src/HOL/Decision_Procs/Polynomial_List.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 45605 a89b4bc311a5 child 49962 a8cc904a6820 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
```     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 Main
```
```     9 begin
```
```    10
```
```    11 text{* Application of polynomial as a real function. *}
```
```    12
```
```    13 primrec poly :: "'a list => 'a  => ('a::{comm_ring})" 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 primrec padd :: "['a list, 'a list] => ('a::comm_ring_1) list"  (infixl "+++" 65) where
```
```    22   padd_Nil:  "[] +++ l2 = l2"
```
```    23 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
```
```    24                             else (h + hd l2)#(t +++ tl l2))"
```
```    25
```
```    26 text{*Multiplication by a constant*}
```
```    27 primrec cmult :: "['a :: comm_ring_1, 'a list] => 'a list"  (infixl "%*" 70) where
```
```    28   cmult_Nil:  "c %* [] = []"
```
```    29 | cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
```
```    30
```
```    31 text{*Multiplication by a polynomial*}
```
```    32 primrec pmult :: "['a list, 'a list] => ('a::comm_ring_1) list"  (infixl "***" 70) where
```
```    33   pmult_Nil:  "[] *** l2 = []"
```
```    34 | pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
```
```    35                               else (h %* l2) +++ ((0) # (t *** l2)))"
```
```    36
```
```    37 text{*Repeated multiplication by a polynomial*}
```
```    38 primrec mulexp :: "[nat, 'a list, 'a  list] => ('a ::comm_ring_1) list" where
```
```    39   mulexp_zero:  "mulexp 0 p q = q"
```
```    40 | mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
```
```    41
```
```    42 text{*Exponential*}
```
```    43 primrec pexp :: "['a list, nat] => ('a::comm_ring_1) list"  (infixl "%^" 80) where
```
```    44   pexp_0:   "p %^ 0 = [1]"
```
```    45 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
```
```    46
```
```    47 text{*Quotient related value of dividing a polynomial by x + a*}
```
```    48 (* Useful for divisor properties in inductive proofs *)
```
```    49 primrec pquot :: "['a list, 'a::field] => 'a list" where
```
```    50   pquot_Nil:  "pquot [] a= []"
```
```    51 | pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
```
```    52                    else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
```
```    53
```
```    54
```
```    55 text{*normalization of polynomials (remove extra 0 coeff)*}
```
```    56 primrec pnormalize :: "('a::comm_ring_1) list => 'a list" where
```
```    57   pnormalize_Nil:  "pnormalize [] = []"
```
```    58 | pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
```
```    59                                      then (if (h = 0) then [] else [h])
```
```    60                                      else (h#(pnormalize p)))"
```
```    61
```
```    62 definition "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
```
```    63 definition "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
```
```    64 text{*Other definitions*}
```
```    65
```
```    66 definition
```
```    67   poly_minus :: "'a list => ('a :: comm_ring_1) list"      ("-- _" [80] 80) where
```
```    68   "-- p = (- 1) %* p"
```
```    69
```
```    70 definition
```
```    71   divides :: "[('a::comm_ring_1) list, 'a list] => bool"  (infixl "divides" 70) where
```
```    72   "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
```
```    73
```
```    74 definition
```
```    75   order :: "('a::comm_ring_1) => 'a list => nat" where
```
```    76     --{*order of a polynomial*}
```
```    77   "order a p = (SOME n. ([-a, 1] %^ n) divides p &
```
```    78                       ~ (([-a, 1] %^ (Suc n)) divides p))"
```
```    79
```
```    80 definition
```
```    81   degree :: "('a::comm_ring_1) list => nat" where
```
```    82      --{*degree of a polynomial*}
```
```    83   "degree p = length (pnormalize p) - 1"
```
```    84
```
```    85 definition
```
```    86   rsquarefree :: "('a::comm_ring_1) list => bool" where
```
```    87      --{*squarefree polynomials --- NB with respect to real roots only.*}
```
```    88   "rsquarefree p = (poly p \<noteq> poly [] &
```
```    89                      (\<forall>a. (order a p = 0) | (order a p = 1)))"
```
```    90
```
```    91 lemma padd_Nil2: "p +++ [] = p"
```
```    92 by (induct p) auto
```
```    93 declare padd_Nil2 [simp]
```
```    94
```
```    95 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
```
```    96 by auto
```
```    97
```
```    98 lemma pminus_Nil: "-- [] = []"
```
```    99 by (simp add: poly_minus_def)
