
```  3258       unfolding IH[THEN conjunct2,symmetric]
```
```  3259       by (simp add: field_simps smult_add_right)
```
```  3260   qed
```
```  3261 qed
```
```  3262
```
```  3263 lemma pseudo_divmod:
```
```  3264   assumes g: "g \<noteq> 0"
```
```  3265     and *: "pseudo_divmod f g = (q,r)"
```
```  3266   shows "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r"  (is ?A)
```
```  3267     and "r = 0 \<or> degree r < degree g"  (is ?B)
```
```  3268 proof -
```
```  3269   from *[unfolded pseudo_divmod_def Let_def]
```
```  3270   have "pseudo_divmod_main (coeff g (degree g)) 0 f g (degree f)
```
```  3271       (1 + length (coeffs f) - length (coeffs g)) = (q, r)"
```
```  3272     by (auto simp: g)
```
```  3273   note main = pseudo_divmod_main[OF _ _ _ this, OF g refl le_refl]
```
```  3274   from g have "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g \<or>
```
```  3275     degree f = 0 \<and> 1 + length (coeffs f) - length (coeffs g) = 0 \<and> f = 0"
```
```  3276     by (cases "f = 0"; cases "coeffs g") (auto simp: degree_eq_length_coeffs)
```
```  3277   note main' = main[OF this]
```
```  3278   then show "r = 0 \<or> degree r < degree g" by auto
```
```  3279   show "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r"
```
```  3280     by (subst main'[THEN conjunct2, symmetric], simp add: degree_eq_length_coeffs,
```
```  3281         cases "f = 0"; cases "coeffs g", use g in auto)
```
```  3282 qed
```
```  3283
```
```  3284 definition "pseudo_mod_main lc r d dr n = snd (pseudo_divmod_main lc 0 r d dr n)"
```
```  3285
```
```  3286 lemma snd_pseudo_divmod_main:
```
```  3287   "snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)"
```
```  3288   by (induct n arbitrary: q q' lc r d dr) (simp_all add: Let_def)
```
```  3289
```
```  3290 definition pseudo_mod :: "'a::{comm_ring_1,semiring_1_no_zero_divisors} poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  3291   where "pseudo_mod f g = snd (pseudo_divmod f g)"
```
```  3292
```
```  3293 lemma pseudo_mod:
```
```  3294   fixes f g :: "'a::{comm_ring_1,semiring_1_no_zero_divisors} poly"
```
```  3295   defines "r \<equiv> pseudo_mod f g"
```
```  3296   assumes g: "g \<noteq> 0"
```
```  3297   shows "\<exists>a q. a \<noteq> 0 \<and> smult a f = g * q + r" "r = 0 \<or> degree r < degree g"
```
```  3298 proof -
```
```  3299   let ?cg = "coeff g (degree g)"
```
```  3300   let ?cge = "?cg ^ (Suc (degree f) - degree g)"
```
```  3301   define a where "a = ?cge"
```
```  3302   from r_def[unfolded pseudo_mod_def] obtain q where pdm: "pseudo_divmod f g = (q, r)"
```
```  3303     by (cases "pseudo_divmod f g") auto
```
```  3304   from pseudo_divmod[OF g pdm] have id: "smult a f = g * q + r" and "r = 0 \<or> degree r < degree g"
```
```  3305     by (auto simp: a_def)
```
```  3306   show "r = 0 \<or> degree r < degree g" by fact
```
```  3307   from g have "a \<noteq> 0"
```
```  3308     by (auto simp: a_def)
```
```  3309   with id show "\<exists>a q. a \<noteq> 0 \<and> smult a f = g * q + r"
```
```  3310     by auto
```
```  3311 qed
```
```  3312
```
```  3313 lemma fst_pseudo_divmod_main_as_divide_poly_main:
```
```  3314   assumes d: "d \<noteq> 0"
```
```  3315   defines lc: "lc \<equiv> coeff d (degree d)"
```
```  3316   shows "fst (pseudo_divmod_main lc q r d dr n) =
```
```  3317     divide_poly_main lc (smult (lc^n) q) (smult (lc^n) r) d dr n"
```
```  3318 proof (induct n arbitrary: q r dr)
```
```  3319   case 0
```
```  3320   then show ?case by simp
```
```  3321 next
```
```  3322   case (Suc n)
```
```  3323   note lc0 = leading_coeff_neq_0[OF d, folded lc]
```
```  3324   then have "pseudo_divmod_main lc q r d dr (Suc n) =
```
```  3325     pseudo_divmod_main lc (smult lc q + monom (coeff r dr) n)
```
```  3326       (smult lc r - monom (coeff r dr) n * d) d (dr - 1) n"
```
```  3327     by (simp add: Let_def ac_simps)
```
```  3328   also have "fst \<dots> = divide_poly_main lc
```
```  3329      (smult (lc^n) (smult lc q + monom (coeff r dr) n))
```
```  3330      (smult (lc^n) (smult lc r - monom (coeff r dr) n * d))
```
```  3331      d (dr - 1) n"
```
```  3332     by (simp only: Suc[unfolded divide_poly_main.simps Let_def])
```
```  3333   also have "\<dots> = divide_poly_main lc (smult (lc ^ Suc n) q) (smult (lc ^ Suc n) r) d dr (Suc n)"
```
```  3334     unfolding smult_monom smult_distribs mult_smult_left[symmetric]
```
```  3335     using lc0 by (simp add: Let_def ac_simps)
```
```  3336   finally show ?case .
```
```  3337 qed
```
```  3338
```
```  3339
```
```  3340 subsubsection \<open>Division in polynomials over fields\<close>
```
```  3341
```
```  3342 lemma pseudo_divmod_field:
```
```  3343   fixes g :: "'a::field poly"
```
```  3344   assumes g: "g \<noteq> 0"
```
```  3345     and *: "pseudo_divmod f g = (q,r)"
```
```  3346   defines "c \<equiv> coeff g (degree g) ^ (Suc (degree f) - degree g)"
```
```  3347   shows "f = g * smult (1/c) q + smult (1/c) r"
```
```  3348 proof -
```
```  3349   from leading_coeff_neq_0[OF g] have c0: "c \<noteq> 0"
```
```  3350     by (auto simp: c_def)
```
```  3351   from pseudo_divmod(1)[OF g *, folded c_def] have "smult c f = g * q + r"
```
```  3352     by auto
```
```  3353   also have "smult (1 / c) \<dots> = g * smult (1 / c) q + smult (1 / c) r"
```
```  3354     by (simp add: smult_add_right)
```
```  3355   finally show ?thesis
```
```  3356     using c0 by auto
```
```  3357 qed
```
```  3358
```
```  3359 lemma divide_poly_main_field:
```
```  3360   fixes d :: "'a::field poly"
```
```  3361   assumes d: "d \<noteq> 0"
```
```  3362   defines lc: "lc \<equiv> coeff d (degree d)"
```
```  3363   shows "divide_poly_main lc q r d dr n =
```
```  3364     fst (pseudo_divmod_main lc (smult ((1 / lc)^n) q) (smult ((1 / lc)^n) r) d dr n)"
```
```  3365   unfolding lc by (subst fst_pseudo_divmod_main_as_divide_poly_main) (auto simp: d power_one_over)
```
```  3366
```
```  3367 lemma divide_poly_field:
```
```  3368   fixes f g :: "'a::field poly"
```
```  3369   defines "f' \<equiv> smult ((1 / coeff g (degree g)) ^ (Suc (degree f) - degree g)) f"
```
```  3370   shows "f div g = fst (pseudo_divmod f' g)"
```
```  3371 proof (cases "g = 0")
```
```  3372   case True
```
```  3373   show ?thesis
```
```  3374     unfolding divide_poly_def pseudo_divmod_def Let_def f'_def True
```
```  3375     by (simp add: divide_poly_main_0)
```
```  3376 next
```
```  3377   case False
```
```  3378   from leading_coeff_neq_0[OF False] have "degree f' = degree f"
```
```  3379     by (auto simp: f'_def)
```
```  3380   then show ?thesis
```
```  3381     using length_coeffs_degree[of f'] length_coeffs_degree[of f]
```
```  3382     unfolding divide_poly_def pseudo_divmod_def Let_def
```
```  3383       divide_poly_main_field[OF False]
```
```  3384       length_coeffs_degree[OF False]
```
```  3385       f'_def
```
```  3386     by force
```
```  3387 qed
```
```  3388
```
```  3389 instantiation poly :: ("{semidom_divide_unit_factor,idom_divide}") normalization_semidom
```
```  3390 begin
```
```  3391
```
```  3392 definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly"
```
```  3393   where "unit_factor_poly p = [:unit_factor (lead_coeff p):]"
```
```  3394
```
```  3395 definition normalize_poly :: "'a poly \<Rightarrow> 'a poly"
```
```  3396   where "normalize p = p div [:unit_factor (lead_coeff p):]"
```
```  3397
```
```  3398 instance
```
```  3399 proof
```
```  3400   fix p :: "'a poly"
```
```  3401   show "unit_factor p * normalize p = p"
```
```  3402   proof (cases "p = 0")
```
```  3403     case True
```
```  3404     then show ?thesis
```
```  3405       by (simp add: unit_factor_poly_def normalize_poly_def)
```
```  3406   next
```
```  3407     case False
```
```  3408     then have "lead_coeff p \<noteq> 0"
```
```  3409       by simp
```
```  3410     then have *: "unit_factor (lead_coeff p) \<noteq> 0"
```
```  3411       using unit_factor_is_unit [of "lead_coeff p"] by auto
```
```  3412     then have "unit_factor (lead_coeff p) dvd 1"
```
```  3413       by (auto intro: unit_factor_is_unit)
```
```  3414     then have **: "unit_factor (lead_coeff p) dvd c" for c
```
```  3415       by (rule dvd_trans) simp
```
```  3416     have ***: "unit_factor (lead_coeff p) * (c div unit_factor (lead_coeff p)) = c" for c
```
```  3417     proof -
```
```  3418       from ** obtain b where "c = unit_factor (lead_coeff p) * b" ..
