src/HOL/Decision_Procs/Polynomial_List.thy
 author haftmann Mon Jun 05 15:59:41 2017 +0200 (2017-06-05) changeset 66010 2f7d39285a1a parent 62390 842917225d56 child 67399 eab6ce8368fa permissions -rw-r--r--
executable domain membership checks
```     1 (*  Title:      HOL/Decision_Procs/Polynomial_List.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 section \<open>Univariate Polynomials as lists\<close>
```
```     6
```
```     7 theory Polynomial_List
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
```
```    11 text \<open>Application of polynomial as a function.\<close>
```
```    12
```
```    13 primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    14 where
```
```    15   poly_Nil: "poly [] x = 0"
```
```    16 | poly_Cons: "poly (h # t) x = h + x * poly t x"
```
```    17
```
```    18
```
```    19 subsection \<open>Arithmetic Operations on Polynomials\<close>
```
```    20
```
```    21 text \<open>Addition\<close>
```
```    22 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
```
```    23 where
```
```    24   padd_Nil: "[] +++ l2 = l2"
```
```    25 | padd_Cons: "(h # t) +++ l2 = (if l2 = [] then h # t else (h + hd l2) # (t +++ tl l2))"
```
```    26
```
```    27 text \<open>Multiplication by a constant\<close>
```
```    28 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
```
```    29   cmult_Nil: "c %* [] = []"
```
```    30 | cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
```
```    31
```
```    32 text \<open>Multiplication by a polynomial\<close>
```
```    33 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
```
```    34 where
```
```    35   pmult_Nil: "[] *** l2 = []"
```
```    36 | pmult_Cons: "(h # t) *** l2 = (if t = [] then h %* l2 else (h %* l2) +++ (0 # (t *** l2)))"
```
```    37
```
```    38 text \<open>Repeated multiplication by a polynomial\<close>
```
```    39 primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list"
```
```    40 where
```
```    41   mulexp_zero: "mulexp 0 p q = q"
```
```    42 | mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
```
```    43
```
```    44 text \<open>Exponential\<close>
```
```    45 primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80)
```
```    46 where
```
```    47   pexp_0: "p %^ 0 = [1]"
```
```    48 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
```
```    49
```
```    50 text \<open>Quotient related value of dividing a polynomial by x + a.
```
```    51   Useful for divisor properties in inductive proofs.\<close>
```
```    52 primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
```
```    53 where
```
```    54   pquot_Nil: "pquot [] a = []"
```
```    55 | pquot_Cons: "pquot (h # t) a =
```
```    56     (if t = [] then [h] else (inverse a * (h - hd( pquot t a))) # pquot t a)"
```
```    57
```
```    58 text \<open>Normalization of polynomials (remove extra 0 coeff).\<close>
```
```    59 primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list"
```
```    60 where
```
```    61   pnormalize_Nil: "pnormalize [] = []"
```
```    62 | pnormalize_Cons: "pnormalize (h # p) =
```
```    63     (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"
```
```    64
```
```    65 definition (in semiring_0) "pnormal p \<longleftrightarrow> pnormalize p = p \<and> p \<noteq> []"
```
```    66 definition (in semiring_0) "nonconstant p \<longleftrightarrow> pnormal p \<and> (\<forall>x. p \<noteq> [x])"
```
```    67
```
```    68 text \<open>Other definitions.\<close>
```
```    69
```
```    70 definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
```
```    71   where "-- p = (- 1) %* p"
```
```    72
```
```    73 definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
```
```    74   where "p1 divides p2 \<longleftrightarrow> (\<exists>q. poly p2 = poly(p1 *** q))"
```
```    75
```
```    76 lemma (in semiring_0) dividesI: "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
```
```    77   by (auto simp add: divides_def)
```
```    78
```
```    79 lemma (in semiring_0) dividesE:
```
```    80   assumes "p1 divides p2"
```
```    81   obtains q where "poly p2 = poly (p1 *** q)"
```
```    82   using assms by (auto simp add: divides_def)
```
```    83
```
```    84 \<comment> \<open>order of a polynomial\<close>
```
```    85 definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat"
```
```    86   where "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ (Suc n)) divides p))"
```
```    87
```
```    88 \<comment> \<open>degree of a polynomial\<close>
```
```    89 definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
```
```    90   where "degree p = length (pnormalize p) - 1"
```
```    91
```
```    92 \<comment> \<open>squarefree polynomials --- NB with respect to real roots only\<close>
```
```    93 definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
```
```    94   where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
```
```    95
```
```    96 context semiring_0
```
```    97 begin
```
```    98
```
```    99 lemma padd_Nil2[simp]: "p +++ [] = p"
```
```   100   by (induct p) auto
```
```   101
```
```   102 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
```
```   103   by auto
```
```   104
```
```   105 lemma pminus_Nil: "-- [] = []"
```
```   106   by (simp add: poly_minus_def)
```
```   107
```
```   108 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
```
```   109
```
```   110 end
```
```   111
```
```   112 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t"
```
```   113   by (induct t) auto
```
```   114
```
```   115 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ (0 # t) = a # t"
```
```   116   by simp
```
```   117
```
```   118
```
```   119 text \<open>Handy general properties.\<close>
```
```   120
```
```   121 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
```
```   122 proof (induct b arbitrary: a)
```
```   123   case Nil
```
```   124   then show ?case
```
```   125     by auto
```
```   126 next
```
```   127   case (Cons b bs a)
```
```   128   then show ?case
```
```   129     by (cases a) (simp_all add: add.commute)
```
```   130 qed
```
```   131
```
```   132 lemma (in comm_semiring_0) padd_assoc: "(a +++ b) +++ c = a +++ (b +++ c)"
```
```   133 proof (induct a arbitrary: b c)
```
```   134   case Nil
```
```   135   then show ?case
```
```   136     by simp
```
```   137 next
```
```   138   case Cons
```
```   139   then show ?case
```
```   140     by (cases b) (simp_all add: ac_simps)
```
```   141 qed
```
```   142
```
```   143 lemma (in semiring_0) poly_cmult_distr: "a %* (p +++ q) = a %* p +++ a %* q"
```
```   144 proof (induct p arbitrary: q)
```
```   145   case Nil
```
```   146   then show ?case
```
```   147     by simp
```
```   148 next
```
```   149   case Cons
```
```   150   then show ?case
```
