src/HOL/Decision_Procs/Polynomial_List.thy
 author haftmann Sat Jul 05 11:01:53 2014 +0200 (2014-07-05) changeset 57514 bdc2c6b40bf2 parent 57512 cc97b347b301 child 58889 5b7a9633cfa8 permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
```     1 (*  Title:      HOL/Decision_Procs/Polynomial_List.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Univariate Polynomials as lists *}
```
```     6
```
```     7 theory Polynomial_List
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
```
```    11 text{* Application of polynomial as a function. *}
```
```    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{*Arithmetic Operations on Polynomials*}
```
```    20
```
```    21 text{*addition*}
```
```    22
```
```    23 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
```
```    24 where
```
```    25   padd_Nil:  "[] +++ l2 = l2"
```
```    26 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t 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"
```
```    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{*normalization of polynomials (remove extra 0 coeff)*}
```
```    59 primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
```
```    60   pnormalize_Nil:  "pnormalize [] = []"
```
```    61 | pnormalize_Cons: "pnormalize (h#p) =
```
```    62     (if pnormalize p = [] then (if h = 0 then [] else [h]) 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) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
```
```    69   where "-- p = (- 1) %* p"
```
```    70
```
```    71 definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
```
```    72   where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
```
```    73
```
```    74 lemma (in semiring_0) dividesI:
```
```    75   "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
```
```    76   by (auto simp add: divides_def)
```
```    77
```
```    78 lemma (in semiring_0) dividesE:
```
```    79   assumes "p1 divides p2"
```
```    80   obtains q where "poly p2 = poly (p1 *** q)"
```
```    81   using assms by (auto simp add: divides_def)
```
```    82
```
```    83     --{*order of a polynomial*}
```
```    84 definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
```
```    85   "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))"
```
```    86
```
```    87      --{*degree of a polynomial*}
```
```    88 definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
```
```    89   where "degree p = length (pnormalize p) - 1"
```
```    90
```
```    91      --{*squarefree polynomials --- NB with respect to real roots only.*}
```
```    92 definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
```
```    93   where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
```
```    94
```
```    95 context semiring_0
```
```    96 begin
```
```    97
```
```    98 lemma padd_Nil2[simp]: "p +++ [] = p"
```
```    99   by (induct p) auto
```
```   100
```
```   101 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
```
```   102   by auto
```
```   103
```
```   104 lemma pminus_Nil: "-- [] = []"
```
```   105   by (simp add: poly_minus_def)
```
```   106
```
```   107 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
```
```   108
```
```   109 end
```
```   110
```
```   111 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto
```
```   112
```
```   113 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
```
```   114   by simp
```
```   115
```
```   116 text{*Handy general properties*}
```
```   117
```
```   118 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
```
```   119 proof (induct b arbitrary: a)
```
```   120   case Nil
```
```   121   thus ?case by auto
```
```   122 next
```
```   123   case (Cons b bs a)
```
```   124   thus ?case by (cases a) (simp_all add: add.commute)
```
```   125 qed
```
```   126
```
```   127 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
```
```   128   apply (induct a)
```
```   129   apply (simp, clarify)
```
```   130   apply (case_tac b, simp_all add: ac_simps)
```
```   131   done
```
```   132
```
```   133 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
```
```   134   apply (induct p arbitrary: q)
```
```   135   apply simp
```
```   136   apply (case_tac q, simp_all add: distrib_left)
```
```   137   done
```
```   138
```
```   139 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
```
```   140   apply (induct t)
```
```   141   apply simp
```
```   142   apply (auto simp add: padd_commut)
```
```   143   apply (case_tac t, auto)
```
```   144   done
```
```   145
```
```   146 text{*properties of evaluation of polynomials.*}
```
```   147
```
```   148 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
```
```   149 proof(induct p1 arbitrary: p2)
```
```   150   case Nil
```
```   151   thus ?case by simp
```
```   152 next
```
```   153   case (Cons a as p2)
```
```   154   thus ?case
```
```   155     by (cases p2) (simp_all  add: ac_simps distrib_left)
```
```   156 qed
```
```   157
```
```   158 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
```
```   159   apply (induct p)
```
```   160   apply (case_tac [2] "x = zero")
```
```   161   apply (auto simp add: distrib_left ac_simps)
```
```   162   done
```
```   163
```
```   164 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
```
```   165   by (induct p) (auto simp add: distrib_left ac_simps)
```
```   166
```
```   167 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
```
```   168   apply (simp add: poly_minus_def)
```
```   169   apply (auto simp add: poly_cmult)
```
```   170   done
```
```   171
```
```   172 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
```
```   173 proof (induct p1 arbitrary: p2)
```
```   174   case Nil
```
```   175   thus ?case by simp
```
```   176 next
```
```   177   case (Cons a as p2)
```
```   178   thus ?case by (cases as)
```
```   179     (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)
```
```   180 qed
```
```   181
```
```   182 class idom_char_0 = idom + ring_char_0
```
```   183
```
```   184 subclass (in field_char_0) idom_char_0 ..