```
```  3419       with False * show ?thesis by simp
```
```  3420     qed
```
```  3421     have "p div [:unit_factor (lead_coeff p):] =
```
```  3422       map_poly (\<lambda>c. c div unit_factor (lead_coeff p)) p"
```
```  3423       by (simp add: const_poly_dvd_iff div_const_poly_conv_map_poly **)
```
```  3424     then show ?thesis
```
```  3425       by (simp add: normalize_poly_def unit_factor_poly_def
```
```  3426         smult_conv_map_poly map_poly_map_poly o_def ***)
```
```  3427   qed
```
```  3428 next
```
```  3429   fix p :: "'a poly"
```
```  3430   assume "is_unit p"
```
```  3431   then obtain c where p: "p = [:c:]" "c dvd 1"
```
```  3432     by (auto simp: is_unit_poly_iff)
```
```  3433   then show "unit_factor p = p"
```
```  3434     by (simp add: unit_factor_poly_def monom_0 is_unit_unit_factor)
```
```  3435 next
```
```  3436   fix p :: "'a poly"
```
```  3437   assume "p \<noteq> 0"
```
```  3438   then show "is_unit (unit_factor p)"
```
```  3439     by (simp add: unit_factor_poly_def monom_0 is_unit_poly_iff unit_factor_is_unit)
```
```  3440 qed (simp_all add: normalize_poly_def unit_factor_poly_def monom_0 lead_coeff_mult unit_factor_mult)
```
```  3441
```
```  3442 end
```
```  3443
```
```  3444 lemma normalize_poly_eq_map_poly: "normalize p = map_poly (\<lambda>x. x div unit_factor (lead_coeff p)) p"
```
```  3445 proof -
```
```  3446   have "[:unit_factor (lead_coeff p):] dvd p"
```
```  3447     by (metis unit_factor_poly_def unit_factor_self)
```
```  3448   then show ?thesis
```
```  3449     by (simp add: normalize_poly_def div_const_poly_conv_map_poly)
```
```  3450 qed
```
```  3451
```
```  3452 lemma coeff_normalize [simp]:
```
```  3453   "coeff (normalize p) n = coeff p n div unit_factor (lead_coeff p)"
```
```  3454   by (simp add: normalize_poly_eq_map_poly coeff_map_poly)
```
```  3455
```
```  3456 class field_unit_factor = field + unit_factor +
```
```  3457   assumes unit_factor_field [simp]: "unit_factor = id"
```
```  3458 begin
```
```  3459
```
```  3460 subclass semidom_divide_unit_factor
```
```  3461 proof
```
```  3462   fix a
```
```  3463   assume "a \<noteq> 0"
```
```  3464   then have "1 = a * inverse a" by simp
```
```  3465   then have "a dvd 1" ..
```
```  3466   then show "unit_factor a dvd 1" by simp
```
```  3467 qed simp_all
```
```  3468
```
```  3469 end
```
```  3470
```
```  3471 lemma unit_factor_pCons:
```
```  3472   "unit_factor (pCons a p) = (if p = 0 then [:unit_factor a:] else unit_factor p)"
```
```  3473   by (simp add: unit_factor_poly_def)
```
```  3474
```
```  3475 lemma normalize_monom [simp]: "normalize (monom a n) = monom (normalize a) n"
```
```  3476   by (cases "a = 0") (simp_all add: map_poly_monom normalize_poly_eq_map_poly degree_monom_eq)
```
```  3477
```
```  3478 lemma unit_factor_monom [simp]: "unit_factor (monom a n) = [:unit_factor a:]"
```
```  3479   by (cases "a = 0") (simp_all add: unit_factor_poly_def degree_monom_eq)
```
```  3480
```
```  3481 lemma normalize_const_poly: "normalize [:c:] = [:normalize c:]"
```
```  3482   by (simp add: normalize_poly_eq_map_poly map_poly_pCons)
```
```  3483
```
```  3484 lemma normalize_smult: "normalize (smult c p) = smult (normalize c) (normalize p)"
```
```  3485 proof -
```
```  3486   have "smult c p = [:c:] * p" by simp
```
```  3487   also have "normalize \<dots> = smult (normalize c) (normalize p)"
```
```  3488     by (subst normalize_mult) (simp add: normalize_const_poly)
```
```  3489   finally show ?thesis .
```
```  3490 qed
```
```  3491
```
```  3492 inductive eucl_rel_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly \<Rightarrow> bool"
```
```  3493   where
```
```  3494     eucl_rel_poly_by0: "eucl_rel_poly x 0 (0, x)"
```
```  3495   | eucl_rel_poly_dividesI: "y \<noteq> 0 \<Longrightarrow> x = q * y \<Longrightarrow> eucl_rel_poly x y (q, 0)"
```
```  3496   | eucl_rel_poly_remainderI:
```
```  3497       "y \<noteq> 0 \<Longrightarrow> degree r < degree y \<Longrightarrow> x = q * y + r \<Longrightarrow> eucl_rel_poly x y (q, r)"
```
```  3498
```
```  3499 lemma eucl_rel_poly_iff:
```
```  3500   "eucl_rel_poly x y (q, r) \<longleftrightarrow>
```
```  3501     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
```
```  3502   by (auto elim: eucl_rel_poly.cases
```
```  3503       intro: eucl_rel_poly_by0 eucl_rel_poly_dividesI eucl_rel_poly_remainderI)
```
```  3504
```
```  3505 lemma eucl_rel_poly_0: "eucl_rel_poly 0 y (0, 0)"
```
```  3506   by (simp add: eucl_rel_poly_iff)
```
```  3507
```
```  3508 lemma eucl_rel_poly_by_0: "eucl_rel_poly x 0 (0, x)"
```
```  3509   by (simp add: eucl_rel_poly_iff)
```
```  3510
```
```  3511 lemma eucl_rel_poly_pCons:
```
```  3512   assumes rel: "eucl_rel_poly x y (q, r)"
```
```  3513   assumes y: "y \<noteq> 0"
```
```  3514   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
```
```  3515   shows "eucl_rel_poly (pCons a x) y (pCons b q, pCons a r - smult b y)"
```
```  3516     (is "eucl_rel_poly ?x y (?q, ?r)")
```
```  3517 proof -
```
```  3518   from assms have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
```
```  3519     by (simp_all add: eucl_rel_poly_iff)
```
```  3520   from b x have "?x = ?q * y + ?r" by simp
```
```  3521   moreover
```
```  3522   have "?r = 0 \<or> degree ?r < degree y"
```
```  3523   proof (rule eq_zero_or_degree_less)
```
```  3524     show "degree ?r \<le> degree y"
```
```  3525     proof (rule degree_diff_le)
```
```  3526       from r show "degree (pCons a r) \<le> degree y"
```
```  3527         by auto
```
```  3528       show "degree (smult b y) \<le> degree y"
```
```  3529         by (rule degree_smult_le)
```
```  3530     qed
```
```  3531     from \<open>y \<noteq> 0\<close> show "coeff ?r (degree y) = 0"
```
```  3532       by (simp add: b)
```
```  3533   qed
```
```  3534   ultimately show ?thesis
```
```  3535     unfolding eucl_rel_poly_iff using \<open>y \<noteq> 0\<close> by simp
```
```  3536 qed
```
```  3537
```
```  3538 lemma eucl_rel_poly_exists: "\<exists>q r. eucl_rel_poly x y (q, r)"
```
```  3539   apply (cases "y = 0")
```
```  3540    apply (fast intro!: eucl_rel_poly_by_0)
```
```  3541   apply (induct x)
```
```  3542    apply (fast intro!: eucl_rel_poly_0)
```
```  3543   apply (fast intro!: eucl_rel_poly_pCons)
```
```  3544   done
```
```  3545
```
```  3546 lemma eucl_rel_poly_unique:
```
```  3547   assumes 1: "eucl_rel_poly x y (q1, r1)"
```
```  3548   assumes 2: "eucl_rel_poly x y (q2, r2)"
```
```  3549   shows "q1 = q2 \<and> r1 = r2"
```
```  3550 proof (cases "y = 0")
```
```  3551   assume "y = 0"
```
```  3552   with assms show ?thesis
```
```  3553     by (simp add: eucl_rel_poly_iff)
```
```  3554 next
```
```  3555   assume [simp]: "y \<noteq> 0"
```
```  3556   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
```
```  3557     unfolding eucl_rel_poly_iff by simp_all
```
```  3558   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
```
```  3559     unfolding eucl_rel_poly_iff by simp_all
```
```  3560   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
```
```  3561     by (simp add: algebra_simps)
```
```  3562   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
```
```  3563     by (auto intro: degree_diff_less)
```
```  3564   show "q1 = q2 \<and> r1 = r2"
```
```  3565   proof (rule classical)
```
```  3566     assume "\<not> ?thesis"
```
```  3567     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
```
```  3568     with r3 have "degree (r2 - r1) < degree y" by simp
```
```  3569     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
```
```  3570     also from \<open>q1 \<noteq> q2\<close> have "\<dots> = degree ((q1 - q2) * y)"
```
```  3571       by (simp add: degree_mult_eq)
```
```  3572     also from q3 have "\<dots> = degree (r2 - r1)"
```
```  3573       by simp
```
```  3574     finally have "degree (r2 - r1) < degree (r2 - r1)" .
