src/HOL/Decision_Procs/Polynomial_List.thy
 author wenzelm Wed Oct 28 00:24:38 2009 +0100 (2009-10-28) changeset 33268 02de0317f66f parent 33153 92080294beb8 child 35028 108662d50512 permissions -rw-r--r--
eliminated hard tabulators, guessing at each author's individual tab-width;
```     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 consts poly :: "'a list => 'a  => ('a::{comm_ring})"
```
```    14 primrec
```
```    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 consts padd :: "['a list, 'a list] => ('a::comm_ring_1) list"  (infixl "+++" 65)
```
```    23 primrec
```
```    24   padd_Nil:  "[] +++ l2 = l2"
```
```    25   padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
```
```    26                             else (h + hd l2)#(t +++ tl l2))"
```
```    27
```
```    28 text{*Multiplication by a constant*}
```
```    29 consts cmult :: "['a :: comm_ring_1, 'a list] => 'a list"  (infixl "%*" 70)
```
```    30 primrec
```
```    31    cmult_Nil:  "c %* [] = []"
```
```    32    cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
```
```    33
```
```    34 text{*Multiplication by a polynomial*}
```
```    35 consts pmult :: "['a list, 'a list] => ('a::comm_ring_1) list"  (infixl "***" 70)
```
```    36 primrec
```
```    37    pmult_Nil:  "[] *** l2 = []"
```
```    38    pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
```
```    39                               else (h %* l2) +++ ((0) # (t *** l2)))"
```
```    40
```
```    41 text{*Repeated multiplication by a polynomial*}
```
```    42 consts mulexp :: "[nat, 'a list, 'a  list] => ('a ::comm_ring_1) list"
```
```    43 primrec
```
```    44    mulexp_zero:  "mulexp 0 p q = q"
```
```    45    mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
```
```    46
```
```    47 text{*Exponential*}
```
```    48 consts pexp :: "['a list, nat] => ('a::comm_ring_1) list"  (infixl "%^" 80)
```
```    49 primrec
```
```    50    pexp_0:   "p %^ 0 = [1]"
```
```    51    pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
```
```    52
```
```    53 text{*Quotient related value of dividing a polynomial by x + a*}
```
```    54 (* Useful for divisor properties in inductive proofs *)
```
```    55 consts "pquot" :: "['a list, 'a::field] => 'a list"
```
```    56 primrec
```
```    57    pquot_Nil:  "pquot [] a= []"
```
```    58    pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
```
```    59                    else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
```
```    60
```
```    61
```
```    62 text{*normalization of polynomials (remove extra 0 coeff)*}
```
```    63 consts pnormalize :: "('a::comm_ring_1) list => 'a list"
```
```    64 primrec
```
```    65    pnormalize_Nil:  "pnormalize [] = []"
```
```    66    pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
```
```    67                                      then (if (h = 0) then [] else [h])
```
```    68                                      else (h#(pnormalize p)))"
```
```    69
```
```    70 definition "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
```
```    71 definition "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
```
```    72 text{*Other definitions*}
```
```    73
```
```    74 definition
```
```    75   poly_minus :: "'a list => ('a :: comm_ring_1) list"      ("-- _" [80] 80) where
```
```    76   "-- p = (- 1) %* p"
```
```    77
```
```    78 definition
```
```    79   divides :: "[('a::comm_ring_1) list, 'a list] => bool"  (infixl "divides" 70) where
```
```    80   "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
```
```    81
```
```    82 definition
```
```    83   order :: "('a::comm_ring_1) => 'a list => nat" where
```
```    84     --{*order of a polynomial*}
```
```    85   "order a p = (SOME n. ([-a, 1] %^ n) divides p &
```
```    86                       ~ (([-a, 1] %^ (Suc n)) divides p))"
```
```    87
```
```    88 definition
```
```    89   degree :: "('a::comm_ring_1) list => nat" where
```
```    90      --{*degree of a polynomial*}
```
```    91   "degree p = length (pnormalize p) - 1"
```
```    92
```
```    93 definition
```
```    94   rsquarefree :: "('a::comm_ring_1) list => bool" where
```
```    95      --{*squarefree polynomials --- NB with respect to real roots only.*}