```
```   185
```
```   186 lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
```
```   187   by (induct n) (auto simp add: poly_cmult poly_mult)
```
```   188
```
```   189 text{*More Polynomial Evaluation Lemmas*}
```
```   190
```
```   191 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
```
```   192   by simp
```
```   193
```
```   194 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
```
```   195   by (simp add: poly_mult mult.assoc)
```
```   196
```
```   197 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
```
```   198   by (induct p) auto
```
```   199
```
```   200 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
```
```   201   by (induct n) (auto simp add: poly_mult mult.assoc)
```
```   202
```
```   203 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
```
```   204  @{term "p(x)"} *}
```
```   205
```
```   206 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   207 proof(induct t)
```
```   208   case Nil
```
```   209   { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
```
```   210   thus ?case by blast
```
```   211 next
```
```   212   case (Cons  x xs)
```
```   213   { fix h
```
```   214     from Cons.hyps[rule_format, of x]
```
```   215     obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
```
```   216     have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
```
```   217       using qr by (cases q) (simp_all add: algebra_simps)
```
```   218     hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
```
```   219   thus ?case by blast
```
```   220 qed
```
```   221
```
```   222 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   223   using lemma_poly_linear_rem [where t = t and a = a] by auto
```
```   224
```
```   225
```
```   226 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
```
```   227 proof -
```
```   228   { assume p: "p = []" hence ?thesis by simp }
```
```   229   moreover
```
```   230   {
```
```   231     fix x xs assume p: "p = x#xs"
```
```   232     {
```
```   233       fix q assume "p = [-a, 1] *** q"
```
```   234       hence "poly p a = 0" by (simp add: poly_add poly_cmult)
```
```   235     }
```
```   236     moreover
```
```   237     { assume p0: "poly p a = 0"
```
```   238       from poly_linear_rem[of x xs a] obtain q r
```
```   239       where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
```
```   240       have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
```
```   241       hence "\<exists>q. p = [- a, 1] *** q"
```
```   242         using p qr
```
```   243         apply -
```
```   244         apply (rule exI[where x=q])
```
```   245         apply auto
```
```   246         apply (cases q)
```
```   247         apply auto
```
```   248         done
```
```   249     }
```
```   250     ultimately have ?thesis using p by blast
```
```   251   }
```
```   252   ultimately show ?thesis by (cases p) auto
```
```   253 qed
```
```   254
```
```   255 lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
```
```   256   by (induct p) auto
```
```   257
```
```   258 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
```
```   259   by (induct p) auto
```
```   260
```
```   261 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
```
```   262   by auto
```
```   263
```
```   264 subsection{*Polynomial length*}
```
```   265
```
```   266 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
```
```   267   by (induct p) auto
```
```   268
```
```   269 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
```
```   270   by (induct p1 arbitrary: p2) (simp_all, arith)
```
```   271
```
```   272 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
```
```   273   by (simp add: poly_add_length)
```
```   274
```
```   275 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
```
```   276   "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
```
```   277   by (auto simp add: poly_mult)
```
```   278
```
```   279 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
```
```   280   by (auto simp add: poly_mult)
```
```   281
```
```   282 text{*Normalisation Properties*}
```
```   283
```
```   284 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
```
```   285   by (induct p) auto
```
```   286
```
```   287 text{*A nontrivial polynomial of degree n has no more than n roots*}
```
```   288 lemma (in idom) poly_roots_index_lemma:
```
```   289    assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
```
```   290   shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
```
```   291   using p n
```
```   292 proof (induct n arbitrary: p x)
```
```   293   case 0
```
```   294   thus ?case by simp
```
```   295 next
```
```   296   case (Suc n p x)
```
```   297   {
```
```   298     assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
```
```   299     from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
```
```   300     from p0(1)[unfolded poly_linear_divides[of p x]]
```
```   301     have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
```
```   302     from C obtain a where a: "poly p a = 0" by blast
```
```   303     from a[unfolded poly_linear_divides[of p a]] p0(2)
```
```   304     obtain q where q: "p = [-a, 1] *** q" by blast
```
```   305     have lg: "length q = n" using q Suc.prems(2) by simp
```
```   306     from q p0 have qx: "poly q x \<noteq> poly [] x"
```
```   307       by (auto simp add: poly_mult poly_add poly_cmult)
