src/HOL/Polynomial.thy
 author huffman Mon Jan 12 08:15:07 2009 -0800 (2009-01-12) changeset 29453 de4f26f59135 parent 29451 5f0cb3fa530d child 29454 b0f586f38dd7 permissions -rw-r--r--
```     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 lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
```
```   104   unfolding Poly_def by (auto split: nat.split)
```
```   105
```
```   106 lemma coeff_pCons:
```
```   107   "coeff (pCons a p) = nat_case a (coeff p)"
```
```   108   unfolding pCons_def
```
```   109   by (simp add: Abs_poly_inverse Poly_nat_case coeff)
```
```   110
```
```   111 lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
```
```   112   by (simp add: coeff_pCons)
```
```   113
```
```   114 lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
```
```   115   by (simp add: coeff_pCons)
```
```   116
```
```   117 lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
```
```   118 by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
```
```   119
```
```   120 lemma degree_pCons_eq:
```
```   121   "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
```
```   122 apply (rule order_antisym [OF degree_pCons_le])
```
```   123 apply (rule le_degree, simp)
```
```   124 done
```
```   125
```
```   126 lemma degree_pCons_0: "degree (pCons a 0) = 0"
```
```   127 apply (rule order_antisym [OF _ le0])
```
```   128 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   129 done
```
```   130
```
```   131 lemma degree_pCons_eq_if:
```
```   132   "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
```
```   133 apply (cases "p = 0", simp_all)
```
```   134 apply (rule order_antisym [OF _ le0])
```
```   135 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
```
```   136 apply (rule order_antisym [OF degree_pCons_le])
```
```   137 apply (rule le_degree, simp)
```
```   138 done
```
```   139
```
```   140 lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
```
```   141 by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   142
```
```   143 lemma pCons_eq_iff [simp]:
```
```   144   "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
```
```   145 proof (safe)
```
```   146   assume "pCons a p = pCons b q"
```
```   147   then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
```
```   148   then show "a = b" by simp
```
```   149 next
```
```   150   assume "pCons a p = pCons b q"
```
```   151   then have "\<forall>n. coeff (pCons a p) (Suc n) =
```
```   152                  coeff (pCons b q) (Suc n)" by simp
```
```   153   then show "p = q" by (simp add: expand_poly_eq)
```
```   154 qed
```
```   155
```
```   156 lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
```
```   157   using pCons_eq_iff [of a p 0 0] by simp
```
```   158
```
```   159 lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
```
```   160   unfolding Poly_def
```
```   161   by (clarify, rule_tac x=n in exI, simp)
```
```   162
```
```   163 lemma pCons_cases [cases type: poly]:
```
```   164   obtains (pCons) a q where "p = pCons a q"
```
```   165 proof
```
```   166   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
```
```   167     by (rule poly_ext)
```
```   168        (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
```
```   169              split: nat.split)
```
```   170 qed
```
```   171
```
```   172 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
```
```   173   assumes zero: "P 0"
```
```   174   assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
```
```   175   shows "P p"
```
```   176 proof (induct p rule: measure_induct_rule [where f=degree])
```
```   177   case (less p)
```
```   178   obtain a q where "p = pCons a q" by (rule pCons_cases)
```
```   179   have "P q"
```
```   180   proof (cases "q = 0")
```
```   181     case True
```
```   182     then show "P q" by (simp add: zero)
```
```   183   next
```
```   184     case False
```
```   185     then have "degree (pCons a q) = Suc (degree q)"
```
```   186       by (rule degree_pCons_eq)
```
```   187     then have "degree q < degree p"
```
```   188       using `p = pCons a q` by simp
```
```   189     then show "P q"
```
```   190       by (rule less.hyps)
```
```   191   qed
```
```   192   then have "P (pCons a q)"
```
```   193     by (rule pCons)
```
```   194   then show ?case
```
```   195     using `p = pCons a q` by simp
```
```   196 qed
```
```   197
```
```   198
```
```   199 subsection {* Monomials *}
