src/HOL/Library/Formal_Power_Series.thy
author wenzelm
Thu Feb 11 23:00:22 2010 +0100 (2010-02-11)
changeset 35115 446c5063e4fd
parent 32960 69916a850301
child 35175 61255c81da01
permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
     1 (*  Title:      Formal_Power_Series.thy
     2     Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header{* A formalization of formal power series *}
     6 
     7 theory Formal_Power_Series
     8 imports Complex_Main Binomial
     9 begin
    10 
    11 
    12 subsection {* The type of formal power series*}
    13 
    14 typedef (open) '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] expand_fun_eq)
    22 
    23 lemma fps_ext: "(\<And>n. p $ n = q $ n) \<Longrightarrow> p = q"
    24   by (simp add: expand_fps_eq)
    25 
    26 lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n"
    27   by (simp add: Abs_fps_inverse)
    28 
    29 text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *}
    30 
    31 instantiation fps :: (zero)  zero
    32 begin
    33 
    34 definition fps_zero_def:
    35   "0 = Abs_fps (\<lambda>n. 0)"
    36 
    37 instance ..
    38 end
    39 
    40 lemma fps_zero_nth [simp]: "0 $ n = 0"
    41   unfolding fps_zero_def by simp
    42 
    43 instantiation fps :: ("{one,zero}")  one
    44 begin
    45 
    46 definition fps_one_def:
    47   "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)"
    48 
    49 instance ..
    50 end
    51 
    52 lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)"
    53   unfolding fps_one_def by simp
    54 
    55 instantiation fps :: (plus)  plus
    56 begin
    57 
    58 definition fps_plus_def:
    59   "op + = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
    60 
    61 instance ..
    62 end
    63 
    64 lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n"
    65   unfolding fps_plus_def by simp
    66 
    67 instantiation fps :: (minus) minus
    68 begin
    69 
    70 definition fps_minus_def:
    71   "op - = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
    72 
    73 instance ..
    74 end
    75 
    76 lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n"
    77   unfolding fps_minus_def by simp
    78 
    79 instantiation fps :: (uminus) uminus
    80 begin
    81 
    82 definition fps_uminus_def:
    83   "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))"
    84 
    85 instance ..
    86 end
    87 
    88 lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)"
    89   unfolding fps_uminus_def by simp
    90 
    91 instantiation fps :: ("{comm_monoid_add, times}")  times
    92 begin
    93 
    94 definition fps_times_def:
    95   "op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
    96 
    97 instance ..
    98 end
    99 
   100 lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n - i))"
   101   unfolding fps_times_def by simp
   102 
   103 declare atLeastAtMost_iff[presburger]
   104 declare Bex_def[presburger]
   105 declare Ball_def[presburger]
   106 
   107 lemma mult_delta_left:
   108   fixes x y :: "'a::mult_zero"
   109   shows "(if b then x else 0) * y = (if b then x * y else 0)"
   110   by simp
   111 
   112 lemma mult_delta_right:
   113   fixes x y :: "'a::mult_zero"
   114   shows "x * (if b then y else 0) = (if b then x * y else 0)"
   115   by simp
   116 
   117 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
   118   by auto
   119 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
   120   by auto
   121 
   122 subsection{* Formal power series form a commutative ring with unity, if the range of sequences
   123   they represent is a commutative ring with unity*}
   124 
   125 instance fps :: (semigroup_add) semigroup_add
   126 proof
   127   fix a b c :: "'a fps" show "a + b + c = a + (b + c)"
   128     by (simp add: fps_ext add_assoc)
   129 qed
   130 
   131 instance fps :: (ab_semigroup_add) ab_semigroup_add
   132 proof
   133   fix a b :: "'a fps" show "a + b = b + a"
   134     by (simp add: fps_ext add_commute)
   135 qed
   136 
   137 lemma fps_mult_assoc_lemma:
   138   fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   139   shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
   140          (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
   141 proof (induct k)
   142   case 0 show ?case by simp
   143 next
   144   case (Suc k) thus ?case
   145     by (simp add: Suc_diff_le setsum_addf add_assoc
