src/HOL/Library/Formal_Power_Series.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 57514 bdc2c6b40bf2 child 58681 a478a0742a8e permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Formal_Power_Series.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 header{* A formalization of formal power series *}
```
```     6
```
```     7 theory Formal_Power_Series
```
```     8 imports "~~/src/HOL/Number_Theory/Binomial"
```
```     9 begin
```
```    10
```
```    11
```
```    12 subsection {* The type of formal power series*}
```
```    13
```
```    14 typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
```
```    15   morphisms fps_nth Abs_fps
```
```    16   by simp
```
```    17
```
```    18 notation fps_nth (infixl "\$" 75)
```
```    19
```
```    20 lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p \$ n = q \$ n)"
```
```    21   by (simp add: fps_nth_inject [symmetric] fun_eq_iff)
```
```    22
```
```    23 lemma fps_ext: "(\<And>n. p \$ n = q \$ n) \<Longrightarrow> p = q"
```
```    24   by (simp add: expand_fps_eq)
```
```    25
```
```    26 lemma fps_nth_Abs_fps [simp]: "Abs_fps f \$ n = f n"
```
```    27   by (simp add: Abs_fps_inverse)
```
```    28
```
```    29 text{* Definition of the basic elements 0 and 1 and the basic operations of addition,
```
```    30   negation and multiplication *}
```
```    31
```
```    32 instantiation fps :: (zero) zero
```
```    33 begin
```
```    34
```
```    35 definition fps_zero_def:
```
```    36   "0 = Abs_fps (\<lambda>n. 0)"
```
```    37
```
```    38 instance ..
```
```    39 end
```
```    40
```
```    41 lemma fps_zero_nth [simp]: "0 \$ n = 0"
```
```    42   unfolding fps_zero_def by simp
```
```    43
```
```    44 instantiation fps :: ("{one, zero}") one
```
```    45 begin
```
```    46
```
```    47 definition fps_one_def:
```
```    48   "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)"
```
```    49
```
```    50 instance ..
```
```    51 end
```
```    52
```
```    53 lemma fps_one_nth [simp]: "1 \$ n = (if n = 0 then 1 else 0)"
```
```    54   unfolding fps_one_def by simp
```
```    55
```
```    56 instantiation fps :: (plus) plus
```
```    57 begin
```
```    58
```
```    59 definition fps_plus_def:
```
```    60   "op + = (\<lambda>f g. Abs_fps (\<lambda>n. f \$ n + g \$ n))"
```
```    61
```
```    62 instance ..
```
```    63 end
```
```    64
```
```    65 lemma fps_add_nth [simp]: "(f + g) \$ n = f \$ n + g \$ n"
```
```    66   unfolding fps_plus_def by simp
```
```    67
```
```    68 instantiation fps :: (minus) minus
```
```    69 begin
```
```    70
```
```    71 definition fps_minus_def:
```
```    72   "op - = (\<lambda>f g. Abs_fps (\<lambda>n. f \$ n - g \$ n))"
```
```    73
```
```    74 instance ..
```
```    75 end
```
```    76
```
```    77 lemma fps_sub_nth [simp]: "(f - g) \$ n = f \$ n - g \$ n"
```
```    78   unfolding fps_minus_def by simp
```
```    79
```
```    80 instantiation fps :: (uminus) uminus
```
```    81 begin
```
```    82
```
```    83 definition fps_uminus_def:
```
```    84   "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f \$ n)))"
```
```    85
```
```    86 instance ..
```
```    87 end
```
```    88
```
```    89 lemma fps_neg_nth [simp]: "(- f) \$ n = - (f \$ n)"
```
```    90   unfolding fps_uminus_def by simp
```
```    91
```
```    92 instantiation fps :: ("{comm_monoid_add, times}") times
```
```    93 begin
```
```    94
```
```    95 definition fps_times_def:
```
```    96   "op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f \$ i * g \$ (n - i)))"
```
```    97
```
```    98 instance ..
```
```    99 end
```
```   100
```
```   101 lemma fps_mult_nth: "(f * g) \$ n = (\<Sum>i=0..n. f\$i * g\$(n - i))"
```
```   102   unfolding fps_times_def by simp
```
```   103
```
```   104 declare atLeastAtMost_iff [presburger]
```
```   105 declare Bex_def [presburger]
```
```   106 declare Ball_def [presburger]
```
```   107
```
```   108 lemma mult_delta_left:
```
```   109   fixes x y :: "'a::mult_zero"
```
```   110   shows "(if b then x else 0) * y = (if b then x * y else 0)"
```
```   111   by simp
```
```   112
```
```   113 lemma mult_delta_right:
```
```   114   fixes x y :: "'a::mult_zero"
```
```   115   shows "x * (if b then y else 0) = (if b then x * y else 0)"
```
```   116   by simp
```
```   117
```
```   118 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
```
```   119   by auto
```
```   120
```
```   121 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```   122   by auto
```
```   123
```
```   124 subsection{* Formal power series form a commutative ring with unity, if the range of sequences
```
```   125   they represent is a commutative ring with unity*}
```
```   126
```
```   127 instance fps :: (semigroup_add) semigroup_add
```
```   128 proof
```
```   129   fix a b c :: "'a fps"
```
```   130   show "a + b + c = a + (b + c)"
```
```   131     by (simp add: fps_ext add.assoc)
```
```   132 qed
```
```   133
```
```   134 instance fps :: (ab_semigroup_add) ab_semigroup_add
```
```   135 proof
```
```   136   fix a b :: "'a fps"
```
```   137   show "a + b = b + a"
```
```   138     by (simp add: fps_ext add.commute)
```
```   139 qed
```
```   140
```
```   141 lemma fps_mult_assoc_lemma:
```
```   142   fixes k :: nat
```
```   143     and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   144   shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
```
```   145          (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
```
```   146   by (induct k) (simp_all add: Suc_diff_le setsum.distrib add.assoc)
```
```   147
```
```   148 instance fps :: (semiring_0) semigroup_mult
```
```   149 proof
```
```   150   fix a b c :: "'a fps"
```
```   151   show "(a * b) * c = a * (b * c)"
```
```   152   proof (rule fps_ext)
```
```   153     fix n :: nat
```
```   154     have "(\<Sum>j=0..n. \<Sum>i=0..j. a\$i * b\$(j - i) * c\$(n - j)) =
```
```   155           (\<Sum>j=0..n. \<Sum>i=0..n - j. a\$j * b\$i * c\$(n - j - i))"
```
```   156       by (rule fps_mult_assoc_lemma)
```
```   157     then show "((a * b) * c) \$ n = (a * (b * c)) \$ n"
```
```   158       by (simp add: fps_mult_nth setsum_right_distrib setsum_left_distrib mult.assoc)
```
```   159   qed
```
```   160 qed
```
```   161
```
```   162 lemma fps_mult_commute_lemma:
```
```   163   fixes n :: nat
```
```   164     and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
```
```   165   shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
```
```   166   by (rule setsum.reindex_bij_witness[where i="op - n" and j="op - n"]) auto
```
```   167
```
```   168 instance fps :: (comm_semiring_0) ab_semigroup_mult
```
```   169 proof
```
```   170   fix a b :: "'a fps"
```
```   171   show "a * b = b * a"
```
```   172   proof (rule fps_ext)
```
```   173     fix n :: nat
```
```   174     have "(\<Sum>i=0..n. a\$i * b\$(n - i)) = (\<Sum>i=0..n. a\$(n - i) * b\$i)"
```
```   175       by (rule fps_mult_commute_lemma)
```
```   176     then show "(a * b) \$ n = (b * a) \$ n"
```
```   177       by (simp add: fps_mult_nth mult.commute)
```
```   178   qed
```
```   179 qed
```
```   180
```
```   181 instance fps :: (monoid_add) monoid_add
```
```   182 proof
```
```   183   fix a :: "'a fps"
```
```   184   show "0 + a = a" by (simp add: fps_ext)
```
```   185   show "a + 0 = a" by (simp add: fps_ext)
```
```   186 qed
```
```   187
```
```   188 instance fps :: (comm_monoid_add) comm_monoid_add
```
```   189 proof
```
```   190   fix a :: "'a fps"
```
```   191   show "0 + a = a" by (simp add: fps_ext)
```
```   192 qed
```
```   193
```
```   194 instance fps :: (semiring_1) monoid_mult
```
```   195 proof
```
```   196   fix a :: "'a fps"
```
```   197   show "1 * a = a" by (simp add: fps_ext fps_mult_nth mult_delta_left setsum.delta)
```
```   198   show "a * 1 = a" by (simp add: fps_ext fps_mult_nth mult_delta_right setsum.delta')
```
```   199 qed
```
```   200
```
```   201 instance fps :: (cancel_semigroup_add) cancel_semigroup_add
```
```   202 proof
```
```   203   fix a b c :: "'a fps"
```
```   204   { assume "a + b = a + c" then show "b = c" by (simp add: expand_fps_eq) }
```
```   205   { assume "b + a = c + a" then show "b = c" by (simp add: expand_fps_eq) }
```
```   206 qed
```
```   207
```
```   208 instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
```
```   209 proof
```
```   210   fix a b c :: "'a fps"
```
```   211   assume "a + b = a + c"
```
```   212   then show "b = c" by (simp add: expand_fps_eq)
```
```   213 qed
```
```   214
```
```   215 instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
```
```   216
```
```   217 instance fps :: (group_add) group_add
```
```   218 proof
```
```   219   fix a b :: "'a fps"
```
```   220   show "- a + a = 0" by (simp add: fps_ext)
```
```   221   show "a + - b = a - b" by (simp add: fps_ext)
```
```   222 qed
```
```   223
```
```   224 instance fps :: (ab_group_add) ab_group_add
```
```   225 proof
```
```   226   fix a b :: "'a fps"
```
```   227   show "- a + a = 0" by (simp add: fps_ext)
```
```   228   show "a - b = a + - b" by (simp add: fps_ext)
```
```   229 qed
```
```   230
```
```   231 instance fps :: (zero_neq_one) zero_neq_one
```
```   232   by default (simp add: expand_fps_eq)
```
```   233
```
```   234 instance fps :: (semiring_0) semiring
```
```   235 proof
```
```   236   fix a b c :: "'a fps"
```
```   237   show "(a + b) * c = a * c + b * c"
```
```   238     by (simp add: expand_fps_eq fps_mult_nth distrib_right setsum.distrib)
```
```   239   show "a * (b + c) = a * b + a * c"
```
```   240     by (simp add: expand_fps_eq fps_mult_nth distrib_left setsum.distrib)
```
```   241 qed
```
```   242
```
```   243 instance fps :: (semiring_0) semiring_0
```
```   244 proof
```
```   245   fix a :: "'a fps"
```
```   246   show "0 * a = 0" by (simp add: fps_ext fps_mult_nth)
```
```   247   show "a * 0 = 0" by (simp add: fps_ext fps_mult_nth)
```
```   248 qed
```
```   249
```
```   250 instance fps :: (semiring_0_cancel) semiring_0_cancel ..
```
```   251
```
```   252 subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
```
```   253
```
```   254 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f \$n \<noteq> 0)"
```
```   255   by (simp add: expand_fps_eq)
```
```   256
```
```   257 lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m < n. f \$ m = 0))"
```
```   258 proof
```
```   259   let ?n = "LEAST n. f \$ n \<noteq> 0"
```
```   260   assume "f \<noteq> 0"
```
```   261   then have "\<exists>n. f \$ n \<noteq> 0"
```
```   262     by (simp add: fps_nonzero_nth)
```
```   263   then have "f \$ ?n \<noteq> 0"
```
```   264     by (rule LeastI_ex)
```
```   265   moreover have "\<forall>m<?n. f \$ m = 0"
```
```   266     by (auto dest: not_less_Least)
```
```   267   ultimately have "f \$ ?n \<noteq> 0 \<and> (\<forall>m<?n. f \$ m = 0)" ..
```
```   268   then show "\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m<n. f \$ m = 0)" ..
```
```   269 next
```
```   270   assume "\<exists>n. f \$ n \<noteq> 0 \<and> (\<forall>m<n. f \$ m = 0)"
```
```   271   then show "f \<noteq> 0" by (auto simp add: expand_fps_eq)
```
```   272 qed
```
```   273
```
```   274 lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f \$ n = g \$n)"
```
```   275   by (rule expand_fps_eq)
```
```   276
```
```   277 lemma fps_setsum_nth: "setsum f S \$ n = setsum (\<lambda>k. (f k) \$ n) S"
```
```   278 proof (cases "finite S")
```
```   279   case True
```
```   280   then show ?thesis by (induct set: finite) auto
```
```   281 next
```
```   282   case False
```
```   283   then show ?thesis by simp
```
```   284 qed
```
```   285
```
```   286 subsection{* Injection of the basic ring elements and multiplication by scalars *}
```
```   287
```
```   288 definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
```
```   289
```
```   290 lemma fps_nth_fps_const [simp]: "fps_const c \$ n = (if n = 0 then c else 0)"
```
```   291   unfolding fps_const_def by simp
```
```   292
```
```   293 lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"
```
```   294   by (simp add: fps_ext)
```
```   295
```
```   296 lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
```
```   297   by (simp add: fps_ext)
```
```   298
```
```   299 lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
```
```   300   by (simp add: fps_ext)
```
```   301
```
```   302 lemma fps_const_add [simp]: "fps_const (c::'a::monoid_add) + fps_const d = fps_const (c + d)"
```
```   303   by (simp add: fps_ext)
```
```   304
```
```   305 lemma fps_const_sub [simp]: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
```
```   306   by (simp add: fps_ext)
```
```   307
```
```   308 lemma fps_const_mult[simp]: "fps_const (c::'a::ring) * fps_const d = fps_const (c * d)"
```
```   309   by (simp add: fps_eq_iff fps_mult_nth setsum.neutral)
```
```   310
```
```   311 lemma fps_const_add_left: "fps_const (c::'a::monoid_add) + f =
```
```   312     Abs_fps (\<lambda>n. if n = 0 then c + f\$0 else f\$n)"
```
```   313   by (simp add: fps_ext)
```
```   314
```
```   315 lemma fps_const_add_right: "f + fps_const (c::'a::monoid_add) =
```
```   316     Abs_fps (\<lambda>n. if n = 0 then f\$0 + c else f\$n)"
```
```   317   by (simp add: fps_ext)
```
```   318
```
```   319 lemma fps_const_mult_left: "fps_const (c::'a::semiring_0) * f = Abs_fps (\<lambda>n. c * f\$n)"
```
```   320   unfolding fps_eq_iff fps_mult_nth
```
```   321   by (simp add: fps_const_def mult_delta_left setsum.delta)
```
```   322
```
```   323 lemma fps_const_mult_right: "f * fps_const (c::'a::semiring_0) = Abs_fps (\<lambda>n. f\$n * c)"
```
```   324   unfolding fps_eq_iff fps_mult_nth
```
```   325   by (simp add: fps_const_def mult_delta_right setsum.delta')
```
```   326
```
```   327 lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)\$n = c* f\$n"
```
```   328   by (simp add: fps_mult_nth mult_delta_left setsum.delta)
```
```   329
```
```   330 lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))\$n = f\$n * c"
```
```   331   by (simp add: fps_mult_nth mult_delta_right setsum.delta')
```
```   332
```
```   333 subsection {* Formal power series form an integral domain*}
```
```   334
```
```   335 instance fps :: (ring) ring ..
```
```   336
```
```   337 instance fps :: (ring_1) ring_1
```
```   338   by (intro_classes, auto simp add: distrib_right)
```
```   339
```
```   340 instance fps :: (comm_ring_1) comm_ring_1
```
```   341   by (intro_classes, auto simp add: distrib_right)
```
```   342
```
```   343 instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
```
```   344 proof
```
```   345   fix a b :: "'a fps"
```
```   346   assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
```
```   347   then obtain i j where i: "a\$i\<noteq>0" "\<forall>k<i. a\$k=0" and j: "b\$j \<noteq>0" "\<forall>k<j. b\$k =0"
```
```   348     unfolding fps_nonzero_nth_minimal
```
```   349     by blast+
```
```   350   have "(a * b) \$ (i+j) = (\<Sum>k=0..i+j. a\$k * b\$(i+j-k))"
```
```   351     by (rule fps_mult_nth)
```
```   352   also have "\<dots> = (a\$i * b\$(i+j-i)) + (\<Sum>k\<in>{0..i+j}-{i}. a\$k * b\$(i+j-k))"
```
```   353     by (rule setsum.remove) simp_all
```
```   354   also have "(\<Sum>k\<in>{0..i+j}-{i}. a\$k * b\$(i+j-k)) = 0"
```
```   355     proof (rule setsum.neutral [rule_format])
```
```   356       fix k assume "k \<in> {0..i+j} - {i}"
```
```   357       then have "k < i \<or> i+j-k < j" by auto
```
```   358       then show "a\$k * b\$(i+j-k) = 0" using i j by auto
```
```   359     qed
```
```   360   also have "a\$i * b\$(i+j-i) + 0 = a\$i * b\$j" by simp
```
```   361   also have "a\$i * b\$j \<noteq> 0" using i j by simp
```
```   362   finally have "(a*b) \$ (i+j) \<noteq> 0" .
```
```   363   then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
```
```   364 qed
```
```   365
```
```   366 instance fps :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
```
```   367
```
```   368 instance fps :: (idom) idom ..
```
```   369
```
```   370 lemma numeral_fps_const: "numeral k = fps_const (numeral k)"
```
```   371   by (induct k) (simp_all only: numeral.simps fps_const_1_eq_1
```
```   372     fps_const_add [symmetric])
```
```   373
```
```   374 lemma neg_numeral_fps_const: "- numeral k = fps_const (- numeral k)"
```
```   375   by (simp only: numeral_fps_const fps_const_neg)
```
```   376
```
```   377 subsection{* The eXtractor series X*}
```
```   378
```
```   379 lemma minus_one_power_iff: "(- (1::'a::comm_ring_1)) ^ n = (if even n then 1 else - 1)"
```
```   380   by (induct n) auto
```
```   381
```
```   382 definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
```
```   383
```
```   384 lemma X_mult_nth [simp]:
```
```   385   "(X * (f :: 'a::semiring_1 fps)) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   386 proof (cases "n = 0")
```
```   387   case False
```
```   388   have "(X * f) \$n = (\<Sum>i = 0..n. X \$ i * f \$ (n - i))"
```
```   389     by (simp add: fps_mult_nth)
```
```   390   also have "\<dots> = f \$ (n - 1)"
```
```   391     using False by (simp add: X_def mult_delta_left setsum.delta)
```
```   392   finally show ?thesis using False by simp
```
```   393 next
```
```   394   case True
```
```   395   then show ?thesis by (simp add: fps_mult_nth X_def)
```
```   396 qed
```
```   397
```
```   398 lemma X_mult_right_nth[simp]:
```
```   399     "((f :: 'a::comm_semiring_1 fps) * X) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   400   by (metis X_mult_nth mult.commute)
```
```   401
```
```   402 lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then 1::'a::comm_ring_1 else 0)"
```
```   403 proof (induct k)
```
```   404   case 0
```
```   405   then show ?case by (simp add: X_def fps_eq_iff)
```
```   406 next
```
```   407   case (Suc k)
```
```   408   {
```
```   409     fix m
```
```   410     have "(X^Suc k) \$ m = (if m = 0 then 0::'a else (X^k) \$ (m - 1))"
```
```   411       by (simp del: One_nat_def)
```
```   412     then have "(X^Suc k) \$ m = (if m = Suc k then 1::'a else 0)"
```
```   413       using Suc.hyps by (auto cong del: if_weak_cong)
```
```   414   }
```
```   415   then show ?case by (simp add: fps_eq_iff)
```
```   416 qed
```
```   417
```
```   418 lemma X_power_mult_nth:
```
```   419     "(X^k * (f :: 'a::comm_ring_1 fps)) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   420   apply (induct k arbitrary: n)
```
```   421   apply simp
```
```   422   unfolding power_Suc mult.assoc
```
```   423   apply (case_tac n)
```
```   424   apply auto
```
```   425   done
```
```   426
```
```   427 lemma X_power_mult_right_nth:
```
```   428     "((f :: 'a::comm_ring_1 fps) * X^k) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   429   by (metis X_power_mult_nth mult.commute)
```
```   430
```
```   431
```
```   432 subsection{* Formal Power series form a metric space *}
```
```   433
```
```   434 definition (in dist) "ball x r = {y. dist y x < r}"
```
```   435
```
```   436 instantiation fps :: (comm_ring_1) dist
```
```   437 begin
```
```   438
```
```   439 definition
```
```   440   dist_fps_def: "dist (a :: 'a fps) b =
```
```   441     (if (\<exists>n. a\$n \<noteq> b\$n) then inverse (2 ^ (LEAST n. a\$n \<noteq> b\$n)) else 0)"
```
```   442
```
```   443 lemma dist_fps_ge0: "dist (a :: 'a fps) b \<ge> 0"
```
```   444   by (simp add: dist_fps_def)
```
```   445
```
```   446 lemma dist_fps_sym: "dist (a :: 'a fps) b = dist b a"
```
```   447   apply (auto simp add: dist_fps_def)
```
```   448   apply (rule cong[OF refl, where x="(\<lambda>n. a \$ n \<noteq> b \$ n)"])
```
```   449   apply (rule ext)
```
```   450   apply auto
```
```   451   done
```
```   452
```
```   453 instance ..
```
```   454
```
```   455 end
```
```   456
```
```   457 instantiation fps :: (comm_ring_1) metric_space
```
```   458 begin
```
```   459
```
```   460 definition open_fps_def: "open (S :: 'a fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> ball a r \<subseteq> S)"
```
```   461
```
```   462 instance
```
```   463 proof
```
```   464   fix S :: "'a fps set"
```
```   465   show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   466     by (auto simp add: open_fps_def ball_def subset_eq)
```
```   467 next
```
```   468   {
```
```   469     fix a b :: "'a fps"
```
```   470     {
```
```   471       assume "a = b"
```
```   472       then have "\<not> (\<exists>n. a \$ n \<noteq> b \$ n)" by simp
```
```   473       then have "dist a b = 0" by (simp add: dist_fps_def)
```
```   474     }
```
```   475     moreover
```
```   476     {
```
```   477       assume d: "dist a b = 0"
```
```   478       then have "\<forall>n. a\$n = b\$n"
```
```   479         by - (rule ccontr, simp add: dist_fps_def)
```
```   480       then have "a = b" by (simp add: fps_eq_iff)
```
```   481     }
```
```   482     ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast
```
```   483   }
```
```   484   note th = this
```
```   485   from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp
```
```   486   fix a b c :: "'a fps"
```
```   487   {
```
```   488     assume "a = b"
```
```   489     then have "dist a b = 0" unfolding th .