```
```   200
```
```   201 definition
```
```   202   monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
```
```   203   "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
```
```   204
```
```   205 lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
```
```   206   unfolding monom_def
```
```   207   by (subst Abs_poly_inverse, auto simp add: Poly_def)
```
```   208
```
```   209 lemma monom_0: "monom a 0 = pCons a 0"
```
```   210   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   211
```
```   212 lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
```
```   213   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   214
```
```   215 lemma monom_eq_0 [simp]: "monom 0 n = 0"
```
```   216   by (rule poly_ext) simp
```
```   217
```
```   218 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
```
```   219   by (simp add: expand_poly_eq)
```
```   220
```
```   221 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
```
```   222   by (simp add: expand_poly_eq)
```
```   223
```
```   224 lemma degree_monom_le: "degree (monom a n) \<le> n"
```
```   225   by (rule degree_le, simp)
```
```   226
```
```   227 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
```
```   228   apply (rule order_antisym [OF degree_monom_le])
```
```   229   apply (rule le_degree, simp)
```
```   230   done
```
```   231
```
```   232
```
```   233 subsection {* Addition and subtraction *}
```
```   234
```
```   235 instantiation poly :: (comm_monoid_add) comm_monoid_add
```
```   236 begin
```
```   237
```
```   238 definition
```
```   239   plus_poly_def [code del]:
```
```   240     "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
```
```   241
```
```   242 lemma Poly_add:
```
```   243   fixes f g :: "nat \<Rightarrow> 'a"
```
```   244   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
```
```   245   unfolding Poly_def
```
```   246   apply (clarify, rename_tac m n)
```
```   247   apply (rule_tac x="max m n" in exI, simp)
```
```   248   done
```
```   249
```
```   250 lemma coeff_add [simp]:
```
```   251   "coeff (p + q) n = coeff p n + coeff q n"
```
```   252   unfolding plus_poly_def
```
```   253   by (simp add: Abs_poly_inverse coeff Poly_add)
```
```   254
```
```   255 instance proof
```
```   256   fix p q r :: "'a poly"
```
```   257   show "(p + q) + r = p + (q + r)"
```
```   258     by (simp add: expand_poly_eq add_assoc)
```
```   259   show "p + q = q + p"
```
```   260     by (simp add: expand_poly_eq add_commute)
```
```   261   show "0 + p = p"
```
```   262     by (simp add: expand_poly_eq)
```
```   263 qed
```
```   264
```
```   265 end
```
```   266
```
```   267 instantiation poly :: (ab_group_add) ab_group_add
```
```   268 begin
```
```   269
```
```   270 definition
```
```   271   uminus_poly_def [code del]:
```
```   272     "- p = Abs_poly (\<lambda>n. - coeff p n)"
```
```   273
```
```   274 definition
```
```   275   minus_poly_def [code del]:
```
```   276     "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
```
```   277
```
```   278 lemma Poly_minus:
```
```   279   fixes f :: "nat \<Rightarrow> 'a"
```
```   280   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
```
```   281   unfolding Poly_def by simp
```
```   282
```
```   283 lemma Poly_diff:
```
```   284   fixes f g :: "nat \<Rightarrow> 'a"
```
```   285   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
```
```   286   unfolding diff_minus by (simp add: Poly_add Poly_minus)
```
```   287
```
```   288 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
```
```   289   unfolding uminus_poly_def
```
```   290   by (simp add: Abs_poly_inverse coeff Poly_minus)
```
```   291
```
```   292 lemma coeff_diff [simp]:
```
```   293   "coeff (p - q) n = coeff p n - coeff q n"
```
```   294   unfolding minus_poly_def
```
```   295   by (simp add: Abs_poly_inverse coeff Poly_diff)
```
```   296
```
```   297 instance proof
```
```   298   fix p q :: "'a poly"
```
```   299   show "- p + p = 0"
```
```   300     by (simp add: expand_poly_eq)
```
```   301   show "p - q = p + - q"
```
```   302     by (simp add: expand_poly_eq diff_minus)
```
```   303 qed
```
```   304
```
```   305 end
```
```   306
```
```   307 lemma add_pCons [simp]:
```
```   308   "pCons a p + pCons b q = pCons (a + b) (p + q)"
```
```   309   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   310
```
```   311 lemma minus_pCons [simp]:
```
```   312   "- pCons a p = pCons (- a) (- p)"
```
```   313   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   314
```
```   315 lemma diff_pCons [simp]:
```
```   316   "pCons a p - pCons b q = pCons (a - b) (p - q)"
```
```   317   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   318
```
```   319 lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
```
```   320   by (rule degree_le, auto simp add: coeff_eq_0)
```
```   321
```
```   322 lemma degree_add_less:
```
```   323   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
```
```   324   by (auto intro: le_less_trans degree_add_le)
```
```   325
```
```   326 lemma degree_add_eq_right:
```
```   327   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
```
```   328   apply (cases "q = 0", simp)
```
```   329   apply (rule order_antisym)
```
```   330   apply (rule ord_le_eq_trans [OF degree_add_le])
```
```   331   apply simp
```
```   332   apply (rule le_degree)
```
```   333   apply (simp add: coeff_eq_0)
```
```   334   done
```
```   335
```
```   336 lemma degree_add_eq_left:
```
```   337   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
```
```   338   using degree_add_eq_right [of q p]
```
```   339   by (simp add: add_commute)
```
```   340
```
```   341 lemma degree_minus [simp]: "degree (- p) = degree p"
```
```   342   unfolding degree_def by simp
```
```   343
```
```   344 lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
```
```   345   using degree_add_le [where p=p and q="-q"]
```
```   346   by (simp add: diff_minus)
```
```   347
```
```   348 lemma degree_diff_less:
```
```   349   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
```
```   350   by (auto intro: le_less_trans degree_diff_le)
```
```   351
```
```   352 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
```
```   353   by (rule poly_ext) simp
```
```   354
```
```   355 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
```
```   356   by (rule poly_ext) simp
```
```   357
```
```   358 lemma minus_monom: "- monom a n = monom (-a) n"
```
```   359   by (rule poly_ext) simp
```
```   360
```
```   361 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
```
```   362   by (cases "finite A", induct set: finite, simp_all)
```
```   363
```
```   364 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
```
```   365   by (rule poly_ext) (simp add: coeff_setsum)
```
```   366
```
```   367
```
```   368 subsection {* Multiplication by a constant *}
```
```   369
```
```   370 definition
```
```   371   smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   372   "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
```
```   373
```
```   374 lemma Poly_smult:
```
```   375   fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
```
```   376   shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
```
```   377   unfolding Poly_def
```
```   378   by (clarify, rule_tac x=n in exI, simp)
```
```   379
```
```   380 lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
```
```   381   unfolding smult_def
```
```   382   by (simp add: Abs_poly_inverse Poly_smult coeff)
```
```   383
```
```   384 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
```
```   385   by (rule degree_le, simp add: coeff_eq_0)
```
```   386
```
```   387 lemma smult_smult: "smult a (smult b p) = smult (a * b) p"
```
```   388   by (rule poly_ext, simp add: mult_assoc)
```
```   389
```
```   390 lemma smult_0_right [simp]: "smult a 0 = 0"
```
```   391   by (rule poly_ext, simp)
```
```   392
```
```   393 lemma smult_0_left [simp]: "smult 0 p = 0"
```
```   394   by (rule poly_ext, simp)
```
```   395
```
```   396 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
```
```   397   by (rule poly_ext, simp)
```
```   398
```
```   399 lemma smult_add_right:
```
```   400   "smult a (p + q) = smult a p + smult a q"
```
```   401   by (rule poly_ext, simp add: ring_simps)
```
```   402
```
```   403 lemma smult_add_left:
```
```   404   "smult (a + b) p = smult a p + smult b p"
```
```   405   by (rule poly_ext, simp add: ring_simps)
```
```   406
```
```   407 lemma smult_minus_right:
```
```   408   "smult (a::'a::comm_ring) (- p) = - smult a p"
```
```   409   by (rule poly_ext, simp)
```
```   410
```
```   411 lemma smult_minus_left:
```
```   412   "smult (- a::'a::comm_ring) p = - smult a p"
```
```   413   by (rule poly_ext, simp)
```
```   414
```