```
```    96   "rsquarefree p = (poly p \<noteq> poly [] &
```
```    97                      (\<forall>a. (order a p = 0) | (order a p = 1)))"
```
```    98
```
```    99 lemma padd_Nil2: "p +++ [] = p"
```
```   100 by (induct p) auto
```
```   101 declare padd_Nil2 [simp]
```
```   102
```
```   103 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
```
```   104 by auto
```
```   105
```
```   106 lemma pminus_Nil: "-- [] = []"
```
```   107 by (simp add: poly_minus_def)
```
```   108 declare pminus_Nil [simp]
```
```   109
```
```   110 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1"
```
```   111 by simp
```
```   112
```
```   113 lemma poly_ident_mult: "1 %* t = t"
```
```   114 by (induct "t", auto)
```
```   115 declare poly_ident_mult [simp]
```
```   116
```
```   117 lemma poly_simple_add_Cons: "[a] +++ ((0)#t) = (a#t)"
```
```   118 by simp
```
```   119 declare poly_simple_add_Cons [simp]
```
```   120
```
```   121 text{*Handy general properties*}
```
```   122
```
```   123 lemma padd_commut: "b +++ a = a +++ b"
```
```   124 apply (subgoal_tac "\<forall>a. b +++ a = a +++ b")
```
```   125 apply (induct_tac [2] "b", auto)
```
```   126 apply (rule padd_Cons [THEN ssubst])
```
```   127 apply (case_tac "aa", auto)
```
```   128 done
```
```   129
```
```   130 lemma padd_assoc [rule_format]: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
```
```   131 apply (induct "a", simp, clarify)
```
```   132 apply (case_tac b, simp_all)
```
```   133 done
```
```   134
```
```   135 lemma poly_cmult_distr [rule_format]:
```
```   136      "\<forall>q. a %* ( p +++ q) = (a %* p +++ a %* q)"
```
```   137 apply (induct "p", simp, clarify)
```
```   138 apply (case_tac "q")
```
```   139 apply (simp_all add: right_distrib)
```
```   140 done
```
```   141
```
```   142 lemma pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
```
```   143 apply (induct "t", simp)
```
```   144 by (auto simp add: mult_zero_left poly_ident_mult padd_commut)
```
```   145
```
```   146
```
```   147 text{*properties of evaluation of polynomials.*}
```
```   148
```
```   149 lemma poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
```
```   150 apply (subgoal_tac "\<forall>p2. poly (p1 +++ p2) x = poly (p1) x + poly (p2) x")
```
```   151 apply (induct_tac [2] "p1", auto)
```
```   152 apply (case_tac "p2")
```
```   153 apply (auto simp add: right_distrib)
```
```   154 done
```
```   155
```
```   156 lemma poly_cmult: "poly (c %* p) x = c * poly p x"
```
```   157 apply (induct "p")
```
```   158 apply (case_tac [2] "x=0")
```
```   159 apply (auto simp add: right_distrib mult_ac)
```
```   160 done
```
```   161
```
```   162 lemma poly_minus: "poly (-- p) x = - (poly p x)"
```
```   163 apply (simp add: poly_minus_def)
```
```   164 apply (auto simp add: poly_cmult)
```
```   165 done
```
```   166
```
```   167 lemma poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
```
```   168 apply (subgoal_tac "\<forall>p2. poly (p1 *** p2) x = poly p1 x * poly p2 x")
```
```   169 apply (simp (no_asm_simp))
```
```   170 apply (induct "p1")
```
```   171 apply (auto simp add: poly_cmult)
```
```   172 apply (case_tac p1)
```
```   173 apply (auto simp add: poly_cmult poly_add left_distrib right_distrib mult_ac)
```
```   174 done
```
```   175
```
```   176 lemma poly_exp: "poly (p %^ n) (x::'a::comm_ring_1) = (poly p x) ^ n"
```
```   177 apply (induct "n")
```
```   178 apply (auto simp add: poly_cmult poly_mult power_Suc)
```
```   179 done
```
```   180
```
```   181 text{*More Polynomial Evaluation Lemmas*}
```
```   182
```
```   183 lemma poly_add_rzero: "poly (a +++ []) x = poly a x"
```
```   184 by simp
```
```   185 declare poly_add_rzero [simp]
```
```   186
```
```   187 lemma poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
```
```   188   by (simp add: poly_mult mult_assoc)
```
```   189
```
```   190 lemma poly_mult_Nil2: "poly (p *** []) x = 0"
```
```   191 by (induct "p", auto)
```
```   192 declare poly_mult_Nil2 [simp]
```
```   193
```
```   194 lemma poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
```
```   195 apply (induct "n")
```
```   196 apply (auto simp add: poly_mult mult_assoc)
