src/HOL/Polynomial.thy
author huffman
Mon Jan 12 12:09:54 2009 -0800 (2009-01-12)
changeset 29460 ad87e5d1488b
parent 29457 2eadbc24de8c
child 29462 dc97c6188a7a
permissions -rw-r--r--
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
     1 (*  Title:      HOL/Polynomial.thy
     2     Author:     Brian Huffman
     3                 Based on an earlier development by Clemens Ballarin
     4 *)
     5 
     6 header {* Univariate Polynomials *}
     7 
     8 theory Polynomial
     9 imports Plain SetInterval
    10 begin
    11 
    12 subsection {* Definition of type @{text poly} *}
    13 
    14 typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
    15   morphisms coeff Abs_poly
    16   by auto
    17 
    18 lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
    19 by (simp add: coeff_inject [symmetric] expand_fun_eq)
    20 
    21 lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
    22 by (simp add: expand_poly_eq)
    23 
    24 
    25 subsection {* Degree of a polynomial *}
    26 
    27 definition
    28   degree :: "'a::zero poly \<Rightarrow> nat" where
    29   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
    30 
    31 lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
    32 proof -
    33   have "coeff p \<in> Poly"
    34     by (rule coeff)
    35   hence "\<exists>n. \<forall>i>n. coeff p i = 0"
    36     unfolding Poly_def by simp
    37   hence "\<forall>i>degree p. coeff p i = 0"
    38     unfolding degree_def by (rule LeastI_ex)
    39   moreover assume "degree p < n"
    40   ultimately show ?thesis by simp
    41 qed
    42 
    43 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
    44   by (erule contrapos_np, rule coeff_eq_0, simp)
    45 
    46 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
    47   unfolding degree_def by (erule Least_le)
    48 
    49 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
    50   unfolding degree_def by (drule not_less_Least, simp)
    51 
    52 
    53 subsection {* The zero polynomial *}
    54 
    55 instantiation poly :: (zero) zero
    56 begin
    57 
    58 definition
    59   zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
    60 
    61 instance ..
    62 end
    63 
    64 lemma coeff_0 [simp]: "coeff 0 n = 0"
    65   unfolding zero_poly_def
    66   by (simp add: Abs_poly_inverse Poly_def)
    67 
    68 lemma degree_0 [simp]: "degree 0 = 0"
    69   by (rule order_antisym [OF degree_le le0]) simp
    70 
    71 lemma leading_coeff_neq_0:
    72   assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
    73 proof (cases "degree p")
    74   case 0
    75   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
    76     by (simp add: expand_poly_eq)
    77   then obtain n where "coeff p n \<noteq> 0" ..
    78   hence "n \<le> degree p" by (rule le_degree)
    79   with `coeff p n \<noteq> 0` and `degree p = 0`
    80   show "coeff p (degree p) \<noteq> 0" by simp
    81 next
    82   case (Suc n)
    83   from `degree p = Suc n` have "n < degree p" by simp
    84   hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
    85   then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
    86   from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
    87   also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
    88   finally have "degree p = i" .
    89   with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
    90 qed
    91 
    92 lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
    93   by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
    94 
    95 
    96 subsection {* List-style constructor for polynomials *}
    97 
    98 definition
    99   pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
   100 where
   101   [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
   102 
   103 syntax
   104   "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
   105 
   106 translations
   107   "[:x, xs:]" == "CONST pCons x [:xs:]"
   108   "[:x:]" == "CONST pCons x 0"
   109 
   110 lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
   111   unfolding Poly_def by (auto split: nat.split)
   112 
   113 lemma coeff_pCons:
   114   "coeff (pCons a p) = nat_case a (coeff p)"
   115   unfolding pCons_def
   116   by (simp add: Abs_poly_inverse Poly_nat_case coeff)
   117 
   118 lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
   119   by (simp add: coeff_pCons)
   120 
   121 lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
   122   by (simp add: coeff_pCons)
   123 
   124 lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
   125 by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
   126 
   127 lemma degree_pCons_eq:
   128   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
   129 apply (rule order_antisym [OF degree_pCons_le])