```   415 lemma smult_diff_right:
```
```   416   "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
```
```   417   by (rule poly_ext, simp add: ring_simps)
```
```   418
```
```   419 lemma smult_diff_left:
```
```   420   "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
```
```   421   by (rule poly_ext, simp add: ring_simps)
```
```   422
```
```   423 lemma smult_pCons [simp]:
```
```   424   "smult a (pCons b p) = pCons (a * b) (smult a p)"
```
```   425   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
```
```   426
```
```   427 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
```
```   428   by (induct n, simp add: monom_0, simp add: monom_Suc)
```
```   429
```
```   430
```
```   431 subsection {* Multiplication of polynomials *}
```
```   432
```
```   433 lemma Poly_mult_lemma:
```
```   434   fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" and m n :: nat
```
```   435   assumes "\<forall>i>m. f i = 0"
```
```   436   assumes "\<forall>j>n. g j = 0"
```
```   437   shows "\<forall>k>m+n. (\<Sum>i\<le>k. f i * g (k-i)) = 0"
```
```   438 proof (clarify)
```
```   439   fix k :: nat
```
```   440   assume "m + n < k"
```
```   441   show "(\<Sum>i\<le>k. f i * g (k - i)) = 0"
```
```   442   proof (rule setsum_0' [rule_format])
```
```   443     fix i :: nat
```
```   444     assume "i \<in> {..k}" hence "i \<le> k" by simp
```
```   445     with `m + n < k` have "m < i \<or> n < k - i" by arith
```
```   446     thus "f i * g (k - i) = 0"
```
```   447       using prems by auto
```
```   448   qed
```
```   449 qed
```
```   450
```
```   451 lemma Poly_mult:
```
```   452   fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0"
```
```   453   shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i * g (n-i)) \<in> Poly"
```
```   454   unfolding Poly_def
```
```   455   by (safe, rule exI, rule Poly_mult_lemma)
```
```   456
```
```   457 lemma poly_mult_assoc_lemma:
```
```   458   fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   459   shows "(\<Sum>j\<le>k. \<Sum>i\<le>j. f i (j - i) (n - j)) =
```
```   460          (\<Sum>j\<le>k. \<Sum>i\<le>k - j. f j i (n - j - i))"
```
```   461 proof (induct k)
```
```   462   case 0 show ?case by simp
```
```   463 next
```
```   464   case (Suc k) thus ?case
```
```   465     by (simp add: Suc_diff_le setsum_addf add_assoc
```
```   466              cong: strong_setsum_cong)
```
```   467 qed
```
```   468
```
```   469 lemma poly_mult_commute_lemma:
```
```   470   fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   471   shows "(\<Sum>i\<le>n. f i (n - i)) = (\<Sum>i\<le>n. f (n - i) i)"
```
```   472 proof (rule setsum_reindex_cong)
```
```   473   show "inj_on (\<lambda>i. n - i) {..n}"
```
```   474     by (rule inj_onI) simp
```
```   475   show "{..n} = (\<lambda>i. n - i) ` {..n}"
```
```   476     by (auto, rule_tac x="n - x" in image_eqI, simp_all)
```
```   477 next
```
```   478   fix i assume "i \<in> {..n}"
```
```   479   hence "n - (n - i) = i" by simp
```
```   480   thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
```
```   481 qed
```
```   482
```
```   483 text {* TODO: move to appropriate theory *}
```
```   484 lemma setsum_atMost_Suc_shift:
```
```   485   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
```
```   486   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   487 proof (induct n)
```
```   488   case 0 show ?case by simp
```
```   489 next
```
```   490   case (Suc n) note IH = this
```
```   491   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
```
```   492     by (rule setsum_atMost_Suc)
```
```   493   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
```
```   494     by (rule IH)
```
```   495   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
```
```   496              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
```
```   497     by (rule add_assoc)
```
```   498   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
```
```   499     by (rule setsum_atMost_Suc [symmetric])
```
```   500   finally show ?case .
```
```   501 qed
```
```   502
```
```   503 instantiation poly :: (comm_semiring_0) comm_semiring_0
```
```   504 begin
```
```   505
```
```   506 definition
```
```   507   times_poly_def:
```
```   508     "p * q = Abs_poly (\<lambda>n. \<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   509
```
```   510 lemma coeff_mult:
```