```
```  3575     then show ?thesis by simp
```
```  3576   qed
```
```  3577 qed
```
```  3578
```
```  3579 lemma eucl_rel_poly_0_iff: "eucl_rel_poly 0 y (q, r) \<longleftrightarrow> q = 0 \<and> r = 0"
```
```  3580   by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_0)
```
```  3581
```
```  3582 lemma eucl_rel_poly_by_0_iff: "eucl_rel_poly x 0 (q, r) \<longleftrightarrow> q = 0 \<and> r = x"
```
```  3583   by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_by_0)
```
```  3584
```
```  3585 lemmas eucl_rel_poly_unique_div = eucl_rel_poly_unique [THEN conjunct1]
```
```  3586
```
```  3587 lemmas eucl_rel_poly_unique_mod = eucl_rel_poly_unique [THEN conjunct2]
```
```  3588
```
```  3589 instantiation poly :: (field) semidom_modulo
```
```  3590 begin
```
```  3591
```
```  3592 definition modulo_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  3593   where mod_poly_def: "f mod g =
```
```  3594     (if g = 0 then f else pseudo_mod (smult ((1 / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)"
```
```  3595
```
```  3596 instance
```
```  3597 proof
```
```  3598   fix x y :: "'a poly"
```
```  3599   show "x div y * y + x mod y = x"
```
```  3600   proof (cases "y = 0")
```
```  3601     case True
```
```  3602     then show ?thesis
```
```  3603       by (simp add: divide_poly_0 mod_poly_def)
```
```  3604   next
```
```  3605     case False
```
```  3606     then have "pseudo_divmod (smult ((1 / lead_coeff y) ^ (Suc (degree x) - degree y)) x) y =
```
```  3607         (x div y, x mod y)"
```
```  3608       by (simp add: divide_poly_field mod_poly_def pseudo_mod_def)
```
```  3609     with False pseudo_divmod [OF False this] show ?thesis
```
```  3610       by (simp add: power_mult_distrib [symmetric] ac_simps)
```
```  3611   qed
```
```  3612 qed
```
```  3613
```
```  3614 end
```
```  3615
```
```  3616 lemma eucl_rel_poly: "eucl_rel_poly x y (x div y, x mod y)"
```
```  3617   unfolding eucl_rel_poly_iff
```
```  3618 proof
```
```  3619   show "x = x div y * y + x mod y"
```
```  3620     by (simp add: div_mult_mod_eq)
```
```  3621   show "if y = 0 then x div y = 0 else x mod y = 0 \<or> degree (x mod y) < degree y"
```
```  3622   proof (cases "y = 0")
```
```  3623     case True
```
```  3624     then show ?thesis by auto
```
```  3625   next
```
```  3626     case False
```
```  3627     with pseudo_mod[OF this] show ?thesis
```
```  3628       by (simp add: mod_poly_def)
```
```  3629   qed
```
```  3630 qed
```
```  3631
```
```  3632 lemma div_poly_eq: "eucl_rel_poly x y (q, r) \<Longrightarrow> x div y = q"
```
```  3633   for x :: "'a::field poly"
```
```  3634   by (rule eucl_rel_poly_unique_div [OF eucl_rel_poly])
```
```  3635
```
```  3636 lemma mod_poly_eq: "eucl_rel_poly x y (q, r) \<Longrightarrow> x mod y = r"
```
```  3637   for x :: "'a::field poly"
```
```  3638   by (rule eucl_rel_poly_unique_mod [OF eucl_rel_poly])
```
```  3639
```
```  3640 instance poly :: (field) idom_modulo ..
```
```  3641
```
```  3642 lemma div_pCons_eq:
```
```  3643   "pCons a p div q =
```
```  3644     (if q = 0 then 0
```
```  3645      else pCons (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) (p div q))"
```
```  3646   using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p]
```
```  3647   by (auto intro: div_poly_eq)
```
```  3648
```
```  3649 lemma mod_pCons_eq:
```
```  3650   "pCons a p mod q =
```
```  3651     (if q = 0 then pCons a p
```
```  3652      else pCons a (p mod q) - smult (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) q)"
```
```  3653   using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p]
```
```  3654   by (auto intro: mod_poly_eq)
```
```  3655
```
```  3656 lemma div_mod_fold_coeffs:
```
```  3657   "(p div q, p mod q) =
```
```  3658     (if q = 0 then (0, p)
```
```  3659      else
```
```  3660       fold_coeffs
```
```  3661         (\<lambda>a (s, r).
```
```  3662           let b = coeff (pCons a r) (degree q) / coeff q (degree q)
```
```  3663           in (pCons b s, pCons a r - smult b q)) p (0, 0))"
```
```  3664   by (rule sym, induct p) (auto simp: div_pCons_eq mod_pCons_eq Let_def)
```
```  3665
```
```  3666 lemma degree_mod_less: "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
```
```  3667   using eucl_rel_poly [of x y] unfolding eucl_rel_poly_iff by simp
```
```  3668
```
```  3669 lemma degree_mod_less': "b \<noteq> 0 \<Longrightarrow> a mod b \<noteq> 0 \<Longrightarrow> degree (a mod b) < degree b"
```
```  3670   using degree_mod_less[of b a] by auto
```
```  3671
```
```  3672 lemma div_poly_less:
```
```  3673   fixes x :: "'a::field poly"
```
```  3674   assumes "degree x < degree y"
```
```  3675   shows "x div y = 0"
```
```  3676 proof -
```
```  3677   from assms have "eucl_rel_poly x y (0, x)"
```
```  3678     by (simp add: eucl_rel_poly_iff)
```
```  3679   then show "x div y = 0"
```
```  3680     by (rule div_poly_eq)
```
```  3681 qed
```
```  3682
```
```  3683 lemma mod_poly_less:
```
```  3684   assumes "degree x < degree y"
```
```  3685   shows "x mod y = x"
```
```  3686 proof -
```
```  3687   from assms have "eucl_rel_poly x y (0, x)"
```
```  3688     by (simp add: eucl_rel_poly_iff)
```
```  3689   then show "x mod y = x"
```
```  3690     by (rule mod_poly_eq)
```
```  3691 qed
```
```  3692
```
```  3693 lemma eucl_rel_poly_smult_left:
```
```  3694   "eucl_rel_poly x y (q, r) \<Longrightarrow> eucl_rel_poly (smult a x) y (smult a q, smult a r)"
```
```  3695   by (simp add: eucl_rel_poly_iff smult_add_right)
```
```  3696
```
```  3697 lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
```
```  3698   for x y :: "'a::field poly"
```
```  3699   by (rule div_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly)
```
```  3700
```
```  3701 lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
```
```  3702   by (rule mod_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly)
```
```  3703
```
```  3704 lemma poly_div_minus_left [simp]: "(- x) div y = - (x div y)"
```
```  3705   for x y :: "'a::field poly"
```
```  3706   using div_smult_left [of "- 1::'a"] by simp
```
```  3707
```
```  3708 lemma poly_mod_minus_left [simp]: "(- x) mod y = - (x mod y)"
```
```  3709   for x y :: "'a::field poly"
```
```  3710   using mod_smult_left [of "- 1::'a"] by simp
```
```  3711
```
```  3712 lemma eucl_rel_poly_add_left:
```
```  3713   assumes "eucl_rel_poly x y (q, r)"
```
```  3714   assumes "eucl_rel_poly x' y (q', r')"
```
```  3715   shows "eucl_rel_poly (x + x') y (q + q', r + r')"
```
```  3716   using assms unfolding eucl_rel_poly_iff
```
```  3717   by (auto simp: algebra_simps degree_add_less)
```
```  3718
```
```  3719 lemma poly_div_add_left: "(x + y) div z = x div z + y div z"
```
```  3720   for x y z :: "'a::field poly"
```
```  3721   using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly]
```
```  3722   by (rule div_poly_eq)
```
```  3723
```
```  3724 lemma poly_mod_add_left: "(x + y) mod z = x mod z + y mod z"
```
```  3725   for x y z :: "'a::field poly"
```
```  3726   using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly]
```
```  3727   by (rule mod_poly_eq)
```
```  3728
```
```  3729 lemma poly_div_diff_left: "(x - y) div z = x div z - y div z"
```
```  3730   for x y z :: "'a::field poly"
```
```  3731   by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)
```
```  3732
```
```  3733 lemma poly_mod_diff_left: "(x - y) mod z = x mod z - y mod z"
```
```  3734   for x y z :: "'a::field poly"
```
```  3735   by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)
```
```  3736
```
```  3737 lemma eucl_rel_poly_smult_right:
```
```  3738   "a \<noteq> 0 \<Longrightarrow> eucl_rel_poly x y (q, r) \<Longrightarrow> eucl_rel_poly x (smult a y) (smult (inverse a) q, r)"
```
```  3739   by (simp add: eucl_rel_poly_iff)
```
```  3740
```
```  3741 lemma div_smult_right: "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
```
```  3742   for x y :: "'a::field poly"
```
```  3743   by (rule div_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly)
```
```  3744
```
```  3745 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
```
```  3746   by (rule mod_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly)
```
```  3747
```
```  3748 lemma poly_div_minus_right [simp]: "x div (- y) = - (x div y)"
```
```  3749   for x y :: "'a::field poly"
```
```  3750   using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)
```
```  3751
```
```  3752 lemma poly_mod_minus_right [simp]: "x mod (- y) = x mod y"
```
```  3753   for x y :: "'a::field poly"
```
```  3754   using mod_smult_right [of "- 1::'a"] by simp
```
```  3755
```
```  3756 lemma eucl_rel_poly_mult:
```
```  3757   "eucl_rel_poly x y (q, r) \<Longrightarrow> eucl_rel_poly q z (q', r') \<Longrightarrow>
```
```  3758     eucl_rel_poly x (y * z) (q', y * r' + r)"
```
```  3759   apply (cases "z = 0", simp add: eucl_rel_poly_iff)
```
```  3760   apply (cases "y = 0", simp add: eucl_rel_poly_by_0_iff eucl_rel_poly_0_iff)
```