```
```   308     from Suc.hyps[OF qx lg] obtain i where
```
```   309       i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
```
```   310     let ?i = "\<lambda>m. if m = Suc n then a else i m"
```
```   311     from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
```
```   312       by blast
```
```   313     from y have "y = a \<or> poly q y = 0"
```
```   314       by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
```
```   315     with i[rule_format, of y] y(1) y(2) have False
```
```   316       apply auto
```
```   317       apply (erule_tac x = "m" in allE)
```
```   318       apply auto
```
```   319       done
```
```   320   }
```
```   321   thus ?case by blast
```
```   322 qed
```
```   323
```
```   324
```
```   325 lemma (in idom) poly_roots_index_length:
```
```   326   "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)"
```
```   327   by (blast intro: poly_roots_index_lemma)
```
```   328
```
```   329 lemma (in idom) poly_roots_finite_lemma1:
```
```   330   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)"
```
```   331   apply (drule poly_roots_index_length, safe)
```
```   332   apply (rule_tac x = "Suc (length p)" in exI)
```
```   333   apply (rule_tac x = i in exI)
```
```   334   apply (simp add: less_Suc_eq_le)
```
```   335   done
```
```   336
```
```   337 lemma (in idom) idom_finite_lemma:
```
```   338   assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
```
```   339   shows "finite {x. P x}"
```
```   340 proof -
```
```   341   let ?M = "{x. P x}"
```
```   342   let ?N = "set j"
```
```   343   have "?M \<subseteq> ?N" using P by auto
```
```   344   thus ?thesis using finite_subset by auto
```
```   345 qed
```
```   346
```
```   347 lemma (in idom) poly_roots_finite_lemma2:
```
```   348   "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
```
```   349   apply (drule poly_roots_index_length, safe)
```
```   350   apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
```
```   351   apply (auto simp add: image_iff)
```
```   352   apply (erule_tac x="x" in allE, clarsimp)
```
```   353   apply (case_tac "n = length p")
```
```   354   apply (auto simp add: order_le_less)
```
```   355   done
```
```   356
```
```   357 lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))"
```
```   358 proof
```
```   359   assume F: "finite (UNIV :: 'a set)"
```
```   360   have "finite (UNIV :: nat set)"
```
```   361   proof (rule finite_imageD)
```
```   362     have "of_nat ` UNIV \<subseteq> UNIV" by simp
```
```   363     then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
```
```   364     show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
```
```   365   qed
```
```   366   with infinite_UNIV_nat show False ..
```
```   367 qed
```
```   368
```
```   369 lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
```
```   370 proof
```
```   371   assume H: "poly p \<noteq> poly []"
```
```   372   show "finite {x. poly p x = (0::'a)}"
```
```   373     using H
```
```   374     apply -
```
```   375     apply (erule contrapos_np, rule ext)
```
```   376     apply (rule ccontr)
```
```   377     apply (clarify dest!: poly_roots_finite_lemma2)
```
```   378     using finite_subset
```
```   379   proof -
```
```   380     fix x i
```
```   381     assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
```
```   382       and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
```
```   383     let ?M= "{x. poly p x = (0\<Colon>'a)}"
```
```   384     from P have "?M \<subseteq> set i" by auto
```
```   385     with finite_subset F show False by auto
```
```   386   qed
```
```   387 next
```
```   388   assume F: "finite {x. poly p x = (0\<Colon>'a)}"
```
```   389   show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
```
```   390 qed
```
```   391
```
```   392 text{*Entirety and Cancellation for polynomials*}
```
```   393
```
```   394 lemma (in idom_char_0) poly_entire_lemma2:
```
```   395   assumes p0: "poly p \<noteq> poly []"
```
```   396     and q0: "poly q \<noteq> poly []"
```
```   397   shows "poly (p***q) \<noteq> poly []"
```
```   398 proof -
```
```   399   let ?S = "\<lambda>p. {x. poly p x = 0}"
```
```   400   have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
```
```   401   with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
```
```   402 qed
```
```   403
```
```   404 lemma (in idom_char_0) poly_entire:
```
```   405   "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
```
```   406   using poly_entire_lemma2[of p q]
```
```   407   by (auto simp add: fun_eq_iff poly_mult)
```
```   408
```
```   409 lemma (in idom_char_0) poly_entire_neg:
```
```   410   "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
```
```   411   by (simp add: poly_entire)
```
```   412
```
```   413 lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
```
```   414   by auto
```
```   415
```
```   416 lemma (in comm_ring_1) poly_add_minus_zero_iff:
```
```   417   "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
```
```   418   by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
```
```   419
```
```   420 lemma (in comm_ring_1) poly_add_minus_mult_eq:
```
```   421   "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
```
```   422   by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult algebra_simps)
```
```   423
```
```   424 subclass (in idom_char_0) comm_ring_1 ..