```   511   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
```
```   512   unfolding times_poly_def
```
```   513   by (simp add: Abs_poly_inverse coeff Poly_mult)
```
```   514
```
```   515 instance proof
```
```   516   fix p q r :: "'a poly"
```
```   517   show 0: "0 * p = 0"
```
```   518     by (simp add: expand_poly_eq coeff_mult)
```
```   519   show "p * 0 = 0"
```
```   520     by (simp add: expand_poly_eq coeff_mult)
```
```   521   show "(p + q) * r = p * r + q * r"
```
```   522     by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf)
```
```   523   show "(p * q) * r = p * (q * r)"
```
```   524   proof (rule poly_ext)
```
```   525     fix n :: nat
```
```   526     have "(\<Sum>j\<le>n. \<Sum>i\<le>j. coeff p i * coeff q (j - i) * coeff r (n - j)) =
```
```   527           (\<Sum>j\<le>n. \<Sum>i\<le>n - j. coeff p j * coeff q i * coeff r (n - j - i))"
```
```   528       by (rule poly_mult_assoc_lemma)
```
```   529     thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n"
```
```   530       by (simp add: coeff_mult setsum_right_distrib
```
```   531                     setsum_left_distrib mult_assoc)
```
```   532   qed
```
```   533   show "p * q = q * p"
```
```   534   proof (rule poly_ext)
```
```   535     fix n :: nat
```
```   536     have "(\<Sum>i\<le>n. coeff p i * coeff q (n - i)) =
```
```   537           (\<Sum>i\<le>n. coeff p (n - i) * coeff q i)"
```
```   538       by (rule poly_mult_commute_lemma)
```
```   539     thus "coeff (p * q) n = coeff (q * p) n"
```
```   540       by (simp add: coeff_mult mult_commute)
```
```   541   qed
```
```   542 qed
```
```   543
```
```   544 end
```
```   545
```
```   546 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
```
```   547 apply (rule degree_le, simp add: coeff_mult)
```
```   548 apply (rule Poly_mult_lemma)
```
```   549 apply (simp_all add: coeff_eq_0)
```
```   550 done
```
```   551
```
```   552 lemma mult_pCons_left [simp]:
```
```   553   "pCons a p * q = smult a q + pCons 0 (p * q)"
```
```   554 apply (rule poly_ext)
```
```   555 apply (case_tac n)
```
```   556 apply (simp add: coeff_mult)
```
```   557 apply (simp add: coeff_mult setsum_atMost_Suc_shift
```
```   558             del: setsum_atMost_Suc)
```
```   559 done
```
```   560
```
```   561 lemma mult_pCons_right [simp]:
```
```   562   "p * pCons a q = smult a p + pCons 0 (p * q)"
```
```   563   using mult_pCons_left [of a q p] by (simp add: mult_commute)
```
```   564
```
```   565 lemma mult_smult_left: "smult a p * q = smult a (p * q)"
```
```   566   by (induct p, simp, simp add: smult_add_right smult_smult)
```
```   567
```
```   568 lemma mult_smult_right: "p * smult a q = smult a (p * q)"
```
```   569   using mult_smult_left [of a q p] by (simp add: mult_commute)
```
```   570
```
```   571 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
```
```   572   by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
```
```   573
```
```   574
```
```   575 subsection {* The unit polynomial and exponentiation *}
```
```   576
```
```   577 instantiation poly :: (comm_semiring_1) comm_semiring_1
```
```   578 begin
```
```   579
```
```   580 definition
```
```   581   one_poly_def:
```
```   582     "1 = pCons 1 0"
```
```   583
```
```   584 instance proof
```
```   585   fix p :: "'a poly" show "1 * p = p"
```
```   586     unfolding one_poly_def
```
```   587     by simp
```
```   588 next
```
```   589   show "0 \<noteq> (1::'a poly)"
```
```   590     unfolding one_poly_def by simp
```
```   591 qed
```
```   592
```
```   593 end
```
```   594
```
```   595 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
```
```   596   unfolding one_poly_def
```
```   597   by (simp add: coeff_pCons split: nat.split)
```
```   598
```
```   599 lemma degree_1 [simp]: "degree 1 = 0"
```
```   600   unfolding one_poly_def
```
```   601   by (rule degree_pCons_0)
```
```   602
```
```   603 instantiation poly :: (comm_semiring_1) recpower
```
```   604 begin
```
```   605
```
```   606 primrec power_poly where
```
```   607   power_poly_0: "(p::'a poly) ^ 0 = 1"
```
```   608 | power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
```
```   609
```
```   610 instance
```
```   611   by default simp_all
```
```   612
```
```   613 end
```
```   614
```
```   615 instance poly :: (comm_ring) comm_ring ..
```
```   616
```
```   617 instance poly :: (comm_ring_1) comm_ring_1 ..
