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