```   151     by (cases q) (simp_all add: distrib_left)
```
```   152 qed
```
```   153
```
```   154 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = 0 # t"
```
```   155 proof (induct t)
```
```   156   case Nil
```
```   157   then show ?case
```
```   158     by simp
```
```   159 next
```
```   160   case (Cons a t)
```
```   161   then show ?case
```
```   162     by (cases t) (auto simp add: padd_commut)
```
```   163 qed
```
```   164
```
```   165 text \<open>Properties of evaluation of polynomials.\<close>
```
```   166
```
```   167 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
```
```   168 proof (induct p1 arbitrary: p2)
```
```   169   case Nil
```
```   170   then show ?case
```
```   171     by simp
```
```   172 next
```
```   173   case (Cons a as p2)
```
```   174   then show ?case
```
```   175     by (cases p2) (simp_all add: ac_simps distrib_left)
```
```   176 qed
```
```   177
```
```   178 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
```
```   179 proof (induct p)
```
```   180   case Nil
```
```   181   then show ?case
```
```   182     by simp
```
```   183 next
```
```   184   case Cons
```
```   185   then show ?case
```
```   186     by (cases "x = zero") (auto simp add: distrib_left ac_simps)
```
```   187 qed
```
```   188
```
```   189 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c * poly p x"
```
```   190   by (induct p) (auto simp add: distrib_left ac_simps)
```
```   191
```
```   192 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
```
```   193   by (simp add: poly_minus_def) (auto simp add: poly_cmult)
```
```   194
```
```   195 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
```
```   196 proof (induct p1 arbitrary: p2)
```
```   197   case Nil
```
```   198   then show ?case
```
```   199     by simp
```
```   200 next
```
```   201   case (Cons a as)
```
```   202   then show ?case
```
```   203     by (cases as) (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)
```
```   204 qed
```
```   205
```
```   206 class idom_char_0 = idom + ring_char_0
```
```   207
```
```   208 subclass (in field_char_0) idom_char_0 ..
```
```   209
```
```   210 lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
```
```   211   by (induct n) (auto simp add: poly_cmult poly_mult)
```
```   212
```
```   213
```
```   214 text \<open>More Polynomial Evaluation lemmas.\<close>
```
```   215
```
```   216 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
```
```   217   by simp
```
```   218
```
```   219 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
```
```   220   by (simp add: poly_mult mult.assoc)
```
```   221
```
```   222 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
```
```   223   by (induct p) auto
```
```   224
```
```   225 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly (p %^ n *** p %^ d) x"
```
```   226   by (induct n) (auto simp add: poly_mult mult.assoc)
```
```   227
```
```   228
```
```   229 subsection \<open>Key Property: if @{term "f a = 0"} then @{term "(x - a)"} divides @{term "p(x)"}.\<close>
```
```   230
```
```   231 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   232 proof (induct t arbitrary: h)
```
```   233   case Nil
```
```   234   have "[h] = [h] +++ [- a, 1] *** []" by simp
```
```   235   then show ?case by blast
```
```   236 next
```
```   237   case (Cons  x xs)
```
```   238   have "\<exists>q r. h # x # xs = [r] +++ [-a, 1] *** q"
```
```   239   proof -
```
```   240     from Cons obtain q r where qr: "x # xs = [r] +++ [- a, 1] *** q"
```
```   241       by blast
```
```   242     have "h # x # xs = [a * r + h] +++ [-a, 1] *** (r # q)"
```
```   243       using qr by (cases q) (simp_all add: algebra_simps)
```
```   244     then show ?thesis by blast
```
```   245   qed
```
```   246   then show ?case by blast
```
```   247 qed
```
```   248
```
```   249 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   250   using lemma_poly_linear_rem [where t = t and a = a] by auto
```
```   251
```
```   252 lemma (in comm_ring_1) poly_linear_divides: "poly p a = 0 \<longleftrightarrow> p = [] \<or> (\<exists>q. p = [-a, 1] *** q)"
```
```   253 proof (cases p)
```
```   254   case Nil
```
```   255   then show ?thesis by simp
```
```   256 next
```
```   257   case (Cons x xs)
```
```   258   have "poly p a = 0" if "p = [-a, 1] *** q" for q
```
```   259     using that by (simp add: poly_add poly_cmult)
```
```   260   moreover
```
```   261   have "\<exists>q. p = [- a, 1] *** q" if p0: "poly p a = 0"
```
```   262   proof -
```
```   263     from poly_linear_rem[of x xs a] obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q"
```
```   264       by blast
```
```   265     have "r = 0"
```
```   266       using p0 by (simp only: Cons qr poly_mult poly_add) simp
```
```   267     with Cons qr show ?thesis
```
```   268       apply -
```
```   269       apply (rule exI[where x = q])
```
```   270       apply auto
```
```   271       apply (cases q)
```
```   272       apply auto
```
```   273       done
```
```   274   qed
```
```   275   ultimately show ?thesis using Cons by blast
```
```   276 qed
```
```   277
```
```   278 lemma (in semiring_0) lemma_poly_length_mult[simp]:
```
```   279   "length (k %* p +++  (h # (a %* p))) = Suc (length p)"
```
```   280   by (induct p arbitrary: h k a) auto
```
```   281
```
```   282 lemma (in semiring_0) lemma_poly_length_mult2[simp]:
```
```   283   "length (k %* p +++  (h # p)) = Suc (length p)"
```
```   284   by (induct p arbitrary: h k) auto
```
```   285
```
```   286 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
```
```   287   by auto
```
```   288
```
```   289
```
```   290 subsection \<open>Polynomial length\<close>
```
```   291
```
```   292 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
```
```   293   by (induct p) auto
```
```   294
```
```   295 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
```
```   296   by (induct p1 arbitrary: p2) auto
```
```   297
```
```   298 lemma (in semiring_0) poly_root_mult_length[simp]: "length ([a, b] *** p) = Suc (length p)"
```
```   299   by (simp add: poly_add_length)
```
```   300
```
```   301 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
```
```   302   "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
```
```   303   by (auto simp add: poly_mult)
```
```   304
```
```   305 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
```
```   306   by (auto simp add: poly_mult)
```
```   307
```
```   308
```
```   309 text \<open>Normalisation Properties.\<close>
```
```   310
```
```   311 lemma (in semiring_0) poly_normalized_nil: "pnormalize p = [] \<longrightarrow> poly p x = 0"
```
```   312   by (induct p) auto
```
```   313
```
```   314 text \<open>A nontrivial polynomial of degree n has no more than n roots.\<close>
```