```
```   490     then have "dist a b \<le> dist a c + dist b c"
```
```   491       using dist_fps_ge0 [of a c] dist_fps_ge0 [of b c] by simp
```
```   492   }
```
```   493   moreover
```
```   494   {
```
```   495     assume "c = a \<or> c = b"
```
```   496     then have "dist a b \<le> dist a c + dist b c"
```
```   497       by (cases "c = a") (simp_all add: th dist_fps_sym)
```
```   498   }
```
```   499   moreover
```
```   500   {
```
```   501     assume ab: "a \<noteq> b" and ac: "a \<noteq> c" and bc: "b \<noteq> c"
```
```   502     def n \<equiv> "\<lambda>a b::'a fps. LEAST n. a\$n \<noteq> b\$n"
```
```   503     then have n': "\<And>m a b. m < n a b \<Longrightarrow> a\$m = b\$m"
```
```   504       by (auto dest: not_less_Least)
```
```   505
```
```   506     from ab ac bc
```
```   507     have dab: "dist a b = inverse (2 ^ n a b)"
```
```   508       and dac: "dist a c = inverse (2 ^ n a c)"
```
```   509       and dbc: "dist b c = inverse (2 ^ n b c)"
```
```   510       by (simp_all add: dist_fps_def n_def fps_eq_iff)
```
```   511     from ab ac bc have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 0"
```
```   512       unfolding th by simp_all
```
```   513     from nz have pos: "dist a b > 0" "dist a c > 0" "dist b c > 0"
```
```   514       using dist_fps_ge0[of a b] dist_fps_ge0[of a c] dist_fps_ge0[of b c]
```
```   515       by auto
```
```   516     have th1: "\<And>n. (2::real)^n >0" by auto
```
```   517     {
```
```   518       assume h: "dist a b > dist a c + dist b c"
```
```   519       then have gt: "dist a b > dist a c" "dist a b > dist b c"
```
```   520         using pos by auto
```
```   521       from gt have gtn: "n a b < n b c" "n a b < n a c"
```
```   522         unfolding dab dbc dac by (auto simp add: th1)
```
```   523       from n'[OF gtn(2)] n'(1)[OF gtn(1)]
```
```   524       have "a \$ n a b = b \$ n a b" by simp
```
```   525       moreover have "a \$ n a b \<noteq> b \$ n a b"
```
```   526          unfolding n_def by (rule LeastI_ex) (insert ab, simp add: fps_eq_iff)
```
```   527       ultimately have False by contradiction
```
```   528     }
```
```   529     then have "dist a b \<le> dist a c + dist b c"
```
```   530       by (auto simp add: not_le[symmetric])
```
```   531   }
```
```   532   ultimately show "dist a b \<le> dist a c + dist b c" by blast
```
```   533 qed
```
```   534
```
```   535 end
```
```   536
```
```   537 text{* The infinite sums and justification of the notation in textbooks*}
```
```   538
```
```   539 lemma reals_power_lt_ex:
```
```   540   fixes x y :: real
```
```   541   assumes xp: "x > 0"
```
```   542     and y1: "y > 1"
```
```   543   shows "\<exists>k>0. (1/y)^k < x"
```
```   544 proof -
```
```   545   have yp: "y > 0"
```
```   546     using y1 by simp
```
```   547   from reals_Archimedean2[of "max 0 (- log y x) + 1"]
```
```   548   obtain k :: nat where k: "real k > max 0 (- log y x) + 1"
```
```   549     by blast
```
```   550   from k have kp: "k > 0"
```
```   551     by simp
```
```   552   from k have "real k > - log y x"
```
```   553     by simp
```
```   554   then have "ln y * real k > - ln x"
```
```   555     unfolding log_def
```
```   556     using ln_gt_zero_iff[OF yp] y1
```
```   557     by (simp add: minus_divide_left field_simps del: minus_divide_left[symmetric])
```
```   558   then have "ln y * real k + ln x > 0"
```
```   559     by simp
```
```   560   then have "exp (real k * ln y + ln x) > exp 0"
```
```   561     by (simp add: ac_simps)
```
```   562   then have "y ^ k * x > 1"
```
```   563     unfolding exp_zero exp_add exp_real_of_nat_mult exp_ln [OF xp] exp_ln [OF yp]
```
```   564     by simp
```
```   565   then have "x > (1 / y)^k" using yp
```
```   566     by (simp add: field_simps nonzero_power_divide)
```
```   567   then show ?thesis
```
```   568     using kp by blast
```
```   569 qed
```
```   570
```
```   571 lemma X_nth[simp]: "X\$n = (if n = 1 then 1 else 0)"
```
```   572   by (simp add: X_def)
```
```   573
```
```   574 lemma X_power_nth[simp]: "(X^k) \$n = (if n = k then 1 else 0::'a::comm_ring_1)"
```
```   575   by (simp add: X_power_iff)
```
```   576
```
```   577 lemma fps_sum_rep_nth: "(setsum (\<lambda>i. fps_const(a\$i)*X^i) {0..m})\$n =
```
```   578     (if n \<le> m then a\$n else 0::'a::comm_ring_1)"
```
```   579   apply (auto simp add: fps_setsum_nth cond_value_iff cong del: if_weak_cong)
```
```   580   apply (simp add: setsum.delta')
```
```   581   done
```
```   582
```
```   583 lemma fps_notation: "(\<lambda>n. setsum (\<lambda>i. fps_const(a\$i) * X^i) {0..n}) ----> a"
```
```   584   (is "?s ----> a")
```
```   585 proof -
```
```   586   {
```
```   587     fix r :: real
```
```   588     assume rp: "r > 0"
```
```   589     have th0: "(2::real) > 1" by simp
```
```   590     from reals_power_lt_ex[OF rp th0]
```
```   591     obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast
```
```   592     {
```
```   593       fix n :: nat
```
```   594       assume nn0: "n \<ge> n0"
```
```   595       then have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0"
```
```   596         by (auto intro: power_decreasing)
```
```   597       {
```
```   598         assume "?s n = a"
```
```   599         then have "dist (?s n) a < r"
```
```   600           unfolding dist_eq_0_iff[of "?s n" a, symmetric]
```
```   601           using rp by (simp del: dist_eq_0_iff)
```
```   602       }
```
```   603       moreover
```
```   604       {
```
```   605         assume neq: "?s n \<noteq> a"
```
```   606         def k \<equiv> "LEAST i. ?s n \$ i \<noteq> a \$ i"
```
```   607         from neq have dth: "dist (?s n) a = (1/2)^k"
```
```   608           by (auto simp add: dist_fps_def inverse_eq_divide power_divide k_def fps_eq_iff)
```
```   609
```
```   610         from neq have kn: "k > n"
```
```   611           by (auto simp: fps_sum_rep_nth not_le k_def fps_eq_iff
```
```   612               split: split_if_asm intro: LeastI2_ex)
```
```   613         then have "dist (?s n) a < (1/2)^n"
```
```   614           unfolding dth by (auto intro: power_strict_decreasing)
```
```   615         also have "\<dots> \<le> (1/2)^n0"
```
```   616           using nn0 by (auto intro: power_decreasing)
```
```   617         also have "\<dots> < r"
```
```   618           using n0 by simp
```
```   619         finally have "dist (?s n) a < r" .
```
```   620       }
```
```   621       ultimately have "dist (?s n) a < r"
```
```   622         by blast
```
```   623     }
```
```   624     then have "\<exists>n0. \<forall> n \<ge> n0. dist (?s n) a < r"
```
```   625       by blast
```
```   626   }
```
```   627   then show ?thesis
```
```   628     unfolding LIMSEQ_def by blast
```
```   629 qed
```
```   630
```
```   631
```
```   632 subsection{* Inverses of formal power series *}
```
```   633
```
```   634 declare setsum.cong[fundef_cong]
```
```   635
```
```   636 instantiation fps :: ("{comm_monoid_add, inverse, times, uminus}") inverse
```
```   637 begin
```
```   638
```
```   639 fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a"
```
```   640 where
```
```   641   "natfun_inverse f 0 = inverse (f\$0)"
```
```   642 | "natfun_inverse f n = - inverse (f\$0) * setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}"
```
```   643
```
```   644 definition
```
```   645   fps_inverse_def: "inverse f = (if f \$ 0 = 0 then 0 else Abs_fps (natfun_inverse f))"
```
```   646
```
```   647 definition
```
```   648   fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
```
```   649
```
```   650 instance ..
```
```   651
```
```   652 end
```
```   653
```
```   654 lemma fps_inverse_zero [simp]:
```
```   655   "inverse (0 :: 'a::{comm_monoid_add,inverse,times,uminus} fps) = 0"
```
```   656   by (simp add: fps_ext fps_inverse_def)
```
```   657
```
```   658 lemma fps_inverse_one [simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
```
```   659   apply (auto simp add: expand_fps_eq fps_inverse_def)
```
```   660   apply (case_tac n)
```
```   661   apply auto
```
```   662   done
```
```   663
```
```   664 lemma inverse_mult_eq_1 [intro]:
```
```   665   assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   666   shows "inverse f * f = 1"
```
```   667 proof -
```
```   668   have c: "inverse f * f = f * inverse f"
```
```   669     by (simp add: mult.commute)
```
```   670   from f0 have ifn: "\<And>n. inverse f \$ n = natfun_inverse f n"
```
```   671     by (simp add: fps_inverse_def)
```
```   672   from f0 have th0: "(inverse f * f) \$ 0 = 1"
```
```   673     by (simp add: fps_mult_nth fps_inverse_def)
```
```   674   {
```
```   675     fix n :: nat
```
```   676     assume np: "n > 0"
```
```   677     from np have eq: "{0..n} = {0} \<union> {1 .. n}"
```
```   678       by auto
```
```   679     have d: "{0} \<inter> {1 .. n} = {}"
```
```   680       by auto
```
```   681     from f0 np have th0: "- (inverse f \$ n) =
```
```   682       (setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}) / (f\$0)"
```
```   683       by (cases n) (simp_all add: divide_inverse fps_inverse_def)
```
```   684     from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
```
```   685     have th1: "setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n} = - (f\$0) * (inverse f)\$n"
```
```   686       by (simp add: field_simps)
```
```   687     have "(f * inverse f) \$ n = (\<Sum>i = 0..n. f \$i * natfun_inverse f (n - i))"
```
```   688       unfolding fps_mult_nth ifn ..
```
```   689     also have "\<dots> = f\$0 * natfun_inverse f n + (\<Sum>i = 1..n. f\$i * natfun_inverse f (n-i))"
```
```   690       by (simp add: eq)
```
```   691     also have "\<dots> = 0"
```
```   692       unfolding th1 ifn by simp
```
```   693     finally have "(inverse f * f)\$n = 0"
```
```   694       unfolding c .
```
```   695   }
```
```   696   with th0 show ?thesis
```
```   697     by (simp add: fps_eq_iff)
```
```   698 qed
```
```   699
```
```   700 lemma fps_inverse_0_iff[simp]: "(inverse f)\$0 = (0::'a::division_ring) \<longleftrightarrow> f\$0 = 0"
```
```   701   by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)
```
```   702
```
```   703 lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f \$0 = 0"
```
```   704 proof -
```
```   705   {
```
```   706     assume "f \$ 0 = 0"
```
```   707     then have "inverse f = 0"
```
```   708       by (simp add: fps_inverse_def)
```
```   709   }
```
```   710   moreover
```
```   711   {
```
```   712     assume h: "inverse f = 0"
```
```   713     assume c: "f \$0 \<noteq> 0"
```
```   714     from inverse_mult_eq_1[OF c] h have False
```
```   715       by simp
```
```   716   }
```
```   717   ultimately show ?thesis by blast
```
```   718 qed
```
```   719
```
```   720 lemma fps_inverse_idempotent[intro]:
```
```   721   assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   722   shows "inverse (inverse f) = f"
```
```   723 proof -
```
```   724   from f0 have if0: "inverse f \$ 0 \<noteq> 0" by simp
```
```   725   from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
```
```   726   have "inverse f * f = inverse f * inverse (inverse f)"
```
```   727     by (simp add: ac_simps)
```
```   728   then show ?thesis
```
```   729     using f0 unfolding mult_cancel_left by simp
```
```   730 qed
```
```   731
```
```   732 lemma fps_inverse_unique:
```
```   733   assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   734     and fg: "f*g = 1"
```
```   735   shows "inverse f = g"
```
```   736 proof -
```
```   737   from inverse_mult_eq_1[OF f0] fg
```
```   738   have th0: "inverse f * f = g * f"
```
```   739     by (simp add: ac_simps)
```
```   740   then show ?thesis
```
```   741     using f0
```
```   742     unfolding mult_cancel_right
```
```   743     by (auto simp add: expand_fps_eq)
```
```   744 qed
```
```   745
```
```   746 lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
```
```   747     = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
```
```   748   apply (rule fps_inverse_unique)
```
```   749   apply simp
```
```   750   apply (simp add: fps_eq_iff fps_mult_nth)
```
```   751   apply clarsimp
```
```   752 proof -
```
```   753   fix n :: nat
```
```   754   assume n: "n > 0"
```
```   755   let ?f = "\<lambda>i. if n = i then (1::'a) else if n - i = 1 then - 1 else 0"
```
```   756   let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
```
```   757   let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
```
```   758   have th1: "setsum ?f {0..n} = setsum ?g {0..n}"
```
```   759     by (rule setsum.cong) auto
```
```   760   have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
```
```   761     apply (insert n)
```
```   762     apply (rule setsum.cong)
```
```   763     apply auto
```
```   764     done
```
```   765   have eq: "{0 .. n} = {0.. n - 1} \<union> {n}"
```
```   766     by auto
```
```   767   from n have d: "{0.. n - 1} \<inter> {n} = {}"
```
```   768     by auto
```
```   769   have f: "finite {0.. n - 1}" "finite {n}"
```
```   770     by auto
```
```   771   show "setsum ?f {0..n} = 0"
```
```   772     unfolding th1
```
```   773     apply (simp add: setsum.union_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
```
```   774     unfolding th2
```
```   775     apply (simp add: setsum.delta)
```
```   776     done
```
```   777 qed
```
```   778
```
```   779
```
```   780 subsection {* Formal Derivatives, and the MacLaurin theorem around 0 *}
```
```   781
```
```   782 definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f \$ (n + 1))"
```
```   783
```
```   784 lemma fps_deriv_nth[simp]: "fps_deriv f \$ n = of_nat (n +1) * f \$ (n + 1)"
```
```   785   by (simp add: fps_deriv_def)
```
```   786
```
```   787 lemma fps_deriv_linear[simp]:
```
```   788   "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
```
```   789     fps_const a * fps_deriv f + fps_const b * fps_deriv g"
```
```   790   unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: field_simps)
```
```   791
```
```   792 lemma fps_deriv_mult[simp]:
```
```   793   fixes f :: "'a::comm_ring_1 fps"
```
```   794   shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
```
```   795 proof -
```
```   796   let ?D = "fps_deriv"
```
```   797   {
```
```   798     fix n :: nat
```
```   799     let ?Zn = "{0 ..n}"
```
```   800     let ?Zn1 = "{0 .. n + 1}"
```
```   801     let ?g = "\<lambda>i. of_nat (i+1) * g \$ (i+1) * f \$ (n - i) +
```
```   802         of_nat (i+1)* f \$ (i+1) * g \$ (n - i)"
```
```   803     let ?h = "\<lambda>i. of_nat i * g \$ i * f \$ ((n+1) - i) +
```
```   804         of_nat i* f \$ i * g \$ ((n + 1) - i)"
```
```   805     have s0: "setsum (\<lambda>i. of_nat i * f \$ i * g \$ (n + 1 - i)) ?Zn1 =
```
```   806       setsum (\<lambda>i. of_nat (n + 1 - i) * f \$ (n + 1 - i) * g \$ i) ?Zn1"
```
```   807        by (rule setsum.reindex_bij_witness[where i="op - (n + 1)" and j="op - (n + 1)"]) auto
```
```   808     have s1: "setsum (\<lambda>i. f \$ i * g \$ (n + 1 - i)) ?Zn1 =
```
```   809       setsum (\<lambda>i. f \$ (n + 1 - i) * g \$ i) ?Zn1"
```
```   810        by (rule setsum.reindex_bij_witness[where i="op - (n + 1)" and j="op - (n + 1)"]) auto
```
```   811     have "(f * ?D g + ?D f * g)\$n = (?D g * f + ?D f * g)\$n"
```
```   812       by (simp only: mult.commute)
```
```   813     also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
```
```   814       by (simp add: fps_mult_nth setsum.distrib[symmetric])
```
```   815     also have "\<dots> = setsum ?h {0..n+1}"
```
```   816       by (rule setsum.reindex_bij_witness_not_neutral
```
```   817             [where S'="{}" and T'="{0}" and j="Suc" and i="\<lambda>i. i - 1"]) auto
```
```   818     also have "\<dots> = (fps_deriv (f * g)) \$ n"
```
```   819       apply (simp only: fps_deriv_nth fps_mult_nth setsum.distrib)
```
```   820       unfolding s0 s1
```
```   821       unfolding setsum.distrib[symmetric] setsum_right_distrib
```
```   822       apply (rule setsum.cong)
```
```   823       apply (auto simp add: of_nat_diff field_simps)
```
```   824       done
```
```   825     finally have "(f * ?D g + ?D f * g) \$ n = ?D (f*g) \$ n" .
```
```   826   }
```
```   827   then show ?thesis unfolding fps_eq_iff by auto
```
```   828 qed
```
```   829
```
```   830 lemma fps_deriv_X[simp]: "fps_deriv X = 1"
```
```   831   by (simp add: fps_deriv_def X_def fps_eq_iff)
```
```   832
```
```   833 lemma fps_deriv_neg[simp]:
```
```   834   "fps_deriv (- (f:: 'a::comm_ring_1 fps)) = - (fps_deriv f)"
```
```   835   by (simp add: fps_eq_iff fps_deriv_def)
```
```   836
```
```   837 lemma fps_deriv_add[simp]:
```
```   838   "fps_deriv ((f:: 'a::comm_ring_1 fps) + g) = fps_deriv f + fps_deriv g"
```
```   839   using fps_deriv_linear[of 1 f 1 g] by simp
```
```   840
```
```   841 lemma fps_deriv_sub[simp]:
```
```   842   "fps_deriv ((f:: 'a::comm_ring_1 fps) - g) = fps_deriv f - fps_deriv g"
```
```   843   using fps_deriv_add [of f "- g"] by simp
```
```   844
```
```   845 lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
```
```   846   by (simp add: fps_ext fps_deriv_def fps_const_def)
```
```   847
```
```   848 lemma fps_deriv_mult_const_left[simp]:
```
```   849   "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
```
```   850   by simp
```
```   851
```
```   852 lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
```
```   853   by (simp add: fps_deriv_def fps_eq_iff)
```
```   854
```
```   855 lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
```
```   856   by (simp add: fps_deriv_def fps_eq_iff )
```
```   857
```
```   858 lemma fps_deriv_mult_const_right[simp]:
```
```   859   "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
```
```   860   by simp
```
```   861
```
```   862 lemma fps_deriv_setsum:
```
```   863   "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: 'a::comm_ring_1 fps)) S"
```
```   864 proof (cases "finite S")
```
```   865   case False
```
```   866   then show ?thesis by simp
```
```   867 next
```
```   868   case True
```
```   869   show ?thesis by (induct rule: finite_induct [OF True]) simp_all
```
```   870 qed
```
```   871
```
```   872 lemma fps_deriv_eq_0_iff [simp]:
```
```   873   "fps_deriv f = 0 \<longleftrightarrow> f = fps_const (f\$0 :: 'a::{idom,semiring_char_0})"
```
```   874 proof -
```
```   875   {
```
```   876     assume "f = fps_const (f\$0)"
```
```   877     then have "fps_deriv f = fps_deriv (fps_const (f\$0))" by simp
```
```   878     then have "fps_deriv f = 0" by simp
```
```   879   }
```
```   880   moreover
```
```   881   {
```
```   882     assume z: "fps_deriv f = 0"
```
```   883     then have "\<forall>n. (fps_deriv f)\$n = 0" by simp
```
```   884     then have "\<forall>n. f\$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
```
```   885     then have "f = fps_const (f\$0)"
```
```   886       apply (clarsimp simp add: fps_eq_iff fps_const_def)
```
```   887       apply (erule_tac x="n - 1" in allE)
```
```   888       apply simp
```
```   889       done
```
```   890   }
```
```   891   ultimately show ?thesis by blast
```
```   892 qed
```
```   893
```
```   894 lemma fps_deriv_eq_iff:
```
```   895   fixes f :: "'a::{idom,semiring_char_0} fps"
```
```   896   shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f\$0 - g\$0) + g)"
```
```   897 proof -
```
```   898   have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0"
```
```   899     by simp
```
```   900   also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f - g) \$ 0)"
```
```   901     unfolding fps_deriv_eq_0_iff ..
```
```   902   finally show ?thesis by (simp add: field_simps)
```
```   903 qed
```
```   904
```
```   905 lemma fps_deriv_eq_iff_ex:
```
```   906   "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>c::'a::{idom,semiring_char_0}. f = fps_const c + g)"
```
```   907   by (auto simp: fps_deriv_eq_iff)
```
```   908
```
```   909
```
```   910 fun fps_nth_deriv :: "nat \<Rightarrow> 'a::semiring_1 fps \<Rightarrow> 'a fps"
```
```   911 where
```
```   912   "fps_nth_deriv 0 f = f"
```
```   913 | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
```
```   914
```
```   915 lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
```
```   916   by (induct n arbitrary: f) auto
```
```   917
```
```   918 lemma fps_nth_deriv_linear[simp]:
```
```   919   "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) =
```
```   920     fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
```
```   921   by (induct n arbitrary: f g) (auto simp add: fps_nth_deriv_commute)
```
```   922
```
```   923 lemma fps_nth_deriv_neg[simp]:
```
```   924   "fps_nth_deriv n (- (f :: 'a::comm_ring_1 fps)) = - (fps_nth_deriv n f)"
```
```   925   by (induct n arbitrary: f) simp_all
```
```   926
```
```   927 lemma fps_nth_deriv_add[simp]:
```
```   928   "fps_nth_deriv n ((f :: 'a::comm_ring_1 fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
```
```   929   using fps_nth_deriv_linear[of n 1 f 1 g] by simp
```
```   930
```
```   931 lemma fps_nth_deriv_sub[simp]:
```
```   932   "fps_nth_deriv n ((f :: 'a::comm_ring_1 fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
```
```   933   using fps_nth_deriv_add [of n f "- g"] by simp
```
```   934
```
```   935 lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
```
```   936   by (induct n) simp_all
```
```   937
```
```   938 lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
```
```   939   by (induct n) simp_all
```
```   940
```
```   941 lemma fps_nth_deriv_const[simp]:
```
```   942   "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
```
```   943   by (cases n) simp_all
```
```   944
```
```   945 lemma fps_nth_deriv_mult_const_left[simp]:
```
```   946   "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f"
```
```   947   using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
```
```   948
```
```   949 lemma fps_nth_deriv_mult_const_right[simp]:
```
```   950   "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
```
```   951   using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult.commute)
```
```   952
```
```   953 lemma fps_nth_deriv_setsum:
```
```   954   "fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: 'a::comm_ring_1 fps)) S"
```
```   955 proof (cases "finite S")
```
```   956   case True
```
```   957   show ?thesis by (induct rule: finite_induct [OF True]) simp_all
```
```   958 next
```
```   959   case False
```
```   960   then show ?thesis by simp
```
```   961 qed
```
```   962
```
```   963 lemma fps_deriv_maclauren_0:
```
```   964   "(fps_nth_deriv k (f :: 'a::comm_semiring_1 fps)) \$ 0 = of_nat (fact k) * f \$ k"
```
```   965   by (induct k arbitrary: f) (auto simp add: field_simps of_nat_mult)
```
```   966
```
```   967
```
```   968 subsection {* Powers *}
```
```   969
```
```   970 lemma fps_power_zeroth_eq_one: "a\$0 =1 \<Longrightarrow> a^n \$ 0 = (1::'a::semiring_1)"
```
```   971   by (induct n) (auto simp add: expand_fps_eq fps_mult_nth)
```
```   972
```
```   973 lemma fps_power_first_eq: "(a :: 'a::comm_ring_1 fps) \$ 0 =1 \<Longrightarrow> a^n \$ 1 = of_nat n * a\$1"
```
```   974 proof (induct n)
```
```   975   case 0
```
```   976   then show ?case by simp
```
```   977 next
```
```   978   case (Suc n)
```
```   979   note h = Suc.hyps[OF `a\$0 = 1`]
```
```   980   show ?case unfolding power_Suc fps_mult_nth
```
```   981     using h `a\$0 = 1` fps_power_zeroth_eq_one[OF `a\$0=1`]
```
```   982     by (simp add: field_simps)
```
```   983 qed
```
```   984
```
```   985 lemma startsby_one_power:"a \$ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n \$ 0 = 1"
```
```   986   by (induct n) (auto simp add: fps_mult_nth)
```
```   987
```
```   988 lemma startsby_zero_power:"a \$0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n \$0 = 0"
```
```   989   by (induct n) (auto simp add: fps_mult_nth)
```
```   990
```
```   991 lemma startsby_power:"a \$0 = (v::'a::comm_ring_1) \<Longrightarrow> a^n \$0 = v^n"
```
```   992   by (induct n) (auto simp add: fps_mult_nth)
```
```   993
```
```   994 lemma startsby_zero_power_iff[simp]: "a^n \$0 = (0::'a::idom) \<longleftrightarrow> n \<noteq> 0 \<and> a\$0 = 0"
```
```   995   apply (rule iffI)
```
```   996   apply (induct n)
```
```   997   apply (auto simp add: fps_mult_nth)
```
```   998   apply (rule startsby_zero_power, simp_all)
```
```   999   done
```
```  1000
```
```  1001 lemma startsby_zero_power_prefix:
```
```  1002   assumes a0: "a \$0 = (0::'a::idom)"
```
```  1003   shows "\<forall>n < k. a ^ k \$ n = 0"
```
```  1004   using a0
```
```  1005 proof (induct k rule: nat_less_induct)
```
```  1006   fix k
```
```  1007   assume H: "\<forall>m<k. a \$0 =  0 \<longrightarrow> (\<forall>n<m. a ^ m \$ n = 0)" and a0: "a \$ 0 = 0"
```
```  1008   let ?ths = "\<forall>m<k. a ^ k \$ m = 0"
```
```  1009   {
```
```  1010     assume "k = 0"
```
```  1011     then have ?ths by simp
```
```  1012   }
```
```  1013   moreover
```
```  1014   {
```
```  1015     fix l
```
```  1016     assume k: "k = Suc l"
```
```  1017     {
```
```  1018       fix m
```
```  1019       assume mk: "m < k"
```
```  1020       {
```
```  1021         assume "m = 0"
```
```  1022         then have "a^k \$ m = 0"
```
```  1023           using startsby_zero_power[of a k] k a0 by simp
```
```  1024       }
```
```  1025       moreover
```
```  1026       {
```
```  1027         assume m0: "m \<noteq> 0"
```
```  1028         have "a ^k \$ m = (a^l * a) \$m"
```
```  1029           by (simp add: k mult.commute)
```
```  1030         also have "\<dots> = (\<Sum>i = 0..m. a ^ l \$ i * a \$ (m - i))"
```
```  1031           by (simp add: fps_mult_nth)
```
```  1032         also have "\<dots> = 0"
```
```  1033           apply (rule setsum.neutral)
```
```  1034           apply auto
```
```  1035           apply (case_tac "x = m")
```
```  1036           using a0 apply simp
```
```  1037           apply (rule H[rule_format])
```
```  1038           using a0 k mk apply auto
```
```  1039           done
```
```  1040         finally have "a^k \$ m = 0" .