```
```   618
```
```   619 instantiation poly :: (comm_ring_1) number_ring
```
```   620 begin
```
```   621
```
```   622 definition
```
```   623   "number_of k = (of_int k :: 'a poly)"
```
```   624
```
```   625 instance
```
```   626   by default (rule number_of_poly_def)
```
```   627
```
```   628 end
```
```   629
```
```   630
```
```   631 subsection {* Polynomials form an integral domain *}
```
```   632
```
```   633 lemma coeff_mult_degree_sum:
```
```   634   "coeff (p * q) (degree p + degree q) =
```
```   635    coeff p (degree p) * coeff q (degree q)"
```
```   636  apply (simp add: coeff_mult)
```
```   637  apply (subst setsum_diff1' [where a="degree p"])
```
```   638    apply simp
```
```   639   apply simp
```
```   640  apply (subst setsum_0' [rule_format])
```
```   641   apply clarsimp
```
```   642   apply (subgoal_tac "degree p < a \<or> degree q < degree p + degree q - a")
```
```   643    apply (force simp add: coeff_eq_0)
```
```   644   apply arith
```
```   645  apply simp
```
```   646 done
```
```   647
```
```   648 instance poly :: (idom) idom
```
```   649 proof
```
```   650   fix p q :: "'a poly"
```
```   651   assume "p \<noteq> 0" and "q \<noteq> 0"
```
```   652   have "coeff (p * q) (degree p + degree q) =
```
```   653         coeff p (degree p) * coeff q (degree q)"
```
```   654     by (rule coeff_mult_degree_sum)
```
```   655   also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
```
```   656     using `p \<noteq> 0` and `q \<noteq> 0` by simp
```
```   657   finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
```
```   658   thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
```
```   659 qed
```
```   660
```
```   661 lemma degree_mult_eq:
```
```   662   fixes p q :: "'a::idom poly"
```
```   663   shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
```
```   664 apply (rule order_antisym [OF degree_mult_le le_degree])
```
```   665 apply (simp add: coeff_mult_degree_sum)
```
```   666 done
```
```   667
```
```   668 lemma dvd_imp_degree_le:
```
```   669   fixes p q :: "'a::idom poly"
```
```   670   shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
```
```   671   by (erule dvdE, simp add: degree_mult_eq)
```
```   672
```
```   673
```
```   674 subsection {* Long division of polynomials *}
```
```   675
```
```   676 definition
```
```   677   divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
```
```   678 where
```
```   679   "divmod_poly_rel x y q r \<longleftrightarrow>
```
```   680     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
```
```   681
```
```   682 lemma divmod_poly_rel_0:
```
```   683   "divmod_poly_rel 0 y 0 0"
```
```   684   unfolding divmod_poly_rel_def by simp
```
```   685
```
```   686 lemma divmod_poly_rel_by_0:
```
```   687   "divmod_poly_rel x 0 0 x"
```
```   688   unfolding divmod_poly_rel_def by simp
```
```   689
```
```   690 lemma eq_zero_or_degree_less:
```
```   691   assumes "degree p \<le> n" and "coeff p n = 0"
```
```   692   shows "p = 0 \<or> degree p < n"
```
```   693 proof (cases n)
```
```   694   case 0
```
```   695   with `degree p \<le> n` and `coeff p n = 0`
```
```   696   have "coeff p (degree p) = 0" by simp
```
```   697   then have "p = 0" by simp
```
```   698   then show ?thesis ..
```
```   699 next
```
```   700   case (Suc m)
```
```   701   have "\<forall>i>n. coeff p i = 0"
```
```   702     using `degree p \<le> n` by (simp add: coeff_eq_0)
```
```   703   then have "\<forall>i\<ge>n. coeff p i = 0"
```
```   704     using `coeff p n = 0` by (simp add: le_less)
```
```   705   then have "\<forall>i>m. coeff p i = 0"
```
```   706     using `n = Suc m` by (simp add: less_eq_Suc_le)
```
```   707   then have "degree p \<le> m"
```
```   708     by (rule degree_le)
```
```   709   then have "degree p < n"
```
```   710     using `n = Suc m` by (simp add: less_Suc_eq_le)
```
```   711   then show ?thesis ..
```
```   712 qed
```
```   713
```
```   714 lemma divmod_poly_rel_pCons:
```
```   715   assumes rel: "divmod_poly_rel x y q r"
```
```   716   assumes y: "y \<noteq> 0"
```
```   717   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
```
```   718   shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
```
```   719     (is "divmod_poly_rel ?x y ?q ?r")
```
```   720 proof -
```
```   721   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
```
```   722     using assms unfolding divmod_poly_rel_def by simp_all
```
```   723
```
```   724   have 1: "?x = ?q * y + ?r"
```
```   725     using b x by simp
```
```   726
```
```   727   have 2: "?r = 0 \<or> degree ?r < degree y"
```
```   728   proof (rule eq_zero_or_degree_less)
```
```   729     have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
```
```   730       by (rule degree_diff_le)
```
```   731     also have "\<dots> \<le> degree y"
```
```   732     proof (rule min_max.le_supI)
```
```   733       show "degree (pCons a r) \<le> degree y"
```
```   734         using r by (auto simp add: degree_pCons_eq_if)
```
```   735       show "degree (smult b y) \<le> degree y"
```
```   736         by (rule degree_smult_le)
```
```   737     qed
```
```   738     finally show "degree ?r \<le> degree y" .
