src/HOL/Library/Univ_Poly.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30738 0842e906300c child 31021 53642251a04f permissions -rw-r--r--
power operation defined generic
```     1 (*  Title:       Univ_Poly.thy
```
```     2     Author:      Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Univariate Polynomials *}
```
```     6
```
```     7 theory Univ_Poly
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text{* Application of polynomial as a function. *}
```
```    12
```
```    13 primrec (in semiring_0) poly :: "'a list => 'a  => 'a" where
```
```    14   poly_Nil:  "poly [] x = 0"
```
```    15 | poly_Cons: "poly (h#t) x = h + x * poly t x"
```
```    16
```
```    17
```
```    18 subsection{*Arithmetic Operations on Polynomials*}
```
```    19
```
```    20 text{*addition*}
```
```    21
```
```    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
```
```    26                             else (h + hd l2)#(t +++ tl l2))"
```
```    27
```
```    28 text{*Multiplication by a constant*}
```
```    29 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
```
```    30    cmult_Nil:  "c %* [] = []"
```
```    31 |  cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
```
```    32
```
```    33 text{*Multiplication by a polynomial*}
```
```    34 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
```
```    35 where
```
```    36    pmult_Nil:  "[] *** l2 = []"
```
```    37 |  pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
```
```    38                               else (h %* l2) +++ ((0) # (t *** l2)))"
```
```    39
```
```    40 text{*Repeated multiplication by a polynomial*}
```
```    41 primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list" where
```
```    42    mulexp_zero:  "mulexp 0 p q = q"
```
```    43 |  mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
```
```    44
```
```    45 text{*Exponential*}
```
```    46 primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80) where
```
```    47    pexp_0:   "p %^ 0 = [1]"
```
```    48 |  pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
```
```    49
```
```    50 text{*Quotient related value of dividing a polynomial by x + a*}
```
```    51 (* Useful for divisor properties in inductive proofs *)
```
```    52 primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
```
```    53    pquot_Nil:  "pquot [] a= []"
```
```    54 |  pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
```
```    55                    else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
```
```    56
```
```    57 text{*normalization of polynomials (remove extra 0 coeff)*}
```
```    58 primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
```
```    59   pnormalize_Nil:  "pnormalize [] = []"
```
```    60 | pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
```
```    61                                      then (if (h = 0) then [] else [h])
```
```    62                                      else (h#(pnormalize p)))"
```
```    63
```
```    64 definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
```
```    65 definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
```
```    66 text{*Other definitions*}
```
```    67
```
```    68 definition (in ring_1)
```
```    69   poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
```
```    70   "-- p = (- 1) %* p"
```
```    71
```
```    72 definition (in semiring_0)
```
```    73   divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70) where
```
```    74   [code del]: "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
```
```    75
```
```    76     --{*order of a polynomial*}
```
```    77 definition (in ring_1) order :: "'a => 'a list => nat" where
```
```    78   "order a p = (SOME n. ([-a, 1] %^ n) divides p &
```
```    79                       ~ (([-a, 1] %^ (Suc n)) divides p))"
```
```    80
```
```    81      --{*degree of a polynomial*}
```
```    82 definition (in semiring_0) degree :: "'a list => nat" where
```
```    83   "degree p = length (pnormalize p) - 1"
```
```    84
```
```    85      --{*squarefree polynomials --- NB with respect to real roots only.*}
```
```    86 definition (in ring_1)
```
```    87   rsquarefree :: "'a list => bool" where
```
```    88   "rsquarefree p = (poly p \<noteq> poly [] &
```
```    89                      (\<forall>a. (order a p = 0) | (order a p = 1)))"
```
```    90
```
```    91 context semiring_0
```
```    92 begin
```
```    93
```
```    94 lemma padd_Nil2[simp]: "p +++ [] = p"
```
```    95 by (induct p) auto
```
```    96
```
```    97 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
```
```    98 by auto
```
```    99
```
```   100 lemma pminus_Nil[simp]: "-- [] = []"
```
```   101 by (simp add: poly_minus_def)
```
```   102
```
```   103 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
```
```   104 end
```
```   105
```
```   106 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct "t", auto)
```
```   107
```
```   108 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
```
```   109 by simp
```
```   110
```
```   111 text{*Handy general properties*}
```
```   112
```
```   113 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
```
```   114 proof(induct b arbitrary: a)
```
```   115   case Nil thus ?case by auto
```
```   116 next
```
```   117   case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute)
```
```   118 qed
```
```   119
```
```   120 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
```
```   121 apply (induct a arbitrary: b c)
```
```   122 apply (simp, clarify)
```
```   123 apply (case_tac b, simp_all add: add_ac)
```
```   124 done
```
```   125
```
```   126 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
```
```   127 apply (induct p arbitrary: q,simp)
```
```   128 apply (case_tac q, simp_all add: right_distrib)
```
```   129 done
```
```   130
```
```   131 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
```
```   132 apply (induct "t", simp)
```
```   133 apply (auto simp add: mult_zero_left poly_ident_mult padd_commut)
```
```   134 apply (case_tac t, auto)
```
```   135 done
```
```   136
```
```   137 text{*properties of evaluation of polynomials.*}
```
```   138
```
```   139 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
```
```   140 proof(induct p1 arbitrary: p2)
```
```   141   case Nil thus ?case by simp
```
```   142 next
```
```   143   case (Cons a as p2) thus ?case
```
```   144     by (cases p2, simp_all  add: add_ac right_distrib)
```
```   145 qed
```
```   146
```
```   147 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
```
```   148 apply (induct "p")
```
```   149 apply (case_tac [2] "x=zero")
```
```   150 apply (auto simp add: right_distrib mult_ac)
```
```   151 done
```
```   152
```
```   153 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
```
```   154   by (induct p, auto simp add: right_distrib mult_ac)
```
```   155
```
```   156 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
```
```   157 apply (simp add: poly_minus_def)
```
```   158 apply (auto simp add: poly_cmult minus_mult_left[symmetric])
```
```   159 done
```
```   160
```
```   161 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
```
```   162 proof(induct p1 arbitrary: p2)
```
```   163   case Nil thus ?case by simp
```
```   164 next
```
```   165   case (Cons a as p2)
```
```   166   thus ?case by (cases as,
```
```   167     simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac)
```
```   168 qed
```
```   169
```
```   170 class recpower_semiring = semiring + recpower
```
```   171 class recpower_semiring_1 = semiring_1 + recpower
```
```   172 class recpower_semiring_0 = semiring_0 + recpower
```
```   173 class recpower_ring = ring + recpower
```
```   174 class recpower_ring_1 = ring_1 + recpower
```
```   175 subclass (in recpower_ring_1) recpower_ring ..