```
```  1041       }
```
```  1042       ultimately have "a^k \$ m = 0"
```
```  1043         by blast
```
```  1044     }
```
```  1045     then have ?ths by blast
```
```  1046   }
```
```  1047   ultimately show ?ths
```
```  1048     by (cases k) auto
```
```  1049 qed
```
```  1050
```
```  1051 lemma startsby_zero_setsum_depends:
```
```  1052   assumes a0: "a \$0 = (0::'a::idom)"
```
```  1053     and kn: "n \<ge> k"
```
```  1054   shows "setsum (\<lambda>i. (a ^ i)\$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)\$k) {0 .. k}"
```
```  1055   apply (rule setsum.mono_neutral_right)
```
```  1056   using kn
```
```  1057   apply auto
```
```  1058   apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```  1059   apply arith
```
```  1060   done
```
```  1061
```
```  1062 lemma startsby_zero_power_nth_same:
```
```  1063   assumes a0: "a\$0 = (0::'a::idom)"
```
```  1064   shows "a^n \$ n = (a\$1) ^ n"
```
```  1065 proof (induct n)
```
```  1066   case 0
```
```  1067   then show ?case by simp
```
```  1068 next
```
```  1069   case (Suc n)
```
```  1070   have "a ^ Suc n \$ (Suc n) = (a^n * a)\$(Suc n)"
```
```  1071     by (simp add: field_simps)
```
```  1072   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {0.. Suc n}"
```
```  1073     by (simp add: fps_mult_nth)
```
```  1074   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {n .. Suc n}"
```
```  1075     apply (rule setsum.mono_neutral_right)
```
```  1076     apply simp
```
```  1077     apply clarsimp
```
```  1078     apply clarsimp
```
```  1079     apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```  1080     apply arith
```
```  1081     done
```
```  1082   also have "\<dots> = a^n \$ n * a\$1"
```
```  1083     using a0 by simp
```
```  1084   finally show ?case
```
```  1085     using Suc.hyps by simp
```
```  1086 qed
```
```  1087
```
```  1088 lemma fps_inverse_power:
```
```  1089   fixes a :: "'a::field fps"
```
```  1090   shows "inverse (a^n) = inverse a ^ n"
```
```  1091 proof -
```
```  1092   {
```
```  1093     assume a0: "a\$0 = 0"
```
```  1094     then have eq: "inverse a = 0"
```
```  1095       by (simp add: fps_inverse_def)
```
```  1096     {
```
```  1097       assume "n = 0"
```
```  1098       then have ?thesis by simp
```
```  1099     }
```
```  1100     moreover
```
```  1101     {
```
```  1102       assume n: "n > 0"
```
```  1103       from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
```
```  1104         by (simp add: fps_inverse_def)
```
```  1105     }
```
```  1106     ultimately have ?thesis by blast
```
```  1107   }
```
```  1108   moreover
```
```  1109   {
```
```  1110     assume a0: "a\$0 \<noteq> 0"
```
```  1111     have ?thesis
```
```  1112       apply (rule fps_inverse_unique)
```
```  1113       apply (simp add: a0)
```
```  1114       unfolding power_mult_distrib[symmetric]
```
```  1115       apply (rule ssubst[where t = "a * inverse a" and s= 1])
```
```  1116       apply simp_all
```
```  1117       apply (subst mult.commute)
```
```  1118       apply (rule inverse_mult_eq_1[OF a0])
```
```  1119       done
```
```  1120   }
```
```  1121   ultimately show ?thesis by blast
```
```  1122 qed
```
```  1123
```
```  1124 lemma fps_deriv_power:
```
```  1125   "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a::comm_ring_1) * fps_deriv a * a ^ (n - 1)"
```
```  1126   apply (induct n)
```
```  1127   apply (auto simp add: field_simps fps_const_add[symmetric] simp del: fps_const_add)
```
```  1128   apply (case_tac n)
```
```  1129   apply (auto simp add: field_simps)
```
```  1130   done
```
```  1131
```
```  1132 lemma fps_inverse_deriv:
```
```  1133   fixes a :: "'a::field fps"
```
```  1134   assumes a0: "a\$0 \<noteq> 0"
```
```  1135   shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
```
```  1136 proof -
```
```  1137   from inverse_mult_eq_1[OF a0]
```
```  1138   have "fps_deriv (inverse a * a) = 0" by simp
```
```  1139   then have "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0"
```
```  1140     by simp
```
```  1141   then have "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"
```
```  1142     by simp
```
```  1143   with inverse_mult_eq_1[OF a0]
```
```  1144   have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) = 0"
```
```  1145     unfolding power2_eq_square
```
```  1146     apply (simp add: field_simps)
```
```  1147     apply (simp add: mult.assoc[symmetric])
```
```  1148     done
```
```  1149   then have "(inverse a)\<^sup>2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * (inverse a)\<^sup>2 =
```
```  1150       0 - fps_deriv a * (inverse a)\<^sup>2"
```
```  1151     by simp
```
```  1152   then show "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
```
```  1153     by (simp add: field_simps)
```
```  1154 qed
```
```  1155
```
```  1156 lemma fps_inverse_mult:
```
```  1157   fixes a :: "'a::field fps"
```
```  1158   shows "inverse (a * b) = inverse a * inverse b"
```
```  1159 proof -
```
```  1160   {
```
```  1161     assume a0: "a\$0 = 0"
```
```  1162     then have ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```  1163     from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
```
```  1164     have ?thesis unfolding th by simp
```
```  1165   }
```
```  1166   moreover
```
```  1167   {
```
```  1168     assume b0: "b\$0 = 0"
```
```  1169     then have ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```  1170     from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
```
```  1171     have ?thesis unfolding th by simp
```
```  1172   }
```
```  1173   moreover
```
```  1174   {
```
```  1175     assume a0: "a\$0 \<noteq> 0" and b0: "b\$0 \<noteq> 0"
```
```  1176     from a0 b0 have ab0:"(a*b) \$ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
```
```  1177     from inverse_mult_eq_1[OF ab0]
```
```  1178     have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
```
```  1179     then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
```
```  1180       by (simp add: field_simps)
```
```  1181     then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp
```
```  1182   }
```
```  1183   ultimately show ?thesis by blast
```
```  1184 qed
```
```  1185
```
```  1186 lemma fps_inverse_deriv':
```
```  1187   fixes a :: "'a::field fps"
```
```  1188   assumes a0: "a\$0 \<noteq> 0"
```
```  1189   shows "fps_deriv (inverse a) = - fps_deriv a / a\<^sup>2"
```
```  1190   using fps_inverse_deriv[OF a0]
```
```  1191   unfolding power2_eq_square fps_divide_def fps_inverse_mult
```
```  1192   by simp
```
```  1193
```
```  1194 lemma inverse_mult_eq_1':
```
```  1195   assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```  1196   shows "f * inverse f= 1"
```
```  1197   by (metis mult.commute inverse_mult_eq_1 f0)
```
```  1198
```
```  1199 lemma fps_divide_deriv:
```
```  1200   fixes a :: "'a::field fps"
```
```  1201   assumes a0: "b\$0 \<noteq> 0"
```
```  1202   shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\<^sup>2"
```
```  1203   using fps_inverse_deriv[OF a0]
```
```  1204   by (simp add: fps_divide_def field_simps
```
```  1205     power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
```
```  1206
```
```  1207
```
```  1208 lemma fps_inverse_gp': "inverse (Abs_fps (\<lambda>n. 1::'a::field)) = 1 - X"
```
```  1209   by (simp add: fps_inverse_gp fps_eq_iff X_def)
```
```  1210
```
```  1211 lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
```
```  1212   by (cases n) simp_all
```
```  1213
```
```  1214
```
```  1215 lemma fps_inverse_X_plus1:
```
```  1216   "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::field)) ^ n)" (is "_ = ?r")
```
```  1217 proof -
```
```  1218   have eq: "(1 + X) * ?r = 1"
```
```  1219     unfolding minus_one_power_iff
```
```  1220     by (auto simp add: field_simps fps_eq_iff)
```
```  1221   show ?thesis
```
```  1222     by (auto simp add: eq intro: fps_inverse_unique)
```
```  1223 qed
```
```  1224
```
```  1225
```
```  1226 subsection{* Integration *}
```
```  1227
```
```  1228 definition fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps"
```
```  1229   where "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a\$(n - 1) / of_nat n))"
```
```  1230
```
```  1231 lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
```
```  1232   unfolding fps_integral_def fps_deriv_def
```
```  1233   by (simp add: fps_eq_iff del: of_nat_Suc)
```
```  1234
```
```  1235 lemma fps_integral_linear:
```
```  1236   "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
```
```  1237     fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
```
```  1238   (is "?l = ?r")
```
```  1239 proof -
```
```  1240   have "fps_deriv ?l = fps_deriv ?r"
```
```  1241     by (simp add: fps_deriv_fps_integral)
```
```  1242   moreover have "?l\$0 = ?r\$0"
```
```  1243     by (simp add: fps_integral_def)
```
```  1244   ultimately show ?thesis
```
```  1245     unfolding fps_deriv_eq_iff by auto
```
```  1246 qed
```
```  1247
```
```  1248
```
```  1249 subsection {* Composition of FPSs *}
```
```  1250
```
```  1251 definition fps_compose :: "'a::semiring_1 fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55)
```
```  1252   where "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n})"
```
```  1253
```
```  1254 lemma fps_compose_nth: "(a oo b)\$n = setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n}"
```
```  1255   by (simp add: fps_compose_def)
```
```  1256
```
```  1257 lemma fps_compose_X[simp]: "a oo X = (a :: 'a::comm_ring_1 fps)"
```
```  1258   by (simp add: fps_ext fps_compose_def mult_delta_right setsum.delta')
```
```  1259
```
```  1260 lemma fps_const_compose[simp]:
```
```  1261   "fps_const (a::'a::comm_ring_1) oo b = fps_const a"
```
```  1262   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum.delta)
```
```  1263
```
```  1264 lemma numeral_compose[simp]: "(numeral k :: 'a::comm_ring_1 fps) oo b = numeral k"
```
```  1265   unfolding numeral_fps_const by simp
```
```  1266
```
```  1267 lemma neg_numeral_compose[simp]: "(- numeral k :: 'a::comm_ring_1 fps) oo b = - numeral k"
```
```  1268   unfolding neg_numeral_fps_const by simp
```
```  1269
```
```  1270 lemma X_fps_compose_startby0[simp]: "a\$0 = 0 \<Longrightarrow> X oo a = (a :: 'a::comm_ring_1 fps)"
```
```  1271   by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum.delta not_le)
```
```  1272
```
```  1273
```
```  1274 subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
```
```  1275
```
```  1276 subsubsection {* Rule 1 *}
```
```  1277   (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
```
```  1278
```
```  1279 lemma fps_power_mult_eq_shift:
```
```  1280   "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) =
```
```  1281     Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a::comm_ring_1) * X^i) {0 .. k}"
```
```  1282   (is "?lhs = ?rhs")
```
```  1283 proof -
```
```  1284   { fix n :: nat
```
```  1285     have "?lhs \$ n = (if n < Suc k then 0 else a n)"
```
```  1286       unfolding X_power_mult_nth by auto
```
```  1287     also have "\<dots> = ?rhs \$ n"
```
```  1288     proof (induct k)
```
```  1289       case 0
```
```  1290       then show ?case by (simp add: fps_setsum_nth)
```
```  1291     next
```
```  1292       case (Suc k)
```
```  1293       note th = Suc.hyps[symmetric]
```
```  1294       have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})\$n =
```
```  1295         (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} -
```
```  1296           fps_const (a (Suc k)) * X^ Suc k) \$ n"
```
```  1297         by (simp add: field_simps)
```
```  1298       also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)\$n"
```
```  1299         using th unfolding fps_sub_nth by simp
```
```  1300       also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
```
```  1301         unfolding X_power_mult_right_nth
```
```  1302         apply (auto simp add: not_less fps_const_def)
```
```  1303         apply (rule cong[of a a, OF refl])
```
```  1304         apply arith
```
```  1305         done
```
```  1306       finally show ?case by simp
```
```  1307     qed
```
```  1308     finally have "?lhs \$ n = ?rhs \$ n" .
```
```  1309   }
```
```  1310   then show ?thesis by (simp add: fps_eq_iff)
```
```  1311 qed
```
```  1312
```
```  1313
```
```  1314 subsubsection {* Rule 2*}
```
```  1315
```
```  1316   (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
```
```  1317   (* If f reprents {a_n} and P is a polynomial, then
```
```  1318         P(xD) f represents {P(n) a_n}*)
```
```  1319
```
```  1320 definition "XD = op * X \<circ> fps_deriv"
```
```  1321
```
```  1322 lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: 'a::comm_ring_1 fps)"
```
```  1323   by (simp add: XD_def field_simps)
```
```  1324
```
```  1325 lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
```
```  1326   by (simp add: XD_def field_simps)
```
```  1327
```
```  1328 lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) =
```
```  1329     fps_const c * XD a + fps_const d * XD (b :: 'a::comm_ring_1 fps)"
```
```  1330   by simp
```
```  1331
```
```  1332 lemma XDN_linear:
```
```  1333   "(XD ^^ n) (fps_const c * a + fps_const d * b) =
```
```  1334     fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: 'a::comm_ring_1 fps)"
```
```  1335   by (induct n) simp_all
```
```  1336
```
```  1337 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a\$n)"
```
```  1338   by (simp add: fps_eq_iff)
```
```  1339
```
```  1340
```
```  1341 lemma fps_mult_XD_shift:
```
```  1342   "(XD ^^ k) (a :: 'a::comm_ring_1 fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a\$n)"
```
```  1343   by (induct k arbitrary: a) (simp_all add: XD_def fps_eq_iff field_simps del: One_nat_def)
```
```  1344
```
```  1345
```
```  1346 subsubsection {* Rule 3 is trivial and is given by @{text fps_times_def} *}
```
```  1347
```
```  1348 subsubsection {* Rule 5 --- summation and "division" by (1 - X) *}
```
```  1349
```
```  1350 lemma fps_divide_X_minus1_setsum_lemma:
```
```  1351   "a = ((1::'a::comm_ring_1 fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1352 proof -
```
```  1353   let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1354   have th0: "\<And>i. (1 - (X::'a fps)) \$ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)"
```
```  1355     by simp
```
```  1356   {
```
```  1357     fix n :: nat
```
```  1358     {
```
```  1359       assume "n = 0"
```
```  1360       then have "a \$ n = ((1 - X) * ?sa) \$ n"
```
```  1361         by (simp add: fps_mult_nth)
```
```  1362     }
```
```  1363     moreover
```
```  1364     {
```
```  1365       assume n0: "n \<noteq> 0"
```
```  1366       then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1} \<union> {2..n} = {1..n}"
```
```  1367         "{0..n - 1} \<union> {n} = {0..n}"
```
```  1368         by (auto simp: set_eq_iff)
```
```  1369       have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}" "{0..n - 1} \<inter> {n} = {}"
```
```  1370         using n0 by simp_all
```
```  1371       have f: "finite {0}" "finite {1}" "finite {2 .. n}"
```
```  1372         "finite {0 .. n - 1}" "finite {n}" by simp_all
```
```  1373       have "((1 - X) * ?sa) \$ n = setsum (\<lambda>i. (1 - X)\$ i * ?sa \$ (n - i)) {0 .. n}"
```
```  1374         by (simp add: fps_mult_nth)
```
```  1375       also have "\<dots> = a\$n"
```
```  1376         unfolding th0
```
```  1377         unfolding setsum.union_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
```
```  1378         unfolding setsum.union_disjoint[OF f(2) f(3) d(2)]
```
```  1379         apply (simp)
```
```  1380         unfolding setsum.union_disjoint[OF f(4,5) d(3), unfolded u(3)]
```
```  1381         apply simp
```
```  1382         done
```
```  1383       finally have "a\$n = ((1 - X) * ?sa) \$ n"
```
```  1384         by simp
```
```  1385     }
```
```  1386     ultimately have "a\$n = ((1 - X) * ?sa) \$ n"
```
```  1387       by blast
```
```  1388   }
```
```  1389   then show ?thesis
```
```  1390     unfolding fps_eq_iff by blast
```
```  1391 qed
```
```  1392
```
```  1393 lemma fps_divide_X_minus1_setsum:
```
```  1394   "a /((1::'a::field fps) - X) = Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1395 proof -
```
```  1396   let ?X = "1 - (X::'a fps)"
```
```  1397   have th0: "?X \$ 0 \<noteq> 0"
```
```  1398     by simp
```
```  1399   have "a /?X = ?X *  Abs_fps (\<lambda>n::nat. setsum (op \$ a) {0..n}) * inverse ?X"
```
```  1400     using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
```
```  1401     by (simp add: fps_divide_def mult.assoc)
```
```  1402   also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n::nat. setsum (op \$ a) {0..n}) "
```
```  1403     by (simp add: ac_simps)
```
```  1404   finally show ?thesis
```
```  1405     by (simp add: inverse_mult_eq_1[OF th0])
```
```  1406 qed
```
```  1407
```
```  1408
```
```  1409 subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
```
```  1410   finite product of FPS, also the relvant instance of powers of a FPS*}
```
```  1411
```
```  1412 definition "natpermute n k = {l :: nat list. length l = k \<and> listsum l = n}"
```
```  1413
```
```  1414 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
```
```  1415   apply (auto simp add: natpermute_def)
```
```  1416   apply (case_tac x)
```
```  1417   apply auto
```
```  1418   done
```
```  1419
```
```  1420 lemma append_natpermute_less_eq:
```
```  1421   assumes "xs @ ys \<in> natpermute n k"
```
```  1422   shows "listsum xs \<le> n"
```
```  1423     and "listsum ys \<le> n"
```
```  1424 proof -
```
```  1425   from assms have "listsum (xs @ ys) = n"
```
```  1426     by (simp add: natpermute_def)
```
```  1427   then have "listsum xs + listsum ys = n"
```
```  1428     by simp
```
```  1429   then show "listsum xs \<le> n" and "listsum ys \<le> n"
```
```  1430     by simp_all
```
```  1431 qed
```
```  1432
```
```  1433 lemma natpermute_split:
```
```  1434   assumes "h \<le> k"
```
```  1435   shows "natpermute n k =
```
```  1436     (\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})"
```
```  1437   (is "?L = ?R" is "?L = (\<Union>m \<in>{0..n}. ?S m)")
```
```  1438 proof -
```
```  1439   {
```
```  1440     fix l
```
```  1441     assume l: "l \<in> ?R"
```
```  1442     from l obtain m xs ys where h: "m \<in> {0..n}"
```
```  1443       and xs: "xs \<in> natpermute m h"
```
```  1444       and ys: "ys \<in> natpermute (n - m) (k - h)"
```
```  1445       and leq: "l = xs@ys" by blast
```
```  1446     from xs have xs': "listsum xs = m"
```
```  1447       by (simp add: natpermute_def)
```
```  1448     from ys have ys': "listsum ys = n - m"
```
```  1449       by (simp add: natpermute_def)
```
```  1450     have "l \<in> ?L" using leq xs ys h
```
```  1451       apply (clarsimp simp add: natpermute_def)
```
```  1452       unfolding xs' ys'
```
```  1453       using assms xs ys
```
```  1454       unfolding natpermute_def
```
```  1455       apply simp
```
```  1456       done
```
```  1457   }
```
```  1458   moreover
```
```  1459   {
```
```  1460     fix l
```
```  1461     assume l: "l \<in> natpermute n k"
```
```  1462     let ?xs = "take h l"
```
```  1463     let ?ys = "drop h l"
```
```  1464     let ?m = "listsum ?xs"
```
```  1465     from l have ls: "listsum (?xs @ ?ys) = n"
```
```  1466       by (simp add: natpermute_def)
```
```  1467     have xs: "?xs \<in> natpermute ?m h" using l assms
```
```  1468       by (simp add: natpermute_def)
```
```  1469     have l_take_drop: "listsum l = listsum (take h l @ drop h l)"
```
```  1470       by simp
```
```  1471     then have ys: "?ys \<in> natpermute (n - ?m) (k - h)"
```
```  1472       using l assms ls by (auto simp add: natpermute_def simp del: append_take_drop_id)
```
```  1473     from ls have m: "?m \<in> {0..n}"
```
```  1474       by (simp add: l_take_drop del: append_take_drop_id)
```
```  1475     from xs ys ls have "l \<in> ?R"
```
```  1476       apply auto
```
```  1477       apply (rule bexI [where x = "?m"])
```
```  1478       apply (rule exI [where x = "?xs"])
```
```  1479       apply (rule exI [where x = "?ys"])
```
```  1480       using ls l
```
```  1481       apply (auto simp add: natpermute_def l_take_drop simp del: append_take_drop_id)
```
```  1482       apply simp
```
```  1483       done
```
```  1484   }
```
```  1485   ultimately show ?thesis by blast
```
```  1486 qed
```
```  1487
```
```  1488 lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
```
```  1489   by (auto simp add: natpermute_def)
```
```  1490
```
```  1491 lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
```
```  1492   apply (auto simp add: set_replicate_conv_if natpermute_def)
```
```  1493   apply (rule nth_equalityI)
```
```  1494   apply simp_all
```
```  1495   done
```
```  1496
```
```  1497 lemma natpermute_finite: "finite (natpermute n k)"
```
```  1498 proof (induct k arbitrary: n)
```
```  1499   case 0
```
```  1500   then show ?case
```
```  1501     apply (subst natpermute_split[of 0 0, simplified])
```
```  1502     apply (simp add: natpermute_0)
```
```  1503     done
```
```  1504 next
```
```  1505   case (Suc k)
```
```  1506   then show ?case unfolding natpermute_split [of k "Suc k", simplified]
```
```  1507     apply -
```
```  1508     apply (rule finite_UN_I)
```
```  1509     apply simp
```
```  1510     unfolding One_nat_def[symmetric] natlist_trivial_1
```
```  1511     apply simp
```
```  1512     done
```
```  1513 qed
```
```  1514
```
```  1515 lemma natpermute_contain_maximal:
```
```  1516   "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
```
```  1517   (is "?A = ?B")
```
```  1518 proof -
```
```  1519   {
```
```  1520     fix xs
```
```  1521     assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
```
```  1522     from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
```
```  1523       unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
```
```  1524     have eqs: "({0..k} - {i}) \<union> {i} = {0..k}"
```
```  1525       using i by auto
```
```  1526     have f: "finite({0..k} - {i})" "finite {i}"
```
```  1527       by auto
```
```  1528     have d: "({0..k} - {i}) \<inter> {i} = {}"
```
```  1529       using i by auto
```
```  1530     from H have "n = setsum (nth xs) {0..k}"
```
```  1531       apply (simp add: natpermute_def)
```
```  1532       apply (auto simp add: atLeastLessThanSuc_atLeastAtMost listsum_setsum_nth)
```
```  1533       done
```
```  1534     also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
```
```  1535       unfolding setsum.union_disjoint[OF f d, unfolded eqs] using i by simp
```
```  1536     finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0"
```
```  1537       by auto
```
```  1538     from H have xsl: "length xs = k+1"
```
```  1539       by (simp add: natpermute_def)
```
```  1540     from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
```
```  1541       unfolding length_replicate by presburger+
```
```  1542     have "xs = replicate (k+1) 0 [i := n]"
```
```  1543       apply (rule nth_equalityI)
```
```  1544       unfolding xsl length_list_update length_replicate
```
```  1545       apply simp
```
```  1546       apply clarify
```
```  1547       unfolding nth_list_update[OF i'(1)]
```
```  1548       using i zxs
```
```  1549       apply (case_tac "ia = i")
```
```  1550       apply (auto simp del: replicate.simps)
```
```  1551       done
```
```  1552     then have "xs \<in> ?B" using i by blast
```
```  1553   }
```
```  1554   moreover
```
```  1555   {
```
```  1556     fix i
```
```  1557     assume i: "i \<in> {0..k}"
```
```  1558     let ?xs = "replicate (k+1) 0 [i:=n]"
```
```  1559     have nxs: "n \<in> set ?xs"
```
```  1560       apply (rule set_update_memI)
```
```  1561       using i apply simp
```
```  1562       done
```
```  1563     have xsl: "length ?xs = k+1"
```
```  1564       by (simp only: length_replicate length_list_update)
```
```  1565     have "listsum ?xs = setsum (nth ?xs) {0..<k+1}"
```
```  1566       unfolding listsum_setsum_nth xsl ..