```   315 lemma (in idom) poly_roots_index_lemma:
```
```   316   assumes "poly p x \<noteq> poly [] x"
```
```   317     and "length p = n"
```
```   318   shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
```
```   319   using assms
```
```   320 proof (induct n arbitrary: p x)
```
```   321   case 0
```
```   322   then show ?case by simp
```
```   323 next
```
```   324   case (Suc n)
```
```   325   have False if C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
```
```   326   proof -
```
```   327     from Suc.prems have p0: "poly p x \<noteq> 0" "p \<noteq> []"
```
```   328       by auto
```
```   329     from p0(1)[unfolded poly_linear_divides[of p x]]
```
```   330     have "\<forall>q. p \<noteq> [- x, 1] *** q"
```
```   331       by blast
```
```   332     from C obtain a where a: "poly p a = 0"
```
```   333       by blast
```
```   334     from a[unfolded poly_linear_divides[of p a]] p0(2) obtain q where q: "p = [-a, 1] *** q"
```
```   335       by blast
```
```   336     have lg: "length q = n"
```
```   337       using q Suc.prems(2) by simp
```
```   338     from q p0 have qx: "poly q x \<noteq> poly [] x"
```
```   339       by (auto simp add: poly_mult poly_add poly_cmult)
```
```   340     from Suc.hyps[OF qx lg] obtain i where i: "\<And>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
```
```   341       by blast
```
```   342     let ?i = "\<lambda>m. if m = Suc n then a else i m"
```
```   343     from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
```
```   344       by blast
```
```   345     from y have "y = a \<or> poly q y = 0"
```
```   346       by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
```
```   347     with i[of y] y(1) y(2) show ?thesis
```
```   348       apply auto
```
```   349       apply (erule_tac x = "m" in allE)
```
```   350       apply auto
```
```   351       done
```
```   352   qed
```
```   353   then show ?case by blast
```
```   354 qed
```
```   355
```
```   356
```
```   357 lemma (in idom) poly_roots_index_length:
```
```   358   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
```
```   359   by (blast intro: poly_roots_index_lemma)
```
```   360
```
```   361 lemma (in idom) poly_roots_finite_lemma1:
```
```   362   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>n::nat. n < N \<and> x = i n)"
```
```   363   apply (drule poly_roots_index_length)
```
```   364   apply safe
```
```   365   apply (rule_tac x = "Suc (length p)" in exI)
```
```   366   apply (rule_tac x = i in exI)
```
```   367   apply (simp add: less_Suc_eq_le)
```
```   368   done
```
```   369
```
```   370 lemma (in idom) idom_finite_lemma:
```
```   371   assumes "\<forall>x. P x \<longrightarrow> (\<exists>n. n < length j \<and> x = j!n)"
```
```   372   shows "finite {x. P x}"
```
```   373 proof -
```
```   374   from assms have "{x. P x} \<subseteq> set j"
```
```   375     by auto
```
```   376   then show ?thesis
```
```   377     using finite_subset by auto
```
```   378 qed
```
```   379
```
```   380 lemma (in idom) poly_roots_finite_lemma2:
```
```   381   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
```
```   382   apply (drule poly_roots_index_length)
```
```   383   apply safe
```
```   384   apply (rule_tac x = "map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
```
```   385   apply (auto simp add: image_iff)
```
```   386   apply (erule_tac x="x" in allE)
```
```   387   apply clarsimp
```
```   388   apply (case_tac "n = length p")
```
```   389   apply (auto simp add: order_le_less)
```
```   390   done
```
```   391
```
```   392 lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> finite (UNIV :: 'a set)"
```
```   393 proof
```
```   394   assume F: "finite (UNIV :: 'a set)"
```
```   395   have "finite (UNIV :: nat set)"
```
```   396   proof (rule finite_imageD)
```
```   397     have "of_nat ` UNIV \<subseteq> UNIV"
```
```   398       by simp
```
```   399     then show "finite (of_nat ` UNIV :: 'a set)"
```
```   400       using F by (rule finite_subset)
```
```   401     show "inj (of_nat :: nat \<Rightarrow> 'a)"
```
```   402       by (simp add: inj_on_def)
```
```   403   qed
```
```   404   with infinite_UNIV_nat show False ..
```
```   405 qed
```
```   406
```
```   407 lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
```
```   408   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   409 proof
```
```   410   show ?rhs if ?lhs
```
```   411     using that
```
```   412     apply -
```
```   413     apply (erule contrapos_np)
```
```   414     apply (rule ext)
```
```   415     apply (rule ccontr)
```
```   416     apply (clarify dest!: poly_roots_finite_lemma2)
```
```   417     using finite_subset
```
```   418   proof -
```
```   419     fix x i
```
```   420     assume F: "\<not> finite {x. poly p x = 0}"
```
```   421       and P: "\<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
```
```   422     from P have "{x. poly p x = 0} \<subseteq> set i"
```
```   423       by auto
```
```   424     with finite_subset F show False
```
```   425       by auto
```
```   426   qed
```
```   427   show ?lhs if ?rhs
```
```   428     using UNIV_ring_char_0_infinte that by auto
```
```   429 qed
```
```   430
```
```   431
```
```   432 text \<open>Entirety and Cancellation for polynomials\<close>
```
```   433
```
```   434 lemma (in idom_char_0) poly_entire_lemma2:
```
```   435   assumes p0: "poly p \<noteq> poly []"
```
```   436     and q0: "poly q \<noteq> poly []"
```
```   437   shows "poly (p***q) \<noteq> poly []"
```
```   438 proof -
```
```   439   let ?S = "\<lambda>p. {x. poly p x = 0}"
```
```   440   have "?S (p *** q) = ?S p \<union> ?S q"
```
```   441     by (auto simp add: poly_mult)
```
```   442   with p0 q0 show ?thesis
```
```   443     unfolding poly_roots_finite by auto
```
```   444 qed
```
```   445
```
```   446 lemma (in idom_char_0) poly_entire:
```
```   447   "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
```
```   448   using poly_entire_lemma2[of p q]
```
```   449   by (auto simp add: fun_eq_iff poly_mult)
```
```   450
```
```   451 lemma (in idom_char_0) poly_entire_neg:
```
```   452   "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
```
```   453   by (simp add: poly_entire)
```
```   454
```
```   455 lemma (in comm_ring_1) poly_add_minus_zero_iff:
```
```   456   "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
```
```   457   by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq_iff poly_cmult)
```
```   458
```
```   459 lemma (in comm_ring_1) poly_add_minus_mult_eq:
```
```   460   "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
```
```   461   by (auto simp add: poly_add poly_minus_def fun_eq_iff poly_mult poly_cmult algebra_simps)
```
```   462
```
```   463 subclass (in idom_char_0) comm_ring_1 ..
