src/HOL/Library/Polynomial.thy
 author haftmann Fri Nov 01 18:51:14 2013 +0100 (2013-11-01) changeset 54230 b1d955791529 parent 52380 3cc46b8cca5e child 54489 03ff4d1e6784 permissions -rw-r--r--
more simplification rules on unary and binary minus
```     1 (*  Title:      HOL/Library/Polynomial.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Clemens Ballarin
```
```     4     Author:     Florian Haftmann
```
```     5 *)
```
```     6
```
```     7 header {* Polynomials as type over a ring structure *}
```
```     8
```
```     9 theory Polynomial
```
```    10 imports Main GCD
```
```    11 begin
```
```    12
```
```    13 subsection {* Auxiliary: operations for lists (later) representing coefficients *}
```
```    14
```
```    15 definition strip_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```    16 where
```
```    17   "strip_while P = rev \<circ> dropWhile P \<circ> rev"
```
```    18
```
```    19 lemma strip_while_Nil [simp]:
```
```    20   "strip_while P [] = []"
```
```    21   by (simp add: strip_while_def)
```
```    22
```
```    23 lemma strip_while_append [simp]:
```
```    24   "\<not> P x \<Longrightarrow> strip_while P (xs @ [x]) = xs @ [x]"
```
```    25   by (simp add: strip_while_def)
```
```    26
```
```    27 lemma strip_while_append_rec [simp]:
```
```    28   "P x \<Longrightarrow> strip_while P (xs @ [x]) = strip_while P xs"
```
```    29   by (simp add: strip_while_def)
```
```    30
```
```    31 lemma strip_while_Cons [simp]:
```
```    32   "\<not> P x \<Longrightarrow> strip_while P (x # xs) = x # strip_while P xs"
```
```    33   by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
```
```    34
```
```    35 lemma strip_while_eq_Nil [simp]:
```
```    36   "strip_while P xs = [] \<longleftrightarrow> (\<forall>x\<in>set xs. P x)"
```
```    37   by (simp add: strip_while_def)
```
```    38
```
```    39 lemma strip_while_eq_Cons_rec:
```
```    40   "strip_while P (x # xs) = x # strip_while P xs \<longleftrightarrow> \<not> (P x \<and> (\<forall>x\<in>set xs. P x))"
```
```    41   by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
```
```    42
```
```    43 lemma strip_while_not_last [simp]:
```
```    44   "\<not> P (last xs) \<Longrightarrow> strip_while P xs = xs"
```
```    45   by (cases xs rule: rev_cases) simp_all
```
```    46
```
```    47 lemma split_strip_while_append:
```
```    48   fixes xs :: "'a list"
```
```    49   obtains ys zs :: "'a list"
```
```    50   where "strip_while P xs = ys" and "\<forall>x\<in>set zs. P x" and "xs = ys @ zs"
```
```    51 proof (rule that)
```
```    52   show "strip_while P xs = strip_while P xs" ..
```
```    53   show "\<forall>x\<in>set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
```
```    54   have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
```
```    55     by (simp add: strip_while_def)
```
```    56   then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
```
```    57     by (simp only: rev_is_rev_conv)
```
```    58 qed
```
```    59
```
```    60
```
```    61 definition nth_default :: "'a \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a"
```
```    62 where
```
```    63   "nth_default x xs n = (if n < length xs then xs ! n else x)"
```
```    64
```
```    65 lemma nth_default_Nil [simp]:
```
```    66   "nth_default y [] n = y"
```
```    67   by (simp add: nth_default_def)
```
```    68
```
```    69 lemma nth_default_Cons_0 [simp]:
```
```    70   "nth_default y (x # xs) 0 = x"
```
```    71   by (simp add: nth_default_def)
```
```    72
```
```    73 lemma nth_default_Cons_Suc [simp]:
```
```    74   "nth_default y (x # xs) (Suc n) = nth_default y xs n"
```
```    75   by (simp add: nth_default_def)
```
```    76
```
```    77 lemma nth_default_map_eq:
```
```    78   "f y = x \<Longrightarrow> nth_default x (map f xs) n = f (nth_default y xs n)"
```
```    79   by (simp add: nth_default_def)
```
```    80
```
```    81 lemma nth_default_strip_while_eq [simp]:
```
```    82   "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
```
```    83 proof -
```
```    84   from split_strip_while_append obtain ys zs
```
```    85     where "strip_while (HOL.eq x) xs = ys" and "\<forall>z\<in>set zs. x = z" and "xs = ys @ zs" by blast
```
```    86   then show ?thesis by (simp add: nth_default_def not_less nth_append)
```
```    87 qed
```
```    88
```
```    89
```
```    90 definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
```
```    91 where
```
```    92   "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
```
```    93
```
```    94 lemma cCons_0_Nil_eq [simp]:
```
```    95   "0 ## [] = []"
```
```    96   by (simp add: cCons_def)
```
```    97
```
```    98 lemma cCons_Cons_eq [simp]:
```
```    99   "x ## y # ys = x # y # ys"
```
```   100   by (simp add: cCons_def)
```
```   101
```
```   102 lemma cCons_append_Cons_eq [simp]:
```
```   103   "x ## xs @ y # ys = x # xs @ y # ys"
```
```   104   by (simp add: cCons_def)
```
```   105
```
```   106 lemma cCons_not_0_eq [simp]:
```
```   107   "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
```
```   108   by (simp add: cCons_def)
```
```   109
```
```   110 lemma strip_while_not_0_Cons_eq [simp]:
```
```   111   "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
```
```   112 proof (cases "x = 0")
```
```   113   case False then show ?thesis by simp
```
```   114 next
```
```   115   case True show ?thesis
```
```   116   proof (induct xs rule: rev_induct)
```
```   117     case Nil with True show ?case by simp
```
```   118   next
```
```   119     case (snoc y ys) then show ?case
```
```   120       by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
```
```   121   qed
```
```   122 qed
```
```   123
```
```   124 lemma tl_cCons [simp]:
```
```   125   "tl (x ## xs) = xs"
```
```   126   by (simp add: cCons_def)
```
```   127
```
```   128
```
```   129 subsection {* Almost everywhere zero functions *}
```
```   130
```
```   131 definition almost_everywhere_zero :: "(nat \<Rightarrow> 'a::zero) \<Rightarrow> bool"
```
```   132 where
```
```   133   "almost_everywhere_zero f \<longleftrightarrow> (\<exists>n. \<forall>i>n. f i = 0)"
```
```   134
```
```   135 lemma almost_everywhere_zeroI:
```
```   136   "(\<And>i. i > n \<Longrightarrow> f i = 0) \<Longrightarrow> almost_everywhere_zero f"
```
```   137   by (auto simp add: almost_everywhere_zero_def)
```
```   138
```
```   139 lemma almost_everywhere_zeroE:
```
```   140   assumes "almost_everywhere_zero f"
```
```   141   obtains n where "\<And>i. i > n \<Longrightarrow> f i = 0"
```
```   142 proof -
```
```   143   from assms have "\<exists>n. \<forall>i>n. f i = 0" by (simp add: almost_everywhere_zero_def)
```
```   144   then obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by blast
```
```   145   with that show thesis .
```
```   146 qed
```
```   147
```
```   148 lemma almost_everywhere_zero_nat_case:
```
```   149   assumes "almost_everywhere_zero f"
```
```   150   shows "almost_everywhere_zero (nat_case a f)"
```
```   151   using assms
```
```   152   by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
```
```   153     blast
```
```   154
```
```   155 lemma almost_everywhere_zero_Suc:
```
```   156   assumes "almost_everywhere_zero f"
```
```   157   shows "almost_everywhere_zero (\<lambda>n. f (Suc n))"
```
```   158 proof -
```
```   159   from assms obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by (erule almost_everywhere_zeroE)
```
```   160   then have "\<And>i. i > n \<Longrightarrow> f (Suc i) = 0" by auto
```
```   161   then show ?thesis by (rule almost_everywhere_zeroI)
```
```   162 qed
```
```   163
```
```   164
```
```   165 subsection {* Definition of type @{text poly} *}
```
```   166
```
```   167 typedef 'a poly = "{f :: nat \<Rightarrow> 'a::zero. almost_everywhere_zero f}"
```
```   168   morphisms coeff Abs_poly
```
```   169   unfolding almost_everywhere_zero_def by auto
```
```   170
```
```   171 setup_lifting (no_code) type_definition_poly
```
```   172
```
```   173 lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
```
```   174   by (simp add: coeff_inject [symmetric] fun_eq_iff)
```
```   175
```
```   176 lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
```
```   177   by (simp add: poly_eq_iff)
```
```   178
```
```   179 lemma coeff_almost_everywhere_zero:
```
```   180   "almost_everywhere_zero (coeff p)"
```
```   181   using coeff [of p] by simp
```
```   182
```
```   183
```
```   184 subsection {* Degree of a polynomial *}
```
```   185
```
```   186 definition degree :: "'a::zero poly \<Rightarrow> nat"
```
```   187 where
```
```   188   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
```
```   189
```
```   190 lemma coeff_eq_0:
```
```   191   assumes "degree p < n"
```
```   192   shows "coeff p n = 0"
```
```   193 proof -
```
```   194   from coeff_almost_everywhere_zero
```
```   195   have "\<exists>n. \<forall>i>n. coeff p i = 0" by (blast intro: almost_everywhere_zeroE)
```
```   196   then have "\<forall>i>degree p. coeff p i = 0"
```
```   197     unfolding degree_def by (rule LeastI_ex)
```
```   198   with assms show ?thesis by simp
```
```   199 qed
```
```   200
```
```   201 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
```
```   202   by (erule contrapos_np, rule coeff_eq_0, simp)
```
```   203
```
```   204 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
```
```   205   unfolding degree_def by (erule Least_le)
```
```   206
```
```   207 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
```
```   208   unfolding degree_def by (drule not_less_Least, simp)
```
```   209
```
```   210
```
```   211 subsection {* The zero polynomial *}
```
```   212
```
```   213 instantiation poly :: (zero) zero
```
```   214 begin
```
```   215
```
```   216 lift_definition zero_poly :: "'a poly"
```
```   217   is "\<lambda>_. 0" by (rule almost_everywhere_zeroI) simp
```
```   218
```
```   219 instance ..