```
```  1567     also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
```
```  1568       by (rule setsum.cong) (simp_all del: replicate.simps)
```
```  1569     also have "\<dots> = n" using i by (simp add: setsum.delta)
```
```  1570     finally have "?xs \<in> natpermute n (k+1)"
```
```  1571       using xsl unfolding natpermute_def mem_Collect_eq by blast
```
```  1572     then have "?xs \<in> ?A"
```
```  1573       using nxs  by blast
```
```  1574   }
```
```  1575   ultimately show ?thesis by auto
```
```  1576 qed
```
```  1577
```
```  1578 text {* The general form *}
```
```  1579 lemma fps_setprod_nth:
```
```  1580   fixes m :: nat
```
```  1581     and a :: "nat \<Rightarrow> 'a::comm_ring_1 fps"
```
```  1582   shows "(setprod a {0 .. m}) \$ n =
```
```  1583     setsum (\<lambda>v. setprod (\<lambda>j. (a j) \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1584   (is "?P m n")
```
```  1585 proof (induct m arbitrary: n rule: nat_less_induct)
```
```  1586   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
```
```  1587   show "?P m n"
```
```  1588   proof (cases m)
```
```  1589     case 0
```
```  1590     then show ?thesis
```
```  1591       apply simp
```
```  1592       unfolding natlist_trivial_1[where n = n, unfolded One_nat_def]
```
```  1593       apply simp
```
```  1594       done
```
```  1595   next
```
```  1596     case (Suc k)
```
```  1597     then have km: "k < m" by arith
```
```  1598     have u0: "{0 .. k} \<union> {m} = {0..m}"
```
```  1599       using Suc by (simp add: set_eq_iff) presburger
```
```  1600     have f0: "finite {0 .. k}" "finite {m}" by auto
```
```  1601     have d0: "{0 .. k} \<inter> {m} = {}" using Suc by auto
```
```  1602     have "(setprod a {0 .. m}) \$ n = (setprod a {0 .. k} * a m) \$ n"
```
```  1603       unfolding setprod.union_disjoint[OF f0 d0, unfolded u0] by simp
```
```  1604     also have "\<dots> = (\<Sum>i = 0..n. (\<Sum>v\<in>natpermute i (k + 1). \<Prod>j\<in>{0..k}. a j \$ v ! j) * a m \$ (n - i))"
```
```  1605       unfolding fps_mult_nth H[rule_format, OF km] ..
```
```  1606     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j \$ v ! j)"
```
```  1607       apply (simp add: Suc)
```
```  1608       unfolding natpermute_split[of m "m + 1", simplified, of n,
```
```  1609         unfolded natlist_trivial_1[unfolded One_nat_def] Suc]
```
```  1610       apply (subst setsum.UNION_disjoint)
```
```  1611       apply simp
```
```  1612       apply simp
```
```  1613       unfolding image_Collect[symmetric]
```
```  1614       apply clarsimp
```
```  1615       apply (rule finite_imageI)
```
```  1616       apply (rule natpermute_finite)
```
```  1617       apply (clarsimp simp add: set_eq_iff)
```
```  1618       apply auto
```
```  1619       apply (rule setsum.cong)
```
```  1620       apply (rule refl)
```
```  1621       unfolding setsum_left_distrib
```
```  1622       apply (rule sym)
```
```  1623       apply (rule_tac l = "\<lambda>xs. xs @ [n - x]" in setsum.reindex_cong)
```
```  1624       apply (simp add: inj_on_def)
```
```  1625       apply auto
```
```  1626       unfolding setprod.union_disjoint[OF f0 d0, unfolded u0, unfolded Suc]
```
```  1627       apply (clarsimp simp add: natpermute_def nth_append)
```
```  1628       done
```
```  1629     finally show ?thesis .
```
```  1630   qed
```
```  1631 qed
```
```  1632
```
```  1633 text{* The special form for powers *}
```
```  1634 lemma fps_power_nth_Suc:
```
```  1635   fixes m :: nat
```
```  1636     and a :: "'a::comm_ring_1 fps"
```
```  1637   shows "(a ^ Suc m)\$n = setsum (\<lambda>v. setprod (\<lambda>j. a \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1638 proof -
```
```  1639   have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}"
```
```  1640     by (simp add: setprod_constant)
```
```  1641   show ?thesis unfolding th0 fps_setprod_nth ..
```
```  1642 qed
```
```  1643
```
```  1644 lemma fps_power_nth:
```
```  1645   fixes m :: nat
```
```  1646     and a :: "'a::comm_ring_1 fps"
```
```  1647   shows "(a ^m)\$n =
```
```  1648     (if m=0 then 1\$n else setsum (\<lambda>v. setprod (\<lambda>j. a \$ (v!j)) {0..m - 1}) (natpermute n m))"
```
```  1649   by (cases m) (simp_all add: fps_power_nth_Suc del: power_Suc)
```
```  1650
```
```  1651 lemma fps_nth_power_0:
```
```  1652   fixes m :: nat
```
```  1653     and a :: "'a::comm_ring_1 fps"
```
```  1654   shows "(a ^m)\$0 = (a\$0) ^ m"
```
```  1655 proof (cases m)
```
```  1656   case 0
```
```  1657   then show ?thesis by simp
```
```  1658 next
```
```  1659   case (Suc n)
```
```  1660   then have c: "m = card {0..n}" by simp
```
```  1661   have "(a ^m)\$0 = setprod (\<lambda>i. a\$0) {0..n}"
```
```  1662     by (simp add: Suc fps_power_nth del: replicate.simps power_Suc)
```
```  1663   also have "\<dots> = (a\$0) ^ m"
```
```  1664    unfolding c by (rule setprod_constant) simp
```
```  1665  finally show ?thesis .
```
```  1666 qed
```
```  1667
```
```  1668 lemma fps_compose_inj_right:
```
```  1669   assumes a0: "a\$0 = (0::'a::idom)"
```
```  1670     and a1: "a\$1 \<noteq> 0"
```
```  1671   shows "(b oo a = c oo a) \<longleftrightarrow> b = c"
```
```  1672   (is "?lhs \<longleftrightarrow>?rhs")
```
```  1673 proof
```
```  1674   assume ?rhs
```
```  1675   then show "?lhs" by simp
```
```  1676 next
```
```  1677   assume h: ?lhs
```
```  1678   {
```
```  1679     fix n
```
```  1680     have "b\$n = c\$n"
```
```  1681     proof (induct n rule: nat_less_induct)
```
```  1682       fix n
```
```  1683       assume H: "\<forall>m<n. b\$m = c\$m"
```
```  1684       {
```
```  1685         assume n0: "n=0"
```
```  1686         from h have "(b oo a)\$n = (c oo a)\$n" by simp
```
```  1687         then have "b\$n = c\$n" using n0 by (simp add: fps_compose_nth)
```
```  1688       }
```
```  1689       moreover
```
```  1690       {
```
```  1691         fix n1 assume n1: "n = Suc n1"
```
```  1692         have f: "finite {0 .. n1}" "finite {n}" by simp_all
```
```  1693         have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
```
```  1694         have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
```
```  1695         have seq: "(\<Sum>i = 0..n1. b \$ i * a ^ i \$ n) = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n)"
```
```  1696           apply (rule setsum.cong)
```
```  1697           using H n1
```
```  1698           apply auto
```
```  1699           done
```
```  1700         have th0: "(b oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + b\$n * (a\$1)^n"
```
```  1701           unfolding fps_compose_nth setsum.union_disjoint[OF f d, unfolded eq] seq
```
```  1702           using startsby_zero_power_nth_same[OF a0]
```
```  1703           by simp
```
```  1704         have th1: "(c oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + c\$n * (a\$1)^n"
```
```  1705           unfolding fps_compose_nth setsum.union_disjoint[OF f d, unfolded eq]
```
```  1706           using startsby_zero_power_nth_same[OF a0]
```
```  1707           by simp
```
```  1708         from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
```
```  1709         have "b\$n = c\$n" by auto
```
```  1710       }
```
```  1711       ultimately show "b\$n = c\$n" by (cases n) auto
```
```  1712     qed}
```
```  1713   then show ?rhs by (simp add: fps_eq_iff)
```
```  1714 qed
```
```  1715
```
```  1716
```
```  1717 subsection {* Radicals *}
```
```  1718
```
```  1719 declare setprod.cong [fundef_cong]
```
```  1720
```
```  1721 function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a::field fps \<Rightarrow> nat \<Rightarrow> 'a"
```
```  1722 where
```
```  1723   "radical r 0 a 0 = 1"
```
```  1724 | "radical r 0 a (Suc n) = 0"
```
```  1725 | "radical r (Suc k) a 0 = r (Suc k) (a\$0)"
```
```  1726 | "radical r (Suc k) a (Suc n) =
```
```  1727     (a\$ Suc n - setsum (\<lambda>xs. setprod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k})
```
```  1728       {xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) /
```
```  1729     (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
```
```  1730   by pat_completeness auto
```
```  1731
```
```  1732 termination radical
```
```  1733 proof
```
```  1734   let ?R = "measure (\<lambda>(r, k, a, n). n)"
```
```  1735   {
```
```  1736     show "wf ?R" by auto
```
```  1737   next
```
```  1738     fix r k a n xs i
```
```  1739     assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
```
```  1740     {
```
```  1741       assume c: "Suc n \<le> xs ! i"
```
```  1742       from xs i have "xs !i \<noteq> Suc n"
```
```  1743         by (auto simp add: in_set_conv_nth natpermute_def)
```
```  1744       with c have c': "Suc n < xs!i" by arith
```
```  1745       have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
```
```  1746         by simp_all
```
```  1747       have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}"
```
```  1748         by auto
```
```  1749       have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})"
```
```  1750         using i by auto
```
```  1751       from xs have "Suc n = listsum xs"
```
```  1752         by (simp add: natpermute_def)
```
```  1753       also have "\<dots> = setsum (nth xs) {0..<Suc k}" using xs
```
```  1754         by (simp add: natpermute_def listsum_setsum_nth)
```
```  1755       also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  1756         unfolding eqs  setsum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  1757         unfolding setsum.union_disjoint[OF fths(2) fths(3) d(2)]
```
```  1758         by simp
```
```  1759       finally have False using c' by simp
```
```  1760     }
```
```  1761     then show "((r, Suc k, a, xs!i), r, Suc k, a, Suc n) \<in> ?R"
```
```  1762       apply auto
```
```  1763       apply (metis not_less)
```
```  1764       done
```
```  1765   next
```
```  1766     fix r k a n
```
```  1767     show "((r, Suc k, a, 0), r, Suc k, a, Suc n) \<in> ?R" by simp
```
```  1768   }
```
```  1769 qed
```
```  1770
```
```  1771 definition "fps_radical r n a = Abs_fps (radical r n a)"
```
```  1772
```
```  1773 lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
```
```  1774   apply (auto simp add: fps_eq_iff fps_radical_def)
```
```  1775   apply (case_tac n)
```
```  1776   apply auto
```
```  1777   done
```
```  1778
```
```  1779 lemma fps_radical_nth_0[simp]: "fps_radical r n a \$ 0 = (if n=0 then 1 else r n (a\$0))"
```
```  1780   by (cases n) (simp_all add: fps_radical_def)
```
```  1781
```
```  1782 lemma fps_radical_power_nth[simp]:
```
```  1783   assumes r: "(r k (a\$0)) ^ k = a\$0"
```
```  1784   shows "fps_radical r k a ^ k \$ 0 = (if k = 0 then 1 else a\$0)"
```
```  1785 proof (cases k)
```
```  1786   case 0
```
```  1787   then show ?thesis by simp
```
```  1788 next
```
```  1789   case (Suc h)
```
```  1790   have eq1: "fps_radical r k a ^ k \$ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a \$ (replicate k 0) ! j)"
```
```  1791     unfolding fps_power_nth Suc by simp
```
```  1792   also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a\$0))"
```
```  1793     apply (rule setprod.cong)
```
```  1794     apply simp
```
```  1795     using Suc
```
```  1796     apply (subgoal_tac "replicate k 0 ! x = 0")
```
```  1797     apply (auto intro: nth_replicate simp del: replicate.simps)
```
```  1798     done
```
```  1799   also have "\<dots> = a\$0" using r Suc by (simp add: setprod_constant)
```
```  1800   finally show ?thesis using Suc by simp
```
```  1801 qed
```
```  1802
```
```  1803 lemma natpermute_max_card:
```
```  1804   assumes n0: "n \<noteq> 0"
```
```  1805   shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k + 1"
```
```  1806   unfolding natpermute_contain_maximal
```
```  1807 proof -
```
```  1808   let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
```
```  1809   let ?K = "{0 ..k}"
```
```  1810   have fK: "finite ?K" by simp
```
```  1811   have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
```
```  1812   have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow>
```
```  1813     {replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
```
```  1814   proof clarify
```
```  1815     fix i j
```
```  1816     assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
```
```  1817     {
```
```  1818       assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
```
```  1819       have "(replicate (k+1) 0 [i:=n] ! i) = n"
```
```  1820         using i by (simp del: replicate.simps)
```
```  1821       moreover
```
```  1822       have "(replicate (k+1) 0 [j:=n] ! i) = 0"
```
```  1823         using i ij by (simp del: replicate.simps)
```
```  1824       ultimately have False
```
```  1825         using eq n0 by (simp del: replicate.simps)
```
```  1826     }
```
```  1827     then show "{replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
```
```  1828       by auto
```
```  1829   qed
```
```  1830   from card_UN_disjoint[OF fK fAK d]
```
```  1831   show "card (\<Union>i\<in>{0..k}. {replicate (k + 1) 0[i := n]}) = k + 1"
```
```  1832     by simp
```
```  1833 qed
```
```  1834
```
```  1835 lemma power_radical:
```
```  1836   fixes a:: "'a::field_char_0 fps"
```
```  1837   assumes a0: "a\$0 \<noteq> 0"
```
```  1838   shows "(r (Suc k) (a\$0)) ^ Suc k = a\$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1839 proof -
```
```  1840   let ?r = "fps_radical r (Suc k) a"
```
```  1841   {
```
```  1842     assume r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
```
```  1843     from a0 r0 have r00: "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1844     {
```
```  1845       fix z
```
```  1846       have "?r ^ Suc k \$ z = a\$z"
```
```  1847       proof (induct z rule: nat_less_induct)
```
```  1848         fix n
```
```  1849         assume H: "\<forall>m<n. ?r ^ Suc k \$ m = a\$m"
```
```  1850         {
```
```  1851           assume "n = 0"
```
```  1852           then have "?r ^ Suc k \$ n = a \$n"
```
```  1853             using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp
```
```  1854         }
```
```  1855         moreover
```
```  1856         {
```
```  1857           fix n1 assume n1: "n = Suc n1"
```
```  1858           have nz: "n \<noteq> 0" using n1 by arith
```
```  1859           let ?Pnk = "natpermute n (k + 1)"
```
```  1860           let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1861           let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1862           have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1863           have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1864           have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1865             using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1866             by (metis natpermute_finite)+
```
```  1867           let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1868           have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
```
```  1869           proof (rule setsum.cong)
```
```  1870             fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
```
```  1871             let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) =
```
```  1872               fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1873             from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1874               unfolding natpermute_contain_maximal by auto
```
```  1875             have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) =
```
```  1876                 (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
```
```  1877               apply (rule setprod.cong, simp)
```
```  1878               using i r0
```
```  1879               apply (simp del: replicate.simps)
```
```  1880               done
```
```  1881             also have "\<dots> = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
```
```  1882               using i r0 by (simp add: setprod_gen_delta)
```
```  1883             finally show ?ths .
```
```  1884           qed rule
```
```  1885           then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1886             by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1887           also have "\<dots> = a\$n - setsum ?f ?Pnknn"
```
```  1888             unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc)
```
```  1889           finally have fn: "setsum ?f ?Pnkn = a\$n - setsum ?f ?Pnknn" .
```
```  1890           have "(?r ^ Suc k)\$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
```
```  1891             unfolding fps_power_nth_Suc setsum.union_disjoint[OF f d, unfolded eq] ..
```
```  1892           also have "\<dots> = a\$n" unfolding fn by simp
```
```  1893           finally have "?r ^ Suc k \$ n = a \$n" .
```
```  1894         }
```
```  1895         ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n) auto
```
```  1896       qed
```
```  1897     }
```
```  1898     then have ?thesis using r0 by (simp add: fps_eq_iff)
```
```  1899   }
```
```  1900   moreover
```
```  1901   {
```
```  1902     assume h: "(fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1903     then have "((fps_radical r (Suc k) a) ^ (Suc k))\$0 = a\$0" by simp
```
```  1904     then have "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
```
```  1905       unfolding fps_power_nth_Suc
```
```  1906       by (simp add: setprod_constant del: replicate.simps)
```
```  1907   }
```
```  1908   ultimately show ?thesis by blast
```
```  1909 qed
```
```  1910
```
```  1911 (*
```
```  1912 lemma power_radical:
```
```  1913   fixes a:: "'a::field_char_0 fps"
```
```  1914   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 \<noteq> 0"
```
```  1915   shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1916 proof-
```
```  1917   let ?r = "fps_radical r (Suc k) a"
```
```  1918   from a0 r0 have r00: "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1919   {fix z have "?r ^ Suc k \$ z = a\$z"
```
```  1920     proof(induct z rule: nat_less_induct)
```
```  1921       fix n assume H: "\<forall>m<n. ?r ^ Suc k \$ m = a\$m"
```
```  1922       {assume "n = 0" then have "?r ^ Suc k \$ n = a \$n"
```
```  1923           using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
```
```  1924       moreover
```
```  1925       {fix n1 assume n1: "n = Suc n1"
```
```  1926         have fK: "finite {0..k}" by simp
```
```  1927         have nz: "n \<noteq> 0" using n1 by arith
```
```  1928         let ?Pnk = "natpermute n (k + 1)"
```
```  1929         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1930         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1931         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1932         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1933         have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1934           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1935           by (metis natpermute_finite)+
```
```  1936         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1937         have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
```
```  1938         proof(rule setsum.cong2)
```
```  1939           fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
```
```  1940           let ?ths = "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = fps_radical r (Suc k) a \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1941           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1942             unfolding natpermute_contain_maximal by auto
```
```  1943           have "(\<Prod>j\<in>{0..k}. fps_radical r (Suc k) a \$ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then fps_radical r (Suc k) a \$ n else r (Suc k) (a\$0))"
```
```  1944             apply (rule setprod.cong, simp)
```
```  1945             using i r0 by (simp del: replicate.simps)
```
```  1946           also have "\<dots> = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
```
```  1947             unfolding setprod_gen_delta[OF fK] using i r0 by simp
```
```  1948           finally show ?ths .
```
```  1949         qed
```
```  1950         then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1951           by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1952         also have "\<dots> = a\$n - setsum ?f ?Pnknn"
```
```  1953           unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
```
```  1954         finally have fn: "setsum ?f ?Pnkn = a\$n - setsum ?f ?Pnknn" .
```
```  1955         have "(?r ^ Suc k)\$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
```
```  1956           unfolding fps_power_nth_Suc setsum.union_disjoint[OF f d, unfolded eq] ..
```
```  1957         also have "\<dots> = a\$n" unfolding fn by simp
```
```  1958         finally have "?r ^ Suc k \$ n = a \$n" .}
```
```  1959       ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n, auto)
```
```  1960   qed }
```
```  1961   then show ?thesis by (simp add: fps_eq_iff)
```
```  1962 qed
```
```  1963
```
```  1964 *)
```
```  1965 lemma eq_divide_imp':
```
```  1966   fixes c :: "'a::field" shows "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
```
```  1967   by (simp add: field_simps)
```
```  1968
```
```  1969 lemma radical_unique:
```
```  1970   assumes r0: "(r (Suc k) (b\$0)) ^ Suc k = b\$0"
```
```  1971     and a0: "r (Suc k) (b\$0 ::'a::field_char_0) = a\$0"
```
```  1972     and b0: "b\$0 \<noteq> 0"
```
```  1973   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
```
```  1974 proof -
```
```  1975   let ?r = "fps_radical r (Suc k) b"
```
```  1976   have r00: "r (Suc k) (b\$0) \<noteq> 0" using b0 r0 by auto
```
```  1977   {
```
```  1978     assume H: "a = ?r"
```
```  1979     from H have "a^Suc k = b"
```
```  1980       using power_radical[OF b0, of r k, unfolded r0] by simp
```
```  1981   }
```
```  1982   moreover
```
```  1983   {
```
```  1984     assume H: "a^Suc k = b"
```
```  1985     have ceq: "card {0..k} = Suc k" by simp
```
```  1986     from a0 have a0r0: "a\$0 = ?r\$0" by simp
```
```  1987     {
```
```  1988       fix n
```
```  1989       have "a \$ n = ?r \$ n"
```
```  1990       proof (induct n rule: nat_less_induct)
```
```  1991         fix n
```
```  1992         assume h: "\<forall>m<n. a\$m = ?r \$m"
```
```  1993         {
```
```  1994           assume "n = 0"
```
```  1995           then have "a\$n = ?r \$n" using a0 by simp
```
```  1996         }
```
```  1997         moreover
```
```  1998         {
```
```  1999           fix n1
```
```  2000           assume n1: "n = Suc n1"
```
```  2001           have fK: "finite {0..k}" by simp
```
```  2002         have nz: "n \<noteq> 0" using n1 by arith
```
```  2003         let ?Pnk = "natpermute n (Suc k)"
```
```  2004         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  2005         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  2006         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  2007         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  2008         have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  2009           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  2010           by (metis natpermute_finite)+
```
```  2011         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  2012         let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a \$ v ! j"
```
```  2013         have "setsum ?g ?Pnkn = setsum (\<lambda>v. a \$ n * (?r\$0)^k) ?Pnkn"
```
```  2014         proof (rule setsum.cong)
```
```  2015           fix v
```
```  2016           assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
```
```  2017           let ?ths = "(\<Prod>j\<in>{0..k}. a \$ v ! j) = a \$ n * (?r\$0)^k"
```
```  2018           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  2019             unfolding Suc_eq_plus1 natpermute_contain_maximal
```
```  2020             by (auto simp del: replicate.simps)
```
```  2021           have "(\<Prod>j\<in>{0..k}. a \$ v ! j) = (\<Prod>j\<in>{0..k}. if j = i then a \$ n else r (Suc k) (b\$0))"
```
```  2022             apply (rule setprod.cong, simp)
```
```  2023             using i a0
```
```  2024             apply (simp del: replicate.simps)
```
```  2025             done
```
```  2026           also have "\<dots> = a \$ n * (?r \$ 0)^k"
```
```  2027             using i by (simp add: setprod_gen_delta)
```
```  2028           finally show ?ths .