```
```   464
```
```   465 lemma (in idom_char_0) poly_mult_left_cancel:
```
```   466   "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
```
```   467 proof -
```
```   468   have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
```
```   469     by (simp only: poly_add_minus_zero_iff)
```
```   470   also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
```
```   471     by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
```
```   472   finally show ?thesis .
```
```   473 qed
```
```   474
```
```   475 lemma (in idom) poly_exp_eq_zero[simp]: "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
```
```   476   apply (simp only: fun_eq_iff add: HOL.all_simps [symmetric])
```
```   477   apply (rule arg_cong [where f = All])
```
```   478   apply (rule ext)
```
```   479   apply (induct n)
```
```   480   apply (auto simp add: poly_exp poly_mult)
```
```   481   done
```
```   482
```
```   483 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a, 1] \<noteq> poly []"
```
```   484   apply (simp add: fun_eq_iff)
```
```   485   apply (rule_tac x = "minus one a" in exI)
```
```   486   apply (simp add: add.commute [of a])
```
```   487   done
```
```   488
```
```   489 lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
```
```   490   by auto
```
```   491
```
```   492
```
```   493 text \<open>A more constructive notion of polynomials being trivial.\<close>
```
```   494
```
```   495 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
```
```   496   apply (simp add: fun_eq_iff)
```
```   497   apply (case_tac "h = zero")
```
```   498   apply (drule_tac [2] x = zero in spec)
```
```   499   apply auto
```
```   500   apply (cases "poly t = poly []")
```
```   501   apply simp
```
```   502 proof -
```
```   503   fix x
```
```   504   assume H: "\<forall>x. x = 0 \<or> poly t x = 0"
```
```   505   assume pnz: "poly t \<noteq> poly []"
```
```   506   let ?S = "{x. poly t x = 0}"
```
```   507   from H have "\<forall>x. x \<noteq> 0 \<longrightarrow> poly t x = 0"
```
```   508     by blast
```
```   509   then have th: "?S \<supseteq> UNIV - {0}"
```
```   510     by auto
```
```   511   from poly_roots_finite pnz have th': "finite ?S"
```
```   512     by blast
```
```   513   from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = 0"
```
```   514     by simp
```
```   515 qed
```
```   516
```
```   517 lemma (in idom_char_0) poly_zero: "poly p = poly [] \<longleftrightarrow> (\<forall>c \<in> set p. c = 0)"
```
```   518 proof (induct p)
```
```   519   case Nil
```
```   520   then show ?case by simp
```
```   521 next
```
```   522   case Cons
```
```   523   show ?case
```
```   524     apply (rule iffI)
```
```   525     apply (drule poly_zero_lemma')
```
```   526     using Cons
```
```   527     apply auto
```
```   528     done
```
```   529 qed
```
```   530
```
```   531 lemma (in idom_char_0) poly_0: "\<forall>c \<in> set p. c = 0 \<Longrightarrow> poly p x = 0"
```
```   532   unfolding poly_zero[symmetric] by simp
```
```   533
```
```   534
```
```   535 text \<open>Basics of divisibility.\<close>
```
```   536
```
```   537 lemma (in idom) poly_primes: "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
```
```   538   apply (auto simp add: divides_def fun_eq_iff poly_mult poly_add poly_cmult distrib_right [symmetric])
```
```   539   apply (drule_tac x = "uminus a" in spec)
```
```   540   apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
```
```   541   apply (cases "p = []")
```
```   542   apply (rule exI[where x="[]"])
```
```   543   apply simp
```
```   544   apply (cases "q = []")
```
```   545   apply (erule allE[where x="[]"])
```
```   546   apply simp
```
```   547
```
```   548   apply clarsimp
```
```   549   apply (cases "\<exists>q. p = a %* q +++ (0 # q)")
```
```   550   apply (clarsimp simp add: poly_add poly_cmult)
```
```   551   apply (rule_tac x = qa in exI)
```
```   552   apply (simp add: distrib_right [symmetric])
```
```   553   apply clarsimp
```
```   554
```
```   555   apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
```
```   556   apply (rule_tac x = "pmult qa q" in exI)
```
```   557   apply (rule_tac [2] x = "pmult p qa" in exI)
```
```   558   apply (auto simp add: poly_add poly_mult poly_cmult ac_simps)
```
```   559   done
```
```   560
```
```   561 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
```
```   562   apply (simp add: divides_def)
```
```   563   apply (rule_tac x = "[one]" in exI)
```
```   564   apply (auto simp add: poly_mult fun_eq_iff)
```
```   565   done
```
```   566
```
```   567 lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
```
```   568   apply (simp add: divides_def)
```
```   569   apply safe
```
```   570   apply (rule_tac x = "pmult qa qaa" in exI)
```
```   571   apply (auto simp add: poly_mult fun_eq_iff mult.assoc)
```
```   572   done
```
```   573
```
```   574 lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
```
```   575   by (auto simp: le_iff_add divides_def poly_exp_add fun_eq_iff)
```
```   576
```
```   577 lemma (in comm_semiring_1) poly_exp_divides: "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
```
```   578   by (blast intro: poly_divides_exp poly_divides_trans)
```
```   579
```
```   580 lemma (in comm_semiring_0) poly_divides_add: "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
```
```   581   apply (auto simp add: divides_def)
```
```   582   apply (rule_tac x = "padd qa qaa" in exI)
```
```   583   apply (auto simp add: poly_add fun_eq_iff poly_mult distrib_left)
```
```   584   done
```
```   585
```
```   586 lemma (in comm_ring_1) poly_divides_diff: "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
```
```   587   apply (auto simp add: divides_def)
```
```   588   apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
```
```   589   apply (auto simp add: poly_add fun_eq_iff poly_mult poly_minus algebra_simps)
```
```   590   done
```
```   591
```
```   592 lemma (in comm_ring_1) poly_divides_diff2: "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
```
```   593   apply (erule poly_divides_diff)
```
```   594   apply (auto simp add: poly_add fun_eq_iff poly_mult divides_def ac_simps)
```
```   595   done
```
```   596
```
```   597 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
```
```   598   apply (simp add: divides_def)
```
```   599   apply (rule exI[where x = "[]"])
```
```   600   apply (auto simp add: fun_eq_iff poly_mult)
```
```   601   done
```
```   602
```
```   603 lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
```
```   604   apply (simp add: divides_def)
```
```   605   apply (rule_tac x = "[]" in exI)
```
```   606   apply (auto simp add: fun_eq_iff)
```
```   607   done
```