```
```   739   next
```
```   740     show "coeff ?r (degree y) = 0"
```
```   741       using `y \<noteq> 0` unfolding b by simp
```
```   742   qed
```
```   743
```
```   744   from 1 2 show ?thesis
```
```   745     unfolding divmod_poly_rel_def
```
```   746     using `y \<noteq> 0` by simp
```
```   747 qed
```
```   748
```
```   749 lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
```
```   750 apply (cases "y = 0")
```
```   751 apply (fast intro!: divmod_poly_rel_by_0)
```
```   752 apply (induct x)
```
```   753 apply (fast intro!: divmod_poly_rel_0)
```
```   754 apply (fast intro!: divmod_poly_rel_pCons)
```
```   755 done
```
```   756
```
```   757 lemma divmod_poly_rel_unique:
```
```   758   assumes 1: "divmod_poly_rel x y q1 r1"
```
```   759   assumes 2: "divmod_poly_rel x y q2 r2"
```
```   760   shows "q1 = q2 \<and> r1 = r2"
```
```   761 proof (cases "y = 0")
```
```   762   assume "y = 0" with assms show ?thesis
```
```   763     by (simp add: divmod_poly_rel_def)
```
```   764 next
```
```   765   assume [simp]: "y \<noteq> 0"
```
```   766   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
```
```   767     unfolding divmod_poly_rel_def by simp_all
```
```   768   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
```
```   769     unfolding divmod_poly_rel_def by simp_all
```
```   770   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
```
```   771     by (simp add: ring_simps)
```
```   772   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
```
```   773     by (auto intro: degree_diff_less)
```
```   774
```
```   775   show "q1 = q2 \<and> r1 = r2"
```
```   776   proof (rule ccontr)
```
```   777     assume "\<not> (q1 = q2 \<and> r1 = r2)"
```
```   778     with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
```
```   779     with r3 have "degree (r2 - r1) < degree y" by simp
```
```   780     also have "degree y \<le> degree (q1 - q2) + degree y" by simp
```
```   781     also have "\<dots> = degree ((q1 - q2) * y)"
```
```   782       using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
```
```   783     also have "\<dots> = degree (r2 - r1)"
```
```   784       using q3 by simp
```
```   785     finally have "degree (r2 - r1) < degree (r2 - r1)" .
```
```   786     then show "False" by simp
```
```   787   qed
```
```   788 qed
```
```   789
```
```   790 lemmas divmod_poly_rel_unique_div =
```
```   791   divmod_poly_rel_unique [THEN conjunct1, standard]
```
```   792
```
```   793 lemmas divmod_poly_rel_unique_mod =
```
```   794   divmod_poly_rel_unique [THEN conjunct2, standard]
```
```   795
```
```   796 instantiation poly :: (field) ring_div
```
```   797 begin
```
```   798
```
```   799 definition div_poly where
```
```   800   [code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
```
```   801
```
```   802 definition mod_poly where
```
```   803   [code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
```
```   804
```
```   805 lemma div_poly_eq:
```
```   806   "divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
```
```   807 unfolding div_poly_def
```
```   808 by (fast elim: divmod_poly_rel_unique_div)
```
```   809
```
```   810 lemma mod_poly_eq:
```
```   811   "divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
```
```   812 unfolding mod_poly_def
```
```   813 by (fast elim: divmod_poly_rel_unique_mod)
```
```   814
```
```   815 lemma divmod_poly_rel:
```
```   816   "divmod_poly_rel x y (x div y) (x mod y)"
```
```   817 proof -
```
```   818   from divmod_poly_rel_exists
```
```   819     obtain q r where "divmod_poly_rel x y q r" by fast
```
```   820   thus ?thesis
```
```   821     by (simp add: div_poly_eq mod_poly_eq)
```
```   822 qed
```
```   823
```
```   824 instance proof
```
```   825   fix x y :: "'a poly"
```
```   826   show "x div y * y + x mod y = x"
```
```   827     using divmod_poly_rel [of x y]
```
```   828     by (simp add: divmod_poly_rel_def)
```
```   829 next
```
```   830   fix x :: "'a poly"
```
```   831   have "divmod_poly_rel x 0 0 x"
```
```   832     by (rule divmod_poly_rel_by_0)
```
