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