```
```   176 class recpower_comm_semiring_1 = recpower + comm_semiring_1
```
```   177 class recpower_comm_ring_1 = recpower + comm_ring_1
```
```   178 subclass (in recpower_comm_ring_1) recpower_comm_semiring_1 ..
```
```   179 class recpower_idom = recpower + idom
```
```   180 subclass (in recpower_idom) recpower_comm_ring_1 ..
```
```   181 class idom_char_0 = idom + ring_char_0
```
```   182 class recpower_idom_char_0 = recpower + idom_char_0
```
```   183 subclass (in recpower_idom_char_0) recpower_idom ..
```
```   184
```
```   185 lemma (in recpower_comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
```
```   186 apply (induct "n")
```
```   187 apply (auto simp add: poly_cmult poly_mult power_Suc)
```
```   188 done
```
```   189
```
```   190 text{*More Polynomial Evaluation Lemmas*}
```
```   191
```
```   192 lemma  (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
```
```   193 by simp
```
```   194
```
```   195 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
```
```   196   by (simp add: poly_mult mult_assoc)
```
```   197
```
```   198 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
```
```   199 by (induct "p", auto)
```
```   200
```
```   201 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
```
```   202 apply (induct "n")
```
```   203 apply (auto simp add: poly_mult mult_assoc)
```
```   204 done
```
```   205
```
```   206 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
```
```   207  @{term "p(x)"} *}
```
```   208
```
```   209 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   210 proof(induct t)
```
```   211   case Nil
```
```   212   {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
```
```   213   thus ?case by blast
```
```   214 next
```
```   215   case (Cons  x xs)
```
```   216   {fix h
```
```   217     from Cons.hyps[rule_format, of x]
```
```   218     obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
```
```   219     have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
```
```   220       using qr by(cases q, simp_all add: algebra_simps diff_def[symmetric]
```
```   221 	minus_mult_left[symmetric] right_minus)
```
```   222     hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
```
```   223   thus ?case by blast
```
```   224 qed
```
```   225
```
```   226 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   227 by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
```
```   228
```
```   229
```
```   230 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
```
```   231 proof-
```
```   232   {assume p: "p = []" hence ?thesis by simp}
```
```   233   moreover
```
```   234   {fix x xs assume p: "p = x#xs"
```
```   235     {fix q assume "p = [-a, 1] *** q" hence "poly p a = 0"
```
```   236 	by (simp add: poly_add poly_cmult minus_mult_left[symmetric])}
```
```   237     moreover
```
```   238     {assume p0: "poly p a = 0"
```
```   239       from poly_linear_rem[of x xs a] obtain q r
```
```   240       where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
```
```   241       have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
```
```   242       hence "\<exists>q. p = [- a, 1] *** q" using p qr  apply - apply (rule exI[where x=q])apply auto apply (cases q) apply auto done}
```
```   243     ultimately have ?thesis using p by blast}
```
```   244   ultimately show ?thesis by (cases p, auto)
```
```   245 qed
```
```   246
```
```   247 lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
```
```   248 by (induct "p", auto)
```
```   249
```
```   250 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
```
```   251 by (induct "p", auto)
```
```   252
```
```   253 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
```
```   254 by auto
```
```   255
```
```   256 subsection{*Polynomial length*}
```
```   257
```
```   258 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
```
```   259 by (induct "p", auto)
```
```   260
```
```   261 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
```
```   262 apply (induct p1 arbitrary: p2, simp_all)
```
```   263 apply arith
```
```   264 done
```
```   265
```
```   266 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
```
```   267 by (simp add: poly_add_length)
```
```   268
```
```   269 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
```
```   270  "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
```
```   271 by (auto simp add: poly_mult)
```
```   272
```
```   273 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
```
```   274 by (auto simp add: poly_mult)
```
```   275
```
```   276 text{*Normalisation Properties*}
```
```   277
```
```   278 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
```
```   279 by (induct "p", auto)
```
```   280
```
```   281 text{*A nontrivial polynomial of degree n has no more than n roots*}
```
```   282 lemma (in idom) poly_roots_index_lemma:
```
```   283    assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
```
```   284   shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
```
```   285   using p n
```
```   286 proof(induct n arbitrary: p x)
```
```   287   case 0 thus ?case by simp
```
```   288 next
```
```   289   case (Suc n p x)
```
```   290   {assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
```
```   291     from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
```
```   292     from p0(1)[unfolded poly_linear_divides[of p x]]
```
```   293     have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
```
```   294     from C obtain a where a: "poly p a = 0" by blast
```
```   295     from a[unfolded poly_linear_divides[of p a]] p0(2)