```
```   220
```
```   221 end
```
```   222
```
```   223 lemma coeff_0 [simp]:
```
```   224   "coeff 0 n = 0"
```
```   225   by transfer rule
```
```   226
```
```   227 lemma degree_0 [simp]:
```
```   228   "degree 0 = 0"
```
```   229   by (rule order_antisym [OF degree_le le0]) simp
```
```   230
```
```   231 lemma leading_coeff_neq_0:
```
```   232   assumes "p \<noteq> 0"
```
```   233   shows "coeff p (degree p) \<noteq> 0"
```
```   234 proof (cases "degree p")
```
```   235   case 0
```
```   236   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
```
```   237     by (simp add: poly_eq_iff)
```
```   238   then obtain n where "coeff p n \<noteq> 0" ..
```
```   239   hence "n \<le> degree p" by (rule le_degree)
```
```   240   with `coeff p n \<noteq> 0` and `degree p = 0`
```
```   241   show "coeff p (degree p) \<noteq> 0" by simp
```
```   242 next
```
```   243   case (Suc n)
```
```   244   from `degree p = Suc n` have "n < degree p" by simp
```
```   245   hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
```
```   246   then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
```
```   247   from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
```
```   248   also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
```
```   249   finally have "degree p = i" .
```
```   250   with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
```
```   251 qed
```
```   252
```
```   253 lemma leading_coeff_0_iff [simp]:
```
```   254   "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
```
```   255   by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
```
```   256
```
```   257
```
```   258 subsection {* List-style constructor for polynomials *}
```
```   259
```
```   260 lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   261   is "\<lambda>a p. nat_case a (coeff p)"
```
```   262   using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_nat_case)
```
```   263
```
```   264 lemmas coeff_pCons = pCons.rep_eq
```
```   265
```
```   266 lemma coeff_pCons_0 [simp]:
```
```   267   "coeff (pCons a p) 0 = a"
```
```   268   by transfer simp
```
```   269
```
```   270 lemma coeff_pCons_Suc [simp]:
```
```   271   "coeff (pCons a p) (Suc n) = coeff p n"
```
```   272   by (simp add: coeff_pCons)
```
```   273
```
```   274 lemma degree_pCons_le:
```
```   275   "degree (pCons a p) \<le> Suc (degree p)"
```
```   276   by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   277
```
```   278 lemma degree_pCons_eq:
```
```   279   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
```
```   280   apply (rule order_antisym [OF degree_pCons_le])
```
```   281   apply (rule le_degree, simp)
```
```   282   done
```
```   283
```
```   284 lemma degree_pCons_0:
```
```   285   "degree (pCons a 0) = 0"
```
```   286   apply (rule order_antisym [OF _ le0])
```
```   287   apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   288   done
```
```   289
```
```   290 lemma degree_pCons_eq_if [simp]:
```
```   291   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
```
```   292   apply (cases "p = 0", simp_all)
```
```   293   apply (rule order_antisym [OF _ le0])
```
```   294   apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   295   apply (rule order_antisym [OF degree_pCons_le])
```
```   296   apply (rule le_degree, simp)
```
```   297   done
```
```   298
```
```   299 lemma pCons_0_0 [simp]:
```
```   300   "pCons 0 0 = 0"
```
```   301   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   302
```
```   303 lemma pCons_eq_iff [simp]:
```
```   304   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
```
```   305 proof safe
```
```   306   assume "pCons a p = pCons b q"
```
```   307   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
```
```   308   then show "a = b" by simp
```
```   309 next
```
```   310   assume "pCons a p = pCons b q"
```
```   311   then have "\<forall>n. coeff (pCons a p) (Suc n) =
```
```   312                  coeff (pCons b q) (Suc n)" by simp
```
```   313   then show "p = q" by (simp add: poly_eq_iff)
```
```   314 qed
```
```   315
```
```   316 lemma pCons_eq_0_iff [simp]:
```
```   317   "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
```
```   318   using pCons_eq_iff [of a p 0 0] by simp
```
```   319
```
```   320 lemma pCons_cases [cases type: poly]:
```
```   321   obtains (pCons) a q where "p = pCons a q"
```
```   322 proof
```
```   323   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
```
```   324     by transfer
```
```   325       (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
```
```   326 qed
```
```   327
```
```   328 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
```
```   329   assumes zero: "P 0"
```
```   330   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
```
```   331   shows "P p"
```
```   332 proof (induct p rule: measure_induct_rule [where f=degree])
```
```   333   case (less p)
```
```   334   obtain a q where "p = pCons a q" by (rule pCons_cases)
```
```   335   have "P q"
```
```   336   proof (cases "q = 0")
```
```   337     case True
```
```   338     then show "P q" by (simp add: zero)
```
```   339   next
```
```   340     case False
```
```   341     then have "degree (pCons a q) = Suc (degree q)"
```
```   342       by (rule degree_pCons_eq)
```
```   343     then have "degree q < degree p"
```
```   344       using `p = pCons a q` by simp
```
```   345     then show "P q"
```
```   346       by (rule less.hyps)
```
```   347   qed
```
```   348   then have "P (pCons a q)"
```
```   349     by (rule pCons)
```
```   350   then show ?case
```
```   351     using `p = pCons a q` by simp
```
```   352 qed
```
```   353
```
```   354
```
```   355 subsection {* List-style syntax for polynomials *}
```
```   356
```
```   357 syntax
```
```   358   "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
```
```   359
```
```   360 translations
```
```   361   "[:x, xs:]" == "CONST pCons x [:xs:]"
```
```   362   "[:x:]" == "CONST pCons x 0"
```
```   363   "[:x:]" <= "CONST pCons x (_constrain 0 t)"
```
```   364
```
```   365
```
```   366 subsection {* Representation of polynomials by lists of coefficients *}
```
```   367
```
```   368 primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
```
```   369 where
```
```   370   "Poly [] = 0"
```
```   371 | "Poly (a # as) = pCons a (Poly as)"
```
```   372
```
```   373 lemma Poly_replicate_0 [simp]:
```
```   374   "Poly (replicate n 0) = 0"
```
```   375   by (induct n) simp_all
```
```   376
```
```   377 lemma Poly_eq_0:
```
```   378   "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
```
```   379   by (induct as) (auto simp add: Cons_replicate_eq)
```
```   380
```
```   381 definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
```
```   382 where
```
```   383   "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
```
```   384
```
```   385 lemma coeffs_eq_Nil [simp]:
```
```   386   "coeffs p = [] \<longleftrightarrow> p = 0"
```
```   387   by (simp add: coeffs_def)
```
```   388
```
```   389 lemma not_0_coeffs_not_Nil:
```
```   390   "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
```
```   391   by simp
```
```   392
```
```   393 lemma coeffs_0_eq_Nil [simp]:
```
```   394   "coeffs 0 = []"
```
```   395   by simp
```
```   396
```
```   397 lemma coeffs_pCons_eq_cCons [simp]:
```
```   398   "coeffs (pCons a p) = a ## coeffs p"
```
```   399 proof -
```
```   400   { fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
```
```   401     assume "\<forall>m\<in>set ms. m > 0"
```
```   402     then have "map (nat_case x f) ms = map f (map (\<lambda>n. n - 1) ms)"
```
```   403       by (induct ms) (auto, metis Suc_pred' nat_case_Suc) }
```
```   404   note * = this
```
```   405   show ?thesis
```
```   406     by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
```
```   407 qed
```
```   408
```
```   409 lemma not_0_cCons_eq [simp]:
```
```   410   "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
```
```   411   by (simp add: cCons_def)
```
```   412
```
```   413 lemma Poly_coeffs [simp, code abstype]:
```
```   414   "Poly (coeffs p) = p"
```
```   415   by (induct p) (simp_all add: cCons_def)
```
```   416
```
```   417 lemma coeffs_Poly [simp]:
```
```   418   "coeffs (Poly as) = strip_while (HOL.eq 0) as"
```
```   419 proof (induct as)
```
```   420   case Nil then show ?case by simp
```
```   421 next
```
```   422   case (Cons a as)
```
```   423   have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
```
```   424     using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
```
```   425   with Cons show ?case by auto
```
```   426 qed
```
```   427
```
```   428 lemma last_coeffs_not_0:
```
```   429   "p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
```
```   430   by (induct p) (auto simp add: cCons_def)
```
```   431
```
```   432 lemma strip_while_coeffs [simp]:
```
```   433   "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
```
```   434   by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)
```
```   435
```
```   436 lemma coeffs_eq_iff:
```
```   437   "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
```
```   438 proof
```
```   439   assume ?P then show ?Q by simp
```
```   440 next
```
```   441   assume ?Q
```
```   442   then have "Poly (coeffs p) = Poly (coeffs q)" by simp
```
```   443   then show ?P by simp
```
```   444 qed
```
```   445
```
```   446 lemma coeff_Poly_eq:
```
```   447   "coeff (Poly xs) n = nth_default 0 xs n"
```
```   448   apply (induct xs arbitrary: n) apply simp_all
```
```   449   by (metis nat_case_0 nat_case_Suc not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)
```
```   450
```
```   451 lemma nth_default_coeffs_eq:
```
```   452   "nth_default 0 (coeffs p) = coeff p"
```
```   453   by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
```
```   454
```
```   455 lemma [code]:
```
```   456   "coeff p = nth_default 0 (coeffs p)"
```
```   457   by (simp add: nth_default_coeffs_eq)
```
```   458
```
```   459 lemma coeffs_eqI:
```
```   460   assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
```
```   461   assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
```
```   462   shows "coeffs p = xs"
```
```   463 proof -
```
```   464   from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
```
```   465   with zero show ?thesis by simp (cases xs, simp_all)
```
```   466 qed
```
```   467
```
```   468 lemma degree_eq_length_coeffs [code]:
```
```   469   "degree p = length (coeffs p) - 1"
```
```   470   by (simp add: coeffs_def)
```
```   471
```
```   472 lemma length_coeffs_degree:
```
```   473   "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
```
```   474   by (induct p) (auto simp add: cCons_def)
```
```   475
```
```   476 lemma [code abstract]:
```
```   477   "coeffs 0 = []"
```
```   478   by (fact coeffs_0_eq_Nil)
```
```   479
```
```   480 lemma [code abstract]:
```
```   481   "coeffs (pCons a p) = a ## coeffs p"
```
```   482   by (fact coeffs_pCons_eq_cCons)
```
```   483
```
```   484 instantiation poly :: ("{zero, equal}") equal
```
```   485 begin
```
```   486
```
```   487 definition
```
```   488   [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
```
```   489
```
```   490 instance proof
```
```   491 qed (simp add: equal equal_poly_def coeffs_eq_iff)
```
```   492
```
```   493 end
```
```   494
```
```   495 lemma [code nbe]:
```
```   496   "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
```
```   497   by (fact equal_refl)
```
```   498
```
```   499 definition is_zero :: "'a::zero poly \<Rightarrow> bool"
```
```   500 where
```
```   501   [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
```
```   502
```
```   503 lemma is_zero_null [code_abbrev]:
```
```   504   "is_zero p \<longleftrightarrow> p = 0"
```
```   505   by (simp add: is_zero_def null_def)
```
```   506
```
```   507
```
```   508 subsection {* Fold combinator for polynomials *}
```
```   509
```
```   510 definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
```
```   511 where
```
```   512   "fold_coeffs f p = foldr f (coeffs p)"
```
```   513
```
```   514 lemma fold_coeffs_0_eq [simp]:
```
```   515   "fold_coeffs f 0 = id"
```
```   516   by (simp add: fold_coeffs_def)
```
```   517
```
```   518 lemma fold_coeffs_pCons_eq [simp]:
```
```   519   "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   520   by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
```
```   521
```
```   522 lemma fold_coeffs_pCons_0_0_eq [simp]:
```
```   523   "fold_coeffs f (pCons 0 0) = id"
```
```   524   by (simp add: fold_coeffs_def)
```
```   525
```
```   526 lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
```
```   527   "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   528   by (simp add: fold_coeffs_def)
```
```   529
```
```   530 lemma fold_coeffs_pCons_not_0_0_eq [simp]:
```
```   531   "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
```
```   532   by (simp add: fold_coeffs_def)
```
```   533
```
```   534
```
```   535 subsection {* Canonical morphism on polynomials -- evaluation *}
```
```   536
```
```   537 definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   538 where
```
```   539   "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" -- {* The Horner Schema *}
```
```   540
```
```   541 lemma poly_0 [simp]:
```
```   542   "poly 0 x = 0"
```
```   543   by (simp add: poly_def)
```
```   544
```
```   545 lemma poly_pCons [simp]:
```
```   546   "poly (pCons a p) x = a + x * poly p x"
```
```   547   by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
```
```   548
```
```   549
```
```   550 subsection {* Monomials *}
```
```   551
```
```   552 lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
```
```   553   is "\<lambda>a m n. if m = n then a else 0"
```
```   554   by (auto intro!: almost_everywhere_zeroI)
```
```   555
```
```   556 lemma coeff_monom [simp]:
```
```   557   "coeff (monom a m) n = (if m = n then a else 0)"
```
```   558   by transfer rule
```
```   559
```
```   560 lemma monom_0:
```
```   561   "monom a 0 = pCons a 0"
```
```   562   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   563
```
```   564 lemma monom_Suc:
```
```   565   "monom a (Suc n) = pCons 0 (monom a n)"
```
```   566   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
```
```   567
```
```   568 lemma monom_eq_0 [simp]: "monom 0 n = 0"
```
```   569   by (rule poly_eqI) simp
```
```   570
```
```   571 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
```
```   572   by (simp add: poly_eq_iff)
```
```   573
```
```   574 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
```
```   575   by (simp add: poly_eq_iff)
```
```   576
```
```   577 lemma degree_monom_le: "degree (monom a n) \<le> n"
```
```   578   by (rule degree_le, simp)
```
```   579
```
```   580 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
```
```   581   apply (rule order_antisym [OF degree_monom_le])
```
```   582   apply (rule le_degree, simp)
```
```   583   done
```
```   584
```
```   585 lemma coeffs_monom [code abstract]:
```
```   586   "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
```
```   587   by (induct n) (simp_all add: monom_0 monom_Suc)
```
```   588
```
```   589 lemma fold_coeffs_monom [simp]:
```
```   590   "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
```
```   591   by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
```
```   592
```
```   593 lemma poly_monom:
```
```   594   fixes a x :: "'a::{comm_semiring_1}"
```
```   595   shows "poly (monom a n) x = a * x ^ n"
```
```   596   by (cases "a = 0", simp_all)
```
```   597     (induct n, simp_all add: mult.left_commute poly_def)
```
```   598
```
```   599
```
```   600 subsection {* Addition and subtraction *}
```
```   601
```
```   602 instantiation poly :: (comm_monoid_add) comm_monoid_add
```
```   603 begin
```
```   604
```
```   605 lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   606   is "\<lambda>p q n. coeff p n + coeff q n"
```
```   607 proof (rule almost_everywhere_zeroI)
```
```   608   fix q p :: "'a poly" and i
```
```   609   assume "max (degree q) (degree p) < i"
```
```   610   then show "coeff p i + coeff q i = 0"
```
```   611     by (simp add: coeff_eq_0)
```
```   612 qed
```
```   613
```
```   614 lemma coeff_add [simp]:
```
```   615   "coeff (p + q) n = coeff p n + coeff q n"
```
```   616   by (simp add: plus_poly.rep_eq)
```
```   617
```
```   618 instance proof
```
```   619   fix p q r :: "'a poly"
```
```   620   show "(p + q) + r = p + (q + r)"
```
```   621     by (simp add: poly_eq_iff add_assoc)
```
```   622   show "p + q = q + p"
```
```   623     by (simp add: poly_eq_iff add_commute)
```
```   624   show "0 + p = p"
```
```   625     by (simp add: poly_eq_iff)
```
```   626 qed
```
```   627
```
```   628 end
```
```   629
```
```   630 instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
```
```   631 proof
```
```   632   fix p q r :: "'a poly"
```
```   633   assume "p + q = p + r" thus "q = r"
```
```   634     by (simp add: poly_eq_iff)
```
```   635 qed
```
```   636
```
```   637 instantiation poly :: (ab_group_add) ab_group_add
```
```   638 begin
```
```   639
```
```   640 lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
```
```   641   is "\<lambda>p n. - coeff p n"
```
```   642 proof (rule almost_everywhere_zeroI)
```
```   643   fix p :: "'a poly" and i
```
```   644   assume "degree p < i"
```
```   645   then show "- coeff p i = 0"
```
```   646     by (simp add: coeff_eq_0)
```
```   647 qed
```
```   648
```
```   649 lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   650   is "\<lambda>p q n. coeff p n - coeff q n"
```
```   651 proof (rule almost_everywhere_zeroI)
```
```   652   fix q p :: "'a poly" and i
```
```   653   assume "max (degree q) (degree p) < i"
```
```   654   then show "coeff p i - coeff q i = 0"
```
```   655     by (simp add: coeff_eq_0)
```
```   656 qed
```
```   657
```
```   658 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
```
```   659   by (simp add: uminus_poly.rep_eq)
```
```   660
```
```   661 lemma coeff_diff [simp]:
```
```   662   "coeff (p - q) n = coeff p n - coeff q n"
```
```   663   by (simp add: minus_poly.rep_eq)
```
```   664
```
```   665 instance proof
```
```   666   fix p q :: "'a poly"
```
```   667   show "- p + p = 0"
```
```   668     by (simp add: poly_eq_iff)
```
```   669   show "p - q = p + - q"
```
```   670     by (simp add: poly_eq_iff)
```
```   671 qed
```
```   672
```
```   673 end
```
```   674
```
```   675 lemma add_pCons [simp]:
```
```   676   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```   677   by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
```
```   678
```
```   679 lemma minus_pCons [simp]:
```
```   680   "- pCons a p = pCons (- a) (- p)"
```
```   681   by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
```
```   682
```
```   683 lemma diff_pCons [simp]:
```
```   684   "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 degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
```
```   688   by (rule degree_le, auto simp add: coeff_eq_0)
```
```   689
```
```   690 lemma degree_add_le:
```
```   691   "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
```
```   692   by (auto intro: order_trans degree_add_le_max)
```
```   693
```
```   694 lemma degree_add_less:
```
```   695   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
```
```   696   by (auto intro: le_less_trans degree_add_le_max)
```
```   697
```
```   698 lemma degree_add_eq_right:
```
```   699   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
```
```   700   apply (cases "q = 0", simp)
```
```   701   apply (rule order_antisym)
```
```   702   apply (simp add: degree_add_le)
```
```   703   apply (rule le_degree)
```
```   704   apply (simp add: coeff_eq_0)
```
```   705   done
```
```   706
```
```   707 lemma degree_add_eq_left:
```
```   708   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
```
```   709   using degree_add_eq_right [of q p]
```
```   710   by (simp add: add_commute)
```
```   711
```
```   712 lemma degree_minus [simp]: "degree (- p) = degree p"
```
```   713   unfolding degree_def by simp
```
```   714
```
```   715 lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
```
```   716   using degree_add_le [where p=p and q="-q"]
```
```   717   by simp
```
```   718
```
```   719 lemma degree_diff_le:
```
```   720   "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
```
```   721   using degree_add_le [of p n "- q"] by simp
```
```   722
```
```   723 lemma degree_diff_less:
```
```   724   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
```
```   725   using degree_add_less [of p n "- q"] by simp
```
```   726
```
```   727 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
```
```   728   by (rule poly_eqI) simp
```
```   729
```