```
```   100 declare pminus_Nil [simp]
```
```   101
```
```   102 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1"
```
```   103 by simp
```
```   104
```
```   105 lemma poly_ident_mult: "1 %* t = t"
```
```   106 by (induct "t", auto)
```
```   107 declare poly_ident_mult [simp]
```
```   108
```
```   109 lemma poly_simple_add_Cons: "[a] +++ ((0)#t) = (a#t)"
```
```   110 by simp
```
```   111 declare poly_simple_add_Cons [simp]
```
```   112
```
```   113 text{*Handy general properties*}
```
```   114
```
```   115 lemma padd_commut: "b +++ a = a +++ b"
```
```   116 apply (subgoal_tac "\<forall>a. b +++ a = a +++ b")
```
```   117 apply (induct_tac [2] "b", auto)
```
```   118 apply (rule padd_Cons [THEN ssubst])
```
```   119 apply (case_tac "aa", auto)
```
```   120 done
```
```   121
```
```   122 lemma padd_assoc [rule_format]: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
```
```   123 apply (induct "a", simp, clarify)
```
```   124 apply (case_tac b, simp_all)
```
```   125 done
```
```   126
```
```   127 lemma poly_cmult_distr [rule_format]:
```
```   128      "\<forall>q. a %* ( p +++ q) = (a %* p +++ a %* q)"
```
```   129 apply (induct "p", simp, clarify)
```
```   130 apply (case_tac "q")
```
```   131 apply (simp_all add: right_distrib)
```
```   132 done
```
```   133
```
```   134 lemma pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
```
```   135 apply (induct "t", simp)
```
```   136 by (auto simp add: mult_zero_left poly_ident_mult padd_commut)
```
```   137
```
```   138
```
```   139 text{*properties of evaluation of polynomials.*}
```
```   140
```
```   141 lemma poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
```
```   142 apply (subgoal_tac "\<forall>p2. poly (p1 +++ p2) x = poly (p1) x + poly (p2) x")
```
```   143 apply (induct_tac [2] "p1", auto)
```
```   144 apply (case_tac "p2")
```
```   145 apply (auto simp add: right_distrib)
```
```   146 done
```
```   147
```
```   148 lemma poly_cmult: "poly (c %* p) x = c * poly p x"
```
```   149 apply (induct "p")
```
```   150 apply (case_tac [2] "x=0")
```
```   151 apply (auto simp add: right_distrib mult_ac)
```
```   152 done
```
```   153
```
```   154 lemma poly_minus: "poly (-- p) x = - (poly p x)"
```
```   155 apply (simp add: poly_minus_def)
```
```   156 apply (auto simp add: poly_cmult)
```
```   157 done
```
```   158
```
```   159 lemma poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
```
```   160 apply (subgoal_tac "\<forall>p2. poly (p1 *** p2) x = poly p1 x * poly p2 x")
```
```   161 apply (simp (no_asm_simp))
```
```   162 apply (induct "p1")
```
```   163 apply (auto simp add: poly_cmult)
```
```   164 apply (case_tac p1)
```
```   165 apply (auto simp add: poly_cmult poly_add left_distrib right_distrib mult_ac)
```
```   166 done
```
```   167
```
```   168 lemma poly_exp: "poly (p %^ n) (x::'a::comm_ring_1) = (poly p x) ^ n"
```
```   169 apply (induct "n")
```
```   170 apply (auto simp add: poly_cmult poly_mult power_Suc)
```
```   171 done
```
```   172
```
```   173 text{*More Polynomial Evaluation Lemmas*}
```
```   174
```
```   175 lemma poly_add_rzero: "poly (a +++ []) x = poly a x"
```
```   176 by simp
```
```   177 declare poly_add_rzero [simp]
```
```   178
```
```   179 lemma poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
```
```   180   by (simp add: poly_mult mult_assoc)
```
```   181
```
```   182 lemma poly_mult_Nil2: "poly (p *** []) x = 0"
```
```   183 by (induct "p", auto)
```
```   184 declare poly_mult_Nil2 [simp]
```
```   185
```
```   186 lemma poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
```
```   187 apply (induct "n")
```
```   188 apply (auto simp add: poly_mult mult_assoc)
```
```   189 done
```
```   190
```
```   191 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
```
```   192  @{term "p(x)"} *}
```
```   193
```
```   194 lemma lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   195 apply (induct "t", safe)
```
```   196 apply (rule_tac x = "[]" in exI)
```
```   197 apply (rule_tac x = h in exI, simp)
```
```   198 apply (drule_tac x = aa in spec, safe)
```
```   199 apply (rule_tac x = "r#q" in exI)
```
```   200 apply (rule_tac x = "a*r + h" in exI)
```
```   201 apply (case_tac "q", auto)
```
```   202 done
```
```   203
```
```   204 lemma poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   205 by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
```
```   206
```
```   207
```
```   208 lemma poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
```
```   209 apply (auto simp add: poly_add poly_cmult right_distrib)