```  3761   apply (cases "r = 0")
```
```  3762    apply (cases "r' = 0")
```
```  3763     apply (simp add: eucl_rel_poly_iff)
```
```  3764    apply (simp add: eucl_rel_poly_iff field_simps degree_mult_eq)
```
```  3765   apply (cases "r' = 0")
```
```  3766    apply (simp add: eucl_rel_poly_iff degree_mult_eq)
```
```  3767   apply (simp add: eucl_rel_poly_iff field_simps)
```
```  3768   apply (simp add: degree_mult_eq degree_add_less)
```
```  3769   done
```
```  3770
```
```  3771 lemma poly_div_mult_right: "x div (y * z) = (x div y) div z"
```
```  3772   for x y z :: "'a::field poly"
```
```  3773   by (rule div_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+)
```
```  3774
```
```  3775 lemma poly_mod_mult_right: "x mod (y * z) = y * (x div y mod z) + x mod y"
```
```  3776   for x y z :: "'a::field poly"
```
```  3777   by (rule mod_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+)
```
```  3778
```
```  3779 lemma mod_pCons:
```
```  3780   fixes a :: "'a::field"
```
```  3781     and x y :: "'a::field poly"
```
```  3782   assumes y: "y \<noteq> 0"
```
```  3783   defines "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
```
```  3784   shows "(pCons a x) mod y = pCons a (x mod y) - smult b y"
```
```  3785   unfolding b_def
```
```  3786   by (rule mod_poly_eq, rule eucl_rel_poly_pCons [OF eucl_rel_poly y refl])
```
```  3787
```
```  3788
```
```  3789 subsubsection \<open>List-based versions for fast implementation\<close>
```
```  3790 (* Subsection by:
```
```  3791       Sebastiaan Joosten
```
```  3792       RenÃ© Thiemann
```
```  3793       Akihisa Yamada
```
```  3794     *)
```
```  3795 fun minus_poly_rev_list :: "'a :: group_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```  3796   where
```
```  3797     "minus_poly_rev_list (x # xs) (y # ys) = (x - y) # (minus_poly_rev_list xs ys)"
```
```  3798   | "minus_poly_rev_list xs [] = xs"
```
```  3799   | "minus_poly_rev_list [] (y # ys) = []"
```
```  3800
```
```  3801 fun pseudo_divmod_main_list ::
```
```  3802   "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list"
```
```  3803   where
```
```  3804     "pseudo_divmod_main_list lc q r d (Suc n) =
```
```  3805       (let
```
```  3806         rr = map (op * lc) r;
```
```  3807         a = hd r;
```
```  3808         qqq = cCons a (map (op * lc) q);
```
```  3809         rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
```
```  3810        in pseudo_divmod_main_list lc qqq rrr d n)"
```
```  3811   | "pseudo_divmod_main_list lc q r d 0 = (q, r)"
```
```  3812
```
```  3813 fun pseudo_mod_main_list :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a list"
```
```  3814   where
```
```  3815     "pseudo_mod_main_list lc r d (Suc n) =
```
```  3816       (let
```
```  3817         rr = map (op * lc) r;
```
```  3818         a = hd r;
```
```  3819         rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d))
```
```  3820        in pseudo_mod_main_list lc rrr d n)"
```
```  3821   | "pseudo_mod_main_list lc r d 0 = r"
```
```  3822
```
```  3823
```
```  3824 fun divmod_poly_one_main_list ::
```
```  3825     "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list"
```
```  3826   where
```
```  3827     "divmod_poly_one_main_list q r d (Suc n) =
```
```  3828       (let
```
```  3829         a = hd r;
```
```  3830         qqq = cCons a q;
```
```  3831         rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
```
```  3832        in divmod_poly_one_main_list qqq rr d n)"
```
```  3833   | "divmod_poly_one_main_list q r d 0 = (q, r)"
```
```  3834
```
```  3835 fun mod_poly_one_main_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a list"
```
```  3836   where
```
```  3837     "mod_poly_one_main_list r d (Suc n) =
```
```  3838       (let
```
```  3839         a = hd r;
```
```  3840         rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d))
```
```  3841        in mod_poly_one_main_list rr d n)"
```
```  3842   | "mod_poly_one_main_list r d 0 = r"
```
```  3843
```
```  3844 definition pseudo_divmod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list \<times> 'a list"
```
```  3845   where "pseudo_divmod_list p q =
```
```  3846     (if q = [] then ([], p)
```
```  3847      else
```
```  3848       (let rq = rev q;
```
```  3849         (qu,re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q)
```
```  3850        in (qu, rev re)))"
```
```  3851
```
```  3852 definition pseudo_mod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```  3853   where "pseudo_mod_list p q =
```
```  3854     (if q = [] then p
```
```  3855      else
```
```  3856       (let
```
```  3857         rq = rev q;
```
```  3858         re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q)
```
```  3859        in rev re))"
```
```  3860
```
```  3861 lemma minus_zero_does_nothing: "minus_poly_rev_list x (map (op * 0) y) = x"
```
```  3862   for x :: "'a::ring list"
```
```  3863   by (induct x y rule: minus_poly_rev_list.induct) auto
```
```  3864
```
```  3865 lemma length_minus_poly_rev_list [simp]: "length (minus_poly_rev_list xs ys) = length xs"
```
```  3866   by (induct xs ys rule: minus_poly_rev_list.induct) auto
```
```  3867
```
```  3868 lemma if_0_minus_poly_rev_list:
```
```  3869   "(if a = 0 then x else minus_poly_rev_list x (map (op * a) y)) =
```
```  3870     minus_poly_rev_list x (map (op * a) y)"
```
```  3871   for a :: "'a::ring"
```
```  3872   by(cases "a = 0") (simp_all add: minus_zero_does_nothing)
```
```  3873
```
```  3874 lemma Poly_append: "Poly (a @ b) = Poly a + monom 1 (length a) * Poly b"
```
```  3875   for a :: "'a::comm_semiring_1 list"
```
```  3876   by (induct a) (auto simp: monom_0 monom_Suc)
```
```  3877
```
```  3878 lemma minus_poly_rev_list: "length p \<ge> length q \<Longrightarrow>
```
```  3879   Poly (rev (minus_poly_rev_list (rev p) (rev q))) =
```
```  3880     Poly p - monom 1 (length p - length q) * Poly q"
```
```  3881   for p q :: "'a :: comm_ring_1 list"
```
```  3882 proof (induct "rev p" "rev q" arbitrary: p q rule: minus_poly_rev_list.induct)
```
```  3883   case (1 x xs y ys)
```
```  3884   then have "length (rev q) \<le> length (rev p)"
```
```  3885     by simp
```
```  3886   from this[folded 1(2,3)] have ys_xs: "length ys \<le> length xs"
```
```  3887     by simp
```
```  3888   then have *: "Poly (rev (minus_poly_rev_list xs ys)) =
```
```  3889       Poly (rev xs) - monom 1 (length xs - length ys) * Poly (rev ys)"
```
```  3890     by (subst "1.hyps"(1)[of "rev xs" "rev ys", unfolded rev_rev_ident length_rev]) auto
```
```  3891   have "Poly p - monom 1 (length p - length q) * Poly q =
```
```  3892     Poly (rev (rev p)) - monom 1 (length (rev (rev p)) - length (rev (rev q))) * Poly (rev (rev q))"
```
```  3893     by simp
```
```  3894   also have "\<dots> =
```
```  3895       Poly (rev (x # xs)) - monom 1 (length (x # xs) - length (y # ys)) * Poly (rev (y # ys))"
```
```  3896     unfolding 1(2,3) by simp
```
```  3897   also from ys_xs have "\<dots> =
```
```  3898     Poly (rev xs) + monom x (length xs) -
```
```  3899       (monom 1 (length xs - length ys) * Poly (rev ys) + monom y (length xs))"
```
```  3900     by (simp add: Poly_append distrib_left mult_monom smult_monom)
```
```  3901   also have "\<dots> = Poly (rev (minus_poly_rev_list xs ys)) + monom (x - y) (length xs)"
```
```  3902     unfolding * diff_monom[symmetric] by simp
```
```  3903   finally show ?case
```
```  3904     by (simp add: 1(2,3)[symmetric] smult_monom Poly_append)
```
```  3905 qed auto
```
```  3906
```
```  3907 lemma smult_monom_mult: "smult a (monom b n * f) = monom (a * b) n * f"
```
```  3908   using smult_monom [of a _ n] by (metis mult_smult_left)
```
```  3909
```
```  3910 lemma head_minus_poly_rev_list:
```
```  3911   "length d \<le> length r \<Longrightarrow> d \<noteq> [] \<Longrightarrow>
```
```  3912     hd (minus_poly_rev_list (map (op * (last d)) r) (map (op * (hd r)) (rev d))) = 0"
```
```  3913   for d r :: "'a::comm_ring list"
```
```  3914 proof (induct r)
```
```  3915   case Nil
```
```  3916   then show ?case by simp
```
```  3917 next
```
```  3918   case (Cons a rs)
```
```  3919   then show ?case by (cases "rev d") (simp_all add: ac_simps)
```
```  3920 qed
```
```  3921
```
```  3922 lemma Poly_map: "Poly (map (op * a) p) = smult a (Poly p)"
```
```  3923 proof (induct p)
```
```  3924   case Nil
```
```  3925   then show ?case by simp
```
```  3926 next
```
```  3927   case (Cons x xs)
```
```  3928   then show ?case by (cases "Poly xs = 0") auto
```
```  3929 qed
```
```  3930
```
```  3931 lemma last_coeff_is_hd: "xs \<noteq> [] \<Longrightarrow> coeff (Poly xs) (length xs - 1) = hd (rev xs)"
```
```  3932   by (simp_all add: hd_conv_nth rev_nth nth_default_nth nth_append)
```
```  3933
```
```  3934 lemma pseudo_divmod_main_list_invar:
```
```  3935   assumes leading_nonzero: "last d \<noteq> 0"
```
```  3936     and lc: "last d = lc"
```
```  3937     and "d \<noteq> []"
```