```
```   197 done
```
```   198
```
```   199 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
```
```   200  @{term "p(x)"} *}
```
```   201
```
```   202 lemma lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   203 apply (induct "t", safe)
```
```   204 apply (rule_tac x = "[]" in exI)
```
```   205 apply (rule_tac x = h in exI, simp)
```
```   206 apply (drule_tac x = aa in spec, safe)
```
```   207 apply (rule_tac x = "r#q" in exI)
```
```   208 apply (rule_tac x = "a*r + h" in exI)
```
```   209 apply (case_tac "q", auto)
```
```   210 done
```
```   211
```
```   212 lemma poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
```
```   213 by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
```
```   214
```
```   215
```
```   216 lemma poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
```
```   217 apply (auto simp add: poly_add poly_cmult right_distrib)
```
```   218 apply (case_tac "p", simp)
```
```   219 apply (cut_tac h = aa and t = list and a = a in poly_linear_rem, safe)
```
```   220 apply (case_tac "q", auto)
```
```   221 apply (drule_tac x = "[]" in spec, simp)
```
```   222 apply (auto simp add: poly_add poly_cmult add_assoc)
```
```   223 apply (drule_tac x = "aa#lista" in spec, auto)
```
```   224 done
```
```   225
```
```   226 lemma lemma_poly_length_mult: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
```
```   227 by (induct "p", auto)
```
```   228 declare lemma_poly_length_mult [simp]
```
```   229
```
```   230 lemma lemma_poly_length_mult2: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
```
```   231 by (induct "p", auto)
```
```   232 declare lemma_poly_length_mult2 [simp]
```
```   233
```
```   234 lemma poly_length_mult: "length([-a,1] *** q) = Suc (length q)"
```
```   235 by auto
```
```   236 declare poly_length_mult [simp]
```
```   237
```
```   238
```
```   239 subsection{*Polynomial length*}
```
```   240
```
```   241 lemma poly_cmult_length: "length (a %* p) = length p"
```
```   242 by (induct "p", auto)
```
```   243 declare poly_cmult_length [simp]
```
```   244
```
```   245 lemma poly_add_length [rule_format]:
```
```   246      "\<forall>p2. length (p1 +++ p2) =
```
```   247              (if (length p1 < length p2) then length p2 else length p1)"
```
```   248 apply (induct "p1", simp_all)
```
```   249 apply arith
```
```   250 done
```
```   251
```
```   252 lemma poly_root_mult_length: "length([a,b] *** p) = Suc (length p)"
```
```   253 by (simp add: poly_cmult_length poly_add_length)
```
```   254 declare poly_root_mult_length [simp]
```
```   255
```
```   256 lemma poly_mult_not_eq_poly_Nil: "(poly (p *** q) x \<noteq> poly [] x) =
```
```   257       (poly p x \<noteq> poly [] x & poly q x \<noteq> poly [] (x::'a::idom))"
```
```   258 apply (auto simp add: poly_mult)
```
```   259 done
```
```   260 declare poly_mult_not_eq_poly_Nil [simp]
```
```   261
```
```   262 lemma poly_mult_eq_zero_disj: "(poly (p *** q) (x::'a::idom) = 0) = (poly p x = 0 | poly q x = 0)"
```
```   263 by (auto simp add: poly_mult)
```
```   264
```
```   265 text{*Normalisation Properties*}
```
```   266
```
```   267 lemma poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
```
```   268 by (induct "p", auto)
```
```   269
```
```   270 text{*A nontrivial polynomial of degree n has no more than n roots*}
```
```   271
```
```   272 lemma poly_roots_index_lemma0 [rule_format]:
```
```   273    "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
```
```   274     --> (\<exists>i. \<forall>x. (poly p x = (0::'a::idom)) --> (\<exists>m. (m \<le> n & x = i m)))"
```
```   275 apply (induct "n", safe)
```
```   276 apply (rule ccontr)
```
```   277 apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
```
```   278 apply (drule poly_linear_divides [THEN iffD1], safe)
```
```   279 apply (drule_tac x = q in spec)
```
```   280 apply (drule_tac x = x in spec)
```
```   281 apply (simp del: poly_Nil pmult_Cons)
```
```   282 apply (erule exE)
```
```   283 apply (drule_tac x = "%m. if m = Suc n then a else i m" in spec, safe)
```
```   284 apply (drule poly_mult_eq_zero_disj [THEN iffD1], safe)
```
```   285 apply (drule_tac x = "Suc (length q)" in spec)
```