```
```   210 apply (case_tac "p", simp)
```
```   211 apply (cut_tac h = aa and t = list and a = a in poly_linear_rem, safe)
```
```   212 apply (case_tac "q", auto)
```
```   213 apply (drule_tac x = "[]" in spec, simp)
```
```   214 apply (auto simp add: poly_add poly_cmult add_assoc)
```
```   215 apply (drule_tac x = "aa#lista" in spec, auto)
```
```   216 done
```
```   217
```
```   218 lemma lemma_poly_length_mult: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
```
```   219 by (induct "p", auto)
```
```   220 declare lemma_poly_length_mult [simp]
```
```   221
```
```   222 lemma lemma_poly_length_mult2: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
```
```   223 by (induct "p", auto)
```
```   224 declare lemma_poly_length_mult2 [simp]
```
```   225
```
```   226 lemma poly_length_mult: "length([-a,1] *** q) = Suc (length q)"
```
```   227 by auto
```
```   228 declare poly_length_mult [simp]
```
```   229
```
```   230
```
```   231 subsection{*Polynomial length*}
```
```   232
```
```   233 lemma poly_cmult_length: "length (a %* p) = length p"
```
```   234 by (induct "p", auto)
```
```   235 declare poly_cmult_length [simp]
```
```   236
```
```   237 lemma poly_add_length [rule_format]:
```
```   238      "\<forall>p2. length (p1 +++ p2) =
```
```   239              (if (length p1 < length p2) then length p2 else length p1)"
```
```   240 apply (induct "p1", simp_all)
```
```   241 apply arith
```
```   242 done
```
```   243
```
```   244 lemma poly_root_mult_length: "length([a,b] *** p) = Suc (length p)"
```
```   245 by (simp add: poly_cmult_length poly_add_length)
```
```   246 declare poly_root_mult_length [simp]
```
```   247
```
```   248 lemma poly_mult_not_eq_poly_Nil: "(poly (p *** q) x \<noteq> poly [] x) =
```
```   249       (poly p x \<noteq> poly [] x & poly q x \<noteq> poly [] (x::'a::idom))"
```
```   250 apply (auto simp add: poly_mult)
```
```   251 done
```
```   252 declare poly_mult_not_eq_poly_Nil [simp]
```
```   253
```
```   254 lemma poly_mult_eq_zero_disj: "(poly (p *** q) (x::'a::idom) = 0) = (poly p x = 0 | poly q x = 0)"
```
```   255 by (auto simp add: poly_mult)
```
```   256
```
```   257 text{*Normalisation Properties*}
```
```   258
```
```   259 lemma poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
```
```   260 by (induct "p", auto)
```
```   261
```
```   262 text{*A nontrivial polynomial of degree n has no more than n roots*}
```
```   263
```
```   264 lemma poly_roots_index_lemma0 [rule_format]:
```
```   265    "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
```
```   266     --> (\<exists>i. \<forall>x. (poly p x = (0::'a::idom)) --> (\<exists>m. (m \<le> n & x = i m)))"
```
```   267 apply (induct "n", safe)
```
```   268 apply (rule ccontr)
```
```   269 apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
```
```   270 apply (drule poly_linear_divides [THEN iffD1], safe)
```
```   271 apply (drule_tac x = q in spec)
```
```   272 apply (drule_tac x = x in spec)
```
```   273 apply (simp del: poly_Nil pmult_Cons)
```
```   274 apply (erule exE)
```
```   275 apply (drule_tac x = "%m. if m = Suc n then a else i m" in spec, safe)
```
```   276 apply (drule poly_mult_eq_zero_disj [THEN iffD1], safe)
```
```   277 apply (drule_tac x = "Suc (length q)" in spec)
```
```   278 apply (auto simp add: field_simps)
```
```   279 apply (drule_tac x = xa in spec)
```
```   280 apply (clarsimp simp add: field_simps)
```
```   281 apply (drule_tac x = m in spec)
```
```   282 apply (auto simp add:field_simps)
```
```   283 done
```
```   284 lemmas poly_roots_index_lemma1 = conjI [THEN poly_roots_index_lemma0]
```
```   285
```
```   286 lemma poly_roots_index_length0: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   287       \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
```
```   288 by (blast intro: poly_roots_index_lemma1)
```
```   289
```
```   290 lemma poly_roots_finite_lemma: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   291       \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
```
```   292 apply (drule poly_roots_index_length0, safe)
```
```   293 apply (rule_tac x = "Suc (length p)" in exI)
```
```   294 apply (rule_tac x = i in exI)
```
```   295 apply (simp add: less_Suc_eq_le)
```
```   296 done
```
```   297
```
```   298
```
```   299 lemma real_finite_lemma:
```
```   300   assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
```
```   301   shows "finite {(x::'a::idom). P x}"
```
```   302 proof-
```
```   303   let ?M = "{x. P x}"
```
```   304   let ?N = "set j"
```
```   305   have "?M \<subseteq> ?N" using P by auto