```  3938     and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q', rev r')"
```
```  3939     and "n = 1 + length r - length d"
```
```  3940   shows "pseudo_divmod_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n =
```
```  3941     (Poly q', Poly r')"
```
```  3942   using assms(4-)
```
```  3943 proof (induct n arbitrary: r q)
```
```  3944   case (Suc n)
```
```  3945   from Suc.prems have *: "\<not> Suc (length r) \<le> length d"
```
```  3946     by simp
```
```  3947   with \<open>d \<noteq> []\<close> have "r \<noteq> []"
```
```  3948     using Suc_leI length_greater_0_conv list.size(3) by fastforce
```
```  3949   let ?a = "(hd (rev r))"
```
```  3950   let ?rr = "map (op * lc) (rev r)"
```
```  3951   let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (op * ?a) (rev d))))"
```
```  3952   let ?qq = "cCons ?a (map (op * lc) q)"
```
```  3953   from * Suc(3) have n: "n = (1 + length r - length d - 1)"
```
```  3954     by simp
```
```  3955   from * have rr_val:"(length ?rrr) = (length r - 1)"
```
```  3956     by auto
```
```  3957   with \<open>r \<noteq> []\<close> * have rr_smaller: "(1 + length r - length d - 1) = (1 + length ?rrr - length d)"
```
```  3958     by auto
```
```  3959   from * have id: "Suc (length r) - length d = Suc (length r - length d)"
```
```  3960     by auto
```
```  3961   from Suc.prems *
```
```  3962   have "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) (1 + length r - length d - 1) = (q', rev r')"
```
```  3963     by (simp add: Let_def if_0_minus_poly_rev_list id)
```
```  3964   with n have v: "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) n = (q', rev r')"
```
```  3965     by auto
```
```  3966   from * have sucrr:"Suc (length r) - length d = Suc (length r - length d)"
```
```  3967     using Suc_diff_le not_less_eq_eq by blast
```
```  3968   from Suc(3) \<open>r \<noteq> []\<close> have n_ok : "n = 1 + (length ?rrr) - length d"
```
```  3969     by simp
```
```  3970   have cong: "\<And>x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow>
```
```  3971       pseudo_divmod_main lc x1 x2 x3 x4 n = pseudo_divmod_main lc y1 y2 y3 y4 n"
```
```  3972     by simp
```
```  3973   have hd_rev: "coeff (Poly r) (length r - Suc 0) = hd (rev r)"
```
```  3974     using last_coeff_is_hd[OF \<open>r \<noteq> []\<close>] by simp
```
```  3975   show ?case
```
```  3976     unfolding Suc.hyps(1)[OF v n_ok, symmetric] pseudo_divmod_main.simps Let_def
```
```  3977   proof (rule cong[OF _ _ refl], goal_cases)
```
```  3978     case 1
```
```  3979     show ?case
```
```  3980       by (simp add: monom_Suc hd_rev[symmetric] smult_monom Poly_map)
```
```  3981   next
```
```  3982     case 2
```
```  3983     show ?case
```
```  3984     proof (subst Poly_on_rev_starting_with_0, goal_cases)
```
```  3985       show "hd (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))) = 0"
```
```  3986         by (fold lc, subst head_minus_poly_rev_list, insert * \<open>d \<noteq> []\<close>, auto)
```
```  3987       from * have "length d \<le> length r"
```
```  3988         by simp
```
```  3989       then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d =
```
```  3990           Poly (rev (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))))"
```
```  3991         by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric]
```
```  3992             minus_poly_rev_list)
```
```  3993     qed
```
```  3994   qed simp
```
```  3995 qed simp
```
```  3996
```
```  3997 lemma pseudo_divmod_impl [code]:
```
```  3998   "pseudo_divmod f g = map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))"
```
```  3999     for f g :: "'a::comm_ring_1 poly"
```
```  4000 proof (cases "g = 0")
```
```  4001   case False
```
```  4002   then have "last (coeffs g) \<noteq> 0"
```
```  4003     and "last (coeffs g) = lead_coeff g"
```
```  4004     and "coeffs g \<noteq> []"
```
```  4005     by (simp_all add: last_coeffs_eq_coeff_degree)
```
```  4006   moreover obtain q r where qr: "pseudo_divmod_main_list
```
```  4007     (last (coeffs g)) (rev [])
```
```  4008     (rev (coeffs f)) (rev (coeffs g))
```
```  4009     (1 + length (coeffs f) -
```
```  4010     length (coeffs g)) = (q, rev (rev r))"
```
```  4011     by force
```
```  4012   ultimately have "(Poly q, Poly (rev r)) = pseudo_divmod_main (lead_coeff g) 0 f g
```
```  4013     (length (coeffs f) - Suc 0) (Suc (length (coeffs f)) - length (coeffs g))"
```
```  4014     by (subst pseudo_divmod_main_list_invar [symmetric]) auto
```
```  4015   moreover have "pseudo_divmod_main_list
```
```  4016     (hd (rev (coeffs g))) []
```
```  4017     (rev (coeffs f)) (rev (coeffs g))
```
```  4018     (1 + length (coeffs f) -
```
```  4019     length (coeffs g)) = (q, r)"
```
```  4020     using qr hd_rev [OF \<open>coeffs g \<noteq> []\<close>] by simp
```
```  4021   ultimately show ?thesis
```
```  4022     by (auto simp: degree_eq_length_coeffs pseudo_divmod_def pseudo_divmod_list_def Let_def)
```
```  4023 next
```
```  4024   case True
```
```  4025   then show ?thesis
```
```  4026     by (auto simp add: pseudo_divmod_def pseudo_divmod_list_def)
```
```  4027 qed
```
```  4028
```
```  4029 lemma pseudo_mod_main_list:
```
```  4030   "snd (pseudo_divmod_main_list l q xs ys n) = pseudo_mod_main_list l xs ys n"
```
```  4031   by (induct n arbitrary: l q xs ys) (auto simp: Let_def)
```
```  4032
```
```  4033 lemma pseudo_mod_impl[code]: "pseudo_mod f g = poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))"
```
```  4034 proof -
```
```  4035   have snd_case: "\<And>f g p. snd ((\<lambda>(x,y). (f x, g y)) p) = g (snd p)"
```
```  4036     by auto
```
```  4037   show ?thesis
```
```  4038     unfolding pseudo_mod_def pseudo_divmod_impl pseudo_divmod_list_def
```
```  4039       pseudo_mod_list_def Let_def
```
```  4040     by (simp add: snd_case pseudo_mod_main_list)
```
```  4041 qed
```
```  4042
```
```  4043
```
```  4044 subsubsection \<open>Improved Code-Equations for Polynomial (Pseudo) Division\<close>
```
```  4045
```
```  4046 lemma pdivmod_pdivmodrel: "eucl_rel_poly p q (r, s) \<longleftrightarrow> (p div q, p mod q) = (r, s)"
```
```  4047   by (metis eucl_rel_poly eucl_rel_poly_unique)
```
```  4048
```
```  4049 lemma pdivmod_via_pseudo_divmod:
```
```  4050   "(f div g, f mod g) =
```
```  4051     (if g = 0 then (0, f)
```
```  4052      else
```
```  4053       let
```
```  4054         ilc = inverse (coeff g (degree g));
```
```  4055         h = smult ilc g;
```
```  4056         (q,r) = pseudo_divmod f h
```
```  4057       in (smult ilc q, r))"
```
```  4058   (is "?l = ?r")
```
```  4059 proof (cases "g = 0")
```
```  4060   case True
```
```  4061   then show ?thesis by simp
```
```  4062 next
```
```  4063   case False
```
```  4064   define lc where "lc = inverse (coeff g (degree g))"
```
```  4065   define h where "h = smult lc g"
```
```  4066   from False have h1: "coeff h (degree h) = 1" and lc: "lc \<noteq> 0"
```
```  4067     by (auto simp: h_def lc_def)
```
```  4068   then have h0: "h \<noteq> 0"
```
```  4069     by auto
```
```  4070   obtain q r where p: "pseudo_divmod f h = (q, r)"
```
```  4071     by force
```
```  4072   from False have id: "?r = (smult lc q, r)"
```
```  4073     by (auto simp: Let_def h_def[symmetric] lc_def[symmetric] p)
```
```  4074   from pseudo_divmod[OF h0 p, unfolded h1]
```
```  4075   have f: "f = h * q + r" and r: "r = 0 \<or> degree r < degree h"
```
```  4076     by auto
```
```  4077   from f r h0 have "eucl_rel_poly f h (q, r)"
```
```  4078     by (auto simp: eucl_rel_poly_iff)
```
```  4079   then have "(f div h, f mod h) = (q, r)"
```
```  4080     by (simp add: pdivmod_pdivmodrel)
```
```  4081   with lc have "(f div g, f mod g) = (smult lc q, r)"
```
```  4082     by (auto simp: h_def div_smult_right[OF lc] mod_smult_right[OF lc])
```
```  4083   with id show ?thesis
```
```  4084     by auto
```
```  4085 qed
```
```  4086
```
```  4087 lemma pdivmod_via_pseudo_divmod_list:
```
```  4088   "(f div g, f mod g) =
```
```  4089     (let cg = coeffs g in
```
```  4090       if cg = [] then (0, f)
```
```  4091       else
```
```  4092         let
```
```  4093           cf = coeffs f;
```
```  4094           ilc = inverse (last cg);
```
```  4095           ch = map (op * ilc) cg;
```
```  4096           (q, r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg)
```
```  4097         in (poly_of_list (map (op * ilc) q), poly_of_list (rev r)))"
```
```  4098 proof -
```
```  4099   note d = pdivmod_via_pseudo_divmod pseudo_divmod_impl pseudo_divmod_list_def
```
```  4100   show ?thesis
```
```  4101   proof (cases "g = 0")
```
```  4102     case True
```
```  4103     with d show ?thesis by auto
```
```  4104   next
```
```  4105     case False
```
```  4106     define ilc where "ilc = inverse (coeff g (degree g))"
```
```  4107     from False have ilc: "ilc \<noteq> 0"
```
```  4108       by (auto simp: ilc_def)
```
```  4109     with False have id: "g = 0 \<longleftrightarrow> False" "coeffs g = [] \<longleftrightarrow> False"