```   833   thus "x div 0 = 0"
```
```   834     by (rule div_poly_eq)
```
```   835 next
```
```   836   fix y :: "'a poly"
```
```   837   have "divmod_poly_rel 0 y 0 0"
```
```   838     by (rule divmod_poly_rel_0)
```
```   839   thus "0 div y = 0"
```
```   840     by (rule div_poly_eq)
```
```   841 next
```
```   842   fix x y z :: "'a poly"
```
```   843   assume "y \<noteq> 0"
```
```   844   hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
```
```   845     using divmod_poly_rel [of x y]
```
```   846     by (simp add: divmod_poly_rel_def left_distrib)
```
```   847   thus "(x + z * y) div y = z + x div y"
```
```   848     by (rule div_poly_eq)
```
```   849 qed
```
```   850
```
```   851 end
```
```   852
```
```   853 lemma degree_mod_less:
```
```   854   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
```
```   855   using divmod_poly_rel [of x y]
```
```   856   unfolding divmod_poly_rel_def by simp
```
```   857
```
```   858 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
```
```   859 proof -
```
```   860   assume "degree x < degree y"
```
```   861   hence "divmod_poly_rel x y 0 x"
```
```   862     by (simp add: divmod_poly_rel_def)
```
```   863   thus "x div y = 0" by (rule div_poly_eq)
```
```   864 qed
```
```   865
```
```   866 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
```
```   867 proof -
```
```   868   assume "degree x < degree y"
```
```   869   hence "divmod_poly_rel x y 0 x"
```
```   870     by (simp add: divmod_poly_rel_def)
```
```   871   thus "x mod y = x" by (rule mod_poly_eq)
```
```   872 qed
```
```   873
```
```   874 lemma mod_pCons:
```
```   875   fixes a and x
```
```   876   assumes y: "y \<noteq> 0"
```
```   877   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
```
```   878   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
```
```   879 unfolding b
```
```   880 apply (rule mod_poly_eq)
```
```   881 apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
```
```   882 done
```
```   883
```
```   884
```
```   885 subsection {* Evaluation of polynomials *}
```
```   886
```
```   887 definition
```
```   888   poly :: "'a::{comm_semiring_1,recpower} poly \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   889   "poly p = (\<lambda>x. \<Sum>n\<le>degree p. coeff p n * x ^ n)"
```
```   890
```
```   891 lemma poly_0 [simp]: "poly 0 x = 0"
```
```   892   unfolding poly_def by simp
```
```   893
```
```   894 lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
```
```   895   unfolding poly_def
```
```   896   by (simp add: degree_pCons_eq_if setsum_atMost_Suc_shift power_Suc
```
```   897                 setsum_left_distrib setsum_right_distrib mult_ac
```
```   898            del: setsum_atMost_Suc)
```
```   899
```
```   900 lemma poly_1 [simp]: "poly 1 x = 1"
```
```   901   unfolding one_poly_def by simp
```
```   902
```
```   903 lemma poly_monom: "poly (monom a n) x = a * x ^ n"
```
```   904   by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
```
```   905
```
```   906 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
```
```   907   apply (induct p arbitrary: q, simp)
```
```   908   apply (case_tac q, simp, simp add: ring_simps)
```
```   909   done
```
```   910
```
```   911 lemma poly_minus [simp]:
```
```   912   fixes x :: "'a::{comm_ring_1,recpower}"
```
```   913   shows "poly (- p) x = - poly p x"
```
```   914   by (induct p, simp_all)
```
```   915
```
```   916 lemma poly_diff [simp]:
```
```   917   fixes x :: "'a::{comm_ring_1,recpower}"
```
```   918   shows "poly (p - q) x = poly p x - poly q x"
```
```   919   by (simp add: diff_minus)
```
```   920
```
```   921 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
```
```   922   by (cases "finite A", induct set: finite, simp_all)
```
```   923
```
```   924 lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
```
```   925   by (induct p, simp, simp add: ring_simps)
```
```   926
```
```   927 lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
```
```   928   by (induct p, simp_all, simp add: ring_simps)
```
```   929
```
```   930 end
```