```
```   296     obtain q where q: "p = [-a, 1] *** q" by blast
```
```   297     have lg: "length q = n" using q Suc.prems(2) by simp
```
```   298     from q p0 have qx: "poly q x \<noteq> poly [] x"
```
```   299       by (auto simp add: poly_mult poly_add poly_cmult)
```
```   300     from Suc.hyps[OF qx lg] obtain i where
```
```   301       i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
```
```   302     let ?i = "\<lambda>m. if m = Suc n then a else i m"
```
```   303     from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
```
```   304       by blast
```
```   305     from y have "y = a \<or> poly q y = 0"
```
```   306       by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
```
```   307     with i[rule_format, of y] y(1) y(2) have False apply auto
```
```   308       apply (erule_tac x="m" in allE)
```
```   309       apply auto
```
```   310       done}
```
```   311   thus ?case by blast
```
```   312 qed
```
```   313
```
```   314
```
```   315 lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
```
```   316       \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
```
```   317 by (blast intro: poly_roots_index_lemma)
```
```   318
```
```   319 lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
```
```   320       \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
```
```   321 apply (drule poly_roots_index_length, safe)
```
```   322 apply (rule_tac x = "Suc (length p)" in exI)
```
```   323 apply (rule_tac x = i in exI)
```
```   324 apply (simp add: less_Suc_eq_le)
```
```   325 done
```
```   326
```
```   327
```
```   328 lemma (in idom) idom_finite_lemma:
```
```   329   assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
```
```   330   shows "finite {x. P x}"
```
```   331 proof-
```
```   332   let ?M = "{x. P x}"
```
```   333   let ?N = "set j"
```
```   334   have "?M \<subseteq> ?N" using P by auto
```
```   335   thus ?thesis using finite_subset by auto
```
```   336 qed
```
```   337
```
```   338
```
```   339 lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
```
```   340       \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
```
```   341 apply (drule poly_roots_index_length, safe)
```
```   342 apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
```
```   343 apply (auto simp add: image_iff)
```
```   344 apply (erule_tac x="x" in allE, clarsimp)
```
```   345 by (case_tac "n=length p", auto simp add: order_le_less)
```
```   346
```
```   347 lemma (in ring_char_0) UNIV_ring_char_0_infinte:
```
```   348   "\<not> (finite (UNIV:: 'a set))"
```
```   349 proof
```
```   350   assume F: "finite (UNIV :: 'a set)"
```
```   351   have "finite (UNIV :: nat set)"
```
```   352   proof (rule finite_imageD)
```
```   353     have "of_nat ` UNIV \<subseteq> UNIV" by simp
```
```   354     then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
```
```   355     show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
```
```   356   qed
```
```   357   with infinite_UNIV_nat show False ..
```
```   358 qed
```
```   359
```
```   360 lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
```
```   361   finite {x. poly p x = 0}"
```
```   362 proof
```
```   363   assume H: "poly p \<noteq> poly []"
```
```   364   show "finite {x. poly p x = (0::'a)}"
```
```   365     using H
```
```   366     apply -
```
```   367     apply (erule contrapos_np, rule ext)
```
```   368     apply (rule ccontr)
```
```   369     apply (clarify dest!: poly_roots_finite_lemma2)
```
```   370     using finite_subset
```
```   371   proof-
```
```   372     fix x i
```
```   373     assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
```
```   374       and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
```
```   375     let ?M= "{x. poly p x = (0\<Colon>'a)}"
```
```   376     from P have "?M \<subseteq> set i" by auto
```
```   377     with finite_subset F show False by auto
```
```   378   qed
```
```   379 next
```
```   380   assume F: "finite {x. poly p x = (0\<Colon>'a)}"
```
```   381   show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
```
```   382 qed
```
```   383
```
```   384 text{*Entirety and Cancellation for polynomials*}
```
```   385
```
```   386 lemma (in idom_char_0) poly_entire_lemma2:
```
```   387   assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
```
```   388   shows "poly (p***q) \<noteq> poly []"
```
```   389 proof-
```
```   390   let ?S = "\<lambda>p. {x. poly p x = 0}"
```
```   391   have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
```
```   392   with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
```
```   393 qed
```
```   394
```
```   395 lemma (in idom_char_0) poly_entire:
```
```   396   "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
```
```   397 using poly_entire_lemma2[of p q]
```
```   398 by (auto simp add: expand_fun_eq poly_mult)
```
```   399
```
```   400 lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
```
```   401 by (simp add: poly_entire)
```
```   402
```
```   403 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
```
```   404 by (auto intro!: ext)
```
```   405
```
```   406 lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
```
```   407 by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
```
```   408
```
```   409 lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
```
```   410 by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib minus_mult_left[symmetric] minus_mult_right[symmetric])
```
```   411
```
```   412 subclass (in idom_char_0) comm_ring_1 ..