```   286 apply (auto simp add: ring_simps)
```
```   287 apply (drule_tac x = xa in spec)
```
```   288 apply (clarsimp simp add: ring_simps)
```
```   289 apply (drule_tac x = m in spec)
```
```   290 apply (auto simp add:ring_simps)
```
```   291 done
```
```   292 lemmas poly_roots_index_lemma1 = conjI [THEN poly_roots_index_lemma0, standard]
```
```   293
```
```   294 lemma poly_roots_index_length0: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   295       \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
```
```   296 by (blast intro: poly_roots_index_lemma1)
```
```   297
```
```   298 lemma poly_roots_finite_lemma: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   299       \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
```
```   300 apply (drule poly_roots_index_length0, safe)
```
```   301 apply (rule_tac x = "Suc (length p)" in exI)
```
```   302 apply (rule_tac x = i in exI)
```
```   303 apply (simp add: less_Suc_eq_le)
```
```   304 done
```
```   305
```
```   306
```
```   307 lemma real_finite_lemma:
```
```   308   assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
```
```   309   shows "finite {(x::'a::idom). P x}"
```
```   310 proof-
```
```   311   let ?M = "{x. P x}"
```
```   312   let ?N = "set j"
```
```   313   have "?M \<subseteq> ?N" using P by auto
```
```   314   thus ?thesis using finite_subset by auto
```
```   315 qed
```
```   316
```
```   317 lemma poly_roots_index_lemma [rule_format]:
```
```   318    "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
```
```   319     --> (\<exists>i. \<forall>x. (poly p x = (0::'a::{idom})) --> x \<in> set i)"
```
```   320 apply (induct "n", safe)
```
```   321 apply (rule ccontr)
```
```   322 apply (subgoal_tac "\<exists>a. poly p a = 0", safe)
```
```   323 apply (drule poly_linear_divides [THEN iffD1], safe)
```
```   324 apply (drule_tac x = q in spec)
```
```   325 apply (drule_tac x = x in spec)
```
```   326 apply (auto simp del: poly_Nil pmult_Cons)
```
```   327 apply (drule_tac x = "a#i" in spec)
```
```   328 apply (auto simp only: poly_mult List.list.size)
```
```   329 apply (drule_tac x = xa in spec)
```
```   330 apply (clarsimp simp add: ring_simps)
```
```   331 done
```
```   332
```
```   333 lemmas poly_roots_index_lemma2 = conjI [THEN poly_roots_index_lemma, standard]
```
```   334
```
```   335 lemma poly_roots_index_length: "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   336       \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
```
```   337 by (blast intro: poly_roots_index_lemma2)
```
```   338
```
```   339 lemma poly_roots_finite_lemma': "poly p (x::'a::idom) \<noteq> poly [] x ==>
```
```   340       \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
```
```   341 by (drule poly_roots_index_length, safe)
```
```   342
```
```   343 lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
```
```   344   unfolding finite_conv_nat_seg_image
```
```   345 proof(auto simp add: expand_set_eq image_iff)
```
```   346   fix n::nat and f:: "nat \<Rightarrow> nat"
```
```   347   let ?N = "{i. i < n}"
```
```   348   let ?fN = "f ` ?N"
```
```   349   let ?y = "Max ?fN + 1"
```
```   350   from nat_seg_image_imp_finite[of "?fN" "f" n]
```
```   351   have thfN: "finite ?fN" by simp
```
```   352   {assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto}
```
```   353   moreover
```
```   354   {assume nz: "n \<noteq> 0"
```
```   355     hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv)
```
```   356     have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
```
```   357     hence "\<forall>x\<in> ?fN. ?y > x" by (auto simp add: less_Suc_eq_le)
```
```   358     hence "?y \<notin> ?fN" by auto
```
```   359     hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
```
```   360   ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
```
```   361 qed
```
```   362
```
```   363 lemma UNIV_ring_char_0_infinte: "\<not> finite (UNIV:: ('a::ring_char_0) set)"
```
```   364 proof
```
```   365   assume F: "finite (UNIV :: 'a set)"
```
```   366   have th0: "of_nat ` UNIV \<subseteq> (UNIV:: 'a set)" by simp
```
```   367   from finite_subset[OF th0 F] have th: "finite (of_nat ` UNIV :: 'a set)" .