   130 apply (rule le_degree, simp)
   131 done
   132 
   133 lemma degree_pCons_0: "degree (pCons a 0) = 0"
   134 apply (rule order_antisym [OF _ le0])
   135 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
   136 done
   137 
   138 lemma degree_pCons_eq_if [simp]:
   139   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
   140 apply (cases "p = 0", simp_all)
   141 apply (rule order_antisym [OF _ le0])
   142 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
   143 apply (rule order_antisym [OF degree_pCons_le])
   144 apply (rule le_degree, simp)
   145 done
   146 
   147 lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
   148 by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   149 
   150 lemma pCons_eq_iff [simp]:
   151   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
   152 proof (safe)
   153   assume "pCons a p = pCons b q"
   154   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
   155   then show "a = b" by simp
   156 next
   157   assume "pCons a p = pCons b q"
   158   then have "\<forall>n. coeff (pCons a p) (Suc n) =
   159                  coeff (pCons b q) (Suc n)" by simp
   160   then show "p = q" by (simp add: expand_poly_eq)
   161 qed
   162 
   163 lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
   164   using pCons_eq_iff [of a p 0 0] by simp
   165 
   166 lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
   167   unfolding Poly_def
   168   by (clarify, rule_tac x=n in exI, simp)
   169 
   170 lemma pCons_cases [cases type: poly]:
   171   obtains (pCons) a q where "p = pCons a q"
   172 proof
   173   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
   174     by (rule poly_ext)
   175        (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
   176              split: nat.split)
   177 qed
   178 
   179 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
   180   assumes zero: "P 0"
   181   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
   182   shows "P p"
   183 proof (induct p rule: measure_induct_rule [where f=degree])
   184   case (less p)
   185   obtain a q where "p = pCons a q" by (rule pCons_cases)
   186   have "P q"
   187   proof (cases "q = 0")
   188     case True
   189     then show "P q" by (simp add: zero)
   190   next
   191     case False
   192     then have "degree (pCons a q) = Suc (degree q)"
   193       by (rule degree_pCons_eq)
   194     then have "degree q < degree p"
   195       using `p = pCons a q` by simp
   196     then show "P q"
   197       by (rule less.hyps)
   198   qed
   199   then have "P (pCons a q)"
   200     by (rule pCons)
   201   then show ?case
   202     using `p = pCons a q` by simp
   203 qed
   204 
   205 
   206 subsection {* Recursion combinator for polynomials *}
   207 
   208 function
   209   poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
   210 where
   211   poly_rec_pCons_eq_if [simp del]:
   212     "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
   213 by (case_tac x, rename_tac q, case_tac q, auto)
   214 
   215 termination poly_rec
   216 by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
   217    (simp add: degree_pCons_eq)
   218 
   219 lemma poly_rec_0:
   220   "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
   221   using poly_rec_pCons_eq_if [of z f 0 0] by simp
   222 
   223 lemma poly_rec_pCons:
   224   "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
   225   by (simp add: poly_rec_pCons_eq_if poly_rec_0)
   226 
   227 
   228 subsection {* Monomials *}
   229 
   230 definition
   231   monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
   232   "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
   233 
   234 lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
   235   unfolding monom_def
   236   by (subst Abs_poly_inverse, auto simp add: Poly_def)
   237 
   238 lemma monom_0: "monom a 0 = pCons a 0"
   239   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   240 
   241 lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
   242   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   243 
   244 lemma monom_eq_0 [simp]: "monom 0 n = 0"
   245   by (rule poly_ext) simp
   246 
   247 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
   248   by (simp add: expand_poly_eq)
   249 
   250 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
   251   by (simp add: expand_poly_eq)
   252 
   253 lemma degree_monom_le: "degree (monom a n) \<le> n"
   254   by (rule degree_le, simp)
   255 
   256 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
   257   apply (rule order_antisym [OF degree_monom_le])
   258   apply (rule le_degree, simp)
   259   done
   260 
   261 
   262 subsection {* Addition and subtraction *}
   263 
   264 instantiation poly :: (comm_monoid_add) comm_monoid_add
   265 begin