```
```   306   thus ?thesis using finite_subset by auto
```
```   307 qed
```
```   308
```
```   309 lemma poly_roots_index_lemma [rule_format]:
```
```   310    "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
```
```   311     --> (\<exists>i. \<forall>x. (poly p x = (0::'a::{idom})) --> x \<in> set i)"
```
```   312 apply (induct "n", safe)
```
```   313 apply (rule ccontr)
```
```   314 apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
```
```   315 apply (drule poly_linear_divides [THEN iffD1], safe)
```
```   316 apply (drule_tac x = q in spec)
```
```   317 apply (drule_tac x = x in spec)
```
```   318 apply (auto simp del: poly_Nil pmult_Cons)
```
```   319 apply (drule_tac x = "a#i" in spec)
```
```   320 apply (auto simp only: poly_mult List.list.size)
```
```   321 apply (drule_tac x = xa in spec)
```
```   322 apply (clarsimp simp add: field_simps)
```
```   323 done
```
```   324
```
```   325 lemmas poly_roots_index_lemma2 = conjI [THEN poly_roots_index_lemma]
```
```   326
```
```   327 lemma poly_roots_index_length: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   328       \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
```
```   329 by (blast intro: poly_roots_index_lemma2)
```
```   330
```
```   331 lemma poly_roots_finite_lemma': "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   332       \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
```
```   333 by (drule poly_roots_index_length, safe)
```
```   334
```
```   335 lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
```
```   336   unfolding finite_conv_nat_seg_image
```
```   337 proof(auto simp add: set_eq_iff image_iff)
```
```   338   fix n::nat and f:: "nat \<Rightarrow> nat"
```
```   339   let ?N = "{i. i < n}"
```
```   340   let ?fN = "f ` ?N"
```
```   341   let ?y = "Max ?fN + 1"
```
```   342   from nat_seg_image_imp_finite[of "?fN" "f" n]
```
```   343   have thfN: "finite ?fN" by simp
```
```   344   {assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto}
```
```   345   moreover
```
```   346   {assume nz: "n \<noteq> 0"
```
```   347     hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv)
```
```   348     have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
```
```   349     hence "\<forall>x\<in> ?fN. ?y > x" by (auto simp add: less_Suc_eq_le)
```
```   350     hence "?y \<notin> ?fN" by auto
```
```   351     hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
```
```   352   ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
```
```   353 qed
```
```   354
```
```   355 lemma UNIV_ring_char_0_infinte: "\<not> finite (UNIV:: ('a::ring_char_0) set)"
```
```   356 proof
```
```   357   assume F: "finite (UNIV :: 'a set)"
```
```   358   have th0: "of_nat ` UNIV \<subseteq> (UNIV:: 'a set)" by simp
```
```   359   from finite_subset[OF th0 F] have th: "finite (of_nat ` UNIV :: 'a set)" .
```
```   360   have th': "inj_on (of_nat::nat \<Rightarrow> 'a) (UNIV)"
```
```   361     unfolding inj_on_def by auto
```
```   362   from finite_imageD[OF th th'] UNIV_nat_infinite
```
```   363   show False by blast
```
```   364 qed
```
```   365
```
```   366 lemma poly_roots_finite: "(poly p \<noteq> poly []) =
```
```   367   finite {x. poly p x = (0::'a::{idom, ring_char_0})}"
```
```   368 proof
```
```   369   assume H: "poly p \<noteq> poly []"
```
```   370   show "finite {x. poly p x = (0::'a)}"
```
```   371     using H
```
```   372     apply -
```
```   373     apply (erule contrapos_np, rule ext)
```
```   374     apply (rule ccontr)
```
```   375     apply (clarify dest!: poly_roots_finite_lemma')
```
```   376     using finite_subset
```
```   377   proof-
```
```   378     fix x i
```
```   379     assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
```
```   380       and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
```
```   381     let ?M= "{x. poly p x = (0\<Colon>'a)}"
```
```   382     from P have "?M \<subseteq> set i" by auto
```
```   383     with finite_subset F show False by auto
```
```   384   qed
```
```   385 next
```
```   386   assume F: "finite {x. poly p x = (0\<Colon>'a)}"
```
```   387   show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
```
```   388 qed
```
```   389
```
```   390 text{*Entirety and Cancellation for polynomials*}
```
```   391
```
```   392 lemma poly_entire_lemma: "[| poly (p:: ('a::{idom,ring_char_0}) list) \<noteq> poly [] ; poly q \<noteq> poly [] |]
```
```   393       ==>  poly (p *** q) \<noteq> poly []"
```
```   394 by (auto simp add: poly_roots_finite poly_mult Collect_disj_eq)
```
```   395
```