```
```  4110       "last (coeffs g) = coeff g (degree g)"
```
```  4111       "coeffs (smult ilc g) = [] \<longleftrightarrow> False"
```
```  4112       by (auto simp: last_coeffs_eq_coeff_degree)
```
```  4113     have id2: "hd (rev (coeffs (smult ilc g))) = 1"
```
```  4114       by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def)
```
```  4115     have id3: "length (coeffs (smult ilc g)) = length (coeffs g)"
```
```  4116       "rev (coeffs (smult ilc g)) = rev (map (op * ilc) (coeffs g))"
```
```  4117       unfolding coeffs_smult using ilc by auto
```
```  4118     obtain q r where pair:
```
```  4119       "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (op * ilc) (coeffs g)))
```
```  4120         (1 + length (coeffs f) - length (coeffs g)) = (q, r)"
```
```  4121       by force
```
```  4122     show ?thesis
```
```  4123       unfolding d Let_def id if_False ilc_def[symmetric] map_prod_def[symmetric] id2
```
```  4124       unfolding id3 pair map_prod_def split
```
```  4125       by (auto simp: Poly_map)
```
```  4126   qed
```
```  4127 qed
```
```  4128
```
```  4129 lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list"
```
```  4130 proof (intro ext, goal_cases)
```
```  4131   case (1 q r d n)
```
```  4132   have *: "map (op * 1) xs = xs" for xs :: "'a list"
```
```  4133     by (induct xs) auto
```
```  4134   show ?case
```
```  4135     by (induct n arbitrary: q r d) (auto simp: * Let_def)
```
```  4136 qed
```
```  4137
```
```  4138 fun divide_poly_main_list :: "'a::idom_divide \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a list"
```
```  4139   where
```
```  4140     "divide_poly_main_list lc q r d (Suc n) =
```
```  4141       (let
```
```  4142         cr = hd r
```
```  4143         in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let
```
```  4144         a = cr div lc;
```
```  4145         qq = cCons a q;
```
```  4146         rr = minus_poly_rev_list r (map (op * a) d)
```
```  4147        in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"
```
```  4148   | "divide_poly_main_list lc q r d 0 = q"
```
```  4149
```
```  4150 lemma divide_poly_main_list_simp [simp]:
```
```  4151   "divide_poly_main_list lc q r d (Suc n) =
```
```  4152     (let
```
```  4153       cr = hd r;
```
```  4154       a = cr div lc;
```
```  4155       qq = cCons a q;
```
```  4156       rr = minus_poly_rev_list r (map (op * a) d)
```
```  4157      in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])"
```
```  4158   by (simp add: Let_def minus_zero_does_nothing)
```
```  4159
```
```  4160 declare divide_poly_main_list.simps(1)[simp del]
```
```  4161
```
```  4162 definition divide_poly_list :: "'a::idom_divide poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  4163   where "divide_poly_list f g =
```
```  4164     (let cg = coeffs g in
```
```  4165       if cg = [] then g
```
```  4166       else
```
```  4167         let
```
```  4168           cf = coeffs f;
```
```  4169           cgr = rev cg
```
```  4170         in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))"
```
```  4171
```
```  4172 lemmas pdivmod_via_divmod_list = pdivmod_via_pseudo_divmod_list[unfolded pseudo_divmod_main_list_1]
```
```  4173
```
```  4174 lemma mod_poly_one_main_list: "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n"
```
```  4175   by (induct n arbitrary: q r d) (auto simp: Let_def)
```
```  4176
```
```  4177 lemma mod_poly_code [code]:
```
```  4178   "f mod g =
```
```  4179     (let cg = coeffs g in
```
```  4180       if cg = [] then f
```
```  4181       else
```
```  4182         let
```
```  4183           cf = coeffs f;
```
```  4184           ilc = inverse (last cg);
```
```  4185           ch = map (op * ilc) cg;
```
```  4186           r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg)
```
```  4187         in poly_of_list (rev r))"
```
```  4188   (is "_ = ?rhs")
```
```  4189 proof -
```
```  4190   have "snd (f div g, f mod g) = ?rhs"
```
```  4191     unfolding pdivmod_via_divmod_list Let_def mod_poly_one_main_list [symmetric, of _ _ _ Nil]
```
```  4192     by (auto split: prod.splits)
```
```  4193   then show ?thesis by simp
```
```  4194 qed
```
```  4195
```
```  4196 definition div_field_poly_impl :: "'a :: field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  4197   where "div_field_poly_impl f g =
```
```  4198     (let cg = coeffs g in
```
```  4199       if cg = [] then 0
```
```  4200       else
```
```  4201         let
```
```  4202           cf = coeffs f;
```
```  4203           ilc = inverse (last cg);
```
```  4204           ch = map (op * ilc) cg;
```
```  4205           q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg))
```
```  4206         in poly_of_list ((map (op * ilc) q)))"
```
```  4207
```
```  4208 text \<open>We do not declare the following lemma as code equation, since then polynomial division
```
```  4209   on non-fields will no longer be executable. However, a code-unfold is possible, since
```
```  4210   \<open>div_field_poly_impl\<close> is a bit more efficient than the generic polynomial division.\<close>
```
```  4211 lemma div_field_poly_impl[code_unfold]: "op div = div_field_poly_impl"
```
```  4212 proof (intro ext)
```
```  4213   fix f g :: "'a poly"
```
```  4214   have "fst (f div g, f mod g) = div_field_poly_impl f g"
```
```  4215     unfolding div_field_poly_impl_def pdivmod_via_divmod_list Let_def
```
```  4216     by (auto split: prod.splits)
```
```  4217   then show "f div g =  div_field_poly_impl f g"
```
```  4218     by simp
```
```  4219 qed
```
```  4220
```
```  4221 lemma divide_poly_main_list:
```
```  4222   assumes lc0: "lc \<noteq> 0"
```
```  4223     and lc: "last d = lc"
```
```  4224     and d: "d \<noteq> []"
```
```  4225     and "n = (1 + length r - length d)"
```
```  4226   shows "Poly (divide_poly_main_list lc q (rev r) (rev d) n) =
```
```  4227     divide_poly_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n"
```
```  4228   using assms(4-)
```
```  4229 proof (induct "n" arbitrary: r q)
```
```  4230   case (Suc n)
```
```  4231   from Suc.prems have ifCond: "\<not> Suc (length r) \<le> length d"
```
```  4232     by simp
```
```  4233   with d have r: "r \<noteq> []"
```
```  4234     using Suc_leI length_greater_0_conv list.size(3) by fastforce
```
```  4235   then obtain rr lcr where r: "r = rr @ [lcr]"
```
```  4236     by (cases r rule: rev_cases) auto
```
```  4237   from d lc obtain dd where d: "d = dd @ [lc]"
```
```  4238     by (cases d rule: rev_cases) auto
```
```  4239   from Suc(2) ifCond have n: "n = 1 + length rr - length d"
```
```  4240     by (auto simp: r)
```
```  4241   from ifCond have len: "length dd \<le> length rr"
```
```  4242     by (simp add: r d)
```
```  4243   show ?case
```
```  4244   proof (cases "lcr div lc * lc = lcr")
```
```  4245     case False
```
```  4246     with r d show ?thesis
```
```  4247       unfolding Suc(2)[symmetric]
```
```  4248       by (auto simp add: Let_def nth_default_append)
```
```  4249   next
```
```  4250     case True
```
```  4251     with r d have id:
```
```  4252       "?thesis \<longleftrightarrow>
```
```  4253         Poly (divide_poly_main_list lc (cCons (lcr div lc) q)
```
```  4254           (rev (rev (minus_poly_rev_list (rev rr) (rev (map (op * (lcr div lc)) dd))))) (rev d) n) =
```
```  4255           divide_poly_main lc
```
```  4256             (monom 1 (Suc n) * Poly q + monom (lcr div lc) n)
```
```  4257             (Poly r - monom (lcr div lc) n * Poly d)
```
```  4258             (Poly d) (length rr - 1) n"
```
```  4259       by (cases r rule: rev_cases; cases "d" rule: rev_cases)
```
```  4260         (auto simp add: Let_def rev_map nth_default_append)
```
```  4261     have cong: "\<And>x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow>
```
```  4262         divide_poly_main lc x1 x2 x3 x4 n = divide_poly_main lc y1 y2 y3 y4 n"
```
```  4263       by simp
```
```  4264     show ?thesis
```
```  4265       unfolding id
```
```  4266     proof (subst Suc(1), simp add: n,
```
```  4267         subst minus_poly_rev_list, force simp: len, rule cong[OF _ _ refl], goal_cases)
```
```  4268       case 2
```
```  4269       have "monom lcr (length rr) = monom (lcr div lc) (length rr - length dd) * monom lc (length dd)"
```
```  4270         by (simp add: mult_monom len True)
```
```  4271       then show ?case unfolding r d Poly_append n ring_distribs
```
```  4272         by (auto simp: Poly_map smult_monom smult_monom_mult)
```
```  4273     qed (auto simp: len monom_Suc smult_monom)
```
```  4274   qed
```
```  4275 qed simp
```
```  4276
```
```  4277 lemma divide_poly_list[code]: "f div g = divide_poly_list f g"
```
```  4278 proof -
```
```  4279   note d = divide_poly_def divide_poly_list_def
```
```  4280   show ?thesis
```
```  4281   proof (cases "g = 0")
```
```  4282     case True
```
```  4283     show ?thesis by (auto simp: d True)
```
```  4284   next
```