   266 
   267 definition
   268   plus_poly_def [code del]:
   269     "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
   270 
   271 lemma Poly_add:
   272   fixes f g :: "nat \<Rightarrow> 'a"
   273   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
   274   unfolding Poly_def
   275   apply (clarify, rename_tac m n)
   276   apply (rule_tac x="max m n" in exI, simp)
   277   done
   278 
   279 lemma coeff_add [simp]:
   280   "coeff (p + q) n = coeff p n + coeff q n"
   281   unfolding plus_poly_def
   282   by (simp add: Abs_poly_inverse coeff Poly_add)
   283 
   284 instance proof
   285   fix p q r :: "'a poly"
   286   show "(p + q) + r = p + (q + r)"
   287     by (simp add: expand_poly_eq add_assoc)
   288   show "p + q = q + p"
   289     by (simp add: expand_poly_eq add_commute)
   290   show "0 + p = p"
   291     by (simp add: expand_poly_eq)
   292 qed
   293 
   294 end
   295 
   296 instantiation poly :: (ab_group_add) ab_group_add
   297 begin
   298 
   299 definition
   300   uminus_poly_def [code del]:
   301     "- p = Abs_poly (\<lambda>n. - coeff p n)"
   302 
   303 definition
   304   minus_poly_def [code del]:
   305     "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
   306 
   307 lemma Poly_minus:
   308   fixes f :: "nat \<Rightarrow> 'a"
   309   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
   310   unfolding Poly_def by simp
   311 
   312 lemma Poly_diff:
   313   fixes f g :: "nat \<Rightarrow> 'a"
   314   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
   315   unfolding diff_minus by (simp add: Poly_add Poly_minus)
   316 
   317 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
   318   unfolding uminus_poly_def
   319   by (simp add: Abs_poly_inverse coeff Poly_minus)
   320 
   321 lemma coeff_diff [simp]:
   322   "coeff (p - q) n = coeff p n - coeff q n"
   323   unfolding minus_poly_def
   324   by (simp add: Abs_poly_inverse coeff Poly_diff)
   325 
   326 instance proof
   327   fix p q :: "'a poly"
   328   show "- p + p = 0"
   329     by (simp add: expand_poly_eq)
   330   show "p - q = p + - q"
   331     by (simp add: expand_poly_eq diff_minus)
   332 qed
   333 
   334 end
   335 
   336 lemma add_pCons [simp]:
   337   "pCons a p + pCons b q = pCons (a + b) (p + q)"
   338   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   339 
   340 lemma minus_pCons [simp]:
   341   "- pCons a p = pCons (- a) (- p)"
   342   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   343 
   344 lemma diff_pCons [simp]:
   345   "pCons a p - pCons b q = pCons (a - b) (p - q)"
   346   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   347 
   348 lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
   349   by (rule degree_le, auto simp add: coeff_eq_0)
   350 
   351 lemma degree_add_less:
   352   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
   353   by (auto intro: le_less_trans degree_add_le)
   354 
   355 lemma degree_add_eq_right:
   356   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
   357   apply (cases "q = 0", simp)
   358   apply (rule order_antisym)
   359   apply (rule ord_le_eq_trans [OF degree_add_le])
   360   apply simp
   361   apply (rule le_degree)
   362   apply (simp add: coeff_eq_0)
   363   done
   364 
   365 lemma degree_add_eq_left:
   366   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
   367   using degree_add_eq_right [of q p]
   368   by (simp add: add_commute)
   369 
   370 lemma degree_minus [simp]: "degree (- p) = degree p"
   371   unfolding degree_def by simp
   372 
   373 lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
   374   using degree_add_le [where p=p and q="-q"]
   375   by (simp add: diff_minus)
   376 
   377 lemma degree_diff_less:
   378   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
   379   by (auto intro: le_less_trans degree_diff_le)
   380 
   381 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
   382   by (rule poly_ext) simp
   383 
   384 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
   385   by (rule poly_ext) simp
   386 
   387 lemma minus_monom: "- monom a n = monom (-a) n"
   388   by (rule poly_ext) simp
   389 
   390 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
   391   by (cases "finite A", induct set: finite, simp_all)
   392 
   393 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
   394   by (rule poly_ext) (simp add: coeff_setsum)
   395 
   396 
   397 subsection {* Multiplication by a constant *}
   398 
   399 definition
   400   smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
   401   "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
   402 
   403 lemma Poly_smult:
   404   fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
   405   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