   146              cong: strong_setsum_cong)
   147 qed
   148 
   149 instance fps :: (semiring_0) semigroup_mult
   150 proof
   151   fix a b c :: "'a fps"
   152   show "(a * b) * c = a * (b * c)"
   153   proof (rule fps_ext)
   154     fix n :: nat
   155     have "(\<Sum>j=0..n. \<Sum>i=0..j. a$i * b$(j - i) * c$(n - j)) =
   156           (\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))"
   157       by (rule fps_mult_assoc_lemma)
   158     thus "((a * b) * c) $ n = (a * (b * c)) $ n"
   159       by (simp add: fps_mult_nth setsum_right_distrib
   160                     setsum_left_distrib mult_assoc)
   161   qed
   162 qed
   163 
   164 lemma fps_mult_commute_lemma:
   165   fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
   166   shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
   167 proof (rule setsum_reindex_cong)
   168   show "inj_on (\<lambda>i. n - i) {0..n}"
   169     by (rule inj_onI) simp
   170   show "{0..n} = (\<lambda>i. n - i) ` {0..n}"
   171     by (auto, rule_tac x="n - x" in image_eqI, simp_all)
   172 next
   173   fix i assume "i \<in> {0..n}"
   174   hence "n - (n - i) = i" by simp
   175   thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
   176 qed
   177 
   178 instance fps :: (comm_semiring_0) ab_semigroup_mult
   179 proof
   180   fix a b :: "'a fps"
   181   show "a * b = b * a"
   182   proof (rule fps_ext)
   183     fix n :: nat
   184     have "(\<Sum>i=0..n. a$i * b$(n - i)) = (\<Sum>i=0..n. a$(n - i) * b$i)"
   185       by (rule fps_mult_commute_lemma)
   186     thus "(a * b) $ n = (b * a) $ n"
   187       by (simp add: fps_mult_nth mult_commute)
   188   qed
   189 qed
   190 
   191 instance fps :: (monoid_add) monoid_add
   192 proof
   193   fix a :: "'a fps" show "0 + a = a "
   194     by (simp add: fps_ext)
   195 next
   196   fix a :: "'a fps" show "a + 0 = a "
   197     by (simp add: fps_ext)
   198 qed
   199 
   200 instance fps :: (comm_monoid_add) comm_monoid_add
   201 proof
   202   fix a :: "'a fps" show "0 + a = a "
   203     by (simp add: fps_ext)
   204 qed
   205 
   206 instance fps :: (semiring_1) monoid_mult
   207 proof
   208   fix a :: "'a fps" show "1 * a = a"
   209     by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta)
   210 next
   211   fix a :: "'a fps" show "a * 1 = a"
   212     by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta')
   213 qed
   214 
   215 instance fps :: (cancel_semigroup_add) cancel_semigroup_add
   216 proof
   217   fix a b c :: "'a fps"
   218   assume "a + b = a + c" then show "b = c"
   219     by (simp add: expand_fps_eq)
   220 next
   221   fix a b c :: "'a fps"
   222   assume "b + a = c + a" then show "b = c"
   223     by (simp add: expand_fps_eq)
   224 qed
   225 
   226 instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
   227 proof
   228   fix a b c :: "'a fps"
   229   assume "a + b = a + c" then show "b = c"
   230     by (simp add: expand_fps_eq)
   231 qed
   232 
   233 instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
   234 
   235 instance fps :: (group_add) group_add
   236 proof
   237   fix a :: "'a fps" show "- a + a = 0"
   238     by (simp add: fps_ext)
   239 next
   240   fix a b :: "'a fps" show "a - b = a + - b"
   241     by (simp add: fps_ext diff_minus)
   242 qed
   243 
   244 instance fps :: (ab_group_add) ab_group_add
   245 proof
   246   fix a :: "'a fps"
   247   show "- a + a = 0"
   248     by (simp add: fps_ext)
   249 next
   250   fix a b :: "'a fps"
   251   show "a - b = a + - b"
   252     by (simp add: fps_ext)
   253 qed
   254 
   255 instance fps :: (zero_neq_one) zero_neq_one
   256   by default (simp add: expand_fps_eq)
   257 
   258 instance fps :: (semiring_0) semiring
   259 proof
   260   fix a b c :: "'a fps"
   261   show "(a + b) * c = a * c + b * c"
   262     by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf)
   263 next
   264   fix a b c :: "'a fps"
   265   show "a * (b + c) = a * b + a * c"
   266     by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf)
   267 qed
   268 
   269 instance fps :: (semiring_0) semiring_0
   270 proof
   271   fix a:: "'a fps" show "0 * a = 0"
   272     by (simp add: fps_ext fps_mult_nth)
   273 next
   274   fix a:: "'a fps" show "a * 0 = 0"
   275     by (simp add: fps_ext fps_mult_nth)
   276 qed
   277 
   278 instance fps :: (semiring_0_cancel) semiring_0_cancel ..