```
```   368   have th': "inj_on (of_nat::nat \<Rightarrow> 'a) (UNIV)"
```
```   369     unfolding inj_on_def by auto
```
```   370   from finite_imageD[OF th th'] UNIV_nat_infinite
```
```   371   show False by blast
```
```   372 qed
```
```   373
```
```   374 lemma poly_roots_finite: "(poly p \<noteq> poly []) =
```
```   375   finite {x. poly p x = (0::'a::{idom, ring_char_0})}"
```
```   376 proof
```
```   377   assume H: "poly p \<noteq> poly []"
```
```   378   show "finite {x. poly p x = (0::'a)}"
```
```   379     using H
```
```   380     apply -
```
```   381     apply (erule contrapos_np, rule ext)
```
```   382     apply (rule ccontr)
```
```   383     apply (clarify dest!: poly_roots_finite_lemma')
```
```   384     using finite_subset
```
```   385   proof-
```
```   386     fix x i
```
```   387     assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
```
```   388       and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
```
```   389     let ?M= "{x. poly p x = (0\<Colon>'a)}"
```
```   390     from P have "?M \<subseteq> set i" by auto
```
```   391     with finite_subset F show False by auto
```
```   392   qed
```
```   393 next
```
```   394   assume F: "finite {x. poly p x = (0\<Colon>'a)}"
```
```   395   show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
```
```   396 qed
```
```   397
```
```   398 text{*Entirety and Cancellation for polynomials*}
```
```   399
```
```   400 lemma poly_entire_lemma: "[| poly (p:: ('a::{idom,ring_char_0}) list) \<noteq> poly [] ; poly q \<noteq> poly [] |]
```
```   401       ==>  poly (p *** q) \<noteq> poly []"
```
```   402 by (auto simp add: poly_roots_finite poly_mult Collect_disj_eq)
```
```   403
```
```   404 lemma poly_entire: "(poly (p *** q) = poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p = poly []) | (poly q = poly []))"
```
```   405 apply (auto intro: ext dest: fun_cong simp add: poly_entire_lemma poly_mult)
```
```   406 apply (blast intro: ccontr dest: poly_entire_lemma poly_mult [THEN subst])
```
```   407 done
```
```   408
```
```   409 lemma poly_entire_neg: "(poly (p *** q) \<noteq> poly ([]::('a::{idom,ring_char_0}) list)) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
```
```   410 by (simp add: poly_entire)
```
```   411
```
```   412 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
```
```   413 by (auto intro!: ext)
```
```   414
```
```   415 lemma poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
```
```   416 by (auto simp add: ring_simps poly_add poly_minus_def fun_eq poly_cmult)
```
```   417
```
```   418 lemma poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
```
```   419 by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib)
```
```   420
```
```   421 lemma poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly ([]::('a::{idom, ring_char_0}) list) | poly q = poly r)"
```
```   422 apply (rule_tac p1 = "p *** q" in poly_add_minus_zero_iff [THEN subst])
```
```   423 apply (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
```
```   424 done
```
```   425
```
```   426 lemma poly_exp_eq_zero:
```
```   427      "(poly (p %^ n) = poly ([]::('a::idom) list)) = (poly p = poly [] & n \<noteq> 0)"
```
```   428 apply (simp only: fun_eq add: all_simps [symmetric])
```
```   429 apply (rule arg_cong [where f = All])
```
```   430 apply (rule ext)
```
```   431 apply (induct_tac "n")
```
```   432 apply (auto simp add: poly_mult)
```
```   433 done
```
```   434 declare poly_exp_eq_zero [simp]
```
```   435
```
```   436 lemma poly_prime_eq_zero: "poly [a,(1::'a::comm_ring_1)] \<noteq> poly []"
```
```   437 apply (simp add: fun_eq)
```
```   438 apply (rule_tac x = "1 - a" in exI, simp)
```
```   439 done
```
```   440 declare poly_prime_eq_zero [simp]
```
```   441
```
```   442 lemma poly_exp_prime_eq_zero: "(poly ([a, (1::'a::idom)] %^ n) \<noteq> poly [])"
```
```   443 by auto
```
```   444 declare poly_exp_prime_eq_zero [simp]
```
```   445
```
```   446 text{*A more constructive notion of polynomials being trivial*}
```
```   447
```
```   448 lemma poly_zero_lemma': "poly (h # t) = poly [] ==> h = (0::'a::{idom,ring_char_0}) & poly t = poly []"
```
```   449 apply(simp add: fun_eq)
```
```   450 apply (case_tac "h = 0")
```
```   451 apply (drule_tac [2] x = 0 in spec, auto)
```
```   452 apply (case_tac "poly t = poly []", simp)
```
```   453 proof-
```
```   454   fix x
```
```   455   assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"  and pnz: "poly t \<noteq> poly []"
```
```   456   let ?S = "{x. poly t x = 0}"
```
```   457   from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
```
```   458   hence th: "?S \<supseteq> UNIV - {0}" by auto
```
```   459   from poly_roots_finite pnz have th': "finite ?S" by blast
```
```   460   from finite_subset[OF th th'] UNIV_ring_char_0_infinte[where ?'a = 'a]
```
```   461   show "poly t x = (0\<Colon>'a)" by simp
```
```   462   qed
```
```   463
```
```   464 lemma poly_zero: "(poly p = poly []) = list_all (%c. c = (0::'a::{idom,ring_char_0})) p"
```