   406   unfolding Poly_def
   407   by (clarify, rule_tac x=n in exI, simp)
   408 
   409 lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
   410   unfolding smult_def
   411   by (simp add: Abs_poly_inverse Poly_smult coeff)
   412 
   413 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
   414   by (rule degree_le, simp add: coeff_eq_0)
   415 
   416 lemma smult_smult: "smult a (smult b p) = smult (a * b) p"
   417   by (rule poly_ext, simp add: mult_assoc)
   418 
   419 lemma smult_0_right [simp]: "smult a 0 = 0"
   420   by (rule poly_ext, simp)
   421 
   422 lemma smult_0_left [simp]: "smult 0 p = 0"
   423   by (rule poly_ext, simp)
   424 
   425 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
   426   by (rule poly_ext, simp)
   427 
   428 lemma smult_add_right:
   429   "smult a (p + q) = smult a p + smult a q"
   430   by (rule poly_ext, simp add: ring_simps)
   431 
   432 lemma smult_add_left:
   433   "smult (a + b) p = smult a p + smult b p"
   434   by (rule poly_ext, simp add: ring_simps)
   435 
   436 lemma smult_minus_right [simp]:
   437   "smult (a::'a::comm_ring) (- p) = - smult a p"
   438   by (rule poly_ext, simp)
   439 
   440 lemma smult_minus_left [simp]:
   441   "smult (- a::'a::comm_ring) p = - smult a p"
   442   by (rule poly_ext, simp)
   443 
   444 lemma smult_diff_right:
   445   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
   446   by (rule poly_ext, simp add: ring_simps)
   447 
   448 lemma smult_diff_left:
   449   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
   450   by (rule poly_ext, simp add: ring_simps)
   451 
   452 lemma smult_pCons [simp]:
   453   "smult a (pCons b p) = pCons (a * b) (smult a p)"
   454   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
   455 
   456 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
   457   by (induct n, simp add: monom_0, simp add: monom_Suc)
   458 
   459 
   460 subsection {* Multiplication of polynomials *}
   461 
   462 lemma Poly_mult_lemma:
   463   fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" and m n :: nat
   464   assumes "\<forall>i>m. f i = 0"
   465   assumes "\<forall>j>n. g j = 0"
   466   shows "\<forall>k>m+n. (\<Sum>i\<le>k. f i * g (k-i)) = 0"
   467 proof (clarify)
   468   fix k :: nat
   469   assume "m + n < k"
   470   show "(\<Sum>i\<le>k. f i * g (k - i)) = 0"
   471   proof (rule setsum_0' [rule_format])
   472     fix i :: nat
   473     assume "i \<in> {..k}" hence "i \<le> k" by simp
   474     with `m + n < k` have "m < i \<or> n < k - i" by arith
   475     thus "f i * g (k - i) = 0"
   476       using prems by auto
   477   qed
   478 qed
   479 
   480 lemma Poly_mult:
   481   fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0"
   482   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i * g (n-i)) \<in> Poly"
   483   unfolding Poly_def
   484   by (safe, rule exI, rule Poly_mult_lemma)
   485 
   486 lemma poly_mult_assoc_lemma:
   487   fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   488   shows "(\<Sum>j\<le>k. \<Sum>i\<le>j. f i (j - i) (n - j)) =
   489          (\<Sum>j\<le>k. \<Sum>i\<le>k - j. f j i (n - j - i))"
   490 proof (induct k)
   491   case 0 show ?case by simp
   492 next
   493   case (Suc k) thus ?case
   494     by (simp add: Suc_diff_le setsum_addf add_assoc
   495              cong: strong_setsum_cong)
   496 qed
   497 
   498 lemma poly_mult_commute_lemma:
   499   fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   500   shows "(\<Sum>i\<le>n. f i (n - i)) = (\<Sum>i\<le>n. f (n - i) i)"
   501 proof (rule setsum_reindex_cong)
   502   show "inj_on (\<lambda>i. n - i) {..n}"
   503     by (rule inj_onI) simp
   504   show "{..n} = (\<lambda>i. n - i) ` {..n}"
   505     by (auto, rule_tac x="n - x" in image_eqI, simp_all)
   506 next
   507   fix i assume "i \<in> {..n}"
   508   hence "n - (n - i) = i" by simp
   509   thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
   510 qed
   511 
   512 text {* TODO: move to appropriate theory *}
   513 lemma setsum_atMost_Suc_shift:
   514   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
   515   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
   516 proof (induct n)
   517   case 0 show ?case by simp
   518 next
   519   case (Suc n) note IH = this
   520   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
   521     by (rule setsum_atMost_Suc)
   522   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
   523     by (rule IH)
   524   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
   525              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
   526     by (rule add_assoc)
   527   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
   528     by (rule setsum_atMost_Suc [symmetric])
   529   finally show ?case .