```   730 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
```
```   731   by (rule poly_eqI) simp
```
```   732
```
```   733 lemma minus_monom: "- monom a n = monom (-a) n"
```
```   734   by (rule poly_eqI) simp
```
```   735
```
```   736 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
```
```   737   by (cases "finite A", induct set: finite, simp_all)
```
```   738
```
```   739 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
```
```   740   by (rule poly_eqI) (simp add: coeff_setsum)
```
```   741
```
```   742 fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   743 where
```
```   744   "plus_coeffs xs [] = xs"
```
```   745 | "plus_coeffs [] ys = ys"
```
```   746 | "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
```
```   747
```
```   748 lemma coeffs_plus_eq_plus_coeffs [code abstract]:
```
```   749   "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
```
```   750 proof -
```
```   751   { fix xs ys :: "'a list" and n
```
```   752     have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
```
```   753     proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
```
```   754       case (3 x xs y ys n) then show ?case by (cases n) (auto simp add: cCons_def)
```
```   755     qed simp_all }
```
```   756   note * = this
```
```   757   { fix xs ys :: "'a list"
```
```   758     assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
```
```   759     moreover assume "plus_coeffs xs ys \<noteq> []"
```
```   760     ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
```
```   761     proof (induct xs ys rule: plus_coeffs.induct)
```
```   762       case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
```
```   763     qed simp_all }
```
```   764   note ** = this
```
```   765   show ?thesis
```
```   766     apply (rule coeffs_eqI)
```
```   767     apply (simp add: * nth_default_coeffs_eq)
```
```   768     apply (rule **)
```
```   769     apply (auto dest: last_coeffs_not_0)
```
```   770     done
```
```   771 qed
```
```   772
```
```   773 lemma coeffs_uminus [code abstract]:
```
```   774   "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
```
```   775   by (rule coeffs_eqI)
```
```   776     (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
```
```   777
```
```   778 lemma [code]:
```
```   779   fixes p q :: "'a::ab_group_add poly"
```
```   780   shows "p - q = p + - q"
```
```   781   by simp
```
```   782
```
```   783 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
```
```   784   apply (induct p arbitrary: q, simp)
```
```   785   apply (case_tac q, simp, simp add: algebra_simps)
```
```   786   done
```
```   787
```
```   788 lemma poly_minus [simp]:
```
```   789   fixes x :: "'a::comm_ring"
```
```   790   shows "poly (- p) x = - poly p x"
```
```   791   by (induct p) simp_all
```
```   792
```
```   793 lemma poly_diff [simp]:
```
```   794   fixes x :: "'a::comm_ring"
```
```   795   shows "poly (p - q) x = poly p x - poly q x"
```
```   796   using poly_add [of p "- q" x] by simp
```
```   797
```
```   798 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
```
```   799   by (induct A rule: infinite_finite_induct) simp_all
```
```   800
```
```   801
```
```   802 subsection {* Multiplication by a constant, polynomial multiplication and the unit polynomial *}
```
```   803
```
```   804 lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   805   is "\<lambda>a p n. a * coeff p n"
```
```   806 proof (rule almost_everywhere_zeroI)
```
```   807   fix a :: 'a and p :: "'a poly" and i
```
```   808   assume "degree p < i"
```
```   809   then show "a * coeff p i = 0"
```
```   810     by (simp add: coeff_eq_0)
```
```   811 qed
```
```   812
```
```   813 lemma coeff_smult [simp]:
```
```   814   "coeff (smult a p) n = a * coeff p n"
```
```   815   by (simp add: smult.rep_eq)
```
```   816
```
```   817 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
```
```   818   by (rule degree_le, simp add: coeff_eq_0)
```
```   819
```
```   820 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
```
```   821   by (rule poly_eqI, simp add: mult_assoc)
```
```   822
```
```   823 lemma smult_0_right [simp]: "smult a 0 = 0"
```
```   824   by (rule poly_eqI, simp)
```
```   825
```
```   826 lemma smult_0_left [simp]: "smult 0 p = 0"
```
```   827   by (rule poly_eqI, simp)
```
```   828
```
```   829 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
```
```   830   by (rule poly_eqI, simp)
```
```   831
```
```   832 lemma smult_add_right:
```
```   833   "smult a (p + q) = smult a p + smult a q"
```
```   834   by (rule poly_eqI, simp add: algebra_simps)
```
```   835
```
```   836 lemma smult_add_left:
```
```   837   "smult (a + b) p = smult a p + smult b p"
```
```   838   by (rule poly_eqI, simp add: algebra_simps)
```
```   839
```
```   840 lemma smult_minus_right [simp]:
```
```   841   "smult (a::'a::comm_ring) (- p) = - smult a p"
```
```   842   by (rule poly_eqI, simp)
```
```   843
```
```   844 lemma smult_minus_left [simp]:
```
```   845   "smult (- a::'a::comm_ring) p = - smult a p"
```
```   846   by (rule poly_eqI, simp)
```
```   847
```
```   848 lemma smult_diff_right:
```
```   849   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
```
```   850   by (rule poly_eqI, simp add: algebra_simps)
```
```   851
```
```   852 lemma smult_diff_left:
```
```   853   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
```
```   854   by (rule poly_eqI, simp add: algebra_simps)
```
```   855
```
```   856 lemmas smult_distribs =
```
```   857   smult_add_left smult_add_right
```
```   858   smult_diff_left smult_diff_right
```
```   859
```
```   860 lemma smult_pCons [simp]:
```
```   861   "smult a (pCons b p) = pCons (a * b) (smult a p)"
```
```   862   by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
```
```   863
```
```   864 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
```
```   865   by (induct n, simp add: monom_0, simp add: monom_Suc)
```
```   866
```
```   867 lemma degree_smult_eq [simp]:
```
```   868   fixes a :: "'a::idom"
```
```   869   shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
```
```   870   by (cases "a = 0", simp, simp add: degree_def)
```
```   871
```
```   872 lemma smult_eq_0_iff [simp]:
```
```   873   fixes a :: "'a::idom"
```
```   874   shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
```
```   875   by (simp add: poly_eq_iff)
```
```   876
```
```   877 lemma coeffs_smult [code abstract]:
```
```   878   fixes p :: "'a::idom poly"
```
```   879   shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
```
```   880   by (rule coeffs_eqI)
```
```   881     (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
```
```   882
```
```   883 instantiation poly :: (comm_semiring_0) comm_semiring_0
```
```   884 begin
```
```   885
```
```   886 definition
```
```   887   "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
```
```   888
```
```   889 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
```
```   890   by (simp add: times_poly_def)
```
```   891
```
```   892 lemma mult_pCons_left [simp]:
```
```   893   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```   894   by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
```
```   895
```
```   896 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
```
```   897   by (induct p) (simp add: mult_poly_0_left, simp)
```
```   898
```
```   899 lemma mult_pCons_right [simp]:
```
```   900   "p * pCons a q = smult a p + pCons 0 (p * q)"
```
```   901   by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps)
```
```   902
```
```   903 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
```
```   904
```
```   905 lemma mult_smult_left [simp]:
```
```   906   "smult a p * q = smult a (p * q)"
```
```   907   by (induct p) (simp add: mult_poly_0, simp add: smult_add_right)
```
```   908
```
```   909 lemma mult_smult_right [simp]:
```
```   910   "p * smult a q = smult a (p * q)"
```
```   911   by (induct q) (simp add: mult_poly_0, simp add: smult_add_right)
```
```   912
```
```   913 lemma mult_poly_add_left:
```
```   914   fixes p q r :: "'a poly"
```
```   915   shows "(p + q) * r = p * r + q * r"
```
```   916   by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps)
```
```   917
```
```   918 instance proof
```
```   919   fix p q r :: "'a poly"
```
```   920   show 0: "0 * p = 0"
```
```   921     by (rule mult_poly_0_left)
```
```   922   show "p * 0 = 0"
```
```   923     by (rule mult_poly_0_right)
```
```   924   show "(p + q) * r = p * r + q * r"
```
```   925     by (rule mult_poly_add_left)
```
```   926   show "(p * q) * r = p * (q * r)"
```
```   927     by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
```
```   928   show "p * q = q * p"
```
```   929     by (induct p, simp add: mult_poly_0, simp)
```
```   930 qed
```
```   931
```
```   932 end
```
```   933
```
```   934 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
```
```   935
```
```   936 lemma coeff_mult:
```
```   937   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   938 proof (induct p arbitrary: n)
```
```   939   case 0 show ?case by simp
```
```   940 next
```
```   941   case (pCons a p n) thus ?case
```
```   942     by (cases n, simp, simp add: setsum_atMost_Suc_shift
```
```   943                             del: setsum_atMost_Suc)
```
```   944 qed
```
```   945
```
```   946 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
```
```   947 apply (rule degree_le)
```
```   948 apply (induct p)
```
```   949 apply simp
```
```   950 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   951 done
```
```   952
```
```   953 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
```
```   954   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
```
```   955
```
```   956 instantiation poly :: (comm_semiring_1) comm_semiring_1
```
```   957 begin
```
```   958
```
```   959 definition one_poly_def:
```
```   960   "1 = pCons 1 0"
```
```   961
```
```   962 instance proof
```
```   963   fix p :: "'a poly" show "1 * p = p"
```
```   964     unfolding one_poly_def by simp
```
```   965 next
```
```   966   show "0 \<noteq> (1::'a poly)"
```
```   967     unfolding one_poly_def by simp
```
```   968 qed
```
```   969
```
```   970 end
```
```   971
```
```   972 instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
```
```   973
```
```   974 instance poly :: (comm_ring) comm_ring ..
```
```   975
```
```   976 instance poly :: (comm_ring_1) comm_ring_1 ..