```
```  2029         qed rule
```
```  2030         then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a \$ n * (?r \$ 0)^k"
```
```  2031           by (simp add: natpermute_max_card[OF nz, simplified])
```
```  2032         have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
```
```  2033         proof (rule setsum.cong, rule refl, rule setprod.cong, simp)
```
```  2034           fix xs i
```
```  2035           assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
```
```  2036           {
```
```  2037             assume c: "n \<le> xs ! i"
```
```  2038             from xs i have "xs !i \<noteq> n"
```
```  2039               by (auto simp add: in_set_conv_nth natpermute_def)
```
```  2040             with c have c': "n < xs!i" by arith
```
```  2041             have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}"
```
```  2042               by simp_all
```
```  2043             have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}"
```
```  2044               by auto
```
```  2045             have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})"
```
```  2046               using i by auto
```
```  2047             from xs have "n = listsum xs"
```
```  2048               by (simp add: natpermute_def)
```
```  2049             also have "\<dots> = setsum (nth xs) {0..<Suc k}"
```
```  2050               using xs by (simp add: natpermute_def listsum_setsum_nth)
```
```  2051             also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  2052               unfolding eqs  setsum.union_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  2053               unfolding setsum.union_disjoint[OF fths(2) fths(3) d(2)]
```
```  2054               by simp
```
```  2055             finally have False using c' by simp
```
```  2056           }
```
```  2057           then have thn: "xs!i < n" by presburger
```
```  2058           from h[rule_format, OF thn] show "a\$(xs !i) = ?r\$(xs!i)" .
```
```  2059         qed
```
```  2060         have th00: "\<And>x::'a. of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
```
```  2061           by (simp add: field_simps del: of_nat_Suc)
```
```  2062         from H have "b\$n = a^Suc k \$ n"
```
```  2063           by (simp add: fps_eq_iff)
```
```  2064         also have "a ^ Suc k\$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
```
```  2065           unfolding fps_power_nth_Suc
```
```  2066           using setsum.union_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
```
```  2067             unfolded eq, of ?g] by simp
```
```  2068         also have "\<dots> = of_nat (k+1) * a \$ n * (?r \$ 0)^k + setsum ?f ?Pnknn"
```
```  2069           unfolding th0 th1 ..
```
```  2070         finally have "of_nat (k+1) * a \$ n * (?r \$ 0)^k = b\$n - setsum ?f ?Pnknn"
```
```  2071           by simp
```
```  2072         then have "a\$n = (b\$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r \$ 0)^k)"
```
```  2073           apply -
```
```  2074           apply (rule eq_divide_imp')
```
```  2075           using r00
```
```  2076           apply (simp del: of_nat_Suc)
```
```  2077           apply (simp add: ac_simps)
```
```  2078           done
```
```  2079         then have "a\$n = ?r \$n"
```
```  2080           apply (simp del: of_nat_Suc)
```
```  2081           unfolding fps_radical_def n1
```
```  2082           apply (simp add: field_simps n1 th00 del: of_nat_Suc)
```
```  2083           done
```
```  2084         }
```
```  2085         ultimately show "a\$n = ?r \$ n" by (cases n) auto
```
```  2086       qed
```
```  2087     }
```
```  2088     then have "a = ?r" by (simp add: fps_eq_iff)
```
```  2089   }
```
```  2090   ultimately show ?thesis by blast
```
```  2091 qed
```
```  2092
```
```  2093
```
```  2094 lemma radical_power:
```
```  2095   assumes r0: "r (Suc k) ((a\$0) ^ Suc k) = a\$0"
```
```  2096     and a0: "(a\$0 :: 'a::field_char_0) \<noteq> 0"
```
```  2097   shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
```
```  2098 proof -
```
```  2099   let ?ak = "a^ Suc k"
```
```  2100   have ak0: "?ak \$ 0 = (a\$0) ^ Suc k"
```
```  2101     by (simp add: fps_nth_power_0 del: power_Suc)
```
```  2102   from r0 have th0: "r (Suc k) (a ^ Suc k \$ 0) ^ Suc k = a ^ Suc k \$ 0"
```
```  2103     using ak0 by auto
```
```  2104   from r0 ak0 have th1: "r (Suc k) (a ^ Suc k \$ 0) = a \$ 0"
```
```  2105     by auto
```
```  2106   from ak0 a0 have ak00: "?ak \$ 0 \<noteq>0 "
```
```  2107     by auto
```
```  2108   from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis
```
```  2109     by metis
```
```  2110 qed
```
```  2111
```
```  2112 lemma fps_deriv_radical:
```
```  2113   fixes a :: "'a::field_char_0 fps"
```
```  2114   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0"
```
```  2115     and a0: "a\$0 \<noteq> 0"
```
```  2116   shows "fps_deriv (fps_radical r (Suc k) a) =
```
```  2117     fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
```
```  2118 proof -
```
```  2119   let ?r = "fps_radical r (Suc k) a"
```
```  2120   let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
```
```  2121   from a0 r0 have r0': "r (Suc k) (a\$0) \<noteq> 0"
```
```  2122     by auto
```
```  2123   from r0' have w0: "?w \$ 0 \<noteq> 0"
```
```  2124     by (simp del: of_nat_Suc)
```
```  2125   note th0 = inverse_mult_eq_1[OF w0]
```
```  2126   let ?iw = "inverse ?w"
```
```  2127   from iffD1[OF power_radical[of a r], OF a0 r0]
```
```  2128   have "fps_deriv (?r ^ Suc k) = fps_deriv a"
```
```  2129     by simp
```
```  2130   then have "fps_deriv ?r * ?w = fps_deriv a"
```
```  2131     by (simp add: fps_deriv_power ac_simps del: power_Suc)
```
```  2132   then have "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a"
```
```  2133     by simp
```
```  2134   then have "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
```
```  2135     by (simp add: fps_divide_def)
```
```  2136   then show ?thesis unfolding th0 by simp
```
```  2137 qed
```
```  2138
```
```  2139 lemma radical_mult_distrib:
```
```  2140   fixes a :: "'a::field_char_0 fps"
```
```  2141   assumes k: "k > 0"
```
```  2142     and ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  2143     and rb0: "r k (b \$ 0) ^ k = b \$ 0"
```
```  2144     and a0: "a\$0 \<noteq> 0"
```
```  2145     and b0: "b\$0 \<noteq> 0"
```
```  2146   shows "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0) \<longleftrightarrow>
```
```  2147     fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
```
```  2148 proof -
```
```  2149   {
```
```  2150     assume  r0': "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  2151     from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
```
```  2152       by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
```
```  2153     {
```
```  2154       assume "k = 0"
```
```  2155       then have ?thesis using r0' by simp
```
```  2156     }
```
```  2157     moreover
```
```  2158     {
```
```  2159       fix h assume k: "k = Suc h"
```
```  2160       let ?ra = "fps_radical r (Suc h) a"
```
```  2161       let ?rb = "fps_radical r (Suc h) b"
```
```  2162       have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
```
```  2163         using r0' k by (simp add: fps_mult_nth)
```
```  2164       have ab0: "(a*b) \$ 0 \<noteq> 0"
```
```  2165         using a0 b0 by (simp add: fps_mult_nth)
```
```  2166       from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
```
```  2167         iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded k]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded k]] k r0'
```
```  2168       have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)
```
```  2169     }
```
```  2170     ultimately have ?thesis by (cases k) auto
```
```  2171   }
```
```  2172   moreover
```
```  2173   {
```
```  2174     assume h: "fps_radical r k (a*b) = fps_radical r k a * fps_radical r k b"
```
```  2175     then have "(fps_radical r k (a*b))\$0 = (fps_radical r k a * fps_radical r k b)\$0"
```
```  2176       by simp
```
```  2177     then have "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  2178       using k by (simp add: fps_mult_nth)
```
```  2179   }
```
```  2180   ultimately show ?thesis by blast
```
```  2181 qed
```
```  2182
```
```  2183 (*
```
```  2184 lemma radical_mult_distrib:
```
```  2185   fixes a:: "'a::field_char_0 fps"
```
```  2186   assumes
```
```  2187   ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  2188   and rb0: "r k (b \$ 0) ^ k = b \$ 0"
```
```  2189   and r0': "r k ((a * b) \$ 0) = r k (a \$ 0) * r k (b \$ 0)"
```
```  2190   and a0: "a\$0 \<noteq> 0"
```
```  2191   and b0: "b\$0 \<noteq> 0"
```
```  2192   shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
```
```  2193 proof-
```
```  2194   from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
```
```  2195     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
```
```  2196   {assume "k=0" then have ?thesis by simp}
```
```  2197   moreover
```
```  2198   {fix h assume k: "k = Suc h"
```
```  2199   let ?ra = "fps_radical r (Suc h) a"
```
```  2200   let ?rb = "fps_radical r (Suc h) b"
```
```  2201   have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
```
```  2202     using r0' k by (simp add: fps_mult_nth)
```
```  2203   have ab0: "(a*b) \$ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
```
```  2204   from radical_unique[of r h "a*b" "fps_radical r (Suc h) a * fps_radical r (Suc h) b", OF r0[unfolded k] th0 ab0, symmetric]
```
```  2205     power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
```
```  2206   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
```
```  2207 ultimately show ?thesis by (cases k, auto)
```
```  2208 qed
```
```  2209 *)
```
```  2210
```
```  2211 lemma fps_divide_1[simp]: "(a :: 'a::field fps) / 1 = a"
```
```  2212   by (simp add: fps_divide_def)
```
```  2213
```
```  2214 lemma radical_divide:
```
```  2215   fixes a :: "'a::field_char_0 fps"
```
```  2216   assumes kp: "k > 0"
```
```  2217     and ra0: "(r k (a \$ 0)) ^ k = a \$ 0"
```
```  2218     and rb0: "(r k (b \$ 0)) ^ k = b \$ 0"
```
```  2219     and a0: "a\$0 \<noteq> 0"
```
```  2220     and b0: "b\$0 \<noteq> 0"
```
```  2221   shows "r k ((a \$ 0) / (b\$0)) = r k (a\$0) / r k (b \$ 0) \<longleftrightarrow>
```
```  2222     fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
```
```  2223   (is "?lhs = ?rhs")
```
```  2224 proof -
```
```  2225   let ?r = "fps_radical r k"
```
```  2226   from kp obtain h where k: "k = Suc h" by (cases k) auto
```
```  2227   have ra0': "r k (a\$0) \<noteq> 0" using a0 ra0 k by auto
```
```  2228   have rb0': "r k (b\$0) \<noteq> 0" using b0 rb0 k by auto
```
```  2229
```
```  2230   {
```
```  2231     assume ?rhs
```
```  2232     then have "?r (a/b) \$ 0 = (?r a / ?r b)\$0" by simp
```
```  2233     then have ?lhs using k a0 b0 rb0'
```
```  2234       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse)
```
```  2235   }
```
```  2236   moreover
```
```  2237   {
```
```  2238     assume h: ?lhs
```
```  2239     from a0 b0 have ab0[simp]: "(a/b)\$0 = a\$0 / b\$0"
```
```  2240       by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
```
```  2241     have th0: "r k ((a/b)\$0) ^ k = (a/b)\$0"
```
```  2242       by (simp add: h nonzero_power_divide[OF rb0'] ra0 rb0)
```
```  2243     from a0 b0 ra0' rb0' kp h
```
```  2244     have th1: "r k ((a / b) \$ 0) = (fps_radical r k a / fps_radical r k b) \$ 0"
```
```  2245       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse)
```
```  2246     from a0 b0 ra0' rb0' kp have ab0': "(a / b) \$ 0 \<noteq> 0"
```
```  2247       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
```
```  2248     note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
```
```  2249     note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
```
```  2250     have th2: "(?r a / ?r b)^k = a/b"
```
```  2251       by (simp add: fps_divide_def power_mult_distrib fps_inverse_power[symmetric])
```
```  2252     from iffD1[OF radical_unique[where r=r and a="?r a / ?r b" and b="a/b" and k=h], symmetric, unfolded k[symmetric], OF th0 th1 ab0' th2]
```
```  2253     have ?rhs .
```
```  2254   }
```
```  2255   ultimately show ?thesis by blast
```
```  2256 qed
```
```  2257
```
```  2258 lemma radical_inverse:
```
```  2259   fixes a :: "'a::field_char_0 fps"
```
```  2260   assumes k: "k > 0"
```
```  2261     and ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  2262     and r1: "(r k 1)^k = 1"
```
```  2263     and a0: "a\$0 \<noteq> 0"
```
```  2264   shows "r k (inverse (a \$ 0)) = r k 1 / (r k (a \$ 0)) \<longleftrightarrow>
```
```  2265     fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
```
```  2266   using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
```
```  2267   by (simp add: divide_inverse fps_divide_def)
```
```  2268
```
```  2269 subsection{* Derivative of composition *}
```
```  2270
```
```  2271 lemma fps_compose_deriv:
```
```  2272   fixes a :: "'a::idom fps"
```
```  2273   assumes b0: "b\$0 = 0"
```
```  2274   shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * fps_deriv b"
```
```  2275 proof -
```
```  2276   {
```
```  2277     fix n
```
```  2278     have "(fps_deriv (a oo b))\$n = setsum (\<lambda>i. a \$ i * (fps_deriv (b^i))\$n) {0.. Suc n}"
```
```  2279       by (simp add: fps_compose_def field_simps setsum_right_distrib del: of_nat_Suc)
```
```  2280     also have "\<dots> = setsum (\<lambda>i. a\$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))\$n) {0.. Suc n}"
```
```  2281       by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
```
```  2282     also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (((b^(i - 1)) * fps_deriv b))\$n) {0.. Suc n}"
```
```  2283       unfolding fps_mult_left_const_nth  by (simp add: field_simps)
```
```  2284     also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (setsum (\<lambda>j. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {0.. Suc n}"
```
```  2285       unfolding fps_mult_nth ..
```
```  2286     also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (setsum (\<lambda>j. (b^ (i - 1))\$j * (fps_deriv b)\$(n - j)) {0..n})) {1.. Suc n}"
```
```  2287       apply (rule setsum.mono_neutral_right)
```
```  2288       apply (auto simp add: mult_delta_left setsum.delta not_le)
```
```  2289       done
```
```  2290     also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
```
```  2291       unfolding fps_deriv_nth
```
```  2292       by (rule setsum.reindex_cong [of Suc]) (auto simp add: mult.assoc)
```
```  2293     finally have th0: "(fps_deriv (a oo b))\$n =
```
```  2294       setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}" .
```
```  2295
```
```  2296     have "(((fps_deriv a) oo b) * (fps_deriv b))\$n = setsum (\<lambda>i. (fps_deriv b)\$ (n - i) * ((fps_deriv a) oo b)\$i) {0..n}"
```
```  2297       unfolding fps_mult_nth by (simp add: ac_simps)
```
```  2298     also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b\$(n - i + 1) * of_nat (j + 1) * a\$(j+1) * (b^j)\$i) {0..n}) {0..n}"
```
```  2299       unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult.assoc
```
```  2300       apply (rule setsum.cong)
```
```  2301       apply (rule refl)
```
```  2302       apply (rule setsum.mono_neutral_left)
```
```  2303       apply (simp_all add: subset_eq)
```
```  2304       apply clarify
```
```  2305       apply (subgoal_tac "b^i\$x = 0")
```
```  2306       apply simp
```
```  2307       apply (rule startsby_zero_power_prefix[OF b0, rule_format])
```
```  2308       apply simp
```
```  2309       done
```
```  2310     also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a\$(i+1) * (setsum (\<lambda>j. (b^ i)\$j * of_nat (n - j + 1) * b\$(n - j + 1)) {0..n})) {0.. n}"
```
```  2311       unfolding setsum_right_distrib
```
```  2312       apply (subst setsum.commute)
```
```  2313       apply (rule setsum.cong, rule refl)+
```
```  2314       apply simp
```
```  2315       done
```
```  2316     finally have "(fps_deriv (a oo b))\$n = (((fps_deriv a) oo b) * (fps_deriv b)) \$n"
```
```  2317       unfolding th0 by simp
```
```  2318   }
```
```  2319   then show ?thesis by (simp add: fps_eq_iff)
```
```  2320 qed
```
```  2321
```
```  2322 lemma fps_mult_X_plus_1_nth:
```
```  2323   "((1+X)*a) \$n = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  2324 proof (cases n)
```
```  2325   case 0
```
```  2326   then show ?thesis
```
```  2327     by (simp add: fps_mult_nth )
```
```  2328 next
```
```  2329   case (Suc m)
```
```  2330   have "((1+X)*a) \$n = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0..n}"
```
```  2331     by (simp add: fps_mult_nth)
```
```  2332   also have "\<dots> = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0.. 1}"
```
```  2333     unfolding Suc by (rule setsum.mono_neutral_right) auto
```
```  2334   also have "\<dots> = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  2335     by (simp add: Suc)
```
```  2336   finally show ?thesis .
```
```  2337 qed
```
```  2338
```
```  2339
```
```  2340 subsection {* Finite FPS (i.e. polynomials) and X *}
```
```  2341
```
```  2342 lemma fps_poly_sum_X:
```
```  2343   assumes z: "\<forall>i > n. a\$i = (0::'a::comm_ring_1)"
```
```  2344   shows "a = setsum (\<lambda>i. fps_const (a\$i) * X^i) {0..n}" (is "a = ?r")
```
```  2345 proof -
```
```  2346   {
```
```  2347     fix i
```
```  2348     have "a\$i = ?r\$i"
```
```  2349       unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
```
```  2350       by (simp add: mult_delta_right setsum.delta' z)
```
```  2351   }
```
```  2352   then show ?thesis unfolding fps_eq_iff by blast
```
```  2353 qed
```
```  2354
```
```  2355
```
```  2356 subsection{* Compositional inverses *}
```
```  2357
```
```  2358 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
```
```  2359 where
```
```  2360   "compinv a 0 = X\$0"
```
```  2361 | "compinv a (Suc n) =
```
```  2362     (X\$ Suc n - setsum (\<lambda>i. (compinv a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"
```
```  2363
```
```  2364 definition "fps_inv a = Abs_fps (compinv a)"
```
```  2365
```
```  2366 lemma fps_inv:
```
```  2367   assumes a0: "a\$0 = 0"
```
```  2368     and a1: "a\$1 \<noteq> 0"
```
```  2369   shows "fps_inv a oo a = X"
```
```  2370 proof -
```
```  2371   let ?i = "fps_inv a oo a"
```
```  2372   {
```
```  2373     fix n
```
```  2374     have "?i \$n = X\$n"
```
```  2375     proof (induct n rule: nat_less_induct)
```
```  2376       fix n
```
```  2377       assume h: "\<forall>m<n. ?i\$m = X\$m"
```
```  2378       show "?i \$ n = X\$n"
```
```  2379       proof (cases n)
```
```  2380         case 0
```
```  2381         then show ?thesis using a0
```
```  2382           by (simp add: fps_compose_nth fps_inv_def)
```
```  2383       next
```
```  2384         case (Suc n1)
```
```  2385         have "?i \$ n = setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} + fps_inv a \$ Suc n1 * (a \$ 1)^ Suc n1"
```
```  2386           by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
```
```  2387         also have "\<dots> = setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} +
```
```  2388           (X\$ Suc n1 - setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1})"
```
```  2389           using a0 a1 Suc by (simp add: fps_inv_def)
```
```  2390         also have "\<dots> = X\$n" using Suc by simp
```
```  2391         finally show ?thesis .
```
```  2392       qed
```
```  2393     qed
```
```  2394   }
```
```  2395   then show ?thesis by (simp add: fps_eq_iff)
```
```  2396 qed
```
```  2397
```
```  2398
```
```  2399 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::field"
```
```  2400 where
```
```  2401   "gcompinv b a 0 = b\$0"
```
```  2402 | "gcompinv b a (Suc n) =
```
```  2403     (b\$ Suc n - setsum (\<lambda>i. (gcompinv b a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"
```
```  2404
```
```  2405 definition "fps_ginv b a = Abs_fps (gcompinv b a)"
```
```  2406
```
```  2407 lemma fps_ginv:
```
```  2408   assumes a0: "a\$0 = 0"
```
```  2409     and a1: "a\$1 \<noteq> 0"
```
```  2410   shows "fps_ginv b a oo a = b"
```
```  2411 proof -
```
```  2412   let ?i = "fps_ginv b a oo a"
```
```  2413   {
```
```  2414     fix n
```
```  2415     have "?i \$n = b\$n"
```
```  2416     proof (induct n rule: nat_less_induct)
```
```  2417       fix n
```
```  2418       assume h: "\<forall>m<n. ?i\$m = b\$m"
```
```  2419       show "?i \$ n = b\$n"
```
```  2420       proof (cases n)
```
```  2421         case 0
```
```  2422         then show ?thesis using a0
```
```  2423           by (simp add: fps_compose_nth fps_ginv_def)
```
```  2424       next
```
```  2425         case (Suc n1)
```
```  2426         have "?i \$ n = setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} + fps_ginv b a \$ Suc n1 * (a \$ 1)^ Suc n1"
```
```  2427           by (simp only: fps_compose_nth) (simp add: Suc startsby_zero_power_nth_same [OF a0] del: power_Suc)
```
```  2428         also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1} +
```
```  2429           (b\$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a \$ i) * (a^i)\$n) {0 .. n1})"
```
```  2430           using a0 a1 Suc by (simp add: fps_ginv_def)
```
```  2431         also have "\<dots> = b\$n" using Suc by simp
```
```  2432         finally show ?thesis .