```   465 apply (induct "p", simp)
```
```   466 apply (rule iffI)
```
```   467 apply (drule poly_zero_lemma', auto)
```
```   468 done
```
```   469
```
```   470
```
```   471
```
```   472 text{*Basics of divisibility.*}
```
```   473
```
```   474 lemma poly_primes: "([a, (1::'a::idom)] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
```
```   475 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
```
```   476 apply (drule_tac x = "-a" in spec)
```
```   477 apply (auto simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
```
```   478 apply (rule_tac x = "qa *** q" in exI)
```
```   479 apply (rule_tac [2] x = "p *** qa" in exI)
```
```   480 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
```
```   481 done
```
```   482
```
```   483 lemma poly_divides_refl: "p divides p"
```
```   484 apply (simp add: divides_def)
```
```   485 apply (rule_tac x = "[1]" in exI)
```
```   486 apply (auto simp add: poly_mult fun_eq)
```
```   487 done
```
```   488 declare poly_divides_refl [simp]
```
```   489
```
```   490 lemma poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
```
```   491 apply (simp add: divides_def, safe)
```
```   492 apply (rule_tac x = "qa *** qaa" in exI)
```
```   493 apply (auto simp add: poly_mult fun_eq mult_assoc)
```
```   494 done
```
```   495
```
```   496 lemma poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
```
```   497 apply (auto simp add: le_iff_add)
```
```   498 apply (induct_tac k)
```
```   499 apply (rule_tac [2] poly_divides_trans)
```
```   500 apply (auto simp add: divides_def)
```
```   501 apply (rule_tac x = p in exI)
```
```   502 apply (auto simp add: poly_mult fun_eq mult_ac)
```
```   503 done
```
```   504
```
```   505 lemma poly_exp_divides: "[| (p %^ n) divides q;  m\<le>n |] ==> (p %^ m) divides q"
```
```   506 by (blast intro: poly_divides_exp poly_divides_trans)
```
```   507
```
```   508 lemma poly_divides_add:
```
```   509    "[| p divides q; p divides r |] ==> p divides (q +++ r)"
```
```   510 apply (simp add: divides_def, auto)
```
```   511 apply (rule_tac x = "qa +++ qaa" in exI)
```
```   512 apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
```
```   513 done
```
```   514
```
```   515 lemma poly_divides_diff:
```
```   516    "[| p divides q; p divides (q +++ r) |] ==> p divides r"
```
```   517 apply (simp add: divides_def, auto)
```
```   518 apply (rule_tac x = "qaa +++ -- qa" in exI)
```
```   519 apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib algebra_simps)
```
```   520 done
```
```   521
```
```   522 lemma poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
```
```   523 apply (erule poly_divides_diff)
```
```   524 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
```
```   525 done
```
```   526
```
```   527 lemma poly_divides_zero: "poly p = poly [] ==> q divides p"
```
```   528 apply (simp add: divides_def)
```
```   529 apply (rule exI[where x="[]"])
```
```   530 apply (auto simp add: fun_eq poly_mult)
```
```   531 done
```
```   532
```
```   533 lemma poly_divides_zero2: "q divides []"
```
```   534 apply (simp add: divides_def)
```
```   535 apply (rule_tac x = "[]" in exI)
```
```   536 apply (auto simp add: fun_eq)
```
```   537 done
```
```   538 declare poly_divides_zero2 [simp]
```
```   539
```
```   540 text{*At last, we can consider the order of a root.*}
```
```   541
```
```   542
```
```   543 lemma poly_order_exists_lemma [rule_format]:
```
```   544      "\<forall>p. length p = d --> poly p \<noteq> poly []
```
```   545              --> (\<exists>n q. p = mulexp n [-a, (1::'a::{idom,ring_char_0})] q & poly q a \<noteq> 0)"
```
```   546 apply (induct "d")
```
```   547 apply (simp add: fun_eq, safe)
```
```   548 apply (case_tac "poly p a = 0")
```
```   549 apply (drule_tac poly_linear_divides [THEN iffD1], safe)
```
```   550 apply (drule_tac x = q in spec)
```
```   551 apply (drule_tac poly_entire_neg [THEN iffD1], safe, force)
```
```   552 apply (rule_tac x = "Suc n" in exI)
```
```   553 apply (rule_tac x = qa in exI)
```
```   554 apply (simp del: pmult_Cons)
```
```   555 apply (rule_tac x = 0 in exI, force)
```
```   556 done
```
```   557
```
```   558 (* FIXME: Tidy up *)
```
```   559 lemma poly_order_exists:
```
```   560      "[| length p = d; poly p \<noteq> poly [] |]
```
```   561       ==> \<exists>n. ([-a, 1] %^ n) divides p &
```
```   562                 ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)"
```
```   563 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
```
```   564 apply (rule_tac x = n in exI, safe)
```
```   565 apply (unfold divides_def)
```
```   566 apply (rule_tac x = q in exI)
```
```   567 apply (induct_tac "n", simp)
```
```   568 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
```
```   569 apply safe
```
```   570 apply (subgoal_tac "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** qa)")
```
```   571 apply simp
```
```   572 apply (induct_tac "n")
```
```   573 apply (simp del: pmult_Cons pexp_Suc)
```
```   574 apply (erule_tac Q = "poly q a = 0" in contrapos_np)