   530 qed
   531 
   532 instantiation poly :: (comm_semiring_0) comm_semiring_0
   533 begin
   534 
   535 definition
   536   times_poly_def:
   537     "p * q = Abs_poly (\<lambda>n. \<Sum>i\<le>n. coeff p i * coeff q (n-i))"
   538 
   539 lemma coeff_mult:
   540   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
   541   unfolding times_poly_def
   542   by (simp add: Abs_poly_inverse coeff Poly_mult)
   543 
   544 instance proof
   545   fix p q r :: "'a poly"
   546   show 0: "0 * p = 0"
   547     by (simp add: expand_poly_eq coeff_mult)
   548   show "p * 0 = 0"
   549     by (simp add: expand_poly_eq coeff_mult)
   550   show "(p + q) * r = p * r + q * r"
   551     by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf)
   552   show "(p * q) * r = p * (q * r)"
   553   proof (rule poly_ext)
   554     fix n :: nat
   555     have "(\<Sum>j\<le>n. \<Sum>i\<le>j. coeff p i * coeff q (j - i) * coeff r (n - j)) =
   556           (\<Sum>j\<le>n. \<Sum>i\<le>n - j. coeff p j * coeff q i * coeff r (n - j - i))"
   557       by (rule poly_mult_assoc_lemma)
   558     thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n"
   559       by (simp add: coeff_mult setsum_right_distrib
   560                     setsum_left_distrib mult_assoc)
   561   qed
   562   show "p * q = q * p"
   563   proof (rule poly_ext)
   564     fix n :: nat
   565     have "(\<Sum>i\<le>n. coeff p i * coeff q (n - i)) =
   566           (\<Sum>i\<le>n. coeff p (n - i) * coeff q i)"
   567       by (rule poly_mult_commute_lemma)
   568     thus "coeff (p * q) n = coeff (q * p) n"
   569       by (simp add: coeff_mult mult_commute)
   570   qed
   571 qed
   572 
   573 end
   574 
   575 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
   576 apply (rule degree_le, simp add: coeff_mult)
   577 apply (rule Poly_mult_lemma)
   578 apply (simp_all add: coeff_eq_0)
   579 done
   580 
   581 lemma mult_pCons_left [simp]:
   582   "pCons a p * q = smult a q + pCons 0 (p * q)"
   583 apply (rule poly_ext)
   584 apply (case_tac n)
   585 apply (simp add: coeff_mult)
   586 apply (simp add: coeff_mult setsum_atMost_Suc_shift
   587             del: setsum_atMost_Suc)
   588 done
   589 
   590 lemma mult_pCons_right [simp]:
   591   "p * pCons a q = smult a p + pCons 0 (p * q)"
   592   using mult_pCons_left [of a q p] by (simp add: mult_commute)
   593 
   594 lemma mult_smult_left: "smult a p * q = smult a (p * q)"
   595   by (induct p, simp, simp add: smult_add_right smult_smult)
   596 
   597 lemma mult_smult_right: "p * smult a q = smult a (p * q)"
   598   using mult_smult_left [of a q p] by (simp add: mult_commute)
   599 
   600 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
   601   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
   602 
   603 
   604 subsection {* The unit polynomial and exponentiation *}
   605 
   606 instantiation poly :: (comm_semiring_1) comm_semiring_1
   607 begin
   608 
   609 definition
   610   one_poly_def:
   611     "1 = pCons 1 0"
   612 
   613 instance proof
   614   fix p :: "'a poly" show "1 * p = p"
   615     unfolding one_poly_def
   616     by simp
   617 next
   618   show "0 \<noteq> (1::'a poly)"
   619     unfolding one_poly_def by simp
   620 qed
   621 
   622 end
   623 
   624 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
   625   unfolding one_poly_def
   626   by (simp add: coeff_pCons split: nat.split)
   627 
   628 lemma degree_1 [simp]: "degree 1 = 0"
   629   unfolding one_poly_def
   630   by (rule degree_pCons_0)
   631 
   632 instantiation poly :: (comm_semiring_1) recpower
   633 begin
   634 
   635 primrec power_poly where
   636   power_poly_0: "(p::'a poly) ^ 0 = 1"
   637 | power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
   638 
   639 instance
   640   by default simp_all
   641 
   642 end
   643 
   644 instance poly :: (comm_ring) comm_ring ..
   645 
   646 instance poly :: (comm_ring_1) comm_ring_1 ..