```
```   977
```
```   978 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
```
```   979   unfolding one_poly_def
```
```   980   by (simp add: coeff_pCons split: nat.split)
```
```   981
```
```   982 lemma degree_1 [simp]: "degree 1 = 0"
```
```   983   unfolding one_poly_def
```
```   984   by (rule degree_pCons_0)
```
```   985
```
```   986 lemma coeffs_1_eq [simp, code abstract]:
```
```   987   "coeffs 1 = "
```
```   988   by (simp add: one_poly_def)
```
```   989
```
```   990 lemma degree_power_le:
```
```   991   "degree (p ^ n) \<le> degree p * n"
```
```   992   by (induct n) (auto intro: order_trans degree_mult_le)
```
```   993
```
```   994 lemma poly_smult [simp]:
```
```   995   "poly (smult a p) x = a * poly p x"
```
```   996   by (induct p, simp, simp add: algebra_simps)
```
```   997
```
```   998 lemma poly_mult [simp]:
```
```   999   "poly (p * q) x = poly p x * poly q x"
```
```  1000   by (induct p, simp_all, simp add: algebra_simps)
```
```  1001
```
```  1002 lemma poly_1 [simp]:
```
```  1003   "poly 1 x = 1"
```
```  1004   by (simp add: one_poly_def)
```
```  1005
```
```  1006 lemma poly_power [simp]:
```
```  1007   fixes p :: "'a::{comm_semiring_1} poly"
```
```  1008   shows "poly (p ^ n) x = poly p x ^ n"
```
```  1009   by (induct n) simp_all
```
```  1010
```
```  1011
```
```  1012 subsection {* Lemmas about divisibility *}
```
```  1013
```
```  1014 lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
```
```  1015 proof -
```
```  1016   assume "p dvd q"
```
```  1017   then obtain k where "q = p * k" ..
```
```  1018   then have "smult a q = p * smult a k" by simp
```
```  1019   then show "p dvd smult a q" ..
```
```  1020 qed
```
```  1021
```
```  1022 lemma dvd_smult_cancel:
```
```  1023   fixes a :: "'a::field"
```
```  1024   shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
```
```  1025   by (drule dvd_smult [where a="inverse a"]) simp
```
```  1026
```
```  1027 lemma dvd_smult_iff:
```
```  1028   fixes a :: "'a::field"
```
```  1029   shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
```
```  1030   by (safe elim!: dvd_smult dvd_smult_cancel)
```
```  1031
```
```  1032 lemma smult_dvd_cancel:
```
```  1033   "smult a p dvd q \<Longrightarrow> p dvd q"
```
```  1034 proof -
```
```  1035   assume "smult a p dvd q"
```
```  1036   then obtain k where "q = smult a p * k" ..
```
```  1037   then have "q = p * smult a k" by simp
```
```  1038   then show "p dvd q" ..
```
```  1039 qed
```
```  1040
```
```  1041 lemma smult_dvd:
```
```  1042   fixes a :: "'a::field"
```
```  1043   shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
```
```  1044   by (rule smult_dvd_cancel [where a="inverse a"]) simp
```
```  1045
```
```  1046 lemma smult_dvd_iff:
```
```  1047   fixes a :: "'a::field"
```
```  1048   shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
```
```  1049   by (auto elim: smult_dvd smult_dvd_cancel)
```
```  1050
```
```  1051
```
```  1052 subsection {* Polynomials form an integral domain *}
```
```  1053
```
```  1054 lemma coeff_mult_degree_sum:
```
```  1055   "coeff (p * q) (degree p + degree q) =
```
```  1056    coeff p (degree p) * coeff q (degree q)"
```
```  1057   by (induct p, simp, simp add: coeff_eq_0)
```
```  1058
```
```  1059 instance poly :: (idom) idom
```
```  1060 proof
```
```  1061   fix p q :: "'a poly"
```
```  1062   assume "p \<noteq> 0" and "q \<noteq> 0"
```
```  1063   have "coeff (p * q) (degree p + degree q) =
```
```  1064         coeff p (degree p) * coeff q (degree q)"
```
```  1065     by (rule coeff_mult_degree_sum)
```
```  1066   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
```
```  1067     using `p \<noteq> 0` and `q \<noteq> 0` by simp
```
```  1068   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
```
```  1069   thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
```
```  1070 qed
```
```  1071
```
```  1072 lemma degree_mult_eq:
```
```  1073   fixes p q :: "'a::idom poly"
```
```  1074   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
```
```  1075 apply (rule order_antisym [OF degree_mult_le le_degree])
```
```  1076 apply (simp add: coeff_mult_degree_sum)
```
```  1077 done
```
```  1078
```
```  1079 lemma dvd_imp_degree_le:
```
```  1080   fixes p q :: "'a::idom poly"
```
```  1081   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
```
```  1082   by (erule dvdE, simp add: degree_mult_eq)
```
```  1083
```
```  1084
```
```  1085 subsection {* Polynomials form an ordered integral domain *}
```
```  1086
```
```  1087 definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
```
```  1088 where
```
```  1089   "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
```
```  1090
```
```  1091 lemma pos_poly_pCons:
```
```  1092   "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
```
```  1093   unfolding pos_poly_def by simp
```
```  1094
```
```  1095 lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
```
```  1096   unfolding pos_poly_def by simp
```
```  1097
```
```  1098 lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
```
```  1099   apply (induct p arbitrary: q, simp)
```
```  1100   apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
```
```  1101   done
```
```  1102
```
```  1103 lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
```
```  1104   unfolding pos_poly_def
```
```  1105   apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
```
```  1106   apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)
```
```  1107   apply auto
```
```  1108   done
```
```  1109
```
```  1110 lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
```
```  1111 by (induct p) (auto simp add: pos_poly_pCons)
```
```  1112
```
```  1113 lemma last_coeffs_eq_coeff_degree:
```
```  1114   "p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"
```
```  1115   by (simp add: coeffs_def)
```
```  1116
```
```  1117 lemma pos_poly_coeffs [code]:
```
```  1118   "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q")
```
```  1119 proof
```
```  1120   assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
```
```  1121 next
```
```  1122   assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def)
```
```  1123   then have "p \<noteq> 0" by auto
```
```  1124   with * show ?Q by (simp add: last_coeffs_eq_coeff_degree)
```
```  1125 qed
```
```  1126
```
```  1127 instantiation poly :: (linordered_idom) linordered_idom
```
```  1128 begin
```
```  1129
```
```  1130 definition
```
```  1131   "x < y \<longleftrightarrow> pos_poly (y - x)"
```
```  1132
```
```  1133 definition
```
```  1134   "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
```
```  1135
```
```  1136 definition
```
```  1137   "abs (x::'a poly) = (if x < 0 then - x else x)"
```
```  1138
```
```  1139 definition
```
```  1140   "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```  1141
```
```  1142 instance proof
```
```  1143   fix x y :: "'a poly"
```
```  1144   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```  1145     unfolding less_eq_poly_def less_poly_def
```
```  1146     apply safe
```
```  1147     apply simp
```
```  1148     apply (drule (1) pos_poly_add)
```
```  1149     apply simp
```
```  1150     done
```
```  1151 next
```
```  1152   fix x :: "'a poly" show "x \<le> x"
```
```  1153     unfolding less_eq_poly_def by simp
```
```  1154 next
```
```  1155   fix x y z :: "'a poly"
```
```  1156   assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
```
```  1157     unfolding less_eq_poly_def
```
```  1158     apply safe
```
```  1159     apply (drule (1) pos_poly_add)
```
```  1160     apply (simp add: algebra_simps)
```
```  1161     done
```
```  1162 next
```
```  1163   fix x y :: "'a poly"
```
```  1164   assume "x \<le> y" and "y \<le> x" thus "x = y"
```
```  1165     unfolding less_eq_poly_def
```
```  1166     apply safe
```
```  1167     apply (drule (1) pos_poly_add)
```
```  1168     apply simp
```
```  1169     done
```
```  1170 next
```
```  1171   fix x y z :: "'a poly"
```
```  1172   assume "x \<le> y" thus "z + x \<le> z + y"
```
```  1173     unfolding less_eq_poly_def
```
```  1174     apply safe
```
```  1175     apply (simp add: algebra_simps)
```
```  1176     done
```
```  1177 next
```
```  1178   fix x y :: "'a poly"
```
```  1179   show "x \<le> y \<or> y \<le> x"
```
```  1180     unfolding less_eq_poly_def
```
```  1181     using pos_poly_total [of "x - y"]
```
```  1182     by auto
```
```  1183 next
```
```  1184   fix x y z :: "'a poly"
```
```  1185   assume "x < y" and "0 < z"
```
```  1186   thus "z * x < z * y"
```
```  1187     unfolding less_poly_def
```
```  1188     by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
```
```  1189 next
```
```  1190   fix x :: "'a poly"
```
```  1191   show "\<bar>x\<bar> = (if x < 0 then - x else x)"
```
```  1192     by (rule abs_poly_def)
```
```  1193 next
```
```  1194   fix x :: "'a poly"
```
```  1195   show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
```
```  1196     by (rule sgn_poly_def)
```
```  1197 qed
```
```  1198
```
```  1199 end
```
```  1200
```
```  1201 text {* TODO: Simplification rules for comparisons *}
```
```  1202
```
```  1203
```
```  1204 subsection {* Synthetic division and polynomial roots *}
```
```  1205
```
```  1206 text {*
```
```  1207   Synthetic division is simply division by the linear polynomial @{term "x - c"}.