```   396 lemma poly_entire: "(poly (p *** q) = poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p = poly []) | (poly q = poly []))"
```
```   397 apply (auto intro: ext dest: fun_cong simp add: poly_entire_lemma poly_mult)
```
```   398 apply (blast intro: ccontr dest: poly_entire_lemma poly_mult [THEN subst])
```
```   399 done
```
```   400
```
```   401 lemma poly_entire_neg: "(poly (p *** q) \<noteq> poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
```
```   402 by (simp add: poly_entire)
```
```   403
```
```   404 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
```
```   405 by (auto intro!: ext)
```
```   406
```
```   407 lemma poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
```
```   408 by (auto simp add: field_simps poly_add poly_minus_def fun_eq poly_cmult)
```
```   409
```
```   410 lemma poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
```
```   411 by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib)
```
```   412
```
```   413 lemma poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly ([]::('a::{idom, ring_char_0}) list) | poly q = poly r)"
```
```   414 apply (rule_tac p1 = "p *** q" in poly_add_minus_zero_iff [THEN subst])
```
```   415 apply (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
```
```   416 done
```
```   417
```
```   418 lemma poly_exp_eq_zero:
```
```   419      "(poly (p %^ n) = poly ([]::('a::idom) list)) = (poly p = poly [] & n \<noteq> 0)"
```
```   420 apply (simp only: fun_eq add: HOL.all_simps [symmetric])
```
```   421 apply (rule arg_cong [where f = All])
```
```   422 apply (rule ext)
```
```   423 apply (induct_tac "n")
```
```   424 apply (auto simp add: poly_mult)
```
```   425 done
```
```   426 declare poly_exp_eq_zero [simp]
```
```   427
```
```   428 lemma poly_prime_eq_zero: "poly [a,(1::'a::comm_ring_1)] \<noteq> poly []"
```
```   429 apply (simp add: fun_eq)
```
```   430 apply (rule_tac x = "1 - a" in exI, simp)
```
```   431 done
```
```   432 declare poly_prime_eq_zero [simp]
```
```   433
```
```   434 lemma poly_exp_prime_eq_zero: "(poly ([a, (1::'a::idom)] %^ n) \<noteq> poly [])"
```
```   435 by auto
```
```   436 declare poly_exp_prime_eq_zero [simp]
```
```   437
```
```   438 text{*A more constructive notion of polynomials being trivial*}
```
```   439
```
```   440 lemma poly_zero_lemma': "poly (h # t) = poly [] ==> h = (0::'a::{idom,ring_char_0}) & poly t = poly []"
```
```   441 apply(simp add: fun_eq)
```
```   442 apply (case_tac "h = 0")
```
```   443 apply (drule_tac [2] x = 0 in spec, auto)
```
```   444 apply (case_tac "poly t = poly []", simp)
```
```   445 proof-
```
```   446   fix x
```
```   447   assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"  and pnz: "poly t \<noteq> poly []"
```
```   448   let ?S = "{x. poly t x = 0}"
```
```   449   from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
```
```   450   hence th: "?S \<supseteq> UNIV - {0}" by auto
```
```   451   from poly_roots_finite pnz have th': "finite ?S" by blast
```
```   452   from finite_subset[OF th th'] UNIV_ring_char_0_infinte[where ?'a = 'a]
```
```   453   show "poly t x = (0\<Colon>'a)" by simp
```
```   454   qed
```
```   455
```
```   456 lemma poly_zero: "(poly p = poly []) = list_all (%c. c = (0::'a::{idom,ring_char_0})) p"
```
```   457 apply (induct "p", simp)
```
```   458 apply (rule iffI)
```
```   459 apply (drule poly_zero_lemma', auto)
```
```   460 done
```
```   461
```
```   462
```
```   463
```
```   464 text{*Basics of divisibility.*}
```
```   465
```
```   466 lemma poly_primes: "([a, (1::'a::idom)] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
```
```   467 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
```
```   468 apply (drule_tac x = "-a" in spec)
```
```   469 apply (auto simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
```
```   470 apply (rule_tac x = "qa *** q" in exI)
```
```   471 apply (rule_tac [2] x = "p *** qa" in exI)
```
```   472 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
```
```   473 done
```
```   474
```
```   475 lemma poly_divides_refl: "p divides p"
```
```   476 apply (simp add: divides_def)
```
```   477 apply (rule_tac x = "[1]" in exI)
```
```   478 apply (auto simp add: poly_mult fun_eq)
```
```   479 done
```
```   480 declare poly_divides_refl [simp]
```
```   481
```
```   482 lemma poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
```
```   483 apply (simp add: divides_def, safe)
```
```   484 apply (rule_tac x = "qa *** qaa" in exI)
```
```   485 apply (auto simp add: poly_mult fun_eq mult_assoc)
```
```   486 done
```
```   487