```
```  2433       qed
```
```  2434     qed
```
```  2435   }
```
```  2436   then show ?thesis by (simp add: fps_eq_iff)
```
```  2437 qed
```
```  2438
```
```  2439 lemma fps_inv_ginv: "fps_inv = fps_ginv X"
```
```  2440   apply (auto simp add: fun_eq_iff fps_eq_iff fps_inv_def fps_ginv_def)
```
```  2441   apply (induct_tac n rule: nat_less_induct)
```
```  2442   apply auto
```
```  2443   apply (case_tac na)
```
```  2444   apply simp
```
```  2445   apply simp
```
```  2446   done
```
```  2447
```
```  2448 lemma fps_compose_1[simp]: "1 oo a = 1"
```
```  2449   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum.delta)
```
```  2450
```
```  2451 lemma fps_compose_0[simp]: "0 oo a = 0"
```
```  2452   by (simp add: fps_eq_iff fps_compose_nth)
```
```  2453
```
```  2454 lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a\$0)"
```
```  2455   by (auto simp add: fps_eq_iff fps_compose_nth power_0_left setsum.neutral)
```
```  2456
```
```  2457 lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
```
```  2458   by (simp add: fps_eq_iff fps_compose_nth field_simps setsum.distrib)
```
```  2459
```
```  2460 lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
```
```  2461 proof (cases "finite S")
```
```  2462   case True
```
```  2463   show ?thesis
```
```  2464   proof (rule finite_induct[OF True])
```
```  2465     show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
```
```  2466   next
```
```  2467     fix x F
```
```  2468     assume fF: "finite F"
```
```  2469       and xF: "x \<notin> F"
```
```  2470       and h: "setsum f F oo a = setsum (\<lambda>i. f i oo a) F"
```
```  2471     show "setsum f (insert x F) oo a  = setsum (\<lambda>i. f i oo a) (insert x F)"
```
```  2472       using fF xF h by (simp add: fps_compose_add_distrib)
```
```  2473   qed
```
```  2474 next
```
```  2475   case False
```
```  2476   then show ?thesis by simp
```
```  2477 qed
```
```  2478
```
```  2479 lemma convolution_eq:
```
```  2480   "setsum (\<lambda>i. a (i :: nat) * b (n - i)) {0 .. n} =
```
```  2481     setsum (\<lambda>(i,j). a i * b j) {(i,j). i \<le> n \<and> j \<le> n \<and> i + j = n}"
```
```  2482   by (rule setsum.reindex_bij_witness[where i=fst and j="\<lambda>i. (i, n - i)"]) auto
```
```  2483
```
```  2484 lemma product_composition_lemma:
```
```  2485   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2486     and d0: "d\$0 = 0"
```
```  2487   shows "((a oo c) * (b oo d))\$n =
```
```  2488     setsum (\<lambda>(k,m). a\$k * b\$m * (c^k * d^m) \$ n) {(k,m). k + m \<le> n}"  (is "?l = ?r")
```
```  2489 proof -
```
```  2490   let ?S = "{(k::nat, m::nat). k + m \<le> n}"
```
```  2491   have s: "?S \<subseteq> {0..n} <*> {0..n}" by (auto simp add: subset_eq)
```
```  2492   have f: "finite {(k::nat, m::nat). k + m \<le> n}"
```
```  2493     apply (rule finite_subset[OF s])
```
```  2494     apply auto
```
```  2495     done
```
```  2496   have "?r =  setsum (\<lambda>i. setsum (\<lambda>(k,m). a\$k * (c^k)\$i * b\$m * (d^m) \$ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
```
```  2497     apply (simp add: fps_mult_nth setsum_right_distrib)
```
```  2498     apply (subst setsum.commute)
```
```  2499     apply (rule setsum.cong)
```
```  2500     apply (auto simp add: field_simps)
```
```  2501     done
```
```  2502   also have "\<dots> = ?l"
```
```  2503     apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
```
```  2504     apply (rule setsum.cong)
```
```  2505     apply (rule refl)
```
```  2506     apply (simp add: setsum.cartesian_product mult.assoc)
```
```  2507     apply (rule setsum.mono_neutral_right[OF f])
```
```  2508     apply (simp add: subset_eq)
```
```  2509     apply presburger
```
```  2510     apply clarsimp
```
```  2511     apply (rule ccontr)
```
```  2512     apply (clarsimp simp add: not_le)
```
```  2513     apply (case_tac "x < aa")
```
```  2514     apply simp
```
```  2515     apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
```
```  2516     apply blast
```
```  2517     apply simp
```
```  2518     apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
```
```  2519     apply blast
```
```  2520     done
```
```  2521   finally show ?thesis by simp
```
```  2522 qed
```
```  2523
```
```  2524 lemma product_composition_lemma':
```
```  2525   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2526     and d0: "d\$0 = 0"
```
```  2527   shows "((a oo c) * (b oo d))\$n =
```
```  2528     setsum (\<lambda>k. setsum (\<lambda>m. a\$k * b\$m * (c^k * d^m) \$ n) {0..n}) {0..n}"  (is "?l = ?r")
```
```  2529   unfolding product_composition_lemma[OF c0 d0]
```
```  2530   unfolding setsum.cartesian_product
```
```  2531   apply (rule setsum.mono_neutral_left)
```
```  2532   apply simp
```
```  2533   apply (clarsimp simp add: subset_eq)
```
```  2534   apply clarsimp
```
```  2535   apply (rule ccontr)
```
```  2536   apply (subgoal_tac "(c^aa * d^ba) \$ n = 0")
```
```  2537   apply simp
```
```  2538   unfolding fps_mult_nth
```
```  2539   apply (rule setsum.neutral)
```
```  2540   apply (clarsimp simp add: not_le)
```
```  2541   apply (case_tac "x < aa")
```
```  2542   apply (rule startsby_zero_power_prefix[OF c0, rule_format])
```
```  2543   apply simp
```
```  2544   apply (subgoal_tac "n - x < ba")
```
```  2545   apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
```
```  2546   apply simp
```
```  2547   apply arith
```
```  2548   done
```
```  2549
```
```  2550
```
```  2551 lemma setsum_pair_less_iff:
```
```  2552   "setsum (\<lambda>((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} =
```
```  2553     setsum (\<lambda>s. setsum (\<lambda>i. a i * b (s - i) * c s) {0..s}) {0..n}"
```
```  2554   (is "?l = ?r")
```
```  2555 proof -
```
```  2556   let ?KM = "{(k,m). k + m \<le> n}"
```
```  2557   let ?f = "\<lambda>s. UNION {(0::nat)..s} (\<lambda>i. {(i,s - i)})"
```
```  2558   have th0: "?KM = UNION {0..n} ?f"
```
```  2559     apply (simp add: set_eq_iff)
```
```  2560     apply presburger (* FIXME: slow! *)
```
```  2561     done
```
```  2562   show "?l = ?r "
```
```  2563     unfolding th0
```
```  2564     apply (subst setsum.UNION_disjoint)
```
```  2565     apply auto
```
```  2566     apply (subst setsum.UNION_disjoint)
```
```  2567     apply auto
```
```  2568     done
```
```  2569 qed
```
```  2570
```
```  2571 lemma fps_compose_mult_distrib_lemma:
```
```  2572   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2573   shows "((a oo c) * (b oo c))\$n =
```
```  2574     setsum (\<lambda>s. setsum (\<lambda>i. a\$i * b\$(s - i) * (c^s) \$ n) {0..s}) {0..n}"
```
```  2575     (is "?l = ?r")
```
```  2576   unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
```
```  2577   unfolding setsum_pair_less_iff[where a = "\<lambda>k. a\$k" and b="\<lambda>m. b\$m" and c="\<lambda>s. (c ^ s)\$n" and n = n] ..
```
```  2578
```
```  2579
```
```  2580 lemma fps_compose_mult_distrib:
```
```  2581   assumes c0: "c \$ 0 = (0::'a::idom)"
```
```  2582   shows "(a * b) oo c = (a oo c) * (b oo c)"
```
```  2583   apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
```
```  2584   apply (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
```
```  2585   done
```
```  2586
```
```  2587 lemma fps_compose_setprod_distrib:
```
```  2588   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2589   shows "setprod a S oo c = setprod (\<lambda>k. a k oo c) S"
```
```  2590   apply (cases "finite S")
```
```  2591   apply simp_all
```
```  2592   apply (induct S rule: finite_induct)
```
```  2593   apply simp
```
```  2594   apply (simp add: fps_compose_mult_distrib[OF c0])
```
```  2595   done
```
```  2596
```
```  2597 lemma fps_compose_power:
```
```  2598   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2599   shows "(a oo c)^n = a^n oo c"
```
```  2600   (is "?l = ?r")
```
```  2601 proof (cases n)
```
```  2602   case 0
```
```  2603   then show ?thesis by simp
```
```  2604 next
```
```  2605   case (Suc m)
```
```  2606   have th0: "a^n = setprod (\<lambda>k. a) {0..m}" "(a oo c) ^ n = setprod (\<lambda>k. a oo c) {0..m}"
```
```  2607     by (simp_all add: setprod_constant Suc)
```
```  2608   then show ?thesis
```
```  2609     by (simp add: fps_compose_setprod_distrib[OF c0])
```
```  2610 qed
```
```  2611
```
```  2612 lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
```
```  2613   by (simp add: fps_eq_iff fps_compose_nth field_simps setsum_negf[symmetric])
```
```  2614
```
```  2615 lemma fps_compose_sub_distrib: "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
```
```  2616   using fps_compose_add_distrib [of a "- b" c] by (simp add: fps_compose_uminus)
```
```  2617
```
```  2618 lemma X_fps_compose: "X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a\$n)"
```
```  2619   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum.delta)
```
```  2620
```
```  2621 lemma fps_inverse_compose:
```
```  2622   assumes b0: "(b\$0 :: 'a::field) = 0"
```
```  2623     and a0: "a\$0 \<noteq> 0"
```
```  2624   shows "inverse a oo b = inverse (a oo b)"
```
```  2625 proof -
```
```  2626   let ?ia = "inverse a"
```
```  2627   let ?ab = "a oo b"
```
```  2628   let ?iab = "inverse ?ab"
```
```  2629
```
```  2630   from a0 have ia0: "?ia \$ 0 \<noteq> 0" by simp
```
```  2631   from a0 have ab0: "?ab \$ 0 \<noteq> 0" by (simp add: fps_compose_def)
```
```  2632   have "(?ia oo b) *  (a oo b) = 1"
```
```  2633     unfolding fps_compose_mult_distrib[OF b0, symmetric]
```
```  2634     unfolding inverse_mult_eq_1[OF a0]
```
```  2635     fps_compose_1 ..
```
```  2636
```
```  2637   then have "(?ia oo b) *  (a oo b) * ?iab  = 1 * ?iab" by simp
```
```  2638   then have "(?ia oo b) *  (?iab * (a oo b))  = ?iab" by simp
```
```  2639   then show ?thesis unfolding inverse_mult_eq_1[OF ab0] by simp
```
```  2640 qed
```
```  2641
```
```  2642 lemma fps_divide_compose:
```
```  2643   assumes c0: "(c\$0 :: 'a::field) = 0"
```
```  2644     and b0: "b\$0 \<noteq> 0"
```
```  2645   shows "(a/b) oo c = (a oo c) / (b oo c)"
```
```  2646     unfolding fps_divide_def fps_compose_mult_distrib[OF c0]
```
```  2647     fps_inverse_compose[OF c0 b0] ..
```
```  2648
```
```  2649 lemma gp:
```
```  2650   assumes a0: "a\$0 = (0::'a::field)"
```
```  2651   shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)"
```
```  2652     (is "?one oo a = _")
```
```  2653 proof -
```
```  2654   have o0: "?one \$ 0 \<noteq> 0" by simp
```
```  2655   have th0: "(1 - X) \$ 0 \<noteq> (0::'a)" by simp
```
```  2656   from fps_inverse_gp[where ?'a = 'a]
```
```  2657   have "inverse ?one = 1 - X" by (simp add: fps_eq_iff)
```
```  2658   then have "inverse (inverse ?one) = inverse (1 - X)" by simp
```
```  2659   then have th: "?one = 1/(1 - X)" unfolding fps_inverse_idempotent[OF o0]
```
```  2660     by (simp add: fps_divide_def)
```
```  2661   show ?thesis
```
```  2662     unfolding th
```
```  2663     unfolding fps_divide_compose[OF a0 th0]
```
```  2664     fps_compose_1 fps_compose_sub_distrib X_fps_compose_startby0[OF a0] ..
```
```  2665 qed
```
```  2666
```
```  2667 lemma fps_const_power [simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)"
```
```  2668   by (induct n) auto
```
```  2669
```
```  2670 lemma fps_compose_radical:
```
```  2671   assumes b0: "b\$0 = (0::'a::field_char_0)"
```
```  2672     and ra0: "r (Suc k) (a\$0) ^ Suc k = a\$0"
```
```  2673     and a0: "a\$0 \<noteq> 0"
```
```  2674   shows "fps_radical r (Suc k)  a oo b = fps_radical r (Suc k) (a oo b)"
```
```  2675 proof -
```
```  2676   let ?r = "fps_radical r (Suc k)"
```
```  2677   let ?ab = "a oo b"
```
```  2678   have ab0: "?ab \$ 0 = a\$0"
```
```  2679     by (simp add: fps_compose_def)
```
```  2680   from ab0 a0 ra0 have rab0: "?ab \$ 0 \<noteq> 0" "r (Suc k) (?ab \$ 0) ^ Suc k = ?ab \$ 0"
```
```  2681     by simp_all
```
```  2682   have th00: "r (Suc k) ((a oo b) \$ 0) = (fps_radical r (Suc k) a oo b) \$ 0"
```
```  2683     by (simp add: ab0 fps_compose_def)
```
```  2684   have th0: "(?r a oo b) ^ (Suc k) = a  oo b"
```
```  2685     unfolding fps_compose_power[OF b0]
```
```  2686     unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  ..
```
```  2687   from iffD1[OF radical_unique[where r=r and k=k and b= ?ab and a = "?r a oo b", OF rab0(2) th00 rab0(1)], OF th0]
```
```  2688   show ?thesis  .
```
```  2689 qed
```
```  2690
```
```  2691 lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
```
```  2692   by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult.assoc)
```
```  2693
```
```  2694 lemma fps_const_mult_apply_right:
```
```  2695   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
```
```  2696   by (auto simp add: fps_const_mult_apply_left mult.commute)
```
```  2697
```
```  2698 lemma fps_compose_assoc:
```
```  2699   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2700     and b0: "b\$0 = 0"
```
```  2701   shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
```
```  2702 proof -
```
```  2703   {
```
```  2704     fix n
```
```  2705     have "?l\$n = (setsum (\<lambda>i. (fps_const (a\$i) * b^i) oo c) {0..n})\$n"
```
```  2706       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
```
```  2707         setsum_right_distrib mult.assoc fps_setsum_nth)
```
```  2708     also have "\<dots> = ((setsum (\<lambda>i. fps_const (a\$i) * b^i) {0..n}) oo c)\$n"
```
```  2709       by (simp add: fps_compose_setsum_distrib)
```
```  2710     also have "\<dots> = ?r\$n"
```
```  2711       apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult.assoc)
```
```  2712       apply (rule setsum.cong)
```
```  2713       apply (rule refl)
```
```  2714       apply (rule setsum.mono_neutral_right)
```
```  2715       apply (auto simp add: not_le)
```
```  2716       apply (erule startsby_zero_power_prefix[OF b0, rule_format])
```
```  2717       done
```
```  2718     finally have "?l\$n = ?r\$n" .
```
```  2719   }
```
```  2720   then show ?thesis by (simp add: fps_eq_iff)
```
```  2721 qed
```
```  2722
```
```  2723
```
```  2724 lemma fps_X_power_compose:
```
```  2725   assumes a0: "a\$0=0"
```
```  2726   shows "X^k oo a = (a::'a::idom fps)^k"
```
```  2727   (is "?l = ?r")
```
```  2728 proof (cases k)
```
```  2729   case 0
```
```  2730   then show ?thesis by simp
```
```  2731 next
```
```  2732   case (Suc h)
```
```  2733   {
```
```  2734     fix n
```
```  2735     {
```
```  2736       assume kn: "k>n"
```
```  2737       then have "?l \$ n = ?r \$n" using a0 startsby_zero_power_prefix[OF a0] Suc
```
```  2738         by (simp add: fps_compose_nth del: power_Suc)
```
```  2739     }
```
```  2740     moreover
```
```  2741     {
```
```  2742       assume kn: "k \<le> n"
```
```  2743       then have "?l\$n = ?r\$n"
```
```  2744         by (simp add: fps_compose_nth mult_delta_left setsum.delta)
```
```  2745     }
```
```  2746     moreover have "k >n \<or> k\<le> n"  by arith
```
```  2747     ultimately have "?l\$n = ?r\$n"  by blast
```
```  2748   }
```
```  2749   then show ?thesis unfolding fps_eq_iff by blast
```
```  2750 qed
```
```  2751
```
```  2752 lemma fps_inv_right:
```
```  2753   assumes a0: "a\$0 = 0"
```
```  2754     and a1: "a\$1 \<noteq> 0"
```
```  2755   shows "a oo fps_inv a = X"
```
```  2756 proof -
```
```  2757   let ?ia = "fps_inv a"
```
```  2758   let ?iaa = "a oo fps_inv a"
```
```  2759   have th0: "?ia \$ 0 = 0" by (simp add: fps_inv_def)
```
```  2760   have th1: "?iaa \$ 0 = 0" using a0 a1
```
```  2761     by (simp add: fps_inv_def fps_compose_nth)
```
```  2762   have th2: "X\$0 = 0" by simp
```
```  2763   from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
```
```  2764   then have "(a oo fps_inv a) oo a = X oo a"
```
```  2765     by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0])
```
```  2766   with fps_compose_inj_right[OF a0 a1]
```
```  2767   show ?thesis by simp
```
```  2768 qed
```
```  2769
```
```  2770 lemma fps_inv_deriv:
```
```  2771   assumes a0:"a\$0 = (0::'a::field)"
```
```  2772     and a1: "a\$1 \<noteq> 0"
```
```  2773   shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
```
```  2774 proof -
```
```  2775   let ?ia = "fps_inv a"
```
```  2776   let ?d = "fps_deriv a oo ?ia"
```
```  2777   let ?dia = "fps_deriv ?ia"
```
```  2778   have ia0: "?ia\$0 = 0" by (simp add: fps_inv_def)
```
```  2779   have th0: "?d\$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth)
```
```  2780   from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
```
```  2781     by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
```
```  2782   then have "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
```
```  2783   with inverse_mult_eq_1 [OF th0]
```
```  2784   show "?dia = inverse ?d" by simp
```
```  2785 qed
```
```  2786
```
```  2787 lemma fps_inv_idempotent:
```
```  2788   assumes a0: "a\$0 = 0"
```
```  2789     and a1: "a\$1 \<noteq> 0"
```
```  2790   shows "fps_inv (fps_inv a) = a"
```
```  2791 proof -
```
```  2792   let ?r = "fps_inv"
```
```  2793   have ra0: "?r a \$ 0 = 0" by (simp add: fps_inv_def)
```
```  2794   from a1 have ra1: "?r a \$ 1 \<noteq> 0" by (simp add: fps_inv_def field_simps)
```
```  2795   have X0: "X\$0 = 0" by simp
```
```  2796   from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = X" .
```
```  2797   then have "?r (?r a) oo ?r a oo a = X oo a" by simp
```
```  2798   then have "?r (?r a) oo (?r a oo a) = a"
```
```  2799     unfolding X_fps_compose_startby0[OF a0]
```
```  2800     unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
```
```  2801   then show ?thesis unfolding fps_inv[OF a0 a1] by simp
```
```  2802 qed
```
```  2803
```
```  2804 lemma fps_ginv_ginv:
```
```  2805   assumes a0: "a\$0 = 0"
```
```  2806     and a1: "a\$1 \<noteq> 0"
```
```  2807     and c0: "c\$0 = 0"
```
```  2808     and  c1: "c\$1 \<noteq> 0"
```
```  2809   shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
```
```  2810 proof -
```
```  2811   let ?r = "fps_ginv"
```
```  2812   from c0 have rca0: "?r c a \$0 = 0" by (simp add: fps_ginv_def)
```
```  2813   from a1 c1 have rca1: "?r c a \$ 1 \<noteq> 0" by (simp add: fps_ginv_def field_simps)
```
```  2814   from fps_ginv[OF rca0 rca1]
```
```  2815   have "?r b (?r c a) oo ?r c a = b" .
```
```  2816   then have "?r b (?r c a) oo ?r c a oo a = b oo a" by simp
```
```  2817   then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
```
```  2818     apply (subst fps_compose_assoc)
```
```  2819     using a0 c0
```
```  2820     apply (auto simp add: fps_ginv_def)
```
```  2821     done
```
```  2822   then have "?r b (?r c a) oo c = b oo a"
```
```  2823     unfolding fps_ginv[OF a0 a1] .
```
```  2824   then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c" by simp
```
```  2825   then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
```
```  2826     apply (subst fps_compose_assoc)
```
```  2827     using a0 c0
```
```  2828     apply (auto simp add: fps_inv_def)
```
```  2829     done
```
```  2830   then show ?thesis unfolding fps_inv_right[OF c0 c1] by simp
```
```  2831 qed
```
```  2832
```
```  2833 lemma fps_ginv_deriv:
```
```  2834   assumes a0:"a\$0 = (0::'a::field)"
```
```  2835     and a1: "a\$1 \<noteq> 0"
```
```  2836   shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv X a"
```
```  2837 proof -
```
```  2838   let ?ia = "fps_ginv b a"
```
```  2839   let ?iXa = "fps_ginv X a"
```
```  2840   let ?d = "fps_deriv"
```
```  2841   let ?dia = "?d ?ia"
```
```  2842   have iXa0: "?iXa \$ 0 = 0" by (simp add: fps_ginv_def)
```
```  2843   have da0: "?d a \$ 0 \<noteq> 0" using a1 by simp
```
```  2844   from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" by simp
```
```  2845   then have "(?d ?ia oo a) * ?d a = ?d b" unfolding fps_compose_deriv[OF a0] .
```
```  2846   then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" by simp
```
```  2847   then have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a"
```
```  2848     by (simp add: fps_divide_def)
```
```  2849   then have "(?d ?ia oo a) oo ?iXa =  (?d b / ?d a) oo ?iXa "
```
```  2850     unfolding inverse_mult_eq_1[OF da0] by simp
```
```  2851   then have "?d ?ia oo (a oo ?iXa) =  (?d b / ?d a) oo ?iXa"
```
```  2852     unfolding fps_compose_assoc[OF iXa0 a0] .
```
```  2853   then show ?thesis unfolding fps_inv_ginv[symmetric]
```
```  2854     unfolding fps_inv_right[OF a0 a1] by simp
```
```  2855 qed
```
```  2856
```
```  2857 subsection{* Elementary series *}
```
```  2858
```
```  2859 subsubsection{* Exponential series *}
```
```  2860
```
```  2861 definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
```
```  2862
```
```  2863 lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::field_char_0) * E a" (is "?l = ?r")
```
```  2864 proof -
```
```  2865   {
```
```  2866     fix n
```
```  2867     have "?l\$n = ?r \$ n"
```
```  2868       apply (auto simp add: E_def field_simps power_Suc[symmetric]
```
```  2869         simp del: fact_Suc of_nat_Suc power_Suc)
```
```  2870       apply (simp add: of_nat_mult field_simps)
```
```  2871       done
```
```  2872   }
```
```  2873   then show ?thesis by (simp add: fps_eq_iff)
```
```  2874 qed
```
```  2875
```
```  2876 lemma E_unique_ODE:
```
```  2877   "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a\$0) * E (c::'a::field_char_0)"
```
```  2878   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2879 proof
```
```  2880   assume d: ?lhs
```
```  2881   from d have th: "\<And>n. a \$ Suc n = c * a\$n / of_nat (Suc n)"
```
```  2882     by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
```
```  2883   {
```
```  2884     fix n
```
```  2885     have "a\$n = a\$0 * c ^ n/ (of_nat (fact n))"
```
```  2886       apply (induct n)
```
```  2887       apply simp
```
```  2888       unfolding th
```
```  2889       using fact_gt_zero_nat
```
```  2890       apply (simp add: field_simps del: of_nat_Suc fact_Suc)
```
```  2891       apply (drule sym)
```
```  2892       apply (simp add: field_simps of_nat_mult)
```
```  2893       done
```
```  2894   }
```
```  2895   note th' = this
```
```  2896   show ?rhs by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro: th')
```
```  2897 next
```
```  2898   assume h: ?rhs
```
```  2899   show ?lhs
```
```  2900     apply (subst h)
```
```  2901     apply simp
```
```  2902     apply (simp only: h[symmetric])
```
```  2903     apply simp
```
```  2904     done
```
```  2905 qed
```
```  2906
```
```  2907 lemma E_add_mult: "E (a + b) = E (a::'a::field_char_0) * E b" (is "?l = ?r")
```
```  2908 proof -
```
```  2909   have "fps_deriv (?r) = fps_const (a+b) * ?r"
```
```  2910     by (simp add: fps_const_add[symmetric] field_simps del: fps_const_add)
```
```  2911   then have "?r = ?l" apply (simp only: E_unique_ODE)
```
```  2912     by (simp add: fps_mult_nth E_def)
```
```  2913   then show ?thesis ..
```
```  2914 qed
```
```  2915
```
```  2916 lemma E_nth[simp]: "E a \$ n = a^n / of_nat (fact n)"
```
```  2917   by (simp add: E_def)
```
```  2918
```
```  2919 lemma E0[simp]: "E (0::'a::field) = 1"
```
```  2920   by (simp add: fps_eq_iff power_0_left)
```
```  2921
```
```  2922 lemma E_neg: "E (- a) = inverse (E (a::'a::field_char_0))"
```
```  2923 proof -
```
```  2924   from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
```
```  2925     by (simp )
```
```  2926   have th1: "E a \$ 0 \<noteq> 0" by simp
```
```  2927   from fps_inverse_unique[OF th1 th0] show ?thesis by simp
```
```  2928 qed
```
```  2929
```
```  2930 lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::field_char_0)) = (fps_const a)^n * (E a)"
```
```  2931   by (induct n) auto
```
```  2932
```
```  2933 lemma X_compose_E[simp]: "X oo E (a::'a::field) = E a - 1"
```
```  2934   by (simp add: fps_eq_iff X_fps_compose)
```
```  2935
```
```  2936 lemma LE_compose:
```
```  2937   assumes a: "a\<noteq>0"
```
```  2938   shows "fps_inv (E a - 1) oo (E a - 1) = X"
```
```  2939     and "(E a - 1) oo fps_inv (E a - 1) = X"
```
```  2940 proof -
```
```  2941   let ?b = "E a - 1"
```
```  2942   have b0: "?b \$ 0 = 0" by simp
```
```  2943   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  2944   from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" .
```
```  2945   from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" .