```
```   425
```
```   426 lemma (in idom_char_0) poly_mult_left_cancel:
```
```   427   "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
```
```   428 proof -
```
```   429   have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
```
```   430     by (simp only: poly_add_minus_zero_iff)
```
```   431   also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
```
```   432     by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
```
```   433   finally show ?thesis .
```
```   434 qed
```
```   435
```
```   436 lemma (in idom) poly_exp_eq_zero[simp]:
```
```   437   "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
```
```   438   apply (simp only: fun_eq add: HOL.all_simps [symmetric])
```
```   439   apply (rule arg_cong [where f = All])
```
```   440   apply (rule ext)
```
```   441   apply (induct n)
```
```   442   apply (auto simp add: poly_exp poly_mult)
```
```   443   done
```
```   444
```
```   445 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
```
```   446   apply (simp add: fun_eq)
```
```   447   apply (rule_tac x = "minus one a" in exI)
```
```   448   apply (simp add: add.commute [of a])
```
```   449   done
```
```   450
```
```   451 lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
```
```   452   by auto
```
```   453
```
```   454 text{*A more constructive notion of polynomials being trivial*}
```
```   455
```
```   456 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
```
```   457   apply (simp add: fun_eq)
```
```   458   apply (case_tac "h = zero")
```
```   459   apply (drule_tac [2] x = zero in spec, auto)
```
```   460   apply (cases "poly t = poly []", simp)
```
```   461 proof -
```
```   462   fix x
```
```   463   assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
```
```   464     and pnz: "poly t \<noteq> poly []"
```
```   465   let ?S = "{x. poly t x = 0}"
```
```   466   from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
```
```   467   hence th: "?S \<supseteq> UNIV - {0}" by auto
```
```   468   from poly_roots_finite pnz have th': "finite ?S" by blast
```
```   469   from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
```
```   470     by simp
```
```   471 qed
```
```   472
```
```   473 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
```
```   474   apply (induct p)
```
```   475   apply simp
```
```   476   apply (rule iffI)
```
```   477   apply (drule poly_zero_lemma', auto)
```
```   478   done
```
```   479
```
```   480 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
```
```   481   unfolding poly_zero[symmetric] by simp
```
```   482
```
```   483
```
```   484
```
```   485 text{*Basics of divisibility.*}
```
```   486
```
```   487 lemma (in idom) poly_primes:
```
```   488   "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
```
```   489   apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
```
```   490   apply (drule_tac x = "uminus a" in spec)
```
```   491   apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
```
```   492   apply (cases "p = []")
```
```   493   apply (rule exI[where x="[]"])
```
```   494   apply simp
```
```   495   apply (cases "q = []")
```
```   496   apply (erule allE[where x="[]"], simp)
```
```   497
```
```   498   apply clarsimp
```
```   499   apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
```
```   500   apply (clarsimp simp add: poly_add poly_cmult)
```
```   501   apply (rule_tac x="qa" in exI)
```
```   502   apply (simp add: distrib_right [symmetric])
```
```   503   apply clarsimp
```
```   504
```
```   505   apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
```
```   506   apply (rule_tac x = "pmult qa q" in exI)
```
```   507   apply (rule_tac [2] x = "pmult p qa" in exI)
```
```   508   apply (auto simp add: poly_add poly_mult poly_cmult ac_simps)
```
```   509   done
```
```   510
```
```   511 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
```
```   512   apply (simp add: divides_def)
```
```   513   apply (rule_tac x = "[one]" in exI)
```
```   514   apply (auto simp add: poly_mult fun_eq)
```
```   515   done
```
```   516
```
```   517 lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
```
```   518   apply (simp add: divides_def, safe)
```
```   519   apply (rule_tac x = "pmult qa qaa" in exI)
```
```   520   apply (auto simp add: poly_mult fun_eq mult.assoc)
```
```   521   done
```
```   522
```
```   523 lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
```
```   524   apply (auto simp add: le_iff_add)
```
```   525   apply (induct_tac k)
```
```   526   apply (rule_tac [2] poly_divides_trans)
```
```   527   apply (auto simp add: divides_def)
```
```   528   apply (rule_tac x = p in exI)
```
```   529   apply (auto simp add: poly_mult fun_eq ac_simps)
```
```   530   done
```
```   531
```
```   532 lemma (in comm_semiring_1) poly_exp_divides:
```
```   533   "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
```
```   534   by (blast intro: poly_divides_exp poly_divides_trans)
```
```   535
```
```   536 lemma (in comm_semiring_0) poly_divides_add:
```
```   537   "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
```
```   538   apply (simp add: divides_def, auto)
```
```   539   apply (rule_tac x = "padd qa qaa" in exI)
```
```   540   apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
```
```   541   done
```
```   542
```
```   543 lemma (in comm_ring_1) poly_divides_diff:
```
```   544   "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
```
```   545   apply (simp add: divides_def, auto)
```
```   546   apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
```
```   547   apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
```
```   548   done
```
```   549
```
```   550 lemma (in comm_ring_1) poly_divides_diff2:
```
```   551   "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
```
```   552   apply (erule poly_divides_diff)
```
```   553   apply (auto simp add: poly_add fun_eq poly_mult divides_def ac_simps)
```
```   554   done
```
```   555
```