```   608
```
```   609
```
```   610 text \<open>At last, we can consider the order of a root.\<close>
```
```   611
```
```   612 lemma (in idom_char_0) poly_order_exists_lemma:
```
```   613   assumes "length p = d"
```
```   614     and "poly p \<noteq> poly []"
```
```   615   shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
```
```   616   using assms
```
```   617 proof (induct d arbitrary: p)
```
```   618   case 0
```
```   619   then show ?case by simp
```
```   620 next
```
```   621   case (Suc n p)
```
```   622   show ?case
```
```   623   proof (cases "poly p a = 0")
```
```   624     case True
```
```   625     from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []"
```
```   626       by auto
```
```   627     then have pN: "p \<noteq> []"
```
```   628       by auto
```
```   629     from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
```
```   630       by blast
```
```   631     from q h True have qh: "length q = n" "poly q \<noteq> poly []"
```
```   632       apply simp_all
```
```   633       apply (simp only: fun_eq_iff)
```
```   634       apply (rule ccontr)
```
```   635       apply (simp add: fun_eq_iff poly_add poly_cmult)
```
```   636       done
```
```   637     from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
```
```   638       by blast
```
```   639     from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0"
```
```   640       by simp
```
```   641     then show ?thesis by blast
```
```   642   next
```
```   643     case False
```
```   644     then show ?thesis
```
```   645       using Suc.prems
```
```   646       apply simp
```
```   647       apply (rule exI[where x="0::nat"])
```
```   648       apply simp
```
```   649       done
```
```   650   qed
```
```   651 qed
```
```   652
```
```   653
```
```   654 lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
```
```   655   by (induct n) (auto simp add: poly_mult ac_simps)
```
```   656
```
```   657 lemma (in comm_semiring_1) divides_left_mult:
```
```   658   assumes "(p *** q) divides r"
```
```   659   shows "p divides r \<and> q divides r"
```
```   660 proof-
```
```   661   from assms obtain t where "poly r = poly (p *** q *** t)"
```
```   662     unfolding divides_def by blast
```
```   663   then have "poly r = poly (p *** (q *** t))" and "poly r = poly (q *** (p *** t))"
```
```   664     by (auto simp add: fun_eq_iff poly_mult ac_simps)
```
```   665   then show ?thesis
```
```   666     unfolding divides_def by blast
```
```   667 qed
```
```   668
```
```   669
```
```   670 (* FIXME: Tidy up *)
```
```   671
```
```   672 lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
```
```   673   by (induct n) simp_all
```
```   674
```
```   675 lemma (in idom_char_0) poly_order_exists:
```
```   676   assumes "length p = d"
```
```   677     and "poly p \<noteq> poly []"
```
```   678   shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p"
```
```   679 proof -
```
```   680   from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0"
```
```   681     by (rule poly_order_exists_lemma)
```
```   682   then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0"
```
```   683     by blast
```
```   684   have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
```
```   685   proof (rule dividesI)
```
```   686     show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
```
```   687       by (induct n) (simp_all add: poly_add poly_cmult poly_mult algebra_simps)
```
```   688   qed
```
```   689   moreover have "\<not> [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
```
```   690   proof
```
```   691     assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
```
```   692     then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"
```
```   693       by (rule dividesE)
```
```   694     moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)"
```
```   695     proof (induct n)
```
```   696       case 0
```
```   697       show ?case
```
```   698       proof (rule ccontr)
```
```   699         assume "\<not> ?thesis"
```
```   700         then have "poly q a = 0"
```
```   701           by (simp add: poly_add poly_cmult)
```
```   702         with \<open>poly q a \<noteq> 0\<close> show False
```
```   703           by simp
```
```   704       qed
```
```   705     next
```
```   706       case (Suc n)
```
```   707       show ?case
```
```   708         by (rule pexp_Suc [THEN ssubst])
```
```   709           (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
```
```   710     qed
```
```   711     ultimately show False by simp
```
```   712   qed
```
```   713   ultimately show ?thesis
```
```   714     by (auto simp add: p)
```
```   715 qed
```
```   716
```
```   717 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
```
```   718   by (auto simp add: divides_def)
```
```   719
```
```   720 lemma (in idom_char_0) poly_order:
```
```   721   "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
```
```   722   apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
```
```   723   apply (cut_tac x = y and y = n in less_linear)
```
```   724   apply (drule_tac m = n in poly_exp_divides)
```
```   725   apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
```
```   726     simp del: pmult_Cons pexp_Suc)
```
```   727   done
```
```   728
```
```   729
```
```   730 text \<open>Order\<close>
```
```   731
```
```   732 lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
```
```   733   by (blast intro: someI2)
```
```   734
```
```   735 lemma (in idom_char_0) order:
```
```   736   "([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p) \<longleftrightarrow>
```
```   737     n = order a p \<and> poly p \<noteq> poly []"
```
```   738   unfolding order_def
```
```   739   apply (rule iffI)
```
```   740   apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
```
```   741   apply (blast intro!: poly_order [THEN [2] some1_equalityD])
```
```   742   done
```
```   743
```
```   744 lemma (in idom_char_0) order2:
```
```   745   "poly p \<noteq> poly [] \<Longrightarrow>
```
```   746     ([-a, 1] %^ (order a p)) divides p \<and> \<not> ([-a, 1] %^ Suc (order a p)) divides p"
```
```   747   by (simp add: order del: pexp_Suc)
```
```   748
```
```   749 lemma (in idom_char_0) order_unique:
```
```   750   "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> \<not> ([-a, 1] %^ (Suc n)) divides p \<Longrightarrow>
```
```   751     n = order a p"
```
```   752   using order [of a n p] by auto
```
```   753
```
```   754 lemma (in idom_char_0) order_unique_lemma:
```
```   755   "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> \<not> ([-a, 1] %^ (Suc n)) divides p \<Longrightarrow>
```
```   756     n = order a p"
```