   279 
   280 subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
   281 
   282 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
   283   by (simp add: expand_fps_eq)
   284 
   285 lemma fps_nonzero_nth_minimal:
   286   "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0))"
   287 proof
   288   let ?n = "LEAST n. f $ n \<noteq> 0"
   289   assume "f \<noteq> 0"
   290   then have "\<exists>n. f $ n \<noteq> 0"
   291     by (simp add: fps_nonzero_nth)
   292   then have "f $ ?n \<noteq> 0"
   293     by (rule LeastI_ex)
   294   moreover have "\<forall>m<?n. f $ m = 0"
   295     by (auto dest: not_less_Least)
   296   ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" ..
   297   then show "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" ..
   298 next
   299   assume "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)"
   300   then show "f \<noteq> 0" by (auto simp add: expand_fps_eq)
   301 qed
   302 
   303 lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
   304   by (rule expand_fps_eq)
   305 
   306 lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S"
   307 proof (cases "finite S")
   308   assume "\<not> finite S" then show ?thesis by simp
   309 next
   310   assume "finite S"
   311   then show ?thesis by (induct set: finite) auto
   312 qed
   313 
   314 subsection{* Injection of the basic ring elements and multiplication by scalars *}
   315 
   316 definition
   317   "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
   318 
   319 lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)"
   320   unfolding fps_const_def by simp
   321 
   322 lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"
   323   by (simp add: fps_ext)
   324 
   325 lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
   326   by (simp add: fps_ext)
   327 
   328 lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
   329   by (simp add: fps_ext)
   330 
   331 lemma fps_const_add [simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
   332   by (simp add: fps_ext)
   333 lemma fps_const_sub [simp]: "fps_const (c::'a\<Colon>group_add) - fps_const d = fps_const (c - d)"
   334   by (simp add: fps_ext)
   335 lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
   336   by (simp add: fps_eq_iff fps_mult_nth setsum_0')
   337 
   338 lemma fps_const_add_left: "fps_const (c::'a\<Colon>monoid_add) + f = Abs_fps (\<lambda>n. if n = 0 then c + f$0 else f$n)"
   339   by (simp add: fps_ext)
   340 
   341 lemma fps_const_add_right: "f + fps_const (c::'a\<Colon>monoid_add) = Abs_fps (\<lambda>n. if n = 0 then f$0 + c else f$n)"
   342   by (simp add: fps_ext)
   343 
   344 lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"
   345   unfolding fps_eq_iff fps_mult_nth
   346   by (simp add: fps_const_def mult_delta_left setsum_delta)
   347 
   348 lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)"
   349   unfolding fps_eq_iff fps_mult_nth
   350   by (simp add: fps_const_def mult_delta_right setsum_delta')
   351 
   352 lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"
   353   by (simp add: fps_mult_nth mult_delta_left setsum_delta)
   354 
   355 lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
   356   by (simp add: fps_mult_nth mult_delta_right setsum_delta')
   357 
   358 subsection {* Formal power series form an integral domain*}
   359 
   360 instance fps :: (ring) ring ..
   361 
   362 instance fps :: (ring_1) ring_1
   363   by (intro_classes, auto simp add: diff_minus left_distrib)
   364 
   365 instance fps :: (comm_ring_1) comm_ring_1
   366   by (intro_classes, auto simp add: diff_minus left_distrib)
   367 
   368 instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
   369 proof
   370   fix a b :: "'a fps"
   371   assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
   372   then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0"
   373     and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal
   374     by blast+
   375   have "(a * b) $ (i+j) = (\<Sum>k=0..i+j. a$k * b$(i+j-k))"
   376     by (rule fps_mult_nth)
   377   also have "\<dots> = (a$i * b$(i+j-i)) + (\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k))"
   378     by (rule setsum_diff1') simp_all
   379   also have "(\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k)) = 0"
   380     proof (rule setsum_0' [rule_format])
   381       fix k assume "k \<in> {0..i+j} - {i}"
   382       then have "k < i \<or> i+j-k < j" by auto
   383       then show "a$k * b$(i+j-k) = 0" using i j by auto
   384     qed
   385   also have "a$i * b$(i+j-i) + 0 = a$i * b$j" by simp
   386   also have "a$i * b$j \<noteq> 0" using i j by simp
   387   finally have "(a*b) $ (i+j) \<noteq> 0" .