```   556 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
```
```   557   apply (simp add: divides_def)
```
```   558   apply (rule exI[where x="[]"])
```
```   559   apply (auto simp add: fun_eq poly_mult)
```
```   560   done
```
```   561
```
```   562 lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
```
```   563   apply (simp add: divides_def)
```
```   564   apply (rule_tac x = "[]" in exI)
```
```   565   apply (auto simp add: fun_eq)
```
```   566   done
```
```   567
```
```   568 text{*At last, we can consider the order of a root.*}
```
```   569
```
```   570 lemma (in idom_char_0) poly_order_exists_lemma:
```
```   571   assumes lp: "length p = d"
```
```   572     and p: "poly p \<noteq> poly []"
```
```   573   shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
```
```   574   using lp p
```
```   575 proof (induct d arbitrary: p)
```
```   576   case 0
```
```   577   thus ?case by simp
```
```   578 next
```
```   579   case (Suc n p)
```
```   580   show ?case
```
```   581   proof (cases "poly p a = 0")
```
```   582     case True
```
```   583     from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
```
```   584     hence pN: "p \<noteq> []" by auto
```
```   585     from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
```
```   586       by blast
```
```   587     from q h True have qh: "length q = n" "poly q \<noteq> poly []"
```
```   588       apply -
```
```   589       apply simp
```
```   590       apply (simp only: fun_eq)
```
```   591       apply (rule ccontr)
```
```   592       apply (simp add: fun_eq poly_add poly_cmult)
```
```   593       done
```
```   594     from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
```
```   595       by blast
```
```   596     from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
```
```   597     then show ?thesis by blast
```
```   598   next
```
```   599     case False
```
```   600     then show ?thesis
```
```   601       using Suc.prems
```
```   602       apply simp
```
```   603       apply (rule exI[where x="0::nat"])
```
```   604       apply simp
```
```   605       done
```
```   606   qed
```
```   607 qed
```
```   608
```
```   609
```
```   610 lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
```
```   611   by (induct n) (auto simp add: poly_mult ac_simps)
```
```   612
```
```   613 lemma (in comm_semiring_1) divides_left_mult:
```
```   614   assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
```
```   615 proof-
```
```   616   from d obtain t where r:"poly r = poly (p***q *** t)"
```
```   617     unfolding divides_def by blast
```
```   618   hence "poly r = poly (p *** (q *** t))"
```
```   619     "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult ac_simps)
```
```   620   thus ?thesis unfolding divides_def by blast
```
```   621 qed
```
```   622
```
```   623
```
```   624 (* FIXME: Tidy up *)
```
```   625
```
```   626 lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
```
```   627   by (induct n) simp_all
```
```   628
```
```   629 lemma (in idom_char_0) poly_order_exists:
```
```   630   assumes "length p = d" and "poly p \<noteq> poly []"
```
```   631   shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p"
```
```   632 proof -
```
```   633   from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0"
```
```   634     by (rule poly_order_exists_lemma)
```
```   635   then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0" by blast
```
```   636   have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
```
```   637   proof (rule dividesI)
```
```   638     show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
```
```   639       by (induct n) (simp_all add: poly_add poly_cmult poly_mult algebra_simps)
```
```   640   qed
```
```   641   moreover have "\<not> [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
```
```   642   proof
```
```   643     assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
```
```   644     then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"
```
```   645       by (rule dividesE)
```
```   646     moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)"
```
```   647     proof (induct n)
```
```   648       case 0 show ?case
```
```   649       proof (rule ccontr)
```
```   650         assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)"
```
```   651         then have "poly q a = 0"
```
```   652           by (simp add: poly_add poly_cmult)
```
```   653         with `poly q a \<noteq> 0` show False by simp
```
```   654       qed
```
```   655     next
```
```   656       case (Suc n) show ?case
```
```   657         by (rule pexp_Suc [THEN ssubst], rule ccontr)
```
```   658           (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
```
```   659     qed
```
```   660     ultimately show False by simp
```
```   661   qed
```
```   662   ultimately show ?thesis by (auto simp add: p)
```
```   663 qed
```
```   664
```
```   665 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
```
```   666   by (auto simp add: divides_def)
```
```   667
```
```   668 lemma (in idom_char_0) poly_order:
```
```   669   "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
```
```   670   apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
```
```   671   apply (cut_tac x = y and y = n in less_linear)
```
```   672   apply (drule_tac m = n in poly_exp_divides)
```
```   673   apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
```
```   674               simp del: pmult_Cons pexp_Suc)
```
```   675   done
```
```   676
```
```   677 text{*Order*}
```
```   678
```
```   679 lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
```
```   680   by (blast intro: someI2)
```
```   681
```
```   682 lemma (in idom_char_0) order:
```
```   683       "(([-a, 1] %^ n) divides p \<and>
```
```   684         ~(([-a, 1] %^ (Suc n)) divides p)) =
```
```   685         ((n = order a p) \<and> ~(poly p = poly []))"
```
```   686   apply (unfold order_def)
```
```   687   apply (rule iffI)
```
```   688   apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