```
```  1208 *}
```
```  1209
```
```  1210 definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
```
```  1211 where
```
```  1212   "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
```
```  1213
```
```  1214 definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
```
```  1215 where
```
```  1216   "synthetic_div p c = fst (synthetic_divmod p c)"
```
```  1217
```
```  1218 lemma synthetic_divmod_0 [simp]:
```
```  1219   "synthetic_divmod 0 c = (0, 0)"
```
```  1220   by (simp add: synthetic_divmod_def)
```
```  1221
```
```  1222 lemma synthetic_divmod_pCons [simp]:
```
```  1223   "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
```
```  1224   by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
```
```  1225
```
```  1226 lemma synthetic_div_0 [simp]:
```
```  1227   "synthetic_div 0 c = 0"
```
```  1228   unfolding synthetic_div_def by simp
```
```  1229
```
```  1230 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
```
```  1231 by (induct p arbitrary: a) simp_all
```
```  1232
```
```  1233 lemma snd_synthetic_divmod:
```
```  1234   "snd (synthetic_divmod p c) = poly p c"
```
```  1235   by (induct p, simp, simp add: split_def)
```
```  1236
```
```  1237 lemma synthetic_div_pCons [simp]:
```
```  1238   "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
```
```  1239   unfolding synthetic_div_def
```
```  1240   by (simp add: split_def snd_synthetic_divmod)
```
```  1241
```
```  1242 lemma synthetic_div_eq_0_iff:
```
```  1243   "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
```
```  1244   by (induct p, simp, case_tac p, simp)
```
```  1245
```
```  1246 lemma degree_synthetic_div:
```
```  1247   "degree (synthetic_div p c) = degree p - 1"
```
```  1248   by (induct p, simp, simp add: synthetic_div_eq_0_iff)
```
```  1249
```
```  1250 lemma synthetic_div_correct:
```
```  1251   "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
```
```  1252   by (induct p) simp_all
```
```  1253
```
```  1254 lemma synthetic_div_unique:
```
```  1255   "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
```
```  1256 apply (induct p arbitrary: q r)
```
```  1257 apply (simp, frule synthetic_div_unique_lemma, simp)
```
```  1258 apply (case_tac q, force)
```
```  1259 done
```
```  1260
```
```  1261 lemma synthetic_div_correct':
```
```  1262   fixes c :: "'a::comm_ring_1"
```
```  1263   shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
```
```  1264   using synthetic_div_correct [of p c]
```
```  1265   by (simp add: algebra_simps)
```
```  1266
```
```  1267 lemma poly_eq_0_iff_dvd:
```
```  1268   fixes c :: "'a::idom"
```
```  1269   shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
```
```  1270 proof
```
```  1271   assume "poly p c = 0"
```
```  1272   with synthetic_div_correct' [of c p]
```
```  1273   have "p = [:-c, 1:] * synthetic_div p c" by simp
```
```  1274   then show "[:-c, 1:] dvd p" ..
```
```  1275 next
```
```  1276   assume "[:-c, 1:] dvd p"
```
```  1277   then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
```
```  1278   then show "poly p c = 0" by simp
```
```  1279 qed
```
```  1280
```
```  1281 lemma dvd_iff_poly_eq_0:
```
```  1282   fixes c :: "'a::idom"
```
```  1283   shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
```
```  1284   by (simp add: poly_eq_0_iff_dvd)
```
```  1285
```
```  1286 lemma poly_roots_finite:
```
```  1287   fixes p :: "'a::idom poly"
```
```  1288   shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
```
```  1289 proof (induct n \<equiv> "degree p" arbitrary: p)
```
```  1290   case (0 p)
```
```  1291   then obtain a where "a \<noteq> 0" and "p = [:a:]"
```
```  1292     by (cases p, simp split: if_splits)
```
```  1293   then show "finite {x. poly p x = 0}" by simp
```
```  1294 next
```
```  1295   case (Suc n p)
```
```  1296   show "finite {x. poly p x = 0}"
```
```  1297   proof (cases "\<exists>x. poly p x = 0")
```
```  1298     case False
```
```  1299     then show "finite {x. poly p x = 0}" by simp
```
```  1300   next
```
```  1301     case True
```
```  1302     then obtain a where "poly p a = 0" ..
```
```  1303     then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
```
```  1304     then obtain k where k: "p = [:-a, 1:] * k" ..
```
```  1305     with `p \<noteq> 0` have "k \<noteq> 0" by auto
```
```  1306     with k have "degree p = Suc (degree k)"
```
```  1307       by (simp add: degree_mult_eq del: mult_pCons_left)
```
```  1308     with `Suc n = degree p` have "n = degree k" by simp
```
```  1309     then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
```
```  1310     then have "finite (insert a {x. poly k x = 0})" by simp
```
```  1311     then show "finite {x. poly p x = 0}"
```
```  1312       by (simp add: k uminus_add_conv_diff Collect_disj_eq
```
```  1313                del: mult_pCons_left)
```
```  1314   qed
```
```  1315 qed
```
```  1316
```
```  1317 lemma poly_eq_poly_eq_iff:
```
```  1318   fixes p q :: "'a::{idom,ring_char_0} poly"
```
```  1319   shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
```
```  1320 proof
```
```  1321   assume ?Q then show ?P by simp
```
```  1322 next
```
```  1323   { fix p :: "'a::{idom,ring_char_0} poly"
```
```  1324     have "poly p = poly 0 \<longleftrightarrow> p = 0"
```
```  1325       apply (cases "p = 0", simp_all)
```
```  1326       apply (drule poly_roots_finite)
```
```  1327       apply (auto simp add: infinite_UNIV_char_0)
```
```  1328       done
```
```  1329   } note this [of "p - q"]
```
```  1330   moreover assume ?P
```
```  1331   ultimately show ?Q by auto
```
```  1332 qed
```
```  1333
```
```  1334 lemma poly_all_0_iff_0:
```
```  1335   fixes p :: "'a::{ring_char_0, idom} poly"
```
```  1336   shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
```
```  1337   by (auto simp add: poly_eq_poly_eq_iff [symmetric])
```
```  1338
```
```  1339
```
```  1340 subsection {* Long division of polynomials *}
```
```  1341
```
```  1342 definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
```
```  1343 where
```
```  1344   "pdivmod_rel x y q r \<longleftrightarrow>
```
```  1345     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
```
```  1346
```
```  1347 lemma pdivmod_rel_0:
```
```  1348   "pdivmod_rel 0 y 0 0"
```
```  1349   unfolding pdivmod_rel_def by simp
```
```  1350
```
```  1351 lemma pdivmod_rel_by_0:
```
```  1352   "pdivmod_rel x 0 0 x"
```
```  1353   unfolding pdivmod_rel_def by simp
```
```  1354
```
```  1355 lemma eq_zero_or_degree_less:
```
```  1356   assumes "degree p \<le> n" and "coeff p n = 0"
```
```  1357   shows "p = 0 \<or> degree p < n"
```
```  1358 proof (cases n)
```
```  1359   case 0
```
```  1360   with `degree p \<le> n` and `coeff p n = 0`
```
```  1361   have "coeff p (degree p) = 0" by simp
```
```  1362   then have "p = 0" by simp
```
```  1363   then show ?thesis ..
```
```  1364 next
```
```  1365   case (Suc m)
```
```  1366   have "\<forall>i>n. coeff p i = 0"
```
```  1367     using `degree p \<le> n` by (simp add: coeff_eq_0)
```
```  1368   then have "\<forall>i\<ge>n. coeff p i = 0"
```
```  1369     using `coeff p n = 0` by (simp add: le_less)
```
```  1370   then have "\<forall>i>m. coeff p i = 0"
```
```  1371     using `n = Suc m` by (simp add: less_eq_Suc_le)
```
```  1372   then have "degree p \<le> m"
```
```  1373     by (rule degree_le)
```
```  1374   then have "degree p < n"
```
```  1375     using `n = Suc m` by (simp add: less_Suc_eq_le)
```
```  1376   then show ?thesis ..
```
```  1377 qed
```
```  1378
```
```  1379 lemma pdivmod_rel_pCons:
```
```  1380   assumes rel: "pdivmod_rel x y q r"
```
```  1381   assumes y: "y \<noteq> 0"
```
```  1382   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
```
```  1383   shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
```
```  1384     (is "pdivmod_rel ?x y ?q ?r")
```
```  1385 proof -
```
```  1386   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
```
```  1387     using assms unfolding pdivmod_rel_def by simp_all
```
```  1388
```
```  1389   have 1: "?x = ?q * y + ?r"
```
```  1390     using b x by simp
```
```  1391
```
```  1392   have 2: "?r = 0 \<or> degree ?r < degree y"
```
```  1393   proof (rule eq_zero_or_degree_less)
```
```  1394     show "degree ?r \<le> degree y"
```
```  1395     proof (rule degree_diff_le)
```
```  1396       show "degree (pCons a r) \<le> degree y"
```
```  1397         using r by auto
```
```  1398       show "degree (smult b y) \<le> degree y"
```
```  1399         by (rule degree_smult_le)
```
```  1400     qed
```
```  1401   next
```
```  1402     show "coeff ?r (degree y) = 0"
```
```  1403       using `y \<noteq> 0` unfolding b by simp
```
```  1404   qed
```
```  1405
```
```  1406   from 1 2 show ?thesis
```
```  1407     unfolding pdivmod_rel_def
```
```  1408     using `y \<noteq> 0` by simp
```
```  1409 qed
```
```  1410
```
```  1411 lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
```
```  1412 apply (cases "y = 0")
```
```  1413 apply (fast intro!: pdivmod_rel_by_0)
```
```  1414 apply (induct x)
```
```  1415 apply (fast intro!: pdivmod_rel_0)
```
```  1416 apply (fast intro!: pdivmod_rel_pCons)
```
```  1417 done
```
```  1418
```
```  1419 lemma pdivmod_rel_unique:
```
```  1420   assumes 1: "pdivmod_rel x y q1 r1"
```
```  1421   assumes 2: "pdivmod_rel x y q2 r2"
```
```  1422   shows "q1 = q2 \<and> r1 = r2"
```
```  1423 proof (cases "y = 0")
```
```  1424   assume "y = 0" with assms show ?thesis
```
```  1425     by (simp add: pdivmod_rel_def)
```
```  1426 next
```
```  1427   assume [simp]: "y \<noteq> 0"
```
```  1428   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
```
```  1429     unfolding pdivmod_rel_def by simp_all
```
```  1430   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
```
```  1431     unfolding pdivmod_rel_def by simp_all
```
```  1432   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
```
```  1433     by (simp add: algebra_simps)
```
```  1434   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
```
```  1435     by (auto intro: degree_diff_less)
```
```  1436
```
```  1437   show "q1 = q2 \<and> r1 = r2"
```
```  1438   proof (rule ccontr)
```
```  1439     assume "\<not> (q1 = q2 \<and> r1 = r2)"
```
```  1440     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
```
```  1441     with r3 have "degree (r2 - r1) < degree y" by simp
```
```  1442     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
```
```  1443     also have "\<dots> = degree ((q1 - q2) * y)"
```
```  1444       using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
```
```  1445     also have "\<dots> = degree (r2 - r1)"
```
```  1446       using q3 by simp
```
```  1447     finally have "degree (r2 - r1) < degree (r2 - r1)" .