```
```   413 lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
```
```   414 proof-
```
```   415   have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
```
```   416   also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
```
```   417     by (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
```
```   418   finally show ?thesis .
```
```   419 qed
```
```   420
```
```   421 lemma (in recpower_idom) poly_exp_eq_zero[simp]:
```
```   422      "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
```
```   423 apply (simp only: fun_eq add: all_simps [symmetric])
```
```   424 apply (rule arg_cong [where f = All])
```
```   425 apply (rule ext)
```
```   426 apply (induct n)
```
```   427 apply (auto simp add: poly_exp poly_mult)
```
```   428 done
```
```   429
```
```   430 lemma (in semiring_1) one_neq_zero[simp]: "1 \<noteq> 0" using zero_neq_one by blast
```
```   431 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
```
```   432 apply (simp add: fun_eq)
```
```   433 apply (rule_tac x = "minus one a" in exI)
```
```   434 apply (unfold diff_minus)
```
```   435 apply (subst add_commute)
```
```   436 apply (subst add_assoc)
```
```   437 apply simp
```
```   438 done
```
```   439
```
```   440 lemma (in recpower_idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
```
```   441 by auto
```
```   442
```
```   443 text{*A more constructive notion of polynomials being trivial*}
```
```   444
```
```   445 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
```
```   446 apply(simp add: fun_eq)
```
```   447 apply (case_tac "h = zero")
```
```   448 apply (drule_tac [2] x = zero in spec, auto)
```
```   449 apply (cases "poly t = poly []", simp)
```
```   450 proof-
```
```   451   fix x
```
```   452   assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"  and pnz: "poly t \<noteq> poly []"
```
```   453   let ?S = "{x. poly t x = 0}"
```
```   454   from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
```
```   455   hence th: "?S \<supseteq> UNIV - {0}" by auto
```
```   456   from poly_roots_finite pnz have th': "finite ?S" by blast
```
```   457   from finite_subset[OF th th'] UNIV_ring_char_0_infinte
```
```   458   show "poly t x = (0\<Colon>'a)" by simp
```
```   459   qed
```
```   460
```
```   461 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
```
```   462 apply (induct "p", simp)
```
```   463 apply (rule iffI)
```
```   464 apply (drule poly_zero_lemma', auto)
```
```   465 done
```
```   466
```
```   467 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
```
```   468   unfolding poly_zero[symmetric] by simp
```
```   469
```
```   470
```
```   471
```
```   472 text{*Basics of divisibility.*}
```
```   473
```
```   474 lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
```
```   475 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
```
```   476 apply (drule_tac x = "uminus a" in spec)
```
```   477 apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
```
```   478 apply (cases "p = []")
```
```   479 apply (rule exI[where x="[]"])
```
```   480 apply simp
```
```   481 apply (cases "q = []")
```
```   482 apply (erule allE[where x="[]"], simp)
```
```   483
```
```   484 apply clarsimp
```
```   485 apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
```
```   486 apply (clarsimp simp add: poly_add poly_cmult)
```
```   487 apply (rule_tac x="qa" in exI)
```
```   488 apply (simp add: left_distrib [symmetric])
```
```   489 apply clarsimp
```
```   490
```
```   491 apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
```
```   492 apply (rule_tac x = "pmult qa q" in exI)
```
```   493 apply (rule_tac [2] x = "pmult p qa" in exI)
```
```   494 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
```
```   495 done
```
```   496
```
```   497 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
```
```   498 apply (simp add: divides_def)
```
```   499 apply (rule_tac x = "[one]" in exI)
```
```   500 apply (auto simp add: poly_mult fun_eq)
```
```   501 done
```
```   502
```
```   503 lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
```
```   504 apply (simp add: divides_def, safe)
```
```   505 apply (rule_tac x = "pmult qa qaa" in exI)
```
```   506 apply (auto simp add: poly_mult fun_eq mult_assoc)
```
```   507 done
```
```   508
```
```   509
```
```   510 lemma (in recpower_comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
```
```   511 apply (auto simp add: le_iff_add)
```
```   512 apply (induct_tac k)
```
```   513 apply (rule_tac [2] poly_divides_trans)
```
```   514 apply (auto simp add: divides_def)
```
```   515 apply (rule_tac x = p in exI)
```
```   516 apply (auto simp add: poly_mult fun_eq mult_ac)
```
```   517 done
```
```   518
```
```   519 lemma (in recpower_comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q;  m\<le>n |] ==> (p %^ m) divides q"
```
```   520 by (blast intro: poly_divides_exp poly_divides_trans)
```
```   521
```
```   522 lemma (in comm_semiring_0) poly_divides_add:
```
```   523    "[| p divides q; p divides r |] ==> p divides (q +++ r)"
```
```   524 apply (simp add: divides_def, auto)
```
```   525 apply (rule_tac x = "padd qa qaa" in exI)
```
```   526 apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
```
```   527 done
```
```   528
```
```   529 lemma (in comm_ring_1) poly_divides_diff:
```
```   530    "[| p divides q; p divides (q +++ r) |] ==> p divides r"
```
```   531 apply (simp add: divides_def, auto)
```
```   532 apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
```
```   533 apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
```
```   534 done
```
```   535
```
```   536 lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
```
```   537 apply (erule poly_divides_diff)
```
```   538 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
```
```   539 done