```
```   488 lemma poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
```
```   489 apply (auto simp add: le_iff_add)
```
```   490 apply (induct_tac k)
```
```   491 apply (rule_tac [2] poly_divides_trans)
```
```   492 apply (auto simp add: divides_def)
```
```   493 apply (rule_tac x = p in exI)
```
```   494 apply (auto simp add: poly_mult fun_eq mult_ac)
```
```   495 done
```
```   496
```
```   497 lemma poly_exp_divides: "[| (p %^ n) divides q;  m\<le>n |] ==> (p %^ m) divides q"
```
```   498 by (blast intro: poly_divides_exp poly_divides_trans)
```
```   499
```
```   500 lemma poly_divides_add:
```
```   501    "[| p divides q; p divides r |] ==> p divides (q +++ r)"
```
```   502 apply (simp add: divides_def, auto)
```
```   503 apply (rule_tac x = "qa +++ qaa" in exI)
```
```   504 apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
```
```   505 done
```
```   506
```
```   507 lemma poly_divides_diff:
```
```   508    "[| p divides q; p divides (q +++ r) |] ==> p divides r"
```
```   509 apply (simp add: divides_def, auto)
```
```   510 apply (rule_tac x = "qaa +++ -- qa" in exI)
```
```   511 apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib algebra_simps)
```
```   512 done
```
```   513
```
```   514 lemma poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
```
```   515 apply (erule poly_divides_diff)
```
```   516 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
```
```   517 done
```
```   518
```
```   519 lemma poly_divides_zero: "poly p = poly [] ==> q divides p"
```
```   520 apply (simp add: divides_def)
```
```   521 apply (rule exI[where x="[]"])
```
```   522 apply (auto simp add: fun_eq poly_mult)
```
```   523 done
```
```   524
```
```   525 lemma poly_divides_zero2: "q divides []"
```
```   526 apply (simp add: divides_def)
```
```   527 apply (rule_tac x = "[]" in exI)
```
```   528 apply (auto simp add: fun_eq)
```
```   529 done
```
```   530 declare poly_divides_zero2 [simp]
```
```   531
```
```   532 text{*At last, we can consider the order of a root.*}
```
```   533
```
```   534
```
```   535 lemma poly_order_exists_lemma [rule_format]:
```
```   536      "\<forall>p. length p = d --> poly p \<noteq> poly []
```
```   537              --> (\<exists>n q. p = mulexp n [-a, (1::'a::{idom,ring_char_0})] q & poly q a \<noteq> 0)"
```
```   538 apply (induct "d")
```
```   539 apply (simp add: fun_eq, safe)
```
```   540 apply (case_tac "poly p a = 0")
```
```   541 apply (drule_tac poly_linear_divides [THEN iffD1], safe)
```
```   542 apply (drule_tac x = q in spec)
```
```   543 apply (drule_tac poly_entire_neg [THEN iffD1], safe, force)
```
```   544 apply (rule_tac x = "Suc n" in exI)
```
```   545 apply (rule_tac x = qa in exI)
```
```   546 apply (simp del: pmult_Cons)
```
```   547 apply (rule_tac x = 0 in exI, force)
```
```   548 done
```
```   549
```
```   550 (* FIXME: Tidy up *)
```
```   551 lemma poly_order_exists:
```
```   552      "[| length p = d; poly p \<noteq> poly [] |]
```
```   553       ==> \<exists>n. ([-a, 1] %^ n) divides p &
```
```   554                 ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)"
```
```   555 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
```
```   556 apply (rule_tac x = n in exI, safe)
```
```   557 apply (unfold divides_def)
```
```   558 apply (rule_tac x = q in exI)
```
```   559 apply (induct_tac "n", simp)
```
```   560 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
```
```   561 apply safe
```
```   562 apply (subgoal_tac "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** qa)")
```
```   563 apply simp
```
```   564 apply (induct_tac "n")
```
```   565 apply (simp del: pmult_Cons pexp_Suc)
```
```   566 apply (erule_tac Q = "poly q a = 0" in contrapos_np)
```
```   567 apply (simp add: poly_add poly_cmult)
```
```   568 apply (rule pexp_Suc [THEN ssubst])
```
```   569 apply (rule ccontr)
```
```   570 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
```
```   571 done
```
```   572
```
```   573 lemma poly_one_divides: "[1] divides p"
```
```   574 by (simp add: divides_def, auto)
```
```   575 declare poly_one_divides [simp]
```
```   576
```
```   577 lemma poly_order: "poly p \<noteq> poly []
```
```   578       ==> EX! n. ([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
```
```   579                  ~(([-a, 1] %^ (Suc n)) divides p)"
```
```   580 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
```
```   581 apply (cut_tac x = y and y = n in less_linear)
```
```   582 apply (drule_tac m = n in poly_exp_divides)
```
```   583 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