   647 
   648 instantiation poly :: (comm_ring_1) number_ring
   649 begin
   650 
   651 definition
   652   "number_of k = (of_int k :: 'a poly)"
   653 
   654 instance
   655   by default (rule number_of_poly_def)
   656 
   657 end
   658 
   659 
   660 subsection {* Polynomials form an integral domain *}
   661 
   662 lemma coeff_mult_degree_sum:
   663   "coeff (p * q) (degree p + degree q) =
   664    coeff p (degree p) * coeff q (degree q)"
   665  apply (simp add: coeff_mult)
   666  apply (subst setsum_diff1' [where a="degree p"])
   667    apply simp
   668   apply simp
   669  apply (subst setsum_0' [rule_format])
   670   apply clarsimp
   671   apply (subgoal_tac "degree p < a \<or> degree q < degree p + degree q - a")
   672    apply (force simp add: coeff_eq_0)
   673   apply arith
   674  apply simp
   675 done
   676 
   677 instance poly :: (idom) idom
   678 proof
   679   fix p q :: "'a poly"
   680   assume "p \<noteq> 0" and "q \<noteq> 0"
   681   have "coeff (p * q) (degree p + degree q) =
   682         coeff p (degree p) * coeff q (degree q)"
   683     by (rule coeff_mult_degree_sum)
   684   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
   685     using `p \<noteq> 0` and `q \<noteq> 0` by simp
   686   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
   687   thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
   688 qed
   689 
   690 lemma degree_mult_eq:
   691   fixes p q :: "'a::idom poly"
   692   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
   693 apply (rule order_antisym [OF degree_mult_le le_degree])
   694 apply (simp add: coeff_mult_degree_sum)
   695 done
   696 
   697 lemma dvd_imp_degree_le:
   698   fixes p q :: "'a::idom poly"
   699   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
   700   by (erule dvdE, simp add: degree_mult_eq)
   701 
   702 
   703 subsection {* Long division of polynomials *}
   704 
   705 definition
   706   divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
   707 where
   708   "divmod_poly_rel x y q r \<longleftrightarrow>
   709     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
   710 
   711 lemma divmod_poly_rel_0:
   712   "divmod_poly_rel 0 y 0 0"
   713   unfolding divmod_poly_rel_def by simp
   714 
   715 lemma divmod_poly_rel_by_0:
   716   "divmod_poly_rel x 0 0 x"
   717   unfolding divmod_poly_rel_def by simp
   718 
   719 lemma eq_zero_or_degree_less:
   720   assumes "degree p \<le> n" and "coeff p n = 0"
   721   shows "p = 0 \<or> degree p < n"
   722 proof (cases n)
   723   case 0
   724   with `degree p \<le> n` and `coeff p n = 0`
   725   have "coeff p (degree p) = 0" by simp
   726   then have "p = 0" by simp
   727   then show ?thesis ..
   728 next
   729   case (Suc m)
   730   have "\<forall>i>n. coeff p i = 0"
   731     using `degree p \<le> n` by (simp add: coeff_eq_0)
   732   then have "\<forall>i\<ge>n. coeff p i = 0"
   733     using `coeff p n = 0` by (simp add: le_less)
   734   then have "\<forall>i>m. coeff p i = 0"
   735     using `n = Suc m` by (simp add: less_eq_Suc_le)
   736   then have "degree p \<le> m"
   737     by (rule degree_le)
   738   then have "degree p < n"
   739     using `n = Suc m` by (simp add: less_Suc_eq_le)
   740   then show ?thesis ..
   741 qed
   742 
   743 lemma divmod_poly_rel_pCons:
   744   assumes rel: "divmod_poly_rel x y q r"
   745   assumes y: "y \<noteq> 0"
   746   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
   747   shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
   748     (is "divmod_poly_rel ?x y ?q ?r")
   749 proof -
   750   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
   751     using assms unfolding divmod_poly_rel_def by simp_all
   752 
   753   have 1: "?x = ?q * y + ?r"
   754     using b x by simp
   755 
   756   have 2: "?r = 0 \<or> degree ?r < degree y"
   757   proof (rule eq_zero_or_degree_less)
   758     have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
   759       by (rule degree_diff_le)
   760     also have "\<dots> \<le> degree y"
   761     proof (rule min_max.le_supI)
   762       show "degree (pCons a r) \<le> degree y"
   763         using r by auto
   764       show "degree (smult b y) \<le> degree y"
   765         by (rule degree_smult_le)
   766     qed
   767     finally show "degree ?r \<le> degree y" .