```
```   540
```
```   541 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
```
```   542 apply (simp add: divides_def)
```
```   543 apply (rule exI[where x="[]"])
```
```   544 apply (auto simp add: fun_eq poly_mult)
```
```   545 done
```
```   546
```
```   547 lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []"
```
```   548 apply (simp add: divides_def)
```
```   549 apply (rule_tac x = "[]" in exI)
```
```   550 apply (auto simp add: fun_eq)
```
```   551 done
```
```   552
```
```   553 text{*At last, we can consider the order of a root.*}
```
```   554
```
```   555 lemma (in idom_char_0)  poly_order_exists_lemma:
```
```   556   assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
```
```   557   shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
```
```   558 using lp p
```
```   559 proof(induct d arbitrary: p)
```
```   560   case 0 thus ?case by simp
```
```   561 next
```
```   562   case (Suc n p)
```
```   563   {assume p0: "poly p a = 0"
```
```   564     from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
```
```   565     hence pN: "p \<noteq> []" by auto
```
```   566     from p0[unfolded poly_linear_divides] pN  obtain q where
```
```   567       q: "p = [-a, 1] *** q" by blast
```
```   568     from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
```
```   569       apply -
```
```   570       apply simp
```
```   571       apply (simp only: fun_eq)
```
```   572       apply (rule ccontr)
```
```   573       apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric])
```
```   574       done
```
```   575     from Suc.hyps[OF qh] obtain m r where
```
```   576       mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast
```
```   577     from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
```
```   578     hence ?case by blast}
```
```   579   moreover
```
```   580   {assume p0: "poly p a \<noteq> 0"
```
```   581     hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
```
```   582   ultimately show ?case by blast
```
```   583 qed
```
```   584
```
```   585
```
```   586 lemma (in recpower_comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
```
```   587 by(induct n, auto simp add: poly_mult power_Suc mult_ac)
```
```   588
```
```   589 lemma (in comm_semiring_1) divides_left_mult:
```
```   590   assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
```
```   591 proof-
```
```   592   from d obtain t where r:"poly r = poly (p***q *** t)"
```
```   593     unfolding divides_def by blast
```
```   594   hence "poly r = poly (p *** (q *** t))"
```
```   595     "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
```
```   596   thus ?thesis unfolding divides_def by blast
```
```   597 qed
```
```   598
```
```   599
```
```   600
```
```   601 (* FIXME: Tidy up *)
```
```   602
```
```   603 lemma (in recpower_semiring_1)
```
```   604   zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
```
```   605   by (induct n, simp_all add: power_Suc)
```
```   606
```
```   607 lemma (in recpower_idom_char_0) poly_order_exists:
```
```   608   assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
```
```   609   shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
```
```   610 proof-
```
```   611 let ?poly = poly
```
```   612 let ?mulexp = mulexp
```
```   613 let ?pexp = pexp
```
```   614 from lp p0
```
```   615 show ?thesis
```
```   616 apply -
```
```   617 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
```
```   618 apply (rule_tac x = n in exI, safe)
```
```   619 apply (unfold divides_def)
```
```   620 apply (rule_tac x = q in exI)
```
```   621 apply (induct_tac "n", simp)
```
```   622 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
```
```   623 apply safe
```
```   624 apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
```
```   625 apply simp
```
```   626 apply (induct_tac "n")
```
```   627 apply (simp del: pmult_Cons pexp_Suc)
```
```   628 apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
```
```   629 apply (simp add: poly_add poly_cmult minus_mult_left[symmetric])
```
```   630 apply (rule pexp_Suc [THEN ssubst])
```
```   631 apply (rule ccontr)
```
```   632 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
```
```   633 done
```
```   634 qed
```
```   635
```
```   636
```
```   637 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
```
```   638 by (simp add: divides_def, auto)
```
```   639
```
```   640 lemma (in recpower_idom_char_0) poly_order: "poly p \<noteq> poly []
```
```   641       ==> EX! n. ([-a, 1] %^ n) divides p &
```
```   642                  ~(([-a, 1] %^ (Suc n)) divides p)"
```
```   643 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
```
```   644 apply (cut_tac x = y and y = n in less_linear)
```
```   645 apply (drule_tac m = n in poly_exp_divides)
```
```   646 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
```
```   647             simp del: pmult_Cons pexp_Suc)
```
```   648 done
```
```   649
```
```   650 text{*Order*}
```
```   651
```
```   652 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
```
```   653 by (blast intro: someI2)
```
```   654
```
```   655 lemma (in recpower_idom_char_0) order:
```
```   656       "(([-a, 1] %^ n) divides p &
```
```   657         ~(([-a, 1] %^ (Suc n)) divides p)) =
```
```   658         ((n = order a p) & ~(poly p = poly []))"
```
```   659 apply (unfold order_def)
```
```   660 apply (rule iffI)
```
```   661 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
```
```   662 apply (blast intro!: poly_order [THEN [2] some1_equalityD])
```
```   663 done
```
```   664
```
```   665 lemma (in recpower_idom_char_0) order2: "[| poly p \<noteq> poly [] |]
```
```   666       ==> ([-a, 1] %^ (order a p)) divides p &
```
```   667               ~(([-a, 1] %^ (Suc(order a p))) divides p)"
```