```
```   584             simp del: pmult_Cons pexp_Suc)
```
```   585 done
```
```   586
```
```   587 text{*Order*}
```
```   588
```
```   589 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
```
```   590 by (blast intro: someI2)
```
```   591
```
```   592 lemma order:
```
```   593       "(([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
```
```   594         ~(([-a, 1] %^ (Suc n)) divides p)) =
```
```   595         ((n = order a p) & ~(poly p = poly []))"
```
```   596 apply (unfold order_def)
```
```   597 apply (rule iffI)
```
```   598 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
```
```   599 apply (blast intro!: poly_order [THEN [2] some1_equalityD])
```
```   600 done
```
```   601
```
```   602 lemma order2: "[| poly p \<noteq> poly [] |]
```
```   603       ==> ([-a, (1::'a::{idom,ring_char_0})] %^ (order a p)) divides p &
```
```   604               ~(([-a, 1] %^ (Suc(order a p))) divides p)"
```
```   605 by (simp add: order del: pexp_Suc)
```
```   606
```
```   607 lemma order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
```
```   608          ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)
```
```   609       |] ==> (n = order a p)"
```
```   610 by (insert order [of a n p], auto)
```
```   611
```
```   612 lemma order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
```
```   613          ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p))
```
```   614       ==> (n = order a p)"
```
```   615 by (blast intro: order_unique)
```
```   616
```
```   617 lemma order_poly: "poly p = poly q ==> order a p = order a q"
```
```   618 by (auto simp add: fun_eq divides_def poly_mult order_def)
```
```   619
```
```   620 lemma pexp_one: "p %^ (Suc 0) = p"
```
```   621 apply (induct "p")
```
```   622 apply (auto simp add: numeral_1_eq_1)
```
```   623 done
```
```   624 declare pexp_one [simp]
```
```   625
```
```   626 lemma lemma_order_root [rule_format]:
```
```   627      "\<forall>p a. 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
```
```   628              --> poly p a = 0"
```
```   629 apply (induct "n", blast)
```
```   630 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
```
```   631 done
```
```   632
```
```   633 lemma order_root: "(poly p a = (0::'a::{idom,ring_char_0})) = ((poly p = poly []) | order a p \<noteq> 0)"
```
```   634 apply (case_tac "poly p = poly []", auto)
```
```   635 apply (simp add: poly_linear_divides del: pmult_Cons, safe)
```
```   636 apply (drule_tac [!] a = a in order2)
```
```   637 apply (rule ccontr)
```
```   638 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
```
```   639 using neq0_conv
```
```   640 apply (blast intro: lemma_order_root)
```
```   641 done
```
```   642
```
```   643 lemma order_divides: "(([-a, 1::'a::{idom,ring_char_0}] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
```
```   644 apply (case_tac "poly p = poly []", auto)
```
```   645 apply (simp add: divides_def fun_eq poly_mult)
```
```   646 apply (rule_tac x = "[]" in exI)
```
```   647 apply (auto dest!: order2 [where a=a]
```
```   648             intro: poly_exp_divides simp del: pexp_Suc)
```
```   649 done
```
```   650
```
```   651 lemma order_decomp:
```
```   652      "poly p \<noteq> poly []
```
```   653       ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
```
```   654                 ~([-a, 1::'a::{idom,ring_char_0}] divides q)"
```
```   655 apply (unfold divides_def)
```
```   656 apply (drule order2 [where a = a])
```
```   657 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
```
```   658 apply (rule_tac x = q in exI, safe)
```
```   659 apply (drule_tac x = qa in spec)
```
```   660 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
```
```   661 done
```
```   662
```
```   663 text{*Important composition properties of orders.*}
```
```   664
```
```   665 lemma order_mult: "poly (p *** q) \<noteq> poly []
```
```   666       ==> order a (p *** q) = order a p + order (a::'a::{idom,ring_char_0}) q"
```
```   667 apply (cut_tac a = a and p = "p***q" and n = "order a p + order a q" in order)
```
```   668 apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   669 apply (drule_tac a = a in order2)+
```
```   670 apply safe
```
```   671 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   672 apply (rule_tac x = "qa *** qaa" in exI)
```
```   673 apply (simp add: poly_mult mult_ac del: pmult_Cons)
```
```   674 apply (drule_tac a = a in order_decomp)+
```
```   675 apply safe
```
```   676 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
```
```   677 apply (simp add: poly_primes del: pmult_Cons)