   768   next
   769     show "coeff ?r (degree y) = 0"
   770       using `y \<noteq> 0` unfolding b by simp
   771   qed
   772 
   773   from 1 2 show ?thesis
   774     unfolding divmod_poly_rel_def
   775     using `y \<noteq> 0` by simp
   776 qed
   777 
   778 lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
   779 apply (cases "y = 0")
   780 apply (fast intro!: divmod_poly_rel_by_0)
   781 apply (induct x)
   782 apply (fast intro!: divmod_poly_rel_0)
   783 apply (fast intro!: divmod_poly_rel_pCons)
   784 done
   785 
   786 lemma divmod_poly_rel_unique:
   787   assumes 1: "divmod_poly_rel x y q1 r1"
   788   assumes 2: "divmod_poly_rel x y q2 r2"
   789   shows "q1 = q2 \<and> r1 = r2"
   790 proof (cases "y = 0")
   791   assume "y = 0" with assms show ?thesis
   792     by (simp add: divmod_poly_rel_def)
   793 next
   794   assume [simp]: "y \<noteq> 0"
   795   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
   796     unfolding divmod_poly_rel_def by simp_all
   797   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
   798     unfolding divmod_poly_rel_def by simp_all
   799   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
   800     by (simp add: ring_simps)
   801   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
   802     by (auto intro: degree_diff_less)
   803 
   804   show "q1 = q2 \<and> r1 = r2"
   805   proof (rule ccontr)
   806     assume "\<not> (q1 = q2 \<and> r1 = r2)"
   807     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
   808     with r3 have "degree (r2 - r1) < degree y" by simp
   809     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
   810     also have "\<dots> = degree ((q1 - q2) * y)"
   811       using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
   812     also have "\<dots> = degree (r2 - r1)"
   813       using q3 by simp
   814     finally have "degree (r2 - r1) < degree (r2 - r1)" .
   815     then show "False" by simp
   816   qed
   817 qed
   818 
   819 lemmas divmod_poly_rel_unique_div =
   820   divmod_poly_rel_unique [THEN conjunct1, standard]
   821 
   822 lemmas divmod_poly_rel_unique_mod =
   823   divmod_poly_rel_unique [THEN conjunct2, standard]
   824 
   825 instantiation poly :: (field) ring_div
   826 begin
   827 
   828 definition div_poly where
   829   [code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
   830 
   831 definition mod_poly where
   832   [code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
   833 
   834 lemma div_poly_eq:
   835   "divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
   836 unfolding div_poly_def
   837 by (fast elim: divmod_poly_rel_unique_div)
   838 
   839 lemma mod_poly_eq:
   840   "divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
   841 unfolding mod_poly_def
   842 by (fast elim: divmod_poly_rel_unique_mod)
   843 
   844 lemma divmod_poly_rel:
   845   "divmod_poly_rel x y (x div y) (x mod y)"
   846 proof -
   847   from divmod_poly_rel_exists
   848     obtain q r where "divmod_poly_rel x y q r" by fast
   849   thus ?thesis
   850     by (simp add: div_poly_eq mod_poly_eq)
   851 qed
   852 
   853 instance proof
   854   fix x y :: "'a poly"
   855   show "x div y * y + x mod y = x"
   856     using divmod_poly_rel [of x y]
   857     by (simp add: divmod_poly_rel_def)
   858 next
   859   fix x :: "'a poly"
   860   have "divmod_poly_rel x 0 0 x"
   861     by (rule divmod_poly_rel_by_0)
   862   thus "x div 0 = 0"
   863     by (rule div_poly_eq)
   864 next
   865   fix y :: "'a poly"
   866   have "divmod_poly_rel 0 y 0 0"
   867     by (rule divmod_poly_rel_0)
   868   thus "0 div y = 0"
   869     by (rule div_poly_eq)
   870 next
   871   fix x y z :: "'a poly"
   872   assume "y \<noteq> 0"
   873   hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
   874     using divmod_poly_rel [of x y]
   875     by (simp add: divmod_poly_rel_def left_distrib)
   876   thus "(x + z * y) div y = z + x div y"
   877     by (rule div_poly_eq)
   878 qed
   879 
   880 end
   881 
   882 lemma degree_mod_less:
   883   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
   884   using divmod_poly_rel [of x y]
   885   unfolding divmod_poly_rel_def by simp
   886 
   887 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
   888 proof -
   889   assume "degree x < degree y"
   890   hence "divmod_poly_rel x y 0 x"
   891     by (simp add: divmod_poly_rel_def)
   892   thus "x div y = 0" by (rule div_poly_eq)
   893 qed
   894 
   895 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
   896 proof -
   897   assume "degree x < degree y"
   898   hence "divmod_poly_rel x y 0 x"
   899     by (simp add: divmod_poly_rel_def)
   900   thus "x mod y = x" by (rule mod_poly_eq)
   901 qed
   902 
   903 lemma mod_pCons:
   904   fixes a and x
   905   assumes y: "y \<noteq> 0"