```
```  2946 qed
```
```  2947
```
```  2948 lemma fps_const_inverse:
```
```  2949   "a \<noteq> 0 \<Longrightarrow> inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
```
```  2950   apply (auto simp add: fps_eq_iff fps_inverse_def)
```
```  2951   apply (case_tac n)
```
```  2952   apply auto
```
```  2953   done
```
```  2954
```
```  2955 lemma inverse_one_plus_X:
```
```  2956   "inverse (1 + X) = Abs_fps (\<lambda>n. (- 1 ::'a::field)^n)"
```
```  2957   (is "inverse ?l = ?r")
```
```  2958 proof -
```
```  2959   have th: "?l * ?r = 1"
```
```  2960     by (auto simp add: field_simps fps_eq_iff minus_one_power_iff)
```
```  2961   have th': "?l \$ 0 \<noteq> 0" by (simp add: )
```
```  2962   from fps_inverse_unique[OF th' th] show ?thesis .
```
```  2963 qed
```
```  2964
```
```  2965 lemma E_power_mult: "(E (c::'a::field_char_0))^n = E (of_nat n * c)"
```
```  2966   by (induct n) (auto simp add: field_simps E_add_mult)
```
```  2967
```
```  2968 lemma radical_E:
```
```  2969   assumes r: "r (Suc k) 1 = 1"
```
```  2970   shows "fps_radical r (Suc k) (E (c::'a::field_char_0)) = E (c / of_nat (Suc k))"
```
```  2971 proof -
```
```  2972   let ?ck = "(c / of_nat (Suc k))"
```
```  2973   let ?r = "fps_radical r (Suc k)"
```
```  2974   have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
```
```  2975     by (simp_all del: of_nat_Suc)
```
```  2976   have th0: "E ?ck ^ (Suc k) = E c" unfolding E_power_mult eq0 ..
```
```  2977   have th: "r (Suc k) (E c \$0) ^ Suc k = E c \$ 0"
```
```  2978     "r (Suc k) (E c \$ 0) = E ?ck \$ 0" "E c \$ 0 \<noteq> 0" using r by simp_all
```
```  2979   from th0 radical_unique[where r=r and k=k, OF th]
```
```  2980   show ?thesis by auto
```
```  2981 qed
```
```  2982
```
```  2983 lemma Ec_E1_eq: "E (1::'a::field_char_0) oo (fps_const c * X) = E c"
```
```  2984   apply (auto simp add: fps_eq_iff E_def fps_compose_def power_mult_distrib)
```
```  2985   apply (simp add: cond_value_iff cond_application_beta setsum.delta' cong del: if_weak_cong)
```
```  2986   done
```
```  2987
```
```  2988 text{* The generalized binomial theorem as a  consequence of @{thm E_add_mult} *}
```
```  2989
```
```  2990 lemma gbinomial_theorem:
```
```  2991   "((a::'a::{field_char_0,field_inverse_zero})+b) ^ n =
```
```  2992     (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
```
```  2993 proof -
```
```  2994   from E_add_mult[of a b]
```
```  2995   have "(E (a + b)) \$ n = (E a * E b)\$n" by simp
```
```  2996   then have "(a + b) ^ n =
```
```  2997     (\<Sum>i::nat = 0::nat..n. a ^ i * b ^ (n - i)  * (of_nat (fact n) / of_nat (fact i * fact (n - i))))"
```
```  2998     by (simp add: field_simps fps_mult_nth of_nat_mult[symmetric] setsum_right_distrib)
```
```  2999   then show ?thesis
```
```  3000     apply simp
```
```  3001     apply (rule setsum.cong)
```
```  3002     apply simp_all
```
```  3003     apply (frule binomial_fact[where ?'a = 'a, symmetric])
```
```  3004     apply (simp add: field_simps of_nat_mult)
```
```  3005     done
```
```  3006 qed
```
```  3007
```
```  3008 text{* And the nat-form -- also available from Binomial.thy *}
```
```  3009 lemma binomial_theorem: "(a+b) ^ n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```  3010   using gbinomial_theorem[of "of_nat a" "of_nat b" n]
```
```  3011   unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric]
```
```  3012     of_nat_setsum[symmetric]
```
```  3013   by simp
```
```  3014
```
```  3015
```
```  3016 subsubsection{* Logarithmic series *}
```
```  3017
```
```  3018 lemma Abs_fps_if_0:
```
```  3019   "Abs_fps(\<lambda>n. if n=0 then (v::'a::ring_1) else f n) = fps_const v + X * Abs_fps (\<lambda>n. f (Suc n))"
```
```  3020   by (auto simp add: fps_eq_iff)
```
```  3021
```
```  3022 definition L :: "'a::field_char_0 \<Rightarrow> 'a fps"
```
```  3023   where "L c = fps_const (1/c) * Abs_fps (\<lambda>n. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"
```
```  3024
```
```  3025 lemma fps_deriv_L: "fps_deriv (L c) = fps_const (1/c) * inverse (1 + X)"
```
```  3026   unfolding inverse_one_plus_X
```
```  3027   by (simp add: L_def fps_eq_iff del: of_nat_Suc)
```
```  3028
```
```  3029 lemma L_nth: "L c \$ n = (if n=0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
```
```  3030   by (simp add: L_def field_simps)
```
```  3031
```
```  3032 lemma L_0[simp]: "L c \$ 0 = 0" by (simp add: L_def)
```
```  3033
```
```  3034 lemma L_E_inv:
```
```  3035   fixes a :: "'a::field_char_0"
```
```  3036   assumes a: "a \<noteq> 0"
```
```  3037   shows "L a = fps_inv (E a - 1)"  (is "?l = ?r")
```
```  3038 proof -
```
```  3039   let ?b = "E a - 1"
```
```  3040   have b0: "?b \$ 0 = 0" by simp
```
```  3041   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  3042   have "fps_deriv (E a - 1) oo fps_inv (E a - 1) =
```
```  3043     (fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)"
```
```  3044     by (simp add: field_simps)
```
```  3045   also have "\<dots> = fps_const a * (X + 1)"
```
```  3046     apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
```
```  3047     apply (simp add: field_simps)
```
```  3048     done
```
```  3049   finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
```
```  3050   from fps_inv_deriv[OF b0 b1, unfolded eq]
```
```  3051   have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
```
```  3052     using a
```
```  3053     by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
```
```  3054   then have "fps_deriv ?l = fps_deriv ?r"
```
```  3055     by (simp add: fps_deriv_L add.commute fps_divide_def divide_inverse)
```
```  3056   then show ?thesis unfolding fps_deriv_eq_iff
```
```  3057     by (simp add: L_nth fps_inv_def)
```
```  3058 qed
```
```  3059
```
```  3060 lemma L_mult_add:
```
```  3061   assumes c0: "c\<noteq>0"
```
```  3062     and d0: "d\<noteq>0"
```
```  3063   shows "L c + L d = fps_const (c+d) * L (c*d)"
```
```  3064   (is "?r = ?l")
```
```  3065 proof-
```
```  3066   from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
```
```  3067   have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + X)"
```
```  3068     by (simp add: fps_deriv_L fps_const_add[symmetric] algebra_simps del: fps_const_add)
```
```  3069   also have "\<dots> = fps_deriv ?l"
```
```  3070     apply (simp add: fps_deriv_L)
```
```  3071     apply (simp add: fps_eq_iff eq)
```
```  3072     done
```
```  3073   finally show ?thesis
```
```  3074     unfolding fps_deriv_eq_iff by simp
```
```  3075 qed
```
```  3076
```
```  3077
```
```  3078 subsubsection{* Binomial series *}
```
```  3079
```
```  3080 definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
```
```  3081
```
```  3082 lemma fps_binomial_nth[simp]: "fps_binomial a \$ n = a gchoose n"
```
```  3083   by (simp add: fps_binomial_def)
```
```  3084
```
```  3085 lemma fps_binomial_ODE_unique:
```
```  3086   fixes c :: "'a::field_char_0"
```
```  3087   shows "fps_deriv a = (fps_const c * a) / (1 + X) \<longleftrightarrow> a = fps_const (a\$0) * fps_binomial c"
```
```  3088   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  3089 proof -
```
```  3090   let ?da = "fps_deriv a"
```
```  3091   let ?x1 = "(1 + X):: 'a fps"
```
```  3092   let ?l = "?x1 * ?da"
```
```  3093   let ?r = "fps_const c * a"
```
```  3094   have x10: "?x1 \$ 0 \<noteq> 0" by simp
```
```  3095   have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
```
```  3096   also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
```
```  3097     apply (simp only: fps_divide_def  mult.assoc[symmetric] inverse_mult_eq_1[OF x10])
```
```  3098     apply (simp add: field_simps)
```
```  3099     done
```
```  3100   finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
```
```  3101   moreover
```
```  3102   {assume h: "?l = ?r"
```
```  3103     {fix n
```
```  3104       from h have lrn: "?l \$ n = ?r\$n" by simp
```
```  3105
```
```  3106       from lrn
```
```  3107       have "a\$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a \$n"
```
```  3108         apply (simp add: field_simps del: of_nat_Suc)
```
```  3109         by (cases n, simp_all add: field_simps del: of_nat_Suc)
```
```  3110     }
```
```  3111     note th0 = this
```
```  3112     {
```
```  3113       fix n
```
```  3114       have "a\$n = (c gchoose n) * a\$0"
```
```  3115       proof (induct n)
```
```  3116         case 0
```
```  3117         then show ?case by simp
```
```  3118       next
```
```  3119         case (Suc m)
```
```  3120         then show ?case unfolding th0
```
```  3121           apply (simp add: field_simps del: of_nat_Suc)
```
```  3122           unfolding mult.assoc[symmetric] gbinomial_mult_1
```
```  3123           apply (simp add: field_simps)
```
```  3124           done
```
```  3125       qed
```
```  3126     }
```
```  3127     note th1 = this
```
```  3128     have ?rhs
```
```  3129       apply (simp add: fps_eq_iff)
```
```  3130       apply (subst th1)
```
```  3131       apply (simp add: field_simps)
```
```  3132       done
```
```  3133   }
```
```  3134   moreover
```
```  3135   {
```
```  3136     assume h: ?rhs
```
```  3137     have th00: "\<And>x y. x * (a\$0 * y) = a\$0 * (x*y)"
```
```  3138       by (simp add: mult.commute)
```
```  3139     have "?l = ?r"
```
```  3140       apply (subst h)
```
```  3141       apply (subst (2) h)
```
```  3142       apply (clarsimp simp add: fps_eq_iff field_simps)
```
```  3143       unfolding mult.assoc[symmetric] th00 gbinomial_mult_1
```
```  3144       apply (simp add: field_simps gbinomial_mult_1)
```
```  3145       done
```
```  3146   }
```
```  3147   ultimately show ?thesis by blast
```
```  3148 qed
```
```  3149
```
```  3150 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
```
```  3151 proof -
```
```  3152   let ?a = "fps_binomial c"
```
```  3153   have th0: "?a = fps_const (?a\$0) * ?a" by (simp)
```
```  3154   from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
```
```  3155 qed
```
```  3156
```
```  3157 lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
```
```  3158 proof -
```
```  3159   let ?P = "?r - ?l"
```
```  3160   let ?b = "fps_binomial"
```
```  3161   let ?db = "\<lambda>x. fps_deriv (?b x)"
```
```  3162   have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
```
```  3163   also have "\<dots> = inverse (1 + X) *
```
```  3164       (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
```
```  3165     unfolding fps_binomial_deriv
```
```  3166     by (simp add: fps_divide_def field_simps)
```
```  3167   also have "\<dots> = (fps_const (c + d)/ (1 + X)) * ?P"
```
```  3168     by (simp add: field_simps fps_divide_def fps_const_add[symmetric] del: fps_const_add)
```
```  3169   finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + X)"
```
```  3170     by (simp add: fps_divide_def)
```
```  3171   have "?P = fps_const (?P\$0) * ?b (c + d)"
```
```  3172     unfolding fps_binomial_ODE_unique[symmetric]
```
```  3173     using th0 by simp
```
```  3174   then have "?P = 0" by (simp add: fps_mult_nth)
```
```  3175   then show ?thesis by simp
```
```  3176 qed
```
```  3177
```
```  3178 lemma fps_minomial_minus_one: "fps_binomial (- 1) = inverse (1 + X)"
```
```  3179   (is "?l = inverse ?r")
```
```  3180 proof-
```
```  3181   have th: "?r\$0 \<noteq> 0" by simp
```
```  3182   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
```
```  3183     by (simp add: fps_inverse_deriv[OF th] fps_divide_def
```
```  3184       power2_eq_square mult.commute fps_const_neg[symmetric] del: fps_const_neg)
```
```  3185   have eq: "inverse ?r \$ 0 = 1"
```
```  3186     by (simp add: fps_inverse_def)
```
```  3187   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
```
```  3188   show ?thesis by (simp add: fps_inverse_def)
```
```  3189 qed
```
```  3190
```
```  3191 text{* Vandermonde's Identity as a consequence *}
```
```  3192 lemma gbinomial_Vandermonde:
```
```  3193   "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
```
```  3194 proof -
```
```  3195   let ?ba = "fps_binomial a"
```
```  3196   let ?bb = "fps_binomial b"
```
```  3197   let ?bab = "fps_binomial (a + b)"
```
```  3198   from fps_binomial_add_mult[of a b] have "?bab \$ n = (?ba * ?bb)\$n" by simp
```
```  3199   then show ?thesis by (simp add: fps_mult_nth)
```
```  3200 qed
```
```  3201
```
```  3202 lemma binomial_Vandermonde:
```
```  3203   "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
```
```  3204   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
```
```  3205   apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
```
```  3206     of_nat_setsum[symmetric] of_nat_add[symmetric])
```
```  3207   apply simp
```
```  3208   done
```
```  3209
```
```  3210 lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2*n) choose n"
```
```  3211   using binomial_Vandermonde[of n n n,symmetric]
```
```  3212   unfolding mult_2
```
```  3213   apply (simp add: power2_eq_square)
```
```  3214   apply (rule setsum.cong)
```
```  3215   apply (auto intro:  binomial_symmetric)
```
```  3216   done
```
```  3217
```
```  3218 lemma Vandermonde_pochhammer_lemma:
```
```  3219   fixes a :: "'a::field_char_0"
```
```  3220   assumes b: "\<forall> j\<in>{0 ..<n}. b \<noteq> of_nat j"
```
```  3221   shows "setsum (\<lambda>k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) /
```
```  3222       (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} =
```
```  3223     pochhammer (- (a + b)) n / pochhammer (- b) n"
```
```  3224   (is "?l = ?r")
```
```  3225 proof -
```
```  3226   let ?m1 = "\<lambda>m. (- 1 :: 'a) ^ m"
```
```  3227   let ?f = "\<lambda>m. of_nat (fact m)"
```
```  3228   let ?p = "\<lambda>(x::'a). pochhammer (- x)"
```
```  3229   from b have bn0: "?p b n \<noteq> 0" unfolding pochhammer_eq_0_iff by simp
```
```  3230   {
```
```  3231     fix k
```
```  3232     assume kn: "k \<in> {0..n}"
```
```  3233     {
```
```  3234       assume c:"pochhammer (b - of_nat n + 1) n = 0"
```
```  3235       then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
```
```  3236         unfolding pochhammer_eq_0_iff by blast
```
```  3237       from j have "b = of_nat n - of_nat j - of_nat 1"
```
```  3238         by (simp add: algebra_simps)
```
```  3239       then have "b = of_nat (n - j - 1)"
```
```  3240         using j kn by (simp add: of_nat_diff)
```
```  3241       with b have False using j by auto
```
```  3242     }
```
```  3243     then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0"
```
```  3244       by (auto simp add: algebra_simps)
```
```  3245
```
```  3246     from nz kn [simplified] have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0"
```
```  3247       by (rule pochhammer_neq_0_mono)
```
```  3248     {
```
```  3249       assume k0: "k = 0 \<or> n =0"
```
```  3250       then have "b gchoose (n - k) =
```
```  3251         (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  3252         using kn
```
```  3253         by (cases "k = 0") (simp_all add: gbinomial_pochhammer)
```
```  3254     }
```
```  3255     moreover
```
```  3256     {
```
```  3257       assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0"
```
```  3258       then obtain m where m: "n = Suc m" by (cases n) auto
```
```  3259       from k0 obtain h where h: "k = Suc h" by (cases k) auto
```
```  3260       {
```
```  3261         assume kn: "k = n"
```
```  3262         then have "b gchoose (n - k) =
```
```  3263           (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  3264           using kn pochhammer_minus'[where k=k and n=n and b=b]
```
```  3265           apply (simp add:  pochhammer_same)
```
```  3266           using bn0
```
```  3267           apply (simp add: field_simps power_add[symmetric])
```
```  3268           done
```
```  3269       }
```
```  3270       moreover
```
```  3271       {
```
```  3272         assume nk: "k \<noteq> n"
```
```  3273         have m1nk: "?m1 n = setprod (\<lambda>i. - 1) {0..m}" "?m1 k = setprod (\<lambda>i. - 1) {0..h}"
```
```  3274           by (simp_all add: setprod_constant m h)
```
```  3275         from kn nk have kn': "k < n" by simp
```
```  3276         have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
```
```  3277           using bn0 kn
```
```  3278           unfolding pochhammer_eq_0_iff
```
```  3279           apply auto
```
```  3280           apply (erule_tac x= "n - ka - 1" in allE)
```
```  3281           apply (auto simp add: algebra_simps of_nat_diff)
```
```  3282           done
```
```  3283         have eq1: "setprod (\<lambda>k. (1::'a) + of_nat m - of_nat k) {0 .. h} =
```
```  3284           setprod of_nat {Suc (m - h) .. Suc m}"
```
```  3285           using kn' h m
```
```  3286           by (intro setprod.reindex_bij_witness[where i="\<lambda>k. Suc m - k" and j="\<lambda>k. Suc m - k"])
```
```  3287              (auto simp: of_nat_diff)
```
```  3288
```
```  3289         have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
```
```  3290           unfolding m1nk
```
```  3291           unfolding m h pochhammer_Suc_setprod
```
```  3292           apply (simp add: field_simps del: fact_Suc)
```
```  3293           unfolding fact_altdef_nat id_def
```
```  3294           unfolding of_nat_setprod
```
```  3295           unfolding setprod.distrib[symmetric]
```
```  3296           apply auto
```
```  3297           unfolding eq1
```
```  3298           apply (subst setprod.union_disjoint[symmetric])
```
```  3299           apply (auto)
```
```  3300           apply (rule setprod.cong)
```
```  3301           apply auto
```
```  3302           done
```
```  3303         have th20: "?m1 n * ?p b n = setprod (\<lambda>i. b - of_nat i) {0..m}"
```
```  3304           unfolding m1nk
```
```  3305           unfolding m h pochhammer_Suc_setprod
```
```  3306           unfolding setprod.distrib[symmetric]
```
```  3307           apply (rule setprod.cong)
```
```  3308           apply auto
```
```  3309           done
```
```  3310         have th21:"pochhammer (b - of_nat n + 1) k = setprod (\<lambda>i. b - of_nat i) {n - k .. n - 1}"
```
```  3311           unfolding h m
```
```  3312           unfolding pochhammer_Suc_setprod
```
```  3313           using kn m h
```
```  3314           by (intro setprod.reindex_bij_witness[where i="\<lambda>k. n - 1 - k" and j="\<lambda>i. m-i"])
```
```  3315              (auto simp: of_nat_diff)
```
```  3316
```
```  3317         have "?m1 n * ?p b n =
```
```  3318           pochhammer (b - of_nat n + 1) k * setprod (\<lambda>i. b - of_nat i) {0.. n - k - 1}"
```
```  3319           unfolding th20 th21
```
```  3320           unfolding h m
```
```  3321           apply (subst setprod.union_disjoint[symmetric])
```
```  3322           using kn' h m
```
```  3323           apply auto
```
```  3324           apply (rule setprod.cong)
```
```  3325           apply auto
```
```  3326           done
```
```  3327         then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k =
```
```  3328           setprod (\<lambda>i. b - of_nat i) {0.. n - k - 1}"
```
```  3329           using nz' by (simp add: field_simps)
```
```  3330         have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) =
```
```  3331           ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
```
```  3332           using bnz0
```
```  3333           by (simp add: field_simps)
```
```  3334         also have "\<dots> = b gchoose (n - k)"
```
```  3335           unfolding th1 th2
```
```  3336           using kn' by (simp add: gbinomial_def)
```
```  3337         finally have "b gchoose (n - k) =
```
```  3338           (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  3339           by simp
```
```  3340       }
```
```  3341       ultimately
```
```  3342       have "b gchoose (n - k) =
```
```  3343         (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  3344         by (cases "k = n") auto
```
```  3345     }
```
```  3346     ultimately have "b gchoose (n - k) =
```
```  3347         (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
```
```  3348       "pochhammer (1 + b - of_nat n) k \<noteq> 0 "
```
```  3349       apply (cases "n = 0")
```
```  3350       using nz'
```
```  3351       apply auto
```
```  3352       apply (cases k)
```
```  3353       apply auto
```
```  3354       done
```
```  3355   }
```
```  3356   note th00 = this
```
```  3357   have "?r = ((a + b) gchoose n) * (of_nat (fact n)/ (?m1 n * pochhammer (- b) n))"
```
```  3358     unfolding gbinomial_pochhammer
```
```  3359     using bn0 by (auto simp add: field_simps)
```
```  3360   also have "\<dots> = ?l"
```
```  3361     unfolding gbinomial_Vandermonde[symmetric]
```
```  3362     apply (simp add: th00)
```
```  3363     unfolding gbinomial_pochhammer
```
```  3364     using bn0
```
```  3365     apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
```
```  3366     apply (rule setsum.cong)
```
```  3367     apply (rule refl)
```
```  3368     apply (drule th00(2))
```
```  3369     apply (simp add: field_simps power_add[symmetric])
```
```  3370     done
```
```  3371   finally show ?thesis by simp
```
```  3372 qed
```
```  3373
```
```  3374 lemma Vandermonde_pochhammer:
```
```  3375   fixes a :: "'a::field_char_0"
```
```  3376   assumes c: "\<forall>i \<in> {0..< n}. c \<noteq> - of_nat i"
```
```  3377   shows "setsum (\<lambda>k. (pochhammer a k * pochhammer (- (of_nat n)) k) /
```
```  3378     (of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"
```
```  3379 proof -
```
```  3380   let ?a = "- a"
```
```  3381   let ?b = "c + of_nat n - 1"
```
```  3382   have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j" using c
```
```  3383     apply (auto simp add: algebra_simps of_nat_diff)
```
```  3384     apply (erule_tac x= "n - j - 1" in ballE)
```
```  3385     apply (auto simp add: of_nat_diff algebra_simps)
```
```  3386     done
```
```  3387   have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
```
```  3388     unfolding pochhammer_minus[OF le_refl]
```
```  3389     by (simp add: algebra_simps)
```
```  3390   have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
```
```  3391     unfolding pochhammer_minus[OF le_refl]
```
```  3392     by simp
```
```  3393   have nz: "pochhammer c n \<noteq> 0" using c
```
```  3394     by (simp add: pochhammer_eq_0_iff)
```
```  3395   from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
```
```  3396   show ?thesis using nz by (simp add: field_simps setsum_right_distrib)
```
```  3397 qed
```
```  3398
```
```  3399
```
```  3400 subsubsection{* Formal trigonometric functions  *}
```
```  3401
```
```  3402 definition "fps_sin (c::'a::field_char_0) =
```
```  3403   Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
```
```  3404
```
```  3405 definition "fps_cos (c::'a::field_char_0) =
```
```  3406   Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
```
```  3407
```
```  3408 lemma fps_sin_deriv:
```
```  3409   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
```
```  3410   (is "?lhs = ?rhs")
```
```  3411 proof (rule fps_ext)
```
```  3412   fix n :: nat
```
```  3413   {
```
```  3414     assume en: "even n"
```
```  3415     have "?lhs\$n = of_nat (n+1) * (fps_sin c \$ (n+1))" by simp
```
```  3416     also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  3417       using en by (simp add: fps_sin_def)
```
```  3418     also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  3419       unfolding fact_Suc of_nat_mult
```
```  3420       by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3421     also have "\<dots> = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
```
```  3422       by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3423     finally have "?lhs \$n = ?rhs\$n" using en
```
```  3424       by (simp add: fps_cos_def field_simps)
```
```  3425   }
```
```  3426   then show "?lhs \$ n = ?rhs \$ n"
```
```  3427     by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
```
```  3428 qed
```
```  3429
```
```  3430 lemma fps_cos_deriv: "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
```
```  3431   (is "?lhs = ?rhs")
```
```  3432 proof (rule fps_ext)
```
```  3433   have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by simp
```
```  3434   have th1: "\<And>n. odd n \<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2"
```
```  3435     by (case_tac n, simp_all)
```
```  3436   fix n::nat
```
```  3437   {
```
```  3438     assume en: "odd n"
```
```  3439     from en have n0: "n \<noteq>0 " by presburger
```
```  3440     have "?lhs\$n = of_nat (n+1) * (fps_cos c \$ (n+1))" by simp
```
```  3441     also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  3442       using en by (simp add: fps_cos_def)
```
```  3443     also have "\<dots> = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  3444       unfolding fact_Suc of_nat_mult
```
```  3445       by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3446     also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
```
```  3447       by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  3448     also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
```
```  3449       unfolding th0 unfolding th1[OF en] by simp
```
```  3450     finally have "?lhs \$n = ?rhs\$n" using en
```
```  3451       by (simp add: fps_sin_def field_simps)
```
```  3452   }
```
```  3453   then show "?lhs \$ n = ?rhs \$ n"
```
```  3454     by (cases "even n") (simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
```
```  3455 qed
```
```  3456
```
```  3457 lemma fps_sin_cos_sum_of_squares:
```
```  3458   "(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1" (is "?lhs = 1")
```
```  3459 proof -
```
```  3460   have "fps_deriv ?lhs = 0"
```
```  3461     apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv)
```
```  3462     apply (simp add: field_simps fps_const_neg[symmetric] del: fps_const_neg)
```
```  3463     done
```
```  3464   then have "?lhs = fps_const (?lhs \$ 0)"
```
```  3465     unfolding fps_deriv_eq_0_iff .