```
```   689   apply (blast intro!: poly_order [THEN [2] some1_equalityD])
```
```   690   done
```
```   691
```
```   692 lemma (in idom_char_0) order2:
```
```   693   "poly p \<noteq> poly [] \<Longrightarrow>
```
```   694     ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)"
```
```   695   by (simp add: order del: pexp_Suc)
```
```   696
```
```   697 lemma (in idom_char_0) order_unique:
```
```   698   "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
```
```   699     n = order a p"
```
```   700   using order [of a n p] by auto
```
```   701
```
```   702 lemma (in idom_char_0) order_unique_lemma:
```
```   703   "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
```
```   704     n = order a p"
```
```   705   by (blast intro: order_unique)
```
```   706
```
```   707 lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
```
```   708   by (auto simp add: fun_eq divides_def poly_mult order_def)
```
```   709
```
```   710 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
```
```   711   by (induct "p") auto
```
```   712
```
```   713 lemma (in comm_ring_1) lemma_order_root:
```
```   714   "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
```
```   715   by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
```
```   716
```
```   717 lemma (in idom_char_0) order_root:
```
```   718   "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
```
```   719   apply (cases "poly p = poly []")
```
```   720   apply auto
```
```   721   apply (simp add: poly_linear_divides del: pmult_Cons, safe)
```
```   722   apply (drule_tac [!] a = a in order2)
```
```   723   apply (rule ccontr)
```
```   724   apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
```
```   725   using neq0_conv
```
```   726   apply (blast intro: lemma_order_root)
```
```   727   done
```
```   728
```
```   729 lemma (in idom_char_0) order_divides:
```
```   730   "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
```
```   731   apply (cases "poly p = poly []")
```
```   732   apply auto
```
```   733   apply (simp add: divides_def fun_eq poly_mult)
```
```   734   apply (rule_tac x = "[]" in exI)
```
```   735   apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
```
```   736   done
```
```   737
```
```   738 lemma (in idom_char_0) order_decomp:
```
```   739   "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)"
```
```   740   apply (unfold divides_def)
```
```   741   apply (drule order2 [where a = a])
```
```   742   apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
```
```   743   apply (rule_tac x = q in exI, safe)
```
```   744   apply (drule_tac x = qa in spec)
```
```   745   apply (auto simp add: poly_mult fun_eq poly_exp ac_simps simp del: pmult_Cons)
```
```   746   done
```
```   747
```
```   748 text{*Important composition properties of orders.*}
```
```   749 lemma order_mult:
```
```   750   "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
```
```   751     order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
```
```   752   apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
```
```   753   apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   754   apply (drule_tac a = a in order2)+
```
```   755   apply safe
```
```   756   apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   757   apply (rule_tac x = "qa *** qaa" in exI)
```
```   758   apply (simp add: poly_mult ac_simps del: pmult_Cons)
```
```   759   apply (drule_tac a = a in order_decomp)+
```
```   760   apply safe
```
```   761   apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
```
```   762   apply (simp add: poly_primes del: pmult_Cons)
```
```   763   apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   764   apply (rule_tac x = qb in exI)
```
```   765   apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
```
```   766   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   767   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))) ")
```
```   768   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   769   apply (simp add: fun_eq poly_exp_add poly_mult ac_simps del: pmult_Cons)
```
```   770   done
```
```   771
```
```   772 lemma (in idom_char_0) order_mult:
```
```   773   assumes "poly (p *** q) \<noteq> poly []"
```
```   774   shows "order a (p *** q) = order a p + order a q"
```
```   775   using assms
```
```   776   apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order)
```
```   777   apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   778   apply (drule_tac a = a in order2)+
```
```   779   apply safe
```
```   780   apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   781   apply (rule_tac x = "pmult qa qaa" in exI)
```
```   782   apply (simp add: poly_mult ac_simps del: pmult_Cons)
```
```   783   apply (drule_tac a = a in order_decomp)+
```
```   784   apply safe
```
```   785   apply (subgoal_tac "[uminus a, one] divides pmult qa qaa")
```
```   786   apply (simp add: poly_primes del: pmult_Cons)
```
```   787   apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   788   apply (rule_tac x = qb in exI)
```
```   789   apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) =
```
```   790     poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
```
```   791   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   792   apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q))
```
```   793       (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) =
```
```   794     poly (pmult (pexp [uminus a, one] (order a q))
```
```   795       (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))")
```
```   796   apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   797   apply (simp add: fun_eq poly_exp_add poly_mult ac_simps del: pmult_Cons)
```
```   798   done
```
```   799
```
```   800 lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
```
```   801   by (rule order_root [THEN ssubst]) auto
```
```   802
```
```   803 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
```
```   804