```   757   by (blast intro: order_unique)
```
```   758
```
```   759 lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
```
```   760   by (auto simp add: fun_eq_iff divides_def poly_mult order_def)
```
```   761
```
```   762 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
```
```   763   by (induct p) auto
```
```   764
```
```   765 lemma (in comm_ring_1) lemma_order_root:
```
```   766   "0 < n \<and> [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
```
```   767   by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
```
```   768
```
```   769 lemma (in idom_char_0) order_root: "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
```
```   770   apply (cases "poly p = poly []")
```
```   771   apply auto
```
```   772   apply (simp add: poly_linear_divides del: pmult_Cons)
```
```   773   apply safe
```
```   774   apply (drule_tac [!] a = a in order2)
```
```   775   apply (rule ccontr)
```
```   776   apply (simp add: divides_def poly_mult fun_eq_iff del: pmult_Cons)
```
```   777   apply blast
```
```   778   using neq0_conv apply (blast intro: lemma_order_root)
```
```   779   done
```
```   780
```
```   781 lemma (in idom_char_0) order_divides:
```
```   782   "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
```
```   783   apply (cases "poly p = poly []")
```
```   784   apply auto
```
```   785   apply (simp add: divides_def fun_eq_iff poly_mult)
```
```   786   apply (rule_tac x = "[]" in exI)
```
```   787   apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
```
```   788   done
```
```   789
```
```   790 lemma (in idom_char_0) order_decomp:
```
```   791   "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ order a p) *** q) \<and> \<not> [-a, 1] divides q"
```
```   792   unfolding divides_def
```
```   793   apply (drule order2 [where a = a])
```
```   794   apply (simp add: divides_def del: pexp_Suc pmult_Cons)
```
```   795   apply safe
```
```   796   apply (rule_tac x = q in exI)
```
```   797   apply safe
```
```   798   apply (drule_tac x = qa in spec)
```
```   799   apply (auto simp add: poly_mult fun_eq_iff poly_exp ac_simps simp del: pmult_Cons)
```
```   800   done
```
```   801
```
```   802 text \<open>Important composition properties of orders.\<close>
```
```   803 lemma order_mult:
```
```   804   fixes a :: "'a::idom_char_0"
```
```   805   shows "poly (p *** q) \<noteq> poly [] \<Longrightarrow> order a (p *** q) = order a p + order a q"
```
```   806   apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
```
```   807   apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   808   apply (drule_tac a = a in order2)+
```
```   809   apply safe
```
```   810   apply (simp add: divides_def fun_eq_iff poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   811   apply (rule_tac x = "qa *** qaa" in exI)
```
```   812   apply (simp add: poly_mult ac_simps del: pmult_Cons)
```
```   813   apply (drule_tac a = a in order_decomp)+
```
```   814   apply safe
```
```   815   apply (subgoal_tac "[-a, 1] divides (qa *** qaa) ")
```
```   816   apply (simp add: poly_primes del: pmult_Cons)
```
```   817   apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   818   apply (rule_tac x = qb in exI)
```
```   819   apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) =
```
```   820     poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
```
```   821   apply (drule poly_mult_left_cancel [THEN iffD1])
```
```   822   apply force
```
```   823   apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) =
```
```   824     poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
```
```   825   apply (drule poly_mult_left_cancel [THEN iffD1])
```
```   826   apply force
```
```   827   apply (simp add: fun_eq_iff poly_exp_add poly_mult ac_simps del: pmult_Cons)
```
```   828   done
```
```   829
```
```   830 lemma (in idom_char_0) order_mult:
```
```   831   assumes "poly (p *** q) \<noteq> poly []"
```
```   832   shows "order a (p *** q) = order a p + order a q"
```
```   833   using assms
```
```   834   apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order)
```
```   835   apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   836   apply (drule_tac a = a in order2)+
```
```   837   apply safe
```
```   838   apply (simp add: divides_def fun_eq_iff poly_exp_add poly_mult del: pmult_Cons)
```
```   839   apply safe
```
```   840   apply (rule_tac x = "pmult qa qaa" in exI)
```
```   841   apply (simp add: poly_mult ac_simps del: pmult_Cons)
```
```   842   apply (drule_tac a = a in order_decomp)+
```
```   843   apply safe
```
```   844   apply (subgoal_tac "[uminus a, one] divides pmult qa qaa")
```
```   845   apply (simp add: poly_primes del: pmult_Cons)
```
```   846   apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   847   apply (rule_tac x = qb in exI)
```
```   848   apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) =
```
```   849     poly (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb))")
```
```   850   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   851   apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q))
```
```   852       (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) =
```
```   853     poly (pmult (pexp [uminus a, one] (order a q))
```
```   854       (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))")
```
```   855   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   856   apply (simp add: fun_eq_iff poly_exp_add poly_mult ac_simps del: pmult_Cons)
```
```   857   done
```
```   858
```
```   859 lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
```
```   860   by (rule order_root [THEN ssubst]) auto
```
```   861
```
```   862 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p"
```
```   863   by auto
```
```   864
```
```   865 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
```
```   866   by (simp add: fun_eq_iff)
```
```   867
```
```   868 lemma (in idom_char_0) rsquarefree_decomp:
```
```   869   "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow> \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
```
```   870   apply (simp add: rsquarefree_def)
```
```   871   apply safe
```
```   872   apply (frule_tac a = a in order_decomp)
```
```   873   apply (drule_tac x = a in spec)
```
```   874   apply (drule_tac a = a in order_root2 [symmetric])
```
```   875   apply (auto simp del: pmult_Cons)
```
```   876   apply (rule_tac x = q in exI, safe)
```
```   877   apply (simp add: poly_mult fun_eq_iff)
```
```   878   apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
```
```   879   apply (simp add: divides_def del: pmult_Cons, safe)
```
```   880   apply (drule_tac x = "[]" in spec)
```
```   881   apply (auto simp add: fun_eq_iff)