   906   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
   907   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
   908 unfolding b
   909 apply (rule mod_poly_eq)
   910 apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
   911 done
   912 
   913 
   914 subsection {* Evaluation of polynomials *}
   915 
   916 definition
   917   poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
   918   "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
   919 
   920 lemma poly_0 [simp]: "poly 0 x = 0"
   921   unfolding poly_def by (simp add: poly_rec_0)
   922 
   923 lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
   924   unfolding poly_def by (simp add: poly_rec_pCons)
   925 
   926 lemma poly_1 [simp]: "poly 1 x = 1"
   927   unfolding one_poly_def by simp
   928 
   929 lemma poly_monom:
   930   fixes a x :: "'a::{comm_semiring_1,recpower}"
   931   shows "poly (monom a n) x = a * x ^ n"
   932   by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
   933 
   934 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
   935   apply (induct p arbitrary: q, simp)
   936   apply (case_tac q, simp, simp add: ring_simps)
   937   done
   938 
   939 lemma poly_minus [simp]:
   940   fixes x :: "'a::comm_ring"
   941   shows "poly (- p) x = - poly p x"
   942   by (induct p, simp_all)
   943 
   944 lemma poly_diff [simp]:
   945   fixes x :: "'a::comm_ring"
   946   shows "poly (p - q) x = poly p x - poly q x"
   947   by (simp add: diff_minus)
   948 
   949 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
   950   by (cases "finite A", induct set: finite, simp_all)
   951 
   952 lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
   953   by (induct p, simp, simp add: ring_simps)
   954 
   955 lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
   956   by (induct p, simp_all, simp add: ring_simps)
   957 
   958 
   959 subsection {* Synthetic division *}
   960 
   961 definition
   962   synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
   963 where
   964   "synthetic_divmod p c =
   965     poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
   966 
   967 definition
   968   synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
   969 where
   970   "synthetic_div p c = fst (synthetic_divmod p c)"
   971 
   972 lemma synthetic_divmod_0 [simp]:
   973   "synthetic_divmod 0 c = (0, 0)"
   974   unfolding synthetic_divmod_def
   975   by (simp add: poly_rec_0)
   976 
   977 lemma synthetic_divmod_pCons [simp]:
   978   "synthetic_divmod (pCons a p) c =
   979     (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
   980   unfolding synthetic_divmod_def
   981   by (simp add: poly_rec_pCons)
   982 
   983 lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
   984   by (induct p, simp, simp add: split_def)
   985 
   986 lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
   987   unfolding synthetic_div_def by simp
   988 
   989 lemma synthetic_div_pCons [simp]:
   990   "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
   991   unfolding synthetic_div_def
   992   by (simp add: split_def snd_synthetic_divmod)
   993 
   994 lemma synthetic_div_eq_0_iff:
   995   "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
   996   by (induct p, simp, case_tac p, simp)
   997 
   998 lemma degree_synthetic_div:
   999   "degree (synthetic_div p c) = degree p - 1"
  1000   by (induct p, simp, simp add: synthetic_div_eq_0_iff)
  1001 
  1002 lemma synthetic_div_correct:
  1003   "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
  1004   by (induct p) simp_all
  1005 
  1006 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
  1007 by (induct p arbitrary: a) simp_all
  1008 
  1009 lemma synthetic_div_unique:
  1010   "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
  1011 apply (induct p arbitrary: q r)
  1012 apply (simp, frule synthetic_div_unique_lemma, simp)
  1013 apply (case_tac q, force)
  1014 done
  1015 
  1016 lemma synthetic_div_correct':
  1017   fixes c :: "'a::comm_ring_1"
  1018   shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
  1019   using synthetic_div_correct [of p c]
  1020   by (simp add: group_simps)
  1021 
  1022 lemma poly_eq_0_iff_dvd:
  1023   fixes c :: "'a::idom"
  1024   shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
  1025 proof
  1026   assume "poly p c = 0"
  1027   with synthetic_div_correct' [of c p]
  1028   have "p = [:-c, 1:] * synthetic_div p c" by simp
  1029   then show "[:-c, 1:] dvd p" ..
  1030 next
  1031   assume "[:-c, 1:] dvd p"
  1032   then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
  1033   then show "poly p c = 0" by simp
  1034 qed
  1035 
  1036 lemma dvd_iff_poly_eq_0:
  1037   fixes c :: "'a::idom"
  1038   shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
  1039   by (simp add: poly_eq_0_iff_dvd)
  1040 
  1041 end