```
```   805 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
```
```   806   by (simp add: fun_eq)
```
```   807
```
```   808 lemma (in idom_char_0) rsquarefree_decomp:
```
```   809   "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow>
```
```   810     \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
```
```   811   apply (simp add: rsquarefree_def, safe)
```
```   812   apply (frule_tac a = a in order_decomp)
```
```   813   apply (drule_tac x = a in spec)
```
```   814   apply (drule_tac a = a in order_root2 [symmetric])
```
```   815   apply (auto simp del: pmult_Cons)
```
```   816   apply (rule_tac x = q in exI, safe)
```
```   817   apply (simp add: poly_mult fun_eq)
```
```   818   apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
```
```   819   apply (simp add: divides_def del: pmult_Cons, safe)
```
```   820   apply (drule_tac x = "[]" in spec)
```
```   821   apply (auto simp add: fun_eq)
```
```   822   done
```
```   823
```
```   824
```
```   825 text{*Normalization of a polynomial.*}
```
```   826
```
```   827 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
```
```   828   by (induct p) (auto simp add: fun_eq)
```
```   829
```
```   830 text{*The degree of a polynomial.*}
```
```   831
```
```   832 lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
```
```   833   by (induct p) auto
```
```   834
```
```   835 lemma (in idom_char_0) degree_zero:
```
```   836   assumes "poly p = poly []"
```
```   837   shows "degree p = 0"
```
```   838   using assms
```
```   839   by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
```
```   840
```
```   841 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
```
```   842   by simp
```
```   843
```
```   844 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
```
```   845   by simp
```
```   846
```
```   847 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
```
```   848   unfolding pnormal_def by simp
```
```   849
```
```   850 lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
```
```   851   unfolding pnormal_def by(auto split: split_if_asm)
```
```   852
```
```   853
```
```   854 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
```
```   855   by (induct p) (simp_all add: pnormal_def split: split_if_asm)
```
```   856
```
```   857 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
```
```   858   unfolding pnormal_def length_greater_0_conv by blast
```
```   859
```
```   860 lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
```
```   861   by (induct p) (auto simp: pnormal_def  split: split_if_asm)
```
```   862
```
```   863
```
```   864 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
```
```   865   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
```
```   866
```
```   867 lemma (in idom_char_0) poly_Cons_eq:
```
```   868   "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
```
```   869   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   870 proof
```
```   871   assume eq: ?lhs
```
```   872   hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
```
```   873     by (simp only: poly_minus poly_add algebra_simps) (simp add: algebra_simps)
```
```   874   hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff)
```
```   875   hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
```
```   876     unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
```
```   877   hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
```
```   878     unfolding poly_zero[symmetric] by simp
```
```   879   then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
```
```   880 next
```
```   881   assume ?rhs
```
```   882   then show ?lhs by(simp add:fun_eq_iff)
```
```   883 qed
```
```   884
```
```   885 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
```
```   886 proof (induct q arbitrary: p)
```
```   887   case Nil
```
```   888   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:
```
```   911   assumes pq: "poly p = poly q"
```
```   912   shows "degree p = degree q"
```
```   913   using pnormalize_unique[OF pq] unfolding degree_def by simp
```
```   914
```
```   915 lemma (in semiring_0) pnormalize_length:
```
```   916   "length (pnormalize p) \<le> length p" by (induct p) auto
```
```   917
```
```   918 lemma (in semiring_0) last_linear_mul_lemma:
```
```   919   "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)"
```
```   920   apply (induct p arbitrary: a x b)
```
```   921   apply auto
```
```   922   apply (rename_tac a p aa x b)
```
```   923   apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
```
```   924   apply simp
```
```   925   apply (induct_tac p)
```
```   926   apply auto
```
```   927   done
```
```   928
```
```   929 lemma (in semiring_1) last_linear_mul:
```
```   930   assumes p: "p \<noteq> []"
```
```   931   shows "last ([a,1] *** p) = last p"
```
```   932 proof -
```
```   933   from p obtain c cs where cs: "p = c#cs" by (cases p) auto
```
```   934   from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
```
```   935     by (simp add: poly_cmult_distr)
```
```   936   show ?thesis using cs
```
```   937     unfolding eq last_linear_mul_lemma by simp
```
```   938 qed
```
```   939
```
```   940 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
```
```   941   by (induct p) (auto split: split_if_asm)
```
```   942
```
```   943 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
```
```   944   by (induct p) auto
```
```   945
```
```   946 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
```
```   947   using pnormalize_eq[of p] unfolding degree_def by simp
```
```   948
```
```   949 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
```
```   950   by (rule ext) simp
```
```   951
```
```   952 lemma (in idom_char_0) linear_mul_degree:
```
```   953   assumes p: "poly p \<noteq> poly []"
```
```   954   shows "degree ([a,1] *** p) = degree p + 1"
```
```   955 proof -
```
```   956   from p have pnz: "pnormalize p \<noteq> []"
```
```   957     unfolding poly_zero lemma_degree_zero .