```   668 by (simp add: order del: pexp_Suc)
```
```   669
```
```   670 lemma (in recpower_idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
```
```   671          ~(([-a, 1] %^ (Suc n)) divides p)
```
```   672       |] ==> (n = order a p)"
```
```   673 by (insert order [of a n p], auto)
```
```   674
```
```   675 lemma (in recpower_idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
```
```   676          ~(([-a, 1] %^ (Suc n)) divides p))
```
```   677       ==> (n = order a p)"
```
```   678 by (blast intro: order_unique)
```
```   679
```
```   680 lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
```
```   681 by (auto simp add: fun_eq divides_def poly_mult order_def)
```
```   682
```
```   683 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
```
```   684 apply (induct "p")
```
```   685 apply (auto simp add: numeral_1_eq_1)
```
```   686 done
```
```   687
```
```   688 lemma (in comm_ring_1) lemma_order_root:
```
```   689      " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
```
```   690              \<Longrightarrow> poly p a = 0"
```
```   691 apply (induct n arbitrary: a p, blast)
```
```   692 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
```
```   693 done
```
```   694
```
```   695 lemma (in recpower_idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"
```
```   696 proof-
```
```   697   let ?poly = poly
```
```   698   show ?thesis
```
```   699 apply (case_tac "?poly p = ?poly []", auto)
```
```   700 apply (simp add: poly_linear_divides del: pmult_Cons, safe)
```
```   701 apply (drule_tac [!] a = a in order2)
```
```   702 apply (rule ccontr)
```
```   703 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
```
```   704 using neq0_conv
```
```   705 apply (blast intro: lemma_order_root)
```
```   706 done
```
```   707 qed
```
```   708
```
```   709 lemma (in recpower_idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
```
```   710 proof-
```
```   711   let ?poly = poly
```
```   712   show ?thesis
```
```   713 apply (case_tac "?poly p = ?poly []", auto)
```
```   714 apply (simp add: divides_def fun_eq poly_mult)
```
```   715 apply (rule_tac x = "[]" in exI)
```
```   716 apply (auto dest!: order2 [where a=a]
```
```   717 	    intro: poly_exp_divides simp del: pexp_Suc)
```
```   718 done
```
```   719 qed
```
```   720
```
```   721 lemma (in recpower_idom_char_0) order_decomp:
```
```   722      "poly p \<noteq> poly []
```
```   723       ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
```
```   724                 ~([-a, 1] divides q)"
```
```   725 apply (unfold divides_def)
```
```   726 apply (drule order2 [where a = a])
```
```   727 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
```
```   728 apply (rule_tac x = q in exI, safe)
```
```   729 apply (drule_tac x = qa in spec)
```
```   730 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
```
```   731 done
```
```   732
```
```   733 text{*Important composition properties of orders.*}
```
```   734 lemma order_mult: "poly (p *** q) \<noteq> poly []
```
```   735       ==> order a (p *** q) = order a p + order (a::'a::{recpower_idom_char_0}) q"
```
```   736 apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
```
```   737 apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   738 apply (drule_tac a = a in order2)+
```
```   739 apply safe
```
```   740 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   741 apply (rule_tac x = "qa *** qaa" in exI)
```
```   742 apply (simp add: poly_mult mult_ac del: pmult_Cons)
```
```   743 apply (drule_tac a = a in order_decomp)+
```
```   744 apply safe
```
```   745 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
```
```   746 apply (simp add: poly_primes del: pmult_Cons)
```
```   747 apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   748 apply (rule_tac x = qb in exI)
```
```   749 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
```
```   750 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   751 apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
```
```   752 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   753 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
```
```   754 done
```
```   755
```
```   756 lemma (in recpower_idom_char_0) order_mult:
```
```   757   assumes pq0: "poly (p *** q) \<noteq> poly []"
```
```   758   shows "order a (p *** q) = order a p + order a q"
```
```   759 proof-
```
```   760   let ?order = order
```
```   761   let ?divides = "op divides"
```
```   762   let ?poly = poly
```
```   763 from pq0
```
```   764 show ?thesis
```
```   765 apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order)
```
```   766 apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   767 apply (drule_tac a = a in order2)+
```
```   768 apply safe
```
```   769 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   770 apply (rule_tac x = "pmult qa qaa" in exI)
```
```   771 apply (simp add: poly_mult mult_ac del: pmult_Cons)
```
```   772 apply (drule_tac a = a in order_decomp)+
```
```   773 apply safe
```
```   774 apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ")
```
```   775 apply (simp add: poly_primes del: pmult_Cons)
```
```   776 apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   777 apply (rule_tac x = qb in exI)
```
```   778 apply (subgoal_tac "?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult qa qaa)) = ?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
```
```   779 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   780 apply (subgoal_tac "?poly (pmult (pexp [uminus a, one ] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) = ?poly (pmult (pexp [uminus a, one] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb))) ")
```
```   781 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   782 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