```
```  3466   also have "\<dots> = 1"
```
```  3467     by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
```
```  3468   finally show ?thesis .
```
```  3469 qed
```
```  3470
```
```  3471 lemma divide_eq_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x / a = y \<longleftrightarrow> x = y * a"
```
```  3472   by auto
```
```  3473
```
```  3474 lemma eq_divide_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x = y / a \<longleftrightarrow> x * a = y"
```
```  3475   by auto
```
```  3476
```
```  3477 lemma fps_sin_nth_0 [simp]: "fps_sin c \$ 0 = 0"
```
```  3478   unfolding fps_sin_def by simp
```
```  3479
```
```  3480 lemma fps_sin_nth_1 [simp]: "fps_sin c \$ 1 = c"
```
```  3481   unfolding fps_sin_def by simp
```
```  3482
```
```  3483 lemma fps_sin_nth_add_2:
```
```  3484   "fps_sin c \$ (n + 2) = - (c * c * fps_sin c \$ n / (of_nat(n+1) * of_nat(n+2)))"
```
```  3485   unfolding fps_sin_def
```
```  3486   apply (cases n, simp)
```
```  3487   apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
```
```  3488   apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
```
```  3489   done
```
```  3490
```
```  3491 lemma fps_cos_nth_0 [simp]: "fps_cos c \$ 0 = 1"
```
```  3492   unfolding fps_cos_def by simp
```
```  3493
```
```  3494 lemma fps_cos_nth_1 [simp]: "fps_cos c \$ 1 = 0"
```
```  3495   unfolding fps_cos_def by simp
```
```  3496
```
```  3497 lemma fps_cos_nth_add_2:
```
```  3498   "fps_cos c \$ (n + 2) = - (c * c * fps_cos c \$ n / (of_nat(n+1) * of_nat(n+2)))"
```
```  3499   unfolding fps_cos_def
```
```  3500   apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
```
```  3501   apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
```
```  3502   done
```
```  3503
```
```  3504 lemma nat_induct2: "P 0 \<Longrightarrow> P 1 \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (n + 2)) \<Longrightarrow> P (n::nat)"
```
```  3505   unfolding One_nat_def numeral_2_eq_2
```
```  3506   apply (induct n rule: nat_less_induct)
```
```  3507   apply (case_tac n)
```
```  3508   apply simp
```
```  3509   apply (rename_tac m)
```
```  3510   apply (case_tac m)
```
```  3511   apply simp
```
```  3512   apply (rename_tac k)
```
```  3513   apply (case_tac k)
```
```  3514   apply simp_all
```
```  3515   done
```
```  3516
```
```  3517 lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
```
```  3518   by simp
```
```  3519
```
```  3520 lemma eq_fps_sin:
```
```  3521   assumes 0: "a \$ 0 = 0"
```
```  3522     and 1: "a \$ 1 = c"
```
```  3523     and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
```
```  3524   shows "a = fps_sin c"
```
```  3525   apply (rule fps_ext)
```
```  3526   apply (induct_tac n rule: nat_induct2)
```
```  3527   apply (simp add: 0)
```
```  3528   apply (simp add: 1 del: One_nat_def)
```
```  3529   apply (rename_tac m, cut_tac f="\<lambda>a. a \$ m" in arg_cong [OF 2])
```
```  3530   apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
```
```  3531               del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
```
```  3532   apply (subst minus_divide_left)
```
```  3533   apply (subst eq_divide_iff)
```
```  3534   apply (simp del: of_nat_add of_nat_Suc)
```
```  3535   apply (simp only: ac_simps)
```
```  3536   done
```
```  3537
```
```  3538 lemma eq_fps_cos:
```
```  3539   assumes 0: "a \$ 0 = 1"
```
```  3540     and 1: "a \$ 1 = 0"
```
```  3541     and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
```
```  3542   shows "a = fps_cos c"
```
```  3543   apply (rule fps_ext)
```
```  3544   apply (induct_tac n rule: nat_induct2)
```
```  3545   apply (simp add: 0)
```
```  3546   apply (simp add: 1 del: One_nat_def)
```
```  3547   apply (rename_tac m, cut_tac f="\<lambda>a. a \$ m" in arg_cong [OF 2])
```
```  3548   apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
```
```  3549               del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
```
```  3550   apply (subst minus_divide_left)
```
```  3551   apply (subst eq_divide_iff)
```
```  3552   apply (simp del: of_nat_add of_nat_Suc)
```
```  3553   apply (simp only: ac_simps)
```
```  3554   done
```
```  3555
```
```  3556 lemma mult_nth_0 [simp]: "(a * b) \$ 0 = a \$ 0 * b \$ 0"
```
```  3557   by (simp add: fps_mult_nth)
```
```  3558
```
```  3559 lemma mult_nth_1 [simp]: "(a * b) \$ 1 = a \$ 0 * b \$ 1 + a \$ 1 * b \$ 0"
```
```  3560   by (simp add: fps_mult_nth)
```
```  3561
```
```  3562 lemma fps_sin_add: "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
```
```  3563   apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
```
```  3564   apply (simp del: fps_const_neg fps_const_add fps_const_mult
```
```  3565               add: fps_const_add [symmetric] fps_const_neg [symmetric]
```
```  3566                    fps_sin_deriv fps_cos_deriv algebra_simps)
```
```  3567   done
```
```  3568
```
```  3569 lemma fps_cos_add: "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
```
```  3570   apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
```
```  3571   apply (simp del: fps_const_neg fps_const_add fps_const_mult
```
```  3572               add: fps_const_add [symmetric] fps_const_neg [symmetric]
```
```  3573                    fps_sin_deriv fps_cos_deriv algebra_simps)
```
```  3574   done
```
```  3575
```
```  3576 lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
```
```  3577   by (auto simp add: fps_eq_iff fps_sin_def)
```
```  3578
```
```  3579 lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
```
```  3580   by (auto simp add: fps_eq_iff fps_cos_def)
```
```  3581
```
```  3582 definition "fps_tan c = fps_sin c / fps_cos c"
```
```  3583
```
```  3584 lemma fps_tan_deriv: "fps_deriv (fps_tan c) = fps_const c / (fps_cos c)\<^sup>2"
```
```  3585 proof -
```
```  3586   have th0: "fps_cos c \$ 0 \<noteq> 0" by (simp add: fps_cos_def)
```
```  3587   show ?thesis
```
```  3588     using fps_sin_cos_sum_of_squares[of c]
```
```  3589     apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv
```
```  3590       fps_const_neg[symmetric] field_simps power2_eq_square del: fps_const_neg)
```
```  3591     unfolding distrib_left[symmetric]
```
```  3592     apply simp
```
```  3593     done
```
```  3594 qed
```
```  3595
```
```  3596 text {* Connection to E c over the complex numbers --- Euler and De Moivre*}
```
```  3597 lemma Eii_sin_cos: "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
```
```  3598   (is "?l = ?r")
```
```  3599 proof -
```
```  3600   { fix n :: nat
```
```  3601     {
```
```  3602       assume en: "even n"
```
```  3603       from en obtain m where m: "n = 2 * m"
```
```  3604         unfolding even_mult_two_ex by blast
```
```  3605
```
```  3606       have "?l \$n = ?r\$n"
```
```  3607         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib power_mult power_minus [of "c ^ 2"])
```
```  3608     }
```
```  3609     moreover
```
```  3610     {
```
```  3611       assume on: "odd n"
```
```  3612       from on obtain m where m: "n = 2*m + 1"
```
```  3613         unfolding odd_nat_equiv_def2 by (auto simp add: mult_2)
```
```  3614       have "?l \$n = ?r\$n"
```
```  3615         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
```
```  3616           power_mult power_minus [of "c ^ 2"])
```
```  3617     }
```
```  3618     ultimately have "?l \$n = ?r\$n"  by blast
```
```  3619   } then show ?thesis by (simp add: fps_eq_iff)
```
```  3620 qed
```
```  3621
```
```  3622 lemma E_minus_ii_sin_cos: "E (- (ii * c)) = fps_cos c - fps_const ii * fps_sin c"
```
```  3623   unfolding minus_mult_right Eii_sin_cos by (simp add: fps_sin_even fps_cos_odd)
```
```  3624
```
```  3625 lemma fps_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
```
```  3626   by (simp add: fps_eq_iff fps_const_def)
```
```  3627
```
```  3628 lemma fps_numeral_fps_const: "numeral i = fps_const (numeral i :: 'a::comm_ring_1)"
```
```  3629   by (fact numeral_fps_const) (* FIXME: duplicate *)
```
```  3630
```
```  3631 lemma fps_cos_Eii: "fps_cos c = (E (ii * c) + E (- ii * c)) / fps_const 2"
```
```  3632 proof -
```
```  3633   have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
```
```  3634     by (simp add: numeral_fps_const)
```
```  3635   show ?thesis
```
```  3636   unfolding Eii_sin_cos minus_mult_commute
```
```  3637   by (simp add: fps_sin_even fps_cos_odd numeral_fps_const fps_divide_def fps_const_inverse th)
```
```  3638 qed
```
```  3639
```
```  3640 lemma fps_sin_Eii: "fps_sin c = (E (ii * c) - E (- ii * c)) / fps_const (2*ii)"
```
```  3641 proof -
```
```  3642   have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * ii)"
```
```  3643     by (simp add: fps_eq_iff numeral_fps_const)
```
```  3644   show ?thesis
```
```  3645     unfolding Eii_sin_cos minus_mult_commute
```
```  3646     by (simp add: fps_sin_even fps_cos_odd fps_divide_def fps_const_inverse th)
```
```  3647 qed
```
```  3648
```
```  3649 lemma fps_tan_Eii:
```
```  3650   "fps_tan c = (E (ii * c) - E (- ii * c)) / (fps_const ii * (E (ii * c) + E (- ii * c)))"
```
```  3651   unfolding fps_tan_def fps_sin_Eii fps_cos_Eii mult_minus_left E_neg
```
```  3652   apply (simp add: fps_divide_def fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
```
```  3653   apply simp
```
```  3654   done
```
```  3655
```
```  3656 lemma fps_demoivre: "(fps_cos a + fps_const ii * fps_sin a)^n = fps_cos (of_nat n * a) + fps_const ii * fps_sin (of_nat n * a)"
```
```  3657   unfolding Eii_sin_cos[symmetric] E_power_mult
```
```  3658   by (simp add: ac_simps)
```
```  3659
```
```  3660
```
```  3661 subsection {* Hypergeometric series *}
```
```  3662
```
```  3663 definition "F as bs (c::'a::{field_char_0,field_inverse_zero}) =
```
```  3664   Abs_fps (\<lambda>n. (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) /
```
```  3665     (foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
```
```  3666
```
```  3667 lemma F_nth[simp]: "F as bs c \$ n =
```
```  3668   (foldl (\<lambda>r a. r* pochhammer a n) 1 as * c^n) /
```
```  3669     (foldl (\<lambda>r b. r * pochhammer b n) 1 bs * of_nat (fact n))"
```
```  3670   by (simp add: F_def)
```
```  3671
```
```  3672 lemma foldl_mult_start:
```
```  3673   fixes v :: "'a::comm_ring_1"
```
```  3674   shows "foldl (\<lambda>r x. r * f x) v as * x = foldl (\<lambda>r x. r * f x) (v * x) as "
```
```  3675   by (induct as arbitrary: x v) (auto simp add: algebra_simps)
```
```  3676
```
```  3677 lemma foldr_mult_foldl:
```
```  3678   fixes v :: "'a::comm_ring_1"
```
```  3679   shows "foldr (\<lambda>x r. r * f x) as v = foldl (\<lambda>r x. r * f x) v as"
```
```  3680   by (induct as arbitrary: v) (auto simp add: foldl_mult_start)
```
```  3681
```
```  3682 lemma F_nth_alt:
```
```  3683   "F as bs c \$ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
```
```  3684     foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))"
```
```  3685   by (simp add: foldl_mult_start foldr_mult_foldl)
```
```  3686
```
```  3687 lemma F_E[simp]: "F [] [] c = E c"
```
```  3688   by (simp add: fps_eq_iff)
```
```  3689
```
```  3690 lemma F_1_0[simp]: "F [1] [] c = 1/(1 - fps_const c * X)"
```
```  3691 proof -
```
```  3692   let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * X)"
```
```  3693   have th0: "(fps_const c * X) \$ 0 = 0" by simp
```
```  3694   show ?thesis unfolding gp[OF th0, symmetric]
```
```  3695     by (auto simp add: fps_eq_iff pochhammer_fact[symmetric]
```
```  3696       fps_compose_nth power_mult_distrib cond_value_iff setsum.delta' cong del: if_weak_cong)
```
```  3697 qed
```
```  3698
```
```  3699 lemma F_B[simp]: "F [-a] [] (- 1) = fps_binomial a"
```
```  3700   by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)
```
```  3701
```
```  3702 lemma F_0[simp]: "F as bs c \$0 = 1"
```
```  3703   apply simp
```
```  3704   apply (subgoal_tac "\<forall>as. foldl (\<lambda>(r::'a) (a::'a). r) 1 as = 1")
```
```  3705   apply auto
```
```  3706   apply (induct_tac as)
```
```  3707   apply auto
```
```  3708   done
```
```  3709
```
```  3710 lemma foldl_prod_prod:
```
```  3711   "foldl (\<lambda>(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (\<lambda>r x. r * g x) w as =
```
```  3712     foldl (\<lambda>r x. r * f x * g x) (v * w) as"
```
```  3713   by (induct as arbitrary: v w) (auto simp add: algebra_simps)
```
```  3714
```
```  3715
```
```  3716 lemma F_rec:
```
```  3717   "F as bs c \$ Suc n = ((foldl (\<lambda>r a. r* (a + of_nat n)) c as) /
```
```  3718     (foldl (\<lambda>r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c \$ n"
```
```  3719   apply (simp del: of_nat_Suc of_nat_add fact_Suc)
```
```  3720   apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
```
```  3721   unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
```
```  3722   apply (simp add: algebra_simps of_nat_mult)
```
```  3723   done
```
```  3724
```
```  3725 lemma XD_nth[simp]: "XD a \$ n = (if n = 0 then 0 else of_nat n * a\$n)"
```
```  3726   by (simp add: XD_def)
```
```  3727
```
```  3728 lemma XD_0th[simp]: "XD a \$ 0 = 0" by simp
```
```  3729 lemma XD_Suc[simp]:" XD a \$ Suc n = of_nat (Suc n) * a \$ Suc n" by simp
```
```  3730
```
```  3731 definition "XDp c a = XD a + fps_const c * a"
```
```  3732
```
```  3733 lemma XDp_nth[simp]: "XDp c a \$ n = (c + of_nat n) * a\$n"
```
```  3734   by (simp add: XDp_def algebra_simps)
```
```  3735
```
```  3736 lemma XDp_commute: "XDp b \<circ> XDp (c::'a::comm_ring_1) = XDp c \<circ> XDp b"
```
```  3737   by (auto simp add: XDp_def fun_eq_iff fps_eq_iff algebra_simps)
```
```  3738
```
```  3739 lemma XDp0 [simp]: "XDp 0 = XD"
```
```  3740   by (simp add: fun_eq_iff fps_eq_iff)
```
```  3741
```
```  3742 lemma XDp_fps_integral [simp]: "XDp 0 (fps_integral a c) = X * a"
```
```  3743   by (simp add: fps_eq_iff fps_integral_def)
```
```  3744
```
```  3745 lemma F_minus_nat:
```
```  3746   "F [- of_nat n] [- of_nat (n + m)] (c::'a::{field_char_0,field_inverse_zero}) \$ k =
```
```  3747     (if k \<le> n then
```
```  3748       pochhammer (- of_nat n) k * c ^ k / (pochhammer (- of_nat (n + m)) k * of_nat (fact k))
```
```  3749      else 0)"
```
```  3750   "F [- of_nat m] [- of_nat (m + n)] (c::'a::{field_char_0,field_inverse_zero}) \$ k =
```
```  3751     (if k \<le> m then
```
```  3752       pochhammer (- of_nat m) k * c ^ k / (pochhammer (- of_nat (m + n)) k * of_nat (fact k))
```
```  3753      else 0)"
```
```  3754   by (auto simp add: pochhammer_eq_0_iff)
```
```  3755
```
```  3756 lemma setsum_eq_if: "setsum f {(n::nat) .. m} = (if m < n then 0 else f n + setsum f {n+1 .. m})"
```
```  3757   apply simp
```
```  3758   apply (subst setsum.insert[symmetric])
```
```  3759   apply (auto simp add: not_less setsum_head_Suc)
```
```  3760   done
```
```  3761
```
```  3762 lemma pochhammer_rec_if: "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
```
```  3763   by (cases n) (simp_all add: pochhammer_rec)
```
```  3764
```
```  3765 lemma XDp_foldr_nth [simp]: "foldr (\<lambda>c r. XDp c \<circ> r) cs (\<lambda>c. XDp c a) c0 \$ n =
```
```  3766     foldr (\<lambda>c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a\$n"
```
```  3767   by (induct cs arbitrary: c0) (auto simp add: algebra_simps)
```
```  3768
```
```  3769 lemma genric_XDp_foldr_nth:
```
```  3770   assumes f: "\<forall>n c a. f c a \$ n = (of_nat n + k c) * a\$n"
```
```  3771   shows "foldr (\<lambda>c r. f c \<circ> r) cs (\<lambda>c. g c a) c0 \$ n =
```
```  3772     foldr (\<lambda>c r. (k c + of_nat n) * r) cs (g c0 a \$ n)"
```
```  3773   by (induct cs arbitrary: c0) (auto simp add: algebra_simps f)
```
```  3774
```
```  3775 lemma dist_less_imp_nth_equal:
```
```  3776   assumes "dist f g < inverse (2 ^ i)"
```
```  3777     and"j \<le> i"
```
```  3778   shows "f \$ j = g \$ j"
```
```  3779 proof (rule ccontr)
```
```  3780   assume "f \$ j \<noteq> g \$ j"
```
```  3781   then have "\<exists>n. f \$ n \<noteq> g \$ n" by auto
```
```  3782   with assms have "i < (LEAST n. f \$ n \<noteq> g \$ n)"
```
```  3783     by (simp add: split_if_asm dist_fps_def)
```
```  3784   also have "\<dots> \<le> j"
```
```  3785     using `f \$ j \<noteq> g \$ j` by (auto intro: Least_le)
```
```  3786   finally show False using `j \<le> i` by simp
```
```  3787 qed
```
```  3788
```
```  3789 lemma nth_equal_imp_dist_less:
```
```  3790   assumes "\<And>j. j \<le> i \<Longrightarrow> f \$ j = g \$ j"
```
```  3791   shows "dist f g < inverse (2 ^ i)"
```
```  3792 proof (cases "f = g")
```
```  3793   case False
```
```  3794   then have "\<exists>n. f \$ n \<noteq> g \$ n" by (simp add: fps_eq_iff)
```
```  3795   with assms have "dist f g = inverse (2 ^ (LEAST n. f \$ n \<noteq> g \$ n))"
```
```  3796     by (simp add: split_if_asm dist_fps_def)
```
```  3797   moreover
```
```  3798   from assms `\<exists>n. f \$ n \<noteq> g \$ n` have "i < (LEAST n. f \$ n \<noteq> g \$ n)"
```
```  3799     by (metis (mono_tags) LeastI not_less)
```
```  3800   ultimately show ?thesis by simp
```
```  3801 qed simp
```
```  3802
```
```  3803 lemma dist_less_eq_nth_equal: "dist f g < inverse (2 ^ i) \<longleftrightarrow> (\<forall>j \<le> i. f \$ j = g \$ j)"
```
```  3804   using dist_less_imp_nth_equal nth_equal_imp_dist_less by blast
```
```  3805
```
```  3806 instance fps :: (comm_ring_1) complete_space
```
```  3807 proof
```
```  3808   fix X :: "nat \<Rightarrow> 'a fps"
```
```  3809   assume "Cauchy X"
```
```  3810   {
```
```  3811     fix i
```
```  3812     have "0 < inverse ((2::real)^i)" by simp
```
```  3813     from metric_CauchyD[OF `Cauchy X` this] dist_less_imp_nth_equal
```
```  3814     have "\<exists>M. \<forall>m \<ge> M. \<forall>j\<le>i. X M \$ j = X m \$ j" by blast
```
```  3815   }
```
```  3816   then obtain M where M: "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. X (M i) \$ j = X m \$ j" by metis
```
```  3817   then have "\<forall>i. \<forall>m \<ge> M i. \<forall>j \<le> i. X (M i) \$ j = X m \$ j" by metis
```
```  3818   show "convergent X"
```
```  3819   proof (rule convergentI)
```
```  3820     show "X ----> Abs_fps (\<lambda>i. X (M i) \$ i)"
```
```  3821       unfolding tendsto_iff
```
```  3822     proof safe
```
```  3823       fix e::real assume "0 < e"
```
```  3824       with LIMSEQ_inverse_realpow_zero[of 2, simplified, simplified filterlim_iff,
```
```  3825         THEN spec, of "\<lambda>x. x < e"]
```
```  3826       have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially"
```
```  3827         apply safe
```
```  3828         apply (auto simp: eventually_nhds)
```
```  3829         done
```
```  3830       then obtain i where "inverse (2 ^ i) < e" by (auto simp: eventually_sequentially)
```
```  3831       have "eventually (\<lambda>x. M i \<le> x) sequentially" by (auto simp: eventually_sequentially)
```
```  3832       then show "eventually (\<lambda>x. dist (X x) (Abs_fps (\<lambda>i. X (M i) \$ i)) < e) sequentially"
```
```  3833       proof eventually_elim
```
```  3834         fix x
```
```  3835         assume "M i \<le> x"
```
```  3836         moreover
```
```  3837         have "\<And>j. j \<le> i \<Longrightarrow> X (M i) \$ j = X (M j) \$ j"
```
```  3838           using M by (metis nat_le_linear)
```
```  3839         ultimately have "dist (X x) (Abs_fps (\<lambda>j. X (M j) \$ j)) < inverse (2 ^ i)"
```
```  3840           using M by (force simp: dist_less_eq_nth_equal)
```
```  3841         also note `inverse (2 ^ i) < e`
```
```  3842         finally show "dist (X x) (Abs_fps (\<lambda>j. X (M j) \$ j)) < e" .
```
```  3843       qed
```
```  3844     qed
```
```  3845   qed
```
```  3846 qed
```
```  3847
```
```  3848 end
```