```
```   958
```
```   959   from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
```
```   960   have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
```
```   961   from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
```
```   962     pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
```
```   963
```
```   964   have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
```
```   965     by simp
```
```   966
```
```   967   have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
```
```   968     by (rule ext) (simp add: poly_mult poly_add poly_cmult)
```
```   969   from degree_unique[OF eqs] th
```
```   970   show ?thesis by (simp add: degree_unique[OF poly_normalize])
```
```   971 qed
```
```   972
```
```   973 lemma (in idom_char_0) linear_pow_mul_degree:
```
```   974   "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
```
```   975 proof (induct n arbitrary: a p)
```
```   976   case (0 a p)
```
```   977   show ?case
```
```   978   proof (cases "poly p = poly []")
```
```   979     case True
```
```   980     then show ?thesis
```
```   981       using degree_unique[OF True] by (simp add: degree_def)
```
```   982   next
```
```   983     case False
```
```   984     then show ?thesis by (auto simp add: poly_Nil_ext)
```
```   985   qed
```
```   986 next
```
```   987   case (Suc n a p)
```
```   988   have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
```
```   989     apply (rule ext)
```
```   990     apply (simp add: poly_mult poly_add poly_cmult)
```
```   991     apply (simp add: ac_simps ac_simps distrib_left)
```
```   992     done
```
```   993   note deq = degree_unique[OF eq]
```
```   994   show ?case
```
```   995   proof (cases "poly p = poly []")
```
```   996     case True
```
```   997     with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
```
```   998       apply -
```
```   999       apply (rule ext)
```
```  1000       apply (simp add: poly_mult poly_cmult poly_add)
```
```  1001       done
```
```  1002     from degree_unique[OF eq'] True show ?thesis
```
```  1003       by (simp add: degree_def)
```
```  1004   next
```
```  1005     case False
```
```  1006     then have ap: "poly ([a,1] *** p) \<noteq> poly []"
```
```  1007       using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
```
```  1008     have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
```
```  1009       by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
```
```  1010     from ap have ap': "(poly ([a,1] *** p) = poly []) = False"
```
```  1011       by blast
```
```  1012     have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
```
```  1013       apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
```
```  1014       apply simp
```
```  1015       done
```
```  1016     from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
```
```  1017     show ?thesis by (auto simp del: poly.simps)
```
```  1018   qed
```
```  1019 qed
```
```  1020
```
```  1021 lemma (in idom_char_0) order_degree:
```
```  1022   assumes p0: "poly p \<noteq> poly []"
```
```  1023   shows "order a p \<le> degree p"
```
```  1024 proof -
```
```  1025   from order2[OF p0, unfolded divides_def]
```
```  1026   obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
```
```  1027   {
```
```  1028     assume "poly q = poly []"
```
```  1029     with q p0 have False by (simp add: poly_mult poly_entire)
```
```  1030   }
```
```  1031   with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
```
```  1032     by auto
```
```  1033 qed
```
```  1034
```
```  1035 text{*Tidier versions of finiteness of roots.*}
```
```  1036
```
```  1037 lemma (in idom_char_0) poly_roots_finite_set:
```
```  1038   "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
```
```  1039   unfolding poly_roots_finite .
```
```  1040
```
```  1041 text{*bound for polynomial.*}
```
```  1042
```
```  1043 lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
```
```  1044   apply (induct p)
```
```  1045   apply auto
```
```  1046   apply (rename_tac a p)
```
```  1047   apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
```
```  1048   apply (rule abs_triangle_ineq)
```
```  1049   apply (auto intro!: mult_mono simp add: abs_mult)
```
```  1050   done
```
```  1051
```
```  1052 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
```
```  1053
```
```  1054 end
```