```
```   783 done
```
```   784 qed
```
```   785
```
```   786 lemma (in recpower_idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
```
```   787 by (rule order_root [THEN ssubst], auto)
```
```   788
```
```   789 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
```
```   790
```
```   791 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
```
```   792 by (simp add: fun_eq)
```
```   793
```
```   794 lemma (in recpower_idom_char_0) rsquarefree_decomp:
```
```   795      "[| rsquarefree p; poly p a = 0 |]
```
```   796       ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
```
```   797 apply (simp add: rsquarefree_def, safe)
```
```   798 apply (frule_tac a = a in order_decomp)
```
```   799 apply (drule_tac x = a in spec)
```
```   800 apply (drule_tac a = a in order_root2 [symmetric])
```
```   801 apply (auto simp del: pmult_Cons)
```
```   802 apply (rule_tac x = q in exI, safe)
```
```   803 apply (simp add: poly_mult fun_eq)
```
```   804 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
```
```   805 apply (simp add: divides_def del: pmult_Cons, safe)
```
```   806 apply (drule_tac x = "[]" in spec)
```
```   807 apply (auto simp add: fun_eq)
```
```   808 done
```
```   809
```
```   810
```
```   811 text{*Normalization of a polynomial.*}
```
```   812
```
```   813 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
```
```   814 apply (induct "p")
```
```   815 apply (auto simp add: fun_eq)
```
```   816 done
```
```   817
```
```   818 text{*The degree of a polynomial.*}
```
```   819
```
```   820 lemma (in semiring_0) lemma_degree_zero:
```
```   821      "list_all (%c. c = 0) p \<longleftrightarrow>  pnormalize p = []"
```
```   822 by (induct "p", auto)
```
```   823
```
```   824 lemma (in idom_char_0) degree_zero:
```
```   825   assumes pN: "poly p = poly []" shows"degree p = 0"
```
```   826 proof-
```
```   827   let ?pn = pnormalize
```
```   828   from pN
```
```   829   show ?thesis
```
```   830     apply (simp add: degree_def)
```
```   831     apply (case_tac "?pn p = []")
```
```   832     apply (auto simp add: poly_zero lemma_degree_zero )
```
```   833     done
```
```   834 qed
```
```   835
```
```   836 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
```
```   837 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
```
```   838 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
```
```   839   unfolding pnormal_def by simp
```
```   840 lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
```
```   841   unfolding pnormal_def
```
```   842   apply (cases "pnormalize p = []", auto)
```
```   843   by (cases "c = 0", auto)
```
```   844
```
```   845
```
```   846 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
```
```   847 proof(induct p)
```
```   848   case Nil thus ?case by (simp add: pnormal_def)
```
```   849 next
```
```   850   case (Cons a as) thus ?case
```
```   851     apply (simp add: pnormal_def)
```
```   852     apply (cases "pnormalize as = []", simp_all)
```
```   853     apply (cases "as = []", simp_all)
```
```   854     apply (cases "a=0", simp_all)
```
```   855     apply (cases "a=0", simp_all)
```
```   856     done
```
```   857 qed
```
```   858
```
```   859 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
```
```   860   unfolding pnormal_def length_greater_0_conv by blast
```
```   861
```
```   862 lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
```
```   863   apply (induct p, auto)
```
```   864   apply (case_tac "p = []", auto)
```
```   865   apply (simp add: pnormal_def)
```
```   866   by (rule pnormal_cons, auto)
```
```   867
```
```   868 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
```
```   869   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
```
```   870
```
```   871 lemma (in idom_char_0) poly_Cons_eq: "poly (c#cs) = poly (d#ds) \<longleftrightarrow> c=d \<and> poly cs = poly ds" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   872 proof
```
```   873   assume eq: ?lhs
```
```   874   hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
```
```   875     by (simp only: poly_minus poly_add algebra_simps) simp
```
```   876   hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add:expand_fun_eq)
```
```   877   hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
```
```   878     unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
```
```   879   hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
```
```   880     unfolding poly_zero[symmetric] by simp
```
```   881   thus ?rhs  by (simp add: poly_minus poly_add algebra_simps expand_fun_eq)
```
```   882 next
```
```   883   assume ?rhs then show ?lhs by(simp add:expand_fun_eq)
```
```   884 qed
```
```   885
```
```   886 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
```
```   887 proof(induct q arbitrary: p)
```
```   888   case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp
```
```   889 next
```
```   890   case (Cons c cs p)
```
```   891   thus ?case
```
```   892   proof(induct p)
```
```   893     case Nil
```
```   894     hence "poly [] = poly (c#cs)" by blast
```
```   895     then have "poly (c#cs) = poly [] " by simp
```
```   896     thus ?case by (simp only: poly_zero lemma_degree_zero) simp
```
```   897   next
```
```   898     case (Cons d ds)
```
```   899     hence eq: "poly (d # ds) = poly (c # cs)" by blast
```
```   900     hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
```
```   901     hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
```
```   902     hence dc: "d = c" by auto
```
```   903     with eq have "poly ds = poly cs"
```
```   904       unfolding  poly_Cons_eq by simp
```
```   905     with Cons.prems have "pnormalize ds = pnormalize cs" by blast
```
```   906     with dc show ?case by simp
```
```   907   qed
```
```   908 qed
```