```
```  1448     then show "False" by simp
```
```  1449   qed
```
```  1450 qed
```
```  1451
```
```  1452 lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
```
```  1453 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
```
```  1454
```
```  1455 lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
```
```  1456 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
```
```  1457
```
```  1458 lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]
```
```  1459
```
```  1460 lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]
```
```  1461
```
```  1462 instantiation poly :: (field) ring_div
```
```  1463 begin
```
```  1464
```
```  1465 definition div_poly where
```
```  1466   "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
```
```  1467
```
```  1468 definition mod_poly where
```
```  1469   "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
```
```  1470
```
```  1471 lemma div_poly_eq:
```
```  1472   "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
```
```  1473 unfolding div_poly_def
```
```  1474 by (fast elim: pdivmod_rel_unique_div)
```
```  1475
```
```  1476 lemma mod_poly_eq:
```
```  1477   "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
```
```  1478 unfolding mod_poly_def
```
```  1479 by (fast elim: pdivmod_rel_unique_mod)
```
```  1480
```
```  1481 lemma pdivmod_rel:
```
```  1482   "pdivmod_rel x y (x div y) (x mod y)"
```
```  1483 proof -
```
```  1484   from pdivmod_rel_exists
```
```  1485     obtain q r where "pdivmod_rel x y q r" by fast
```
```  1486   thus ?thesis
```
```  1487     by (simp add: div_poly_eq mod_poly_eq)
```
```  1488 qed
```
```  1489
```
```  1490 instance proof
```
```  1491   fix x y :: "'a poly"
```
```  1492   show "x div y * y + x mod y = x"
```
```  1493     using pdivmod_rel [of x y]
```
```  1494     by (simp add: pdivmod_rel_def)
```
```  1495 next
```
```  1496   fix x :: "'a poly"
```
```  1497   have "pdivmod_rel x 0 0 x"
```
```  1498     by (rule pdivmod_rel_by_0)
```
```  1499   thus "x div 0 = 0"
```
```  1500     by (rule div_poly_eq)
```
```  1501 next
```
```  1502   fix y :: "'a poly"
```
```  1503   have "pdivmod_rel 0 y 0 0"
```
```  1504     by (rule pdivmod_rel_0)
```
```  1505   thus "0 div y = 0"
```
```  1506     by (rule div_poly_eq)
```
```  1507 next
```
```  1508   fix x y z :: "'a poly"
```
```  1509   assume "y \<noteq> 0"
```
```  1510   hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
```
```  1511     using pdivmod_rel [of x y]
```
```  1512     by (simp add: pdivmod_rel_def distrib_right)
```
```  1513   thus "(x + z * y) div y = z + x div y"
```
```  1514     by (rule div_poly_eq)
```
```  1515 next
```
```  1516   fix x y z :: "'a poly"
```
```  1517   assume "x \<noteq> 0"
```
```  1518   show "(x * y) div (x * z) = y div z"
```
```  1519   proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
```
```  1520     have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
```
```  1521       by (rule pdivmod_rel_by_0)
```
```  1522     then have [simp]: "\<And>x::'a poly. x div 0 = 0"
```
```  1523       by (rule div_poly_eq)
```
```  1524     have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
```
```  1525       by (rule pdivmod_rel_0)
```
```  1526     then have [simp]: "\<And>x::'a poly. 0 div x = 0"
```
```  1527       by (rule div_poly_eq)
```
```  1528     case False then show ?thesis by auto
```
```  1529   next
```
```  1530     case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
```
```  1531     with `x \<noteq> 0`
```
```  1532     have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
```
```  1533       by (auto simp add: pdivmod_rel_def algebra_simps)
```
```  1534         (rule classical, simp add: degree_mult_eq)
```
```  1535     moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
```
```  1536     ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
```
```  1537     then show ?thesis by (simp add: div_poly_eq)
```
```  1538   qed
```
```  1539 qed
```
```  1540
```
```  1541 end
```
```  1542
```
```  1543 lemma degree_mod_less:
```
```  1544   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
```
```  1545   using pdivmod_rel [of x y]
```
```  1546   unfolding pdivmod_rel_def by simp
```
```  1547
```
```  1548 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
```
```  1549 proof -
```
```  1550   assume "degree x < degree y"
```
```  1551   hence "pdivmod_rel x y 0 x"
```
```  1552     by (simp add: pdivmod_rel_def)
```
```  1553   thus "x div y = 0" by (rule div_poly_eq)
```
```  1554 qed
```
```  1555
```
```  1556 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
```
```  1557 proof -
```
```  1558   assume "degree x < degree y"
```
```  1559   hence "pdivmod_rel x y 0 x"
```
```  1560     by (simp add: pdivmod_rel_def)
```
```  1561   thus "x mod y = x" by (rule mod_poly_eq)
```
```  1562 qed
```
```  1563
```
```  1564 lemma pdivmod_rel_smult_left:
```
```  1565   "pdivmod_rel x y q r
```
```  1566     \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
```
```  1567   unfolding pdivmod_rel_def by (simp add: smult_add_right)
```
```  1568
```
```  1569 lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
```
```  1570   by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
```
```  1571
```
```  1572 lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
```
```  1573   by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
```
```  1574
```
```  1575 lemma poly_div_minus_left [simp]:
```
```  1576   fixes x y :: "'a::field poly"
```
```  1577   shows "(- x) div y = - (x div y)"
```
```  1578   using div_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
```
```  1579
```
```  1580 lemma poly_mod_minus_left [simp]:
```
```  1581   fixes x y :: "'a::field poly"
```
```  1582   shows "(- x) mod y = - (x mod y)"
```
```  1583   using mod_smult_left [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
```
```  1584
```
```  1585 lemma pdivmod_rel_smult_right:
```
```  1586   "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
```
```  1587     \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
```
```  1588   unfolding pdivmod_rel_def by simp
```
```  1589
```
```  1590 lemma div_smult_right:
```
```  1591   "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
```
```  1592   by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
```
```  1593
```
```  1594 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
```
```  1595   by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
```
```  1596
```
```  1597 lemma poly_div_minus_right [simp]:
```
```  1598   fixes x y :: "'a::field poly"
```
```  1599   shows "x div (- y) = - (x div y)"
```
```  1600   using div_smult_right [of "- 1::'a"]
```
```  1601   by (simp add: nonzero_inverse_minus_eq del: minus_one) (* FIXME *)
```
```  1602
```
```  1603 lemma poly_mod_minus_right [simp]:
```
```  1604   fixes x y :: "'a::field poly"
```
```  1605   shows "x mod (- y) = x mod y"
```
```  1606   using mod_smult_right [of "- 1::'a"] by (simp del: minus_one) (* FIXME *)
```
```  1607
```
```  1608 lemma pdivmod_rel_mult:
```
```  1609   "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
```
```  1610     \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
```
```  1611 apply (cases "z = 0", simp add: pdivmod_rel_def)
```
```  1612 apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
```
```  1613 apply (cases "r = 0")
```
```  1614 apply (cases "r' = 0")
```
```  1615 apply (simp add: pdivmod_rel_def)
```
```  1616 apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)
```
```  1617 apply (cases "r' = 0")
```
```  1618 apply (simp add: pdivmod_rel_def degree_mult_eq)
```
```  1619 apply (simp add: pdivmod_rel_def field_simps)
```
```  1620 apply (simp add: degree_mult_eq degree_add_less)
```
```  1621 done
```
```  1622
```
```  1623 lemma poly_div_mult_right:
```
```  1624   fixes x y z :: "'a::field poly"
```
```  1625   shows "x div (y * z) = (x div y) div z"
```
```  1626   by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
```
```  1627
```
```  1628 lemma poly_mod_mult_right:
```
```  1629   fixes x y z :: "'a::field poly"
```
```  1630   shows "x mod (y * z) = y * (x div y mod z) + x mod y"
```
```  1631   by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
```
```  1632
```
```  1633 lemma mod_pCons:
```
```  1634   fixes a and x
```
```  1635   assumes y: "y \<noteq> 0"
```
```  1636   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
```
```  1637   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
```
```  1638 unfolding b
```
```  1639 apply (rule mod_poly_eq)
```
```  1640 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
```
```  1641 done
```
```  1642
```
```  1643 definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
```
```  1644 where
```
```  1645   "pdivmod p q = (p div q, p mod q)"
```
```  1646
```
```  1647 lemma div_poly_code [code]:
```
```  1648   "p div q = fst (pdivmod p q)"
```
```  1649   by (simp add: pdivmod_def)
```
```  1650
```
```  1651 lemma mod_poly_code [code]:
```
```  1652   "p mod q = snd (pdivmod p q)"
```
```  1653   by (simp add: pdivmod_def)
```
```  1654
```
```  1655 lemma pdivmod_0:
```
```  1656   "pdivmod 0 q = (0, 0)"
```
```  1657   by (simp add: pdivmod_def)
```
```  1658
```
```  1659 lemma pdivmod_pCons:
```
```  1660   "pdivmod (pCons a p) q =
```
```  1661     (if q = 0 then (0, pCons a p) else
```
```  1662       (let (s, r) = pdivmod p q;
```
```  1663            b = coeff (pCons a r) (degree q) / coeff q (degree q)
```
```  1664         in (pCons b s, pCons a r - smult b q)))"
```
```  1665   apply (simp add: pdivmod_def Let_def, safe)
```
```  1666   apply (rule div_poly_eq)
```
```  1667   apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1668   apply (rule mod_poly_eq)
```
```  1669   apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
```
```  1670   done
```
```  1671
```
```  1672 lemma pdivmod_fold_coeffs [code]:
```
```  1673   "pdivmod p q = (if q = 0 then (0, p)
```
```  1674     else fold_coeffs (\<lambda>a (s, r).