```
```   882   done
```
```   883
```
```   884
```
```   885 text \<open>Normalization of a polynomial.\<close>
```
```   886
```
```   887 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
```
```   888   by (induct p) (auto simp add: fun_eq_iff)
```
```   889
```
```   890 text \<open>The degree of a polynomial.\<close>
```
```   891
```
```   892 lemma (in semiring_0) lemma_degree_zero: "(\<forall>c \<in> set p. c = 0) \<longleftrightarrow> pnormalize p = []"
```
```   893   by (induct p) auto
```
```   894
```
```   895 lemma (in idom_char_0) degree_zero:
```
```   896   assumes "poly p = poly []"
```
```   897   shows "degree p = 0"
```
```   898   using assms
```
```   899   by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
```
```   900
```
```   901 lemma (in semiring_0) pnormalize_sing: "pnormalize [x] = [x] \<longleftrightarrow> x \<noteq> 0"
```
```   902   by simp
```
```   903
```
```   904 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> pnormalize [x, y] = [x, y]"
```
```   905   by simp
```
```   906
```
```   907 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c # p)"
```
```   908   unfolding pnormal_def by simp
```
```   909
```
```   910 lemma (in semiring_0) pnormal_tail: "p \<noteq> [] \<Longrightarrow> pnormal (c # p) \<Longrightarrow> pnormal p"
```
```   911   unfolding pnormal_def by (auto split: if_split_asm)
```
```   912
```
```   913 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
```
```   914   by (induct p) (simp_all add: pnormal_def split: if_split_asm)
```
```   915
```
```   916 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
```
```   917   unfolding pnormal_def length_greater_0_conv by blast
```
```   918
```
```   919 lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
```
```   920   by (induct p) (auto simp: pnormal_def  split: if_split_asm)
```
```   921
```
```   922 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
```
```   923   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
```
```   924
```
```   925 lemma (in idom_char_0) poly_Cons_eq: "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
```
```   926   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   927 proof
```
```   928   show ?rhs if ?lhs
```
```   929   proof -
```
```   930     from that have "poly ((c # cs) +++ -- (d # ds)) x = 0" for x
```
```   931       by (simp only: poly_minus poly_add algebra_simps) (simp add: algebra_simps)
```
```   932     then have "poly ((c # cs) +++ -- (d # ds)) = poly []"
```
```   933       by (simp add: fun_eq_iff)
```
```   934     then have "c = d" and "\<forall>x \<in> set (cs +++ -- ds). x = 0"
```
```   935       unfolding poly_zero by (simp_all add: poly_minus_def algebra_simps)
```
```   936     from this(2) have "poly (cs +++ -- ds) x = 0" for x
```
```   937       unfolding poly_zero[symmetric] by simp
```
```   938     with \<open>c = d\<close> show ?thesis
```
```   939       by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
```
```   940   qed
```
```   941   show ?lhs if ?rhs
```
```   942     using that by (simp add:fun_eq_iff)
```
```   943 qed
```
```   944
```
```   945 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
```
```   946 proof (induct q arbitrary: p)
```
```   947   case Nil
```
```   948   then show ?case
```
```   949     by (simp only: poly_zero lemma_degree_zero) simp
```
```   950 next
```
```   951   case (Cons c cs p)
```
```   952   then show ?case
```
```   953   proof (induct p)
```
```   954     case Nil
```
```   955     then have "poly [] = poly (c # cs)"
```
```   956       by blast
```
```   957     then have "poly (c#cs) = poly []"
```
```   958       by simp
```
```   959     then show ?case
```
```   960       by (simp only: poly_zero lemma_degree_zero) simp
```
```   961   next
```
```   962     case (Cons d ds)
```
```   963     then have eq: "poly (d # ds) = poly (c # cs)"
```
```   964       by blast
```
```   965     then have eq': "\<And>x. poly (d # ds) x = poly (c # cs) x"
```
```   966       by simp
```
```   967     then have "poly (d # ds) 0 = poly (c # cs) 0"
```
```   968       by blast
```
```   969     then have dc: "d = c"
```
```   970       by auto
```
```   971     with eq have "poly ds = poly cs"
```
```   972       unfolding  poly_Cons_eq by simp
```
```   973     with Cons.prems have "pnormalize ds = pnormalize cs"
```
```   974       by blast
```
```   975     with dc show ?case
```
```   976       by simp
```
```   977   qed
```
```   978 qed
```
```   979
```
```   980 lemma (in idom_char_0) degree_unique:
```
```   981   assumes pq: "poly p = poly q"
```
```   982   shows "degree p = degree q"
```
```   983   using pnormalize_unique[OF pq] unfolding degree_def by simp
```
```   984
```
```   985 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p"
```
```   986   by (induct p) auto
```
```   987
```
```   988 lemma (in semiring_0) last_linear_mul_lemma:
```
```   989   "last ((a %* p) +++ (x # (b %* p))) = (if p = [] then x else b * last p)"
```
```   990   apply (induct p arbitrary: a x b)
```
```   991   apply auto
```
```   992   subgoal for a p c x b
```
```   993     apply (subgoal_tac "padd (cmult c p) (times b a # cmult b p) \<noteq> []")
```
```   994     apply simp
```
```   995     apply (induct p)
```
```   996     apply auto
```
```   997     done
```
```   998   done
```
```   999
```
```  1000 lemma (in semiring_1) last_linear_mul:
```
```  1001   assumes p: "p \<noteq> []"
```
```  1002   shows "last ([a, 1] *** p) = last p"
```
```  1003 proof -
```
```  1004   from p obtain c cs where cs: "p = c # cs"
```
```  1005     by (cases p) auto
```
```  1006   from cs have eq: "[a, 1] *** p = (a %* (c # cs)) +++ (0 # (1 %* (c # cs)))"
```
```  1007     by (simp add: poly_cmult_distr)
```
```  1008   show ?thesis
```
```  1009     using cs unfolding eq last_linear_mul_lemma by simp
```
```  1010 qed
```
```  1011
```
```  1012 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
```
```  1013   by (induct p) (auto split: if_split_asm)
```
```  1014
```
```  1015 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
```
```  1016   by (induct p) auto
```
```  1017
```
```  1018 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
```
```  1019   using pnormalize_eq[of p] unfolding degree_def by simp
```
```  1020
```
```  1021 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
```
```  1022   by auto
```
```  1023
```
```  1024 lemma (in idom_char_0) linear_mul_degree:
```
```  1025   assumes p: "poly p \<noteq> poly []"
```
```  1026   shows "degree ([a, 1] *** p) = degree p + 1"
```
```  1027 proof -
```
```  1028   from p have pnz: "pnormalize p \<noteq> []"
```
```  1029     unfolding poly_zero lemma_degree_zero .