```   909
```
```   910 lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
```
```   911   shows "degree p = degree q"
```
```   912 using pnormalize_unique[OF pq] unfolding degree_def by simp
```
```   913
```
```   914 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto)
```
```   915
```
```   916 lemma (in semiring_0) last_linear_mul_lemma:
```
```   917   "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)"
```
```   918
```
```   919 apply (induct p arbitrary: a x b, auto)
```
```   920 apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
```
```   921 apply (induct_tac p, auto)
```
```   922 done
```
```   923
```
```   924 lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p"
```
```   925 proof-
```
```   926   from p obtain c cs where cs: "p = c#cs" by (cases p, auto)
```
```   927   from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
```
```   928     by (simp add: poly_cmult_distr)
```
```   929   show ?thesis using cs
```
```   930     unfolding eq last_linear_mul_lemma by simp
```
```   931 qed
```
```   932
```
```   933 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
```
```   934   apply (induct p, auto)
```
```   935   apply (case_tac p, auto)+
```
```   936   done
```
```   937
```
```   938 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
```
```   939   by (induct p, auto)
```
```   940
```
```   941 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
```
```   942   using pnormalize_eq[of p] unfolding degree_def by simp
```
```   943
```
```   944 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp
```
```   945
```
```   946 lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []"
```
```   947   shows "degree ([a,1] *** p) = degree p + 1"
```
```   948 proof-
```
```   949   from p have pnz: "pnormalize p \<noteq> []"
```
```   950     unfolding poly_zero lemma_degree_zero .
```
```   951
```
```   952   from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
```
```   953   have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
```
```   954   from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
```
```   955     pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
```
```   956
```
```   957
```
```   958   have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
```
```   959     by (auto simp add: poly_length_mult)
```
```   960
```
```   961   have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
```
```   962     by (rule ext) (simp add: poly_mult poly_add poly_cmult)
```
```   963   from degree_unique[OF eqs] th
```
```   964   show ?thesis by (simp add: degree_unique[OF poly_normalize])
```
```   965 qed
```
```   966
```
```   967 lemma (in idom_char_0) linear_pow_mul_degree:
```
```   968   "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
```
```   969 proof(induct n arbitrary: a p)
```
```   970   case (0 a p)
```
```   971   {assume p: "poly p = poly []"
```
```   972     hence ?case using degree_unique[OF p] by (simp add: degree_def)}
```
```   973   moreover
```
```   974   {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
```
```   975   ultimately show ?case by blast
```
```   976 next
```
```   977   case (Suc n a p)
```
```   978   have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
```
```   979     apply (rule ext, simp add: poly_mult poly_add poly_cmult)
```
```   980     by (simp add: mult_ac add_ac right_distrib)
```
```   981   note deq = degree_unique[OF eq]
```
```   982   {assume p: "poly p = poly []"
```
```   983     with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
```
```   984       by - (rule ext,simp add: poly_mult poly_cmult poly_add)
```
```   985     from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
```
```   986   moreover
```
```   987   {assume p: "poly p \<noteq> poly []"
```
```   988     from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
```
```   989       using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
```
```   990     have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
```
```   991      by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
```
```   992    from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
```
```   993    have  th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
```
```   994      apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
```
```   995      by simp
```
```   996
```
```   997    from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
```
```   998    have ?case by (auto simp del: poly.simps)}
```
```   999   ultimately show ?case by blast
```
```  1000 qed
```
```  1001
```
```  1002 lemma (in recpower_idom_char_0) order_degree:
```
```  1003   assumes p0: "poly p \<noteq> poly []"
```
```  1004   shows "order a p \<le> degree p"
```
```  1005 proof-
```
```  1006   from order2[OF p0, unfolded divides_def]
```
```  1007   obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
```
```  1008   {assume "poly q = poly []"
```
```  1009     with q p0 have False by (simp add: poly_mult poly_entire)}
```
```  1010   with degree_unique[OF q, unfolded linear_pow_mul_degree]
```
```  1011   show ?thesis by auto
```
```  1012 qed
```
```  1013
```
```  1014 text{*Tidier versions of finiteness of roots.*}
```
```  1015
```
```  1016 lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
```
```  1017 unfolding poly_roots_finite .
```
```  1018
```
```  1019 text{*bound for polynomial.*}
```
```  1020
```
```  1021 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
```
```  1022 apply (induct "p", auto)
```
```  1023 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
```
```  1024 apply (rule abs_triangle_ineq)
```
```  1025 apply (auto intro!: mult_mono simp add: abs_mult)
```
```  1026 done
```
```  1027
```
```  1028 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
```
```  1029
```
```  1030 end
```