```
```   678 apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   679 apply (rule_tac x = qb in exI)
```
```   680 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
```
```   681 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   682 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))) ")
```
```   683 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   684 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
```
```   685 done
```
```   686
```
```   687
```
```   688
```
```   689 lemma order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order (a::'a::{idom,ring_char_0}) p \<noteq> 0)"
```
```   690 by (rule order_root [THEN ssubst], auto)
```
```   691
```
```   692
```
```   693 lemma pmult_one: "[1] *** p = p"
```
```   694 by auto
```
```   695 declare pmult_one [simp]
```
```   696
```
```   697 lemma poly_Nil_zero: "poly [] = poly [0]"
```
```   698 by (simp add: fun_eq)
```
```   699
```
```   700 lemma rsquarefree_decomp:
```
```   701      "[| rsquarefree p; poly p a = (0::'a::{idom,ring_char_0}) |]
```
```   702       ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
```
```   703 apply (simp add: rsquarefree_def, safe)
```
```   704 apply (frule_tac a = a in order_decomp)
```
```   705 apply (drule_tac x = a in spec)
```
```   706 apply (drule_tac a = a in order_root2 [symmetric])
```
```   707 apply (auto simp del: pmult_Cons)
```
```   708 apply (rule_tac x = q in exI, safe)
```
```   709 apply (simp add: poly_mult fun_eq)
```
```   710 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
```
```   711 apply (simp add: divides_def del: pmult_Cons, safe)
```
```   712 apply (drule_tac x = "[]" in spec)
```
```   713 apply (auto simp add: fun_eq)
```
```   714 done
```
```   715
```
```   716
```
```   717 text{*Normalization of a polynomial.*}
```
```   718
```
```   719 lemma poly_normalize: "poly (pnormalize p) = poly p"
```
```   720 apply (induct "p")
```
```   721 apply (auto simp add: fun_eq)
```
```   722 done
```
```   723 declare poly_normalize [simp]
```
```   724
```
```   725
```
```   726 text{*The degree of a polynomial.*}
```
```   727
```
```   728 lemma lemma_degree_zero:
```
```   729      "list_all (%c. c = 0) p \<longleftrightarrow>  pnormalize p = []"
```
```   730 by (induct "p", auto)
```
```   731
```
```   732 lemma degree_zero: "(poly p = poly ([]:: (('a::{idom,ring_char_0}) list))) \<Longrightarrow> (degree p = 0)"
```
```   733 apply (simp add: degree_def)
```
```   734 apply (case_tac "pnormalize p = []")
```
```   735 apply (auto simp add: poly_zero lemma_degree_zero )
```
```   736 done
```
```   737
```
```   738 lemma pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
```
```   739 lemma pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
```
```   740 lemma pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
```
```   741   unfolding pnormal_def by simp
```
```   742 lemma pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
```
```   743   unfolding pnormal_def
```
```   744   apply (cases "pnormalize p = []", auto)
```
```   745   by (cases "c = 0", auto)
```
```   746 lemma pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
```
```   747   apply (induct p, auto simp add: pnormal_def)
```
```   748   apply (case_tac "pnormalize p = []", auto)
```
```   749   by (case_tac "a=0", auto)
```
```   750 lemma  pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
```
```   751   unfolding pnormal_def length_greater_0_conv by blast
```
```   752 lemma pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
```
```   753   apply (induct p, auto)
```
```   754   apply (case_tac "p = []", auto)
```
```   755   apply (simp add: pnormal_def)
```
```   756   by (rule pnormal_cons, auto)
```
```   757 lemma pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
```
```   758   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
```
```   759
```
```   760 text{*Tidier versions of finiteness of roots.*}
```
```   761
```
```   762 lemma poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x::'a::{idom,ring_char_0}. poly p x = 0}"
```
```   763 unfolding poly_roots_finite .
```
```   764
```
```   765 text{*bound for polynomial.*}
```
```   766
```
```   767 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
```
```   768 apply (induct "p", auto)
```
```   769 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
```
```   770 apply (rule abs_triangle_ineq)
```
```   771 apply (auto intro!: mult_mono simp add: abs_mult)
```
```   772 done
```
```   773
```
```   774 lemma poly_Sing: "poly [c] x = c" by simp
```
```   775
```
```   776 end
```