```
```  1030
```
```  1031   from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
```
```  1032   have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
```
```  1033
```
```  1034   from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
```
```  1035     pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
```
```  1036   have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
```
```  1037     by simp
```
```  1038
```
```  1039   have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
```
```  1040     by (rule ext) (simp add: poly_mult poly_add poly_cmult)
```
```  1041   from degree_unique[OF eqs] th show ?thesis
```
```  1042     by (simp add: degree_unique[OF poly_normalize])
```
```  1043 qed
```
```  1044
```
```  1045 lemma (in idom_char_0) linear_pow_mul_degree:
```
```  1046   "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
```
```  1047 proof (induct n arbitrary: a p)
```
```  1048   case (0 a p)
```
```  1049   show ?case
```
```  1050   proof (cases "poly p = poly []")
```
```  1051     case True
```
```  1052     then show ?thesis
```
```  1053       using degree_unique[OF True] by (simp add: degree_def)
```
```  1054   next
```
```  1055     case False
```
```  1056     then show ?thesis
```
```  1057       by (auto simp add: poly_Nil_ext)
```
```  1058   qed
```
```  1059 next
```
```  1060   case (Suc n a p)
```
```  1061   have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1] %^ n *** ([a, 1] *** p))"
```
```  1062     apply (rule ext)
```
```  1063     apply (simp add: poly_mult poly_add poly_cmult)
```
```  1064     apply (simp add: ac_simps distrib_left)
```
```  1065     done
```
```  1066   note deq = degree_unique[OF eq]
```
```  1067   show ?case
```
```  1068   proof (cases "poly p = poly []")
```
```  1069     case True
```
```  1070     with eq have eq': "poly ([a, 1] %^(Suc n) *** p) = poly []"
```
```  1071       by (auto simp add: poly_mult poly_cmult poly_add)
```
```  1072     from degree_unique[OF eq'] True show ?thesis
```
```  1073       by (simp add: degree_def)
```
```  1074   next
```
```  1075     case False
```
```  1076     then have ap: "poly ([a,1] *** p) \<noteq> poly []"
```
```  1077       using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
```
```  1078     have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1]%^n *** ([a, 1] *** p))"
```
```  1079       by (auto simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
```
```  1080     from ap have ap': "poly ([a, 1] *** p) = poly [] \<longleftrightarrow> False"
```
```  1081       by blast
```
```  1082     have th0: "degree ([a, 1]%^n *** ([a, 1] *** p)) = degree ([a, 1] *** p) + n"
```
```  1083       apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
```
```  1084       apply simp
```
```  1085       done
```
```  1086     from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
```
```  1087     show ?thesis
```
```  1088       by (auto simp del: poly.simps)
```
```  1089   qed
```
```  1090 qed
```
```  1091
```
```  1092 lemma (in idom_char_0) order_degree:
```
```  1093   assumes p0: "poly p \<noteq> poly []"
```
```  1094   shows "order a p \<le> degree p"
```
```  1095 proof -
```
```  1096   from order2[OF p0, unfolded divides_def]
```
```  1097   obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)"
```
```  1098     by blast
```
```  1099   with q p0 have "poly q \<noteq> poly []"
```
```  1100     by (simp add: poly_mult poly_entire)
```
```  1101   with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
```
```  1102     by auto
```
```  1103 qed
```
```  1104
```
```  1105
```
```  1106 text \<open>Tidier versions of finiteness of roots.\<close>
```
```  1107 lemma (in idom_char_0) poly_roots_finite_set:
```
```  1108   "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
```
```  1109   unfolding poly_roots_finite .
```
```  1110
```
```  1111
```
```  1112 text \<open>Bound for polynomial.\<close>
```
```  1113 lemma poly_mono:
```
```  1114   fixes x :: "'a::linordered_idom"
```
```  1115   shows "\<bar>x\<bar> \<le> k \<Longrightarrow> \<bar>poly p x\<bar> \<le> poly (map abs p) k"
```
```  1116 proof (induct p)
```
```  1117   case Nil
```
```  1118   then show ?case by simp
```
```  1119 next
```
```  1120   case (Cons a p)
```
```  1121   then show ?case
```
```  1122     apply auto
```
```  1123     apply (rule_tac y = "\<bar>a\<bar> + \<bar>x * poly p x\<bar>" in order_trans)
```
```  1124     apply (rule abs_triangle_ineq)
```
```  1125     apply (auto intro!: mult_mono simp add: abs_mult)
```
```  1126     done
```
```  1127 qed
```
```  1128
```
```  1129 lemma (in semiring_0) poly_Sing: "poly [c] x = c"
```
```  1130   by simp
```
```  1131
```
```  1132 end
```