```
```   575 apply (simp add: poly_add poly_cmult)
```
```   576 apply (rule pexp_Suc [THEN ssubst])
```
```   577 apply (rule ccontr)
```
```   578 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
```
```   579 done
```
```   580
```
```   581 lemma poly_one_divides: "[1] divides p"
```
```   582 by (simp add: divides_def, auto)
```
```   583 declare poly_one_divides [simp]
```
```   584
```
```   585 lemma poly_order: "poly p \<noteq> poly []
```
```   586       ==> EX! n. ([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
```
```   587                  ~(([-a, 1] %^ (Suc n)) divides p)"
```
```   588 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
```
```   589 apply (cut_tac x = y and y = n in less_linear)
```
```   590 apply (drule_tac m = n in poly_exp_divides)
```
```   591 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
```
```   592             simp del: pmult_Cons pexp_Suc)
```
```   593 done
```
```   594
```
```   595 text{*Order*}
```
```   596
```
```   597 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
```
```   598 by (blast intro: someI2)
```
```   599
```
```   600 lemma order:
```
```   601       "(([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
```
```   602         ~(([-a, 1] %^ (Suc n)) divides p)) =
```
```   603         ((n = order a p) & ~(poly p = poly []))"
```
```   604 apply (unfold order_def)
```
```   605 apply (rule iffI)
```
```   606 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
```
```   607 apply (blast intro!: poly_order [THEN [2] some1_equalityD])
```
```   608 done
```
```   609
```
```   610 lemma order2: "[| poly p \<noteq> poly [] |]
```
```   611       ==> ([-a, (1::'a::{idom,ring_char_0})] %^ (order a p)) divides p &
```
```   612               ~(([-a, 1] %^ (Suc(order a p))) divides p)"
```
```   613 by (simp add: order del: pexp_Suc)
```
```   614
```
```   615 lemma order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
```
```   616          ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)
```
```   617       |] ==> (n = order a p)"
```
```   618 by (insert order [of a n p], auto)
```
```   619
```
```   620 lemma order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
```
```   621          ~(([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p))
```
```   622       ==> (n = order a p)"
```
```   623 by (blast intro: order_unique)
```
```   624
```
```   625 lemma order_poly: "poly p = poly q ==> order a p = order a q"
```
```   626 by (auto simp add: fun_eq divides_def poly_mult order_def)
```
```   627
```
```   628 lemma pexp_one: "p %^ (Suc 0) = p"
```
```   629 apply (induct "p")
```
```   630 apply (auto simp add: numeral_1_eq_1)
```
```   631 done
```
```   632 declare pexp_one [simp]
```
```   633
```
```   634 lemma lemma_order_root [rule_format]:
```
```   635      "\<forall>p a. 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
```
```   636              --> poly p a = 0"
```
```   637 apply (induct "n", blast)
```
```   638 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
```
```   639 done
```
```   640
```
```   641 lemma order_root: "(poly p a = (0::'a::{idom,ring_char_0})) = ((poly p = poly []) | order a p \<noteq> 0)"
```
```   642 apply (case_tac "poly p = poly []", auto)
```
```   643 apply (simp add: poly_linear_divides del: pmult_Cons, safe)
```
```   644 apply (drule_tac [!] a = a in order2)
```
```   645 apply (rule ccontr)
```
```   646 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
```
```   647 using neq0_conv
```
```   648 apply (blast intro: lemma_order_root)
```
```   649 done
```
```   650
```
```   651 lemma order_divides: "(([-a, 1::'a::{idom,ring_char_0}] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
```
```   652 apply (case_tac "poly p = poly []", auto)
```
```   653 apply (simp add: divides_def fun_eq poly_mult)
```
```   654 apply (rule_tac x = "[]" in exI)
```
```   655 apply (auto dest!: order2 [where a=a]
```
```   656             intro: poly_exp_divides simp del: pexp_Suc)
```
```   657 done
```
```   658
```
```   659 lemma order_decomp:
```
```   660      "poly p \<noteq> poly []
```
```   661       ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
```
```   662                 ~([-a, 1::'a::{idom,ring_char_0}] divides q)"
```
```   663 apply (unfold divides_def)
```
```   664 apply (drule order2 [where a = a])
```
```   665 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
```
```   666 apply (rule_tac x = q in exI, safe)
```
```   667 apply (drule_tac x = qa in spec)
```
```   668 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
```
```   669 done
```
```   670
```
```   671 text{*Important composition properties of orders.*}
```
```   672
```
```   673 lemma order_mult: "poly (p *** q) \<noteq> poly []
```
```   674       ==> order a (p *** q) = order a p + order (a::'a::{idom,ring_char_0}) q"
```
```   675 apply (cut_tac a = a and p = "p***q" and n = "order a p + order a q" in order)
```
```   676 apply (auto simp add: poly_entire simp del: pmult_Cons)
```
```   677 apply (drule_tac a = a in order2)+