```
```  1675       let b = coeff (pCons a r) (degree q) / coeff q (degree q)
```
```  1676       in (pCons b s, pCons a r - smult b q)
```
```  1677    ) p (0, 0))"
```
```  1678   apply (cases "q = 0")
```
```  1679   apply (simp add: pdivmod_def)
```
```  1680   apply (rule sym)
```
```  1681   apply (induct p)
```
```  1682   apply (simp_all add: pdivmod_0 pdivmod_pCons)
```
```  1683   apply (case_tac "a = 0 \<and> p = 0")
```
```  1684   apply (auto simp add: pdivmod_def)
```
```  1685   done
```
```  1686
```
```  1687
```
```  1688 subsection {* Order of polynomial roots *}
```
```  1689
```
```  1690 definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
```
```  1691 where
```
```  1692   "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
```
```  1693
```
```  1694 lemma coeff_linear_power:
```
```  1695   fixes a :: "'a::comm_semiring_1"
```
```  1696   shows "coeff ([:a, 1:] ^ n) n = 1"
```
```  1697 apply (induct n, simp_all)
```
```  1698 apply (subst coeff_eq_0)
```
```  1699 apply (auto intro: le_less_trans degree_power_le)
```
```  1700 done
```
```  1701
```
```  1702 lemma degree_linear_power:
```
```  1703   fixes a :: "'a::comm_semiring_1"
```
```  1704   shows "degree ([:a, 1:] ^ n) = n"
```
```  1705 apply (rule order_antisym)
```
```  1706 apply (rule ord_le_eq_trans [OF degree_power_le], simp)
```
```  1707 apply (rule le_degree, simp add: coeff_linear_power)
```
```  1708 done
```
```  1709
```
```  1710 lemma order_1: "[:-a, 1:] ^ order a p dvd p"
```
```  1711 apply (cases "p = 0", simp)
```
```  1712 apply (cases "order a p", simp)
```
```  1713 apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
```
```  1714 apply (drule not_less_Least, simp)
```
```  1715 apply (fold order_def, simp)
```
```  1716 done
```
```  1717
```
```  1718 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1719 unfolding order_def
```
```  1720 apply (rule LeastI_ex)
```
```  1721 apply (rule_tac x="degree p" in exI)
```
```  1722 apply (rule notI)
```
```  1723 apply (drule (1) dvd_imp_degree_le)
```
```  1724 apply (simp only: degree_linear_power)
```
```  1725 done
```
```  1726
```
```  1727 lemma order:
```
```  1728   "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
```
```  1729 by (rule conjI [OF order_1 order_2])
```
```  1730
```
```  1731 lemma order_degree:
```
```  1732   assumes p: "p \<noteq> 0"
```
```  1733   shows "order a p \<le> degree p"
```
```  1734 proof -
```
```  1735   have "order a p = degree ([:-a, 1:] ^ order a p)"
```
```  1736     by (simp only: degree_linear_power)
```
```  1737   also have "\<dots> \<le> degree p"
```
```  1738     using order_1 p by (rule dvd_imp_degree_le)
```
```  1739   finally show ?thesis .
```
```  1740 qed
```
```  1741
```
```  1742 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
```
```  1743 apply (cases "p = 0", simp_all)
```
```  1744 apply (rule iffI)
```
```  1745 apply (rule ccontr, simp)
```
```  1746 apply (frule order_2 [where a=a], simp)
```
```  1747 apply (simp add: poly_eq_0_iff_dvd)
```
```  1748 apply (simp add: poly_eq_0_iff_dvd)
```
```  1749 apply (simp only: order_def)
```
```  1750 apply (drule not_less_Least, simp)
```
```  1751 done
```
```  1752
```
```  1753
```
```  1754 subsection {* GCD of polynomials *}
```
```  1755
```
```  1756 instantiation poly :: (field) gcd
```
```  1757 begin
```
```  1758
```
```  1759 function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  1760 where
```
```  1761   "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
```
```  1762 | "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
```
```  1763 by auto
```
```  1764
```
```  1765 termination "gcd :: _ poly \<Rightarrow> _"
```
```  1766 by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
```
```  1767    (auto dest: degree_mod_less)
```
```  1768
```
```  1769 declare gcd_poly.simps [simp del]
```
```  1770
```
```  1771 instance ..
```
```  1772
```
```  1773 end
```
```  1774
```
```  1775 lemma
```
```  1776   fixes x y :: "_ poly"
```
```  1777   shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
```
```  1778     and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
```
```  1779   apply (induct x y rule: gcd_poly.induct)
```
```  1780   apply (simp_all add: gcd_poly.simps)
```
```  1781   apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
```
```  1782   apply (blast dest: dvd_mod_imp_dvd)
```
```  1783   done
```
```  1784
```
```  1785 lemma poly_gcd_greatest:
```
```  1786   fixes k x y :: "_ poly"
```
```  1787   shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
```
```  1788   by (induct x y rule: gcd_poly.induct)
```
```  1789      (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
```
```  1790
```
```  1791 lemma dvd_poly_gcd_iff [iff]:
```
```  1792   fixes k x y :: "_ poly"
```
```  1793   shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
```
```  1794   by (blast intro!: poly_gcd_greatest intro: dvd_trans)
```
```  1795
```
```  1796 lemma poly_gcd_monic:
```
```  1797   fixes x y :: "_ poly"
```
```  1798   shows "coeff (gcd x y) (degree (gcd x y)) =
```
```  1799     (if x = 0 \<and> y = 0 then 0 else 1)"
```
```  1800   by (induct x y rule: gcd_poly.induct)
```
```  1801      (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
```
```  1802
```
```  1803 lemma poly_gcd_zero_iff [simp]:
```
```  1804   fixes x y :: "_ poly"
```
```  1805   shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```  1806   by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
```
```  1807
```
```  1808 lemma poly_gcd_0_0 [simp]:
```
```  1809   "gcd (0::_ poly) 0 = 0"
```
```  1810   by simp
```
```  1811
```
```  1812 lemma poly_dvd_antisym:
```
```  1813   fixes p q :: "'a::idom poly"
```
```  1814   assumes coeff: "coeff p (degree p) = coeff q (degree q)"
```
```  1815   assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
```
```  1816 proof (cases "p = 0")
```
```  1817   case True with coeff show "p = q" by simp
```
```  1818 next
```
```  1819   case False with coeff have "q \<noteq> 0" by auto
```
```  1820   have degree: "degree p = degree q"
```
```  1821     using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
```
```  1822     by (intro order_antisym dvd_imp_degree_le)
```
```  1823
```
```  1824   from `p dvd q` obtain a where a: "q = p * a" ..
```
```  1825   with `q \<noteq> 0` have "a \<noteq> 0" by auto
```
```  1826   with degree a `p \<noteq> 0` have "degree a = 0"
```
```  1827     by (simp add: degree_mult_eq)
```
```  1828   with coeff a show "p = q"
```
```  1829     by (cases a, auto split: if_splits)
```
```  1830 qed
```
```  1831
```
```  1832 lemma poly_gcd_unique:
```
```  1833   fixes d x y :: "_ poly"
```
```  1834   assumes dvd1: "d dvd x" and dvd2: "d dvd y"
```
```  1835     and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
```
```  1836     and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
```
```  1837   shows "gcd x y = d"
```
```  1838 proof -
```
```  1839   have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
```
```  1840     by (simp_all add: poly_gcd_monic monic)
```
```  1841   moreover have "gcd x y dvd d"
```
```  1842     using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
```
```  1843   moreover have "d dvd gcd x y"
```
```  1844     using dvd1 dvd2 by (rule poly_gcd_greatest)
```
```  1845   ultimately show ?thesis
```
```  1846     by (rule poly_dvd_antisym)
```
```  1847 qed
```
```  1848
```
```  1849 interpretation gcd_poly!: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
```
```  1850 proof
```
```  1851   fix x y z :: "'a poly"
```
```  1852   show "gcd (gcd x y) z = gcd x (gcd y z)"
```
```  1853     by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
```
```  1854   show "gcd x y = gcd y x"
```
```  1855     by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1856 qed
```
```  1857
```
```  1858 lemmas poly_gcd_assoc = gcd_poly.assoc
```
```  1859 lemmas poly_gcd_commute = gcd_poly.commute
```
```  1860 lemmas poly_gcd_left_commute = gcd_poly.left_commute
```
```  1861
```
```  1862 lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
```
```  1863
```
```  1864 lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
```
```  1865 by (rule poly_gcd_unique) simp_all
```
```  1866
```
```  1867 lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
```
```  1868 by (rule poly_gcd_unique) simp_all
```
```  1869
```
```  1870 lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
```
```  1871 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1872
```
```  1873 lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
```
```  1874 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
```
```  1875
```
```  1876 lemma poly_gcd_code [code]:
```
```  1877   "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
```
```  1878   by (simp add: gcd_poly.simps)
```
```  1879
```
```  1880
```
```  1881 subsection {* Composition of polynomials *}
```
```  1882
```
```  1883 definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```  1884 where
```
```  1885   "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
```
```  1886
```
```  1887 lemma pcompose_0 [simp]:
```
```  1888   "pcompose 0 q = 0"
```
```  1889   by (simp add: pcompose_def)
```
```  1890
```
```  1891 lemma pcompose_pCons:
```
```  1892   "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
```
```  1893   by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
```
```  1894
```
```  1895 lemma poly_pcompose:
```
```  1896   "poly (pcompose p q) x = poly p (poly q x)"
```
```  1897   by (induct p) (simp_all add: pcompose_pCons)
```
```  1898
```
```  1899 lemma degree_pcompose_le:
```
```  1900   "degree (pcompose p q) \<le> degree p * degree q"
```
```  1901 apply (induct p, simp)
```
```  1902 apply (simp add: pcompose_pCons, clarify)
```
```  1903 apply (rule degree_add_le, simp)
```
```  1904 apply (rule order_trans [OF degree_mult_le], simp)
```
```  1905 done
```
```  1906
```
```  1907
```
```  1908 no_notation cCons (infixr "##" 65)
```
```  1909
```
```  1910 end
```
```  1911
```