   388   then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
   389 qed
   390 
   391 instance fps :: (idom) idom ..
   392 
   393 instantiation fps :: (comm_ring_1) number_ring
   394 begin
   395 definition number_of_fps_def: "(number_of k::'a fps) = of_int k"
   396 
   397 instance proof
   398 qed (rule number_of_fps_def)
   399 end
   400 
   401 lemma number_of_fps_const: "(number_of k::('a::comm_ring_1) fps) = fps_const (of_int k)"
   402   
   403 proof(induct k rule: int_induct[where k=0])
   404   case base thus ?case unfolding number_of_fps_def of_int_0 by simp
   405 next
   406   case (step1 i) thus ?case unfolding number_of_fps_def 
   407     by (simp add: fps_const_add[symmetric] del: fps_const_add)
   408 next
   409   case (step2 i) thus ?case unfolding number_of_fps_def 
   410     by (simp add: fps_const_sub[symmetric] del: fps_const_sub)
   411 qed
   412 subsection{* The eXtractor series X*}
   413 
   414 lemma minus_one_power_iff: "(- (1::'a :: {comm_ring_1})) ^ n = (if even n then 1 else - 1)"
   415   by (induct n, auto)
   416 
   417 definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
   418 lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n - 1))"
   419 proof-
   420   {assume n: "n \<noteq> 0"
   421     have fN: "finite {0 .. n}" by simp
   422     have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n - i))" by (simp add: fps_mult_nth)
   423     also have "\<dots> = f $ (n - 1)"
   424       using n by (simp add: X_def mult_delta_left setsum_delta [OF fN])
   425   finally have ?thesis using n by simp }
   426   moreover
   427   {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
   428   ultimately show ?thesis by blast
   429 qed
   430 
   431 lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))"
   432   by (metis X_mult_nth mult_commute)
   433 
   434 lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
   435 proof(induct k)
   436   case 0 thus ?case by (simp add: X_def fps_eq_iff)
   437 next
   438   case (Suc k)
   439   {fix m
   440     have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))"
   441       by (simp add: power_Suc del: One_nat_def)
   442     then     have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"
   443       using Suc.hyps by (auto cong del: if_weak_cong)}
   444   then show ?case by (simp add: fps_eq_iff)
   445 qed
   446 
   447 lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n - k))"
   448   apply (induct k arbitrary: n)
   449   apply (simp)
   450   unfolding power_Suc mult_assoc
   451   by (case_tac n, auto)
   452 
   453 lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
   454   by (metis X_power_mult_nth mult_commute)
   455 
   456 
   457 
   458   
   459 subsection{* Formal Power series form a metric space *}
   460 
   461 definition (in dist) ball_def: "ball x r = {y. dist y x < r}"
   462 instantiation fps :: (comm_ring_1) dist
   463 begin
   464 
   465 definition dist_fps_def: "dist (a::'a fps) b = (if (\<exists>n. a$n \<noteq> b$n) then inverse (2 ^ The (leastP (\<lambda>n. a$n \<noteq> b$n))) else 0)"
   466 
   467 lemma dist_fps_ge0: "dist (a::'a fps) b \<ge> 0"
   468   by (simp add: dist_fps_def)
   469 
   470 lemma dist_fps_sym: "dist (a::'a fps) b = dist b a"
   471   apply (auto simp add: dist_fps_def)
   472   apply (rule cong[OF refl, where x="(\<lambda>n\<Colon>nat. a $ n \<noteq> b $ n)"])
   473   apply (rule ext)
   474   by auto
   475 instance ..
   476 end
   477 
   478 lemma fps_nonzero_least_unique: assumes a0: "a \<noteq