```
```   678 apply safe
```
```   679 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
```
```   680 apply (rule_tac x = "qa *** qaa" in exI)
```
```   681 apply (simp add: poly_mult mult_ac del: pmult_Cons)
```
```   682 apply (drule_tac a = a in order_decomp)+
```
```   683 apply safe
```
```   684 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
```
```   685 apply (simp add: poly_primes del: pmult_Cons)
```
```   686 apply (auto simp add: divides_def simp del: pmult_Cons)
```
```   687 apply (rule_tac x = qb in exI)
```
```   688 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
```
```   689 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   690 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))) ")
```
```   691 apply (drule poly_mult_left_cancel [THEN iffD1], force)
```
```   692 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
```
```   693 done
```
```   694
```
```   695
```
```   696
```
```   697 lemma order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order (a::'a::{idom,ring_char_0}) p \<noteq> 0)"
```
```   698 by (rule order_root [THEN ssubst], auto)
```
```   699
```
```   700
```
```   701 lemma pmult_one: "[1] *** p = p"
```
```   702 by auto
```
```   703 declare pmult_one [simp]
```
```   704
```
```   705 lemma poly_Nil_zero: "poly [] = poly [0]"
```
```   706 by (simp add: fun_eq)
```
```   707
```
```   708 lemma rsquarefree_decomp:
```
```   709      "[| rsquarefree p; poly p a = (0::'a::{idom,ring_char_0}) |]
```
```   710       ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
```
```   711 apply (simp add: rsquarefree_def, safe)
```
```   712 apply (frule_tac a = a in order_decomp)
```
```   713 apply (drule_tac x = a in spec)
```
```   714 apply (drule_tac a = a in order_root2 [symmetric])
```
```   715 apply (auto simp del: pmult_Cons)
```
```   716 apply (rule_tac x = q in exI, safe)
```
```   717 apply (simp add: poly_mult fun_eq)
```
```   718 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
```
```   719 apply (simp add: divides_def del: pmult_Cons, safe)
```
```   720 apply (drule_tac x = "[]" in spec)
```
```   721 apply (auto simp add: fun_eq)
```
```   722 done
```
```   723
```
```   724
```
```   725 text{*Normalization of a polynomial.*}
```
```   726
```
```   727 lemma poly_normalize: "poly (pnormalize p) = poly p"
```
```   728 apply (induct "p")
```
```   729 apply (auto simp add: fun_eq)
```
```   730 done
```
```   731 declare poly_normalize [simp]
```
```   732
```
```   733
```
```   734 text{*The degree of a polynomial.*}
```
```   735
```
```   736 lemma lemma_degree_zero:
```
```   737      "list_all (%c. c = 0) p \<longleftrightarrow>  pnormalize p = []"
```
```   738 by (induct "p", auto)
```
```   739
```
```   740 lemma degree_zero: "(poly p = poly ([]:: (('a::{idom,ring_char_0}) list))) \<Longrightarrow> (degree p = 0)"
```
```   741 apply (simp add: degree_def)
```
```   742 apply (case_tac "pnormalize p = []")
```
```   743 apply (auto simp add: poly_zero lemma_degree_zero )
```
```   744 done
```
```   745
```
```   746 lemma pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
```
```   747 lemma pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
```
```   748 lemma pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
```
```   749   unfolding pnormal_def by simp
```
```   750 lemma pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
```
```   751   unfolding pnormal_def
```
```   752   apply (cases "pnormalize p = []", auto)
```
```   753   by (cases "c = 0", auto)
```
```   754 lemma pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
```
```   755   apply (induct p, auto simp add: pnormal_def)
```
```   756   apply (case_tac "pnormalize p = []", auto)
```
```   757   by (case_tac "a=0", auto)
```
```   758 lemma  pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
```
```   759   unfolding pnormal_def length_greater_0_conv by blast
```
```   760 lemma pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
```
```   761   apply (induct p, auto)
```
```   762   apply (case_tac "p = []", auto)
```
```   763   apply (simp add: pnormal_def)
```
```   764   by (rule pnormal_cons, auto)
```
```   765 lemma pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
```
```   766   using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
```
```   767
```
```   768 text{*Tidier versions of finiteness of roots.*}
```
```   769
```
```   770 lemma poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x::'a::{idom,ring_char_0}. poly p x = 0}"
```
```   771 unfolding poly_roots_finite .
```
```   772
```
```   773 text{*bound for polynomial.*}
```
```   774
```
```   775 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
```
```   776 apply (induct "p", auto)
```
```   777 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
```
```   778 apply (rule abs_triangle_ineq)
```
```   779 apply (auto intro!: mult_mono simp add: abs_mult)
```
```   780 done
```
```   781
```
```   782 lemma poly_Sing: "poly [c] x = c" by simp
```
```   783
```
```   784 end
```