```  4285     case False
```
```  4286     then obtain cg lcg where cg: "coeffs g = cg @ [lcg]"
```
```  4287       by (cases "coeffs g" rule: rev_cases) auto
```
```  4288     with False have id: "(g = 0) = False" "(cg @ [lcg] = []) = False"
```
```  4289       by auto
```
```  4290     from cg False have lcg: "coeff g (degree g) = lcg"
```
```  4291       using last_coeffs_eq_coeff_degree last_snoc by force
```
```  4292     with False have "lcg \<noteq> 0" by auto
```
```  4293     from cg Poly_coeffs [of g] have ltp: "Poly (cg @ [lcg]) = g"
```
```  4294       by auto
```
```  4295     show ?thesis
```
```  4296       unfolding d cg Let_def id if_False poly_of_list_def
```
```  4297       by (subst divide_poly_main_list, insert False cg \<open>lcg \<noteq> 0\<close>)
```
```  4298         (auto simp: lcg ltp, simp add: degree_eq_length_coeffs)
```
```  4299   qed
```
```  4300 qed
```
```  4301
```
```  4302
```
```  4303 subsection \<open>Primality and irreducibility in polynomial rings\<close>
```
```  4304
```
```  4305 lemma prod_mset_const_poly: "(\<Prod>x\<in>#A. [:f x:]) = [:prod_mset (image_mset f A):]"
```
```  4306   by (induct A) (simp_all add: ac_simps)
```
```  4307
```
```  4308 lemma irreducible_const_poly_iff:
```
```  4309   fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}"
```
```  4310   shows "irreducible [:c:] \<longleftrightarrow> irreducible c"
```
```  4311 proof
```
```  4312   assume A: "irreducible c"
```
```  4313   show "irreducible [:c:]"
```
```  4314   proof (rule irreducibleI)
```
```  4315     fix a b assume ab: "[:c:] = a * b"
```
```  4316     hence "degree [:c:] = degree (a * b)" by (simp only: )
```
```  4317     also from A ab have "a \<noteq> 0" "b \<noteq> 0" by auto
```
```  4318     hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq)
```
```  4319     finally have "degree a = 0" "degree b = 0" by auto
```
```  4320     then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE)
```
```  4321     from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: )
```
```  4322     hence "c = a' * b'" by (simp add: ab' mult_ac)
```
```  4323     from A and this have "a' dvd 1 \<or> b' dvd 1" by (rule irreducibleD)
```
```  4324     with ab' show "a dvd 1 \<or> b dvd 1"
```
```  4325       by (auto simp add: is_unit_const_poly_iff)
```
```  4326   qed (insert A, auto simp: irreducible_def is_unit_poly_iff)
```
```  4327 next
```
```  4328   assume A: "irreducible [:c:]"
```
```  4329   then have "c \<noteq> 0" and "\<not> c dvd 1"
```
```  4330     by (auto simp add: irreducible_def is_unit_const_poly_iff)
```
```  4331   then show "irreducible c"
```
```  4332   proof (rule irreducibleI)
```
```  4333     fix a b assume ab: "c = a * b"
```
```  4334     hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac)
```
```  4335     from A and this have "[:a:] dvd 1 \<or> [:b:] dvd 1" by (rule irreducibleD)
```
```  4336     then show "a dvd 1 \<or> b dvd 1"
```
```  4337       by (auto simp add: is_unit_const_poly_iff)
```
```  4338   qed
```
```  4339 qed
```
```  4340
```
```  4341 lemma lift_prime_elem_poly:
```
```  4342   assumes "prime_elem (c :: 'a :: semidom)"
```
```  4343   shows   "prime_elem [:c:]"
```
```  4344 proof (rule prime_elemI)
```
```  4345   fix a b assume *: "[:c:] dvd a * b"
```
```  4346   from * have dvd: "c dvd coeff (a * b) n" for n
```
```  4347     by (subst (asm) const_poly_dvd_iff) blast
```
```  4348   {
```
```  4349     define m where "m = (GREATEST m. \<not>c dvd coeff b m)"
```
```  4350     assume "\<not>[:c:] dvd b"
```
```  4351     hence A: "\<exists>i. \<not>c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
```
```  4352     have B: "\<forall>i. \<not>c dvd coeff b i \<longrightarrow> i \<le> degree b"
```
```  4353       by (auto intro: le_degree)
```
```  4354     have coeff_m: "\<not>c dvd coeff b m" unfolding m_def by (rule GreatestI_ex_nat[OF A B])
```
```  4355     have "i \<le> m" if "\<not>c dvd coeff b i" for i
```
```  4356       unfolding m_def by (rule Greatest_le_nat[OF that B])
```
```  4357     hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force
```
```  4358
```
```  4359     have "c dvd coeff a i" for i
```
```  4360     proof (induction i rule: nat_descend_induct[of "degree a"])
```
```  4361       case (base i)
```
```  4362       thus ?case by (simp add: coeff_eq_0)
```
```  4363     next
```
```  4364       case (descend i)
```
```  4365       let ?A = "{..i+m} - {i}"
```
```  4366       have "c dvd coeff (a * b) (i + m)" by (rule dvd)
```
```  4367       also have "coeff (a * b) (i + m) = (\<Sum>k\<le>i + m. coeff a k * coeff b (i + m - k))"
```
```  4368         by (simp add: coeff_mult)
```
```  4369       also have "{..i+m} = insert i ?A" by auto
```
```  4370       also have "(\<Sum>k\<in>\<dots>. coeff a k * coeff b (i + m - k)) =
```
```  4371                    coeff a i * coeff b m + (\<Sum>k\<in>?A. coeff a k * coeff b (i + m - k))"
```
```  4372         (is "_ = _ + ?S")
```
```  4373         by (subst sum.insert) simp_all
```
```  4374       finally have eq: "c dvd coeff a i * coeff b m + ?S" .
```
```  4375       moreover have "c dvd ?S"
```
```  4376       proof (rule dvd_sum)
```
```  4377         fix k assume k: "k \<in> {..i+m} - {i}"
```
```  4378         show "c dvd coeff a k * coeff b (i + m - k)"
```
```  4379         proof (cases "k < i")
```
```  4380           case False
```
```  4381           with k have "c dvd coeff a k" by (intro descend.IH) simp
```
```  4382           thus ?thesis by simp
```
```  4383         next
```
```  4384           case True
```
```  4385           hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp
```
```  4386           thus ?thesis by simp
```
```  4387         qed
```
```  4388       qed
```
```  4389       ultimately have "c dvd coeff a i * coeff b m"
```
```  4390         by (simp add: dvd_add_left_iff)
```
```  4391       with assms coeff_m show "c dvd coeff a i"
```
```  4392         by (simp add: prime_elem_dvd_mult_iff)
```
```  4393     qed
```
```  4394     hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast
```
```  4395   }
```
```  4396   then show "[:c:] dvd a \<or> [:c:] dvd b" by blast
```
```  4397 next
```
```  4398   from assms show "[:c:] \<noteq> 0" and "\<not> [:c:] dvd 1"
```
```  4399     by (simp_all add: prime_elem_def is_unit_const_poly_iff)
```
```  4400 qed
```
```  4401
```
```  4402 lemma prime_elem_const_poly_iff:
```
```  4403   fixes c :: "'a :: semidom"
```
```  4404   shows   "prime_elem [:c:] \<longleftrightarrow> prime_elem c"
```
```  4405 proof
```
```  4406   assume A: "prime_elem [:c:]"
```
```  4407   show "prime_elem c"
```
```  4408   proof (rule prime_elemI)
```
```  4409     fix a b assume "c dvd a * b"
```
```  4410     hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac)
```
```  4411     from A and this have "[:c:] dvd [:a:] \<or> [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
```
```  4412     thus "c dvd a \<or> c dvd b" by simp
```
```  4413   qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
```
```  4414 qed (auto intro: lift_prime_elem_poly)
```
```  4415
```
```  4416
```
```  4417 subsection \<open>Content and primitive part of a polynomial\<close>
```
```  4418
```
```  4419 definition content :: "'a::semiring_gcd poly \<Rightarrow> 'a"
```
```  4420   where "content p = gcd_list (coeffs p)"
```
```  4421
```
```  4422 lemma content_eq_fold_coeffs [code]: "content p = fold_coeffs gcd p 0"
```
```  4423   by (simp add: content_def Gcd_fin.set_eq_fold fold_coeffs_def foldr_fold fun_eq_iff ac_simps)
```
```  4424
```
```  4425 lemma content_0 [simp]: "content 0 = 0"
```
```  4426   by (simp add: content_def)
```
```  4427
```
```  4428 lemma content_1 [simp]: "content 1 = 1"
```
```  4429   by (simp add: content_def)
```
```  4430
```
```  4431 lemma content_const [simp]: "content [:c:] = normalize c"
```
```  4432   by (simp add: content_def cCons_def)
```
```  4433
```
```  4434 lemma const_poly_dvd_iff_dvd_content: "[:c:] dvd p \<longleftrightarrow> c dvd content p"
```
```  4435   for c :: "'a::semiring_gcd"
```
```  4436 proof (cases "p = 0")
```
```  4437   case True
```
```  4438   then show ?thesis by simp
```
```  4439 next
```
```  4440   case False
```
```  4441   have "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)"
```
```  4442     by (rule const_poly_dvd_iff)
```
```  4443   also have "\<dots> \<longleftrightarrow> (\<forall>a\<in>set (coeffs p). c dvd a)"
```
```  4444   proof safe
```
```  4445     fix n :: nat
```
```  4446     assume "\<forall>a\<in>set (coeffs p). c dvd a"
```
```  4447     then show "c dvd coeff p n"
```
```  4448       by (cases "n \<le> degree p") (auto simp: coeff_eq_0 coeffs_def split: if_splits)
```
```  4449   qed (auto simp: coeffs_def simp del: upt_Suc split: if_splits)
```
```  4450   also have "\<dots> \<longleftrightarrow> c dvd content p"
```
```  4451     by (simp add: content_def dvd_Gcd_fin_iff dvd_mult_unit_iff)
```
```  4452   finally show ?thesis .
```
```  4453 qed
```
```  4454
```
```  4455 lemma content_dvd [simp]: "[:content p:] dvd p"
```
```  4456   by (subst const_poly_dvd_iff_dvd_content) simp_all
```
```  4457
```
```  4458 lemma content_dvd_coeff [simp]: "content p dvd coeff p n"
```
```  4459 proof (cases "p = 0")
```
```  4460   case True
```
```  4461   then show ?thesis
```
```  4462     by simp
```
```  4463 next
```
```  4464   case False
```
```  4465   then show ?thesis
```
```  4466     by (cases "n \<le> degree p")
```
```  4467       (auto simp add: content_def not_le coeff_eq_0 coeff_in_coeffs intro: Gcd_fin_dvd)
```
```  4468 qed
```
```  4469
```
```  4470 lemma content_dvd_coeffs: "c \<in> set (coeffs p) \<Longrightarrow> content p dvd c"
```
```  4471   by (simp add: content_def Gcd_fin_dvd)
```
```  4472
```
```  4473 lemma normalize_content [simp]: "normalize (content p) = content p"
```
```  4474   by (simp add: content_def)
```
```  4475
```
```  4476 lemma is_unit_content_iff [simp]: "is_unit (content p) \<longleftrightarrow> content p = 1"
```
```  4477 proof
```
```  4478   assume "is_unit (content p)"
```
```  4479   then have "normalize (content p) = 1" by (simp add: is_unit_normalize del: normalize_content)
```
```  4480   then show "content p = 1" by simp
```
```  4481 qed auto
```
```  4482
```
```  4483 lemma content_smult [simp]: "content (smult c p) = normalize c * content p"
```
```  4484   by (simp add: content_def coeffs_smult Gcd_fin_mult)
```
```  4485
```
```  4486 lemma content_eq_zero_iff [simp]: "content p = 0 \<longleftrightarrow> p = 0"
```
```  4487   by (auto simp: content_def simp: poly_eq_iff coeffs_def)
```
```  4488
```
```  4489 definition primitive_part :: "'a :: semiring_gcd poly \<Rightarrow> 'a poly"
```
```  4490   where "primitive_part p = map_poly (\<lambda>x. x div content p) p"
```
```  4491
```
```  4492 lemma primitive_part_0 [simp]: "primitive_part 0 = 0"
```
```  4493   by (simp add: primitive_part_def)
```
```  4494
```
```  4495 lemma content_times_primitive_part [simp]: "smult (content p) (primitive_part p) = p"
```
```  4496   for p :: "'a :: semiring_gcd poly"
```
```  4497 proof (cases "p = 0")
```
```  4498   case True
```
```  4499   then show ?thesis by simp
```
```  4500 next
```
```  4501   case False
```
```  4502   then show ?thesis
```
```  4503   unfolding primitive_part_def
```
```  4504   by (auto simp: smult_conv_map_poly map_poly_map_poly o_def content_dvd_coeffs
```
```  4505       intro: map_poly_idI)
```
```  4506 qed
```
```  4507
```
```  4508 lemma primitive_part_eq_0_iff [simp]: "primitive_part p = 0 \<longleftrightarrow> p = 0"
```
```  4509 proof (cases "p = 0")
```
```  4510   case True
```
```  4511   then show ?thesis by simp
```
```  4512 next
```
```  4513   case False
```
```  4514   then have "primitive_part p = map_poly (\<lambda>x. x div content p) p"
```
```  4515     by (simp add:  primitive_part_def)
```
```  4516   also from False have "\<dots> = 0 \<longleftrightarrow> p = 0"
```
```  4517     by (intro map_poly_eq_0_iff) (auto simp: dvd_div_eq_0_iff content_dvd_coeffs)
```
```  4518   finally show ?thesis
```
```  4519     using False by simp
```
```  4520 qed
```
```  4521
```
```  4522 lemma content_primitive_part [simp]:
```
```  4523   assumes "p \<noteq> 0"
```
```  4524   shows "content (primitive_part p) = 1"
```
```  4525 proof -
```
```  4526   have "p = smult (content p) (primitive_part p)"
```
```  4527     by simp
```
```  4528   also have "content \<dots> = content (primitive_part p) * content p"
```
```  4529     by (simp del: content_times_primitive_part add: ac_simps)
```
```  4530   finally have "1 * content p = content (primitive_part p) * content p"
```
```  4531     by simp
```
```  4532   then have "1 * content p div content p = content (primitive_part p) * content p div content p"
```
```  4533     by simp
```
```  4534   with assms show ?thesis
```
```  4535     by simp
```
```  4536 qed
```
```  4537
```
```  4538 lemma content_decompose:
```
```  4539   obtains p' :: "'a::semiring_gcd poly" where "p = smult (content p) p'" "content p' = 1"
```
```  4540 proof (cases "p = 0")
```
```  4541   case True
```
```  4542   then show ?thesis by (intro that[of 1]) simp_all
```
```  4543 next
```
```  4544   case False
```
```  4545   from content_dvd[of p] obtain r where r: "p = [:content p:] * r"
```
```  4546     by (rule dvdE)
```
```  4547   have "content p * 1 = content p * content r"
```
```  4548     by (subst r) simp
```
```  4549   with False have "content r = 1"
```
```  4550     by (subst (asm) mult_left_cancel) simp_all
```
```  4551   with r show ?thesis
```
```  4552     by (intro that[of r]) simp_all
```
```  4553 qed
```
```  4554
```
```  4555 lemma content_dvd_contentI [intro]: "p dvd q \<Longrightarrow> content p dvd content q"
```
```  4556   using const_poly_dvd_iff_dvd_content content_dvd dvd_trans by blast
```
```  4557
```
```  4558 lemma primitive_part_const_poly [simp]: "primitive_part [:x:] = [:unit_factor x:]"
```
```  4559   by (simp add: primitive_part_def map_poly_pCons)
```
```  4560
```
```  4561 lemma primitive_part_prim: "content p = 1 \<Longrightarrow> primitive_part p = p"
```
```  4562   by (auto simp: primitive_part_def)
```
```  4563
```
```  4564 lemma degree_primitive_part [simp]: "degree (primitive_part p) = degree p"
```
```  4565 proof (cases "p = 0")
```
```  4566   case True
```
```  4567   then show ?thesis by simp
```
```  4568 next
```
```  4569   case False
```
```  4570   have "p = smult (content p) (primitive_part p)"
```
```  4571     by simp
```
```  4572   also from False have "degree \<dots> = degree (primitive_part p)"
```
```  4573     by (subst degree_smult_eq) simp_all
```
```  4574   finally show ?thesis ..
```
```  4575 qed
```
```  4576
```
```  4577 lemma smult_content_normalize_primitive_part [simp]:
```
```  4578   "smult (content p) (normalize (primitive_part p)) = normalize p"
```
```  4579 proof -
```
```  4580   have "smult (content p) (normalize (primitive_part p)) =
```
```  4581       normalize ([:content p:] * primitive_part p)"
```
```  4582     by (subst normalize_mult) (simp_all add: normalize_const_poly)
```
```  4583   also have "[:content p:] * primitive_part p = p" by simp
```
```  4584   finally show ?thesis .
```
```  4585 qed
```
```  4586
```
```  4587 lemma content_mult:
```
```  4588   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
```
```  4589   shows "content (p * q) = content p * content q"
```
```  4590 proof -
```
```  4591   from content_decompose[of p] guess p' . note p = this
```
```  4592   from content_decompose[of q] guess q' . note q = this
```
```  4593   have "content (p * q) = content p * content q * content (p' * q')"
```
```  4594     by (subst p, subst q) (simp add: mult_ac normalize_mult)
```
```  4595   also have "content (p' * q') = 1"
```
```  4596   proof (cases "p' * q' = 0")
```
```  4597     case True
```
```  4598     with \<open>content p' = 1\<close> \<open>content q' = 1\<close> show ?thesis
```
```  4599       by auto
```
```  4600   next
```
```  4601     case False
```
```  4602     from \<open>content p' = 1\<close> \<open>content q' = 1\<close>
```
```  4603     have "p' \<noteq> 0" "q' \<noteq> 0"
```
```  4604       by auto
```
```  4605     then have "p' * q' \<noteq> 0"
```
```  4606       by auto
```
```  4607     have "is_unit (content (p' * q'))"
```
```  4608     proof (rule ccontr)
```
```  4609       assume "\<not> is_unit (content (p' * q'))"
```
```  4610       with False have "\<exists>p. p dvd content (p' * q') \<and> prime p"
```
```  4611         by (intro prime_divisor_exists) simp_all
```
```  4612       then obtain p where "p dvd content (p' * q')" "prime p"
```
```  4613         by blast
```
```  4614       from \<open>p dvd content (p' * q')\<close> have "[:p:] dvd p' * q'"
```
```  4615         by (simp add: const_poly_dvd_iff_dvd_content)
```
```  4616       moreover from \<open>prime p\<close> have "prime_elem [:p:]"
```
```  4617         by (simp add: lift_prime_elem_poly)
```
```  4618       ultimately have "[:p:] dvd p' \<or> [:p:] dvd q'"
```
```  4619         by (simp add: prime_elem_dvd_mult_iff)
```
```  4620       with \<open>content p' = 1\<close> \<open>content q' = 1\<close> have "is_unit p"
```
```  4621         by (simp add: const_poly_dvd_iff_dvd_content)
```
```  4622       with \<open>prime p\<close> show False
```
```  4623         by simp
```
```  4624     qed
```
```  4625     then have "normalize (content (p' * q')) = 1"
```
```  4626       by (simp add: is_unit_normalize del: normalize_content)
```
```  4627     then show ?thesis
```
```  4628       by simp
```
```  4629   qed
```
```  4630   finally show ?thesis
```
```  4631     by simp
```
```  4632 qed
```
```  4633
```
```  4634 lemma primitive_part_mult:
```
```  4635   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```  4636   shows "primitive_part (p * q) = primitive_part p * primitive_part q"
```
```  4637 proof -
```
```  4638   have "primitive_part (p * q) = p * q div [:content (p * q):]"
```
```  4639     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
```
```  4640   also have "\<dots> = (p div [:content p:]) * (q div [:content q:])"
```
```  4641     by (subst div_mult_div_if_dvd) (simp_all add: content_mult mult_ac)
```
```  4642   also have "\<dots> = primitive_part p * primitive_part q"
```
```  4643     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
```
```  4644   finally show ?thesis .
```
```  4645 qed
```
```  4646
```
```  4647 lemma primitive_part_smult:
```
```  4648   fixes p :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```  4649   shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
```
```  4650 proof -
```
```  4651   have "smult a p = [:a:] * p" by simp
```
```  4652   also have "primitive_part \<dots> = smult (unit_factor a) (primitive_part p)"
```
```  4653     by (subst primitive_part_mult) simp_all
```
```  4654   finally show ?thesis .
```
```  4655 qed
```
```  4656
```
```  4657 lemma primitive_part_dvd_primitive_partI [intro]:
```
```  4658   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```  4659   shows "p dvd q \<Longrightarrow> primitive_part p dvd primitive_part q"
```
```  4660   by (auto elim!: dvdE simp: primitive_part_mult)
```
```  4661
```
```  4662 lemma content_prod_mset:
```
```  4663   fixes A :: "'a :: {factorial_semiring, semiring_Gcd} poly multiset"
```
```  4664   shows "content (prod_mset A) = prod_mset (image_mset content A)"
```
```  4665   by (induction A) (simp_all add: content_mult mult_ac)
```
```  4666
```
```  4667 lemma content_prod_eq_1_iff:
```
```  4668   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
```
```  4669   shows "content (p * q) = 1 \<longleftrightarrow> content p = 1 \<and> content q = 1"
```
```  4670 proof safe
```
```  4671   assume A: "content (p * q) = 1"
```
```  4672   {
```
```  4673     fix p q :: "'a poly" assume "content p * content q = 1"
```
```  4674     hence "1 = content p * content q" by simp
```
```  4675     hence "content p dvd 1" by (rule dvdI)
```
```  4676     hence "content p = 1" by simp
```
```  4677   } note B = this
```
```  4678   from A B[of p q] B [of q p] show "content p = 1" "content q = 1"
```
```  4679     by (simp_all add: content_mult mult_ac)
```
```  4680 qed (auto simp: content_mult)
```
```  4681
```
```  4682
```
```  4683 no_notation cCons (infixr "##" 65)
```
```  4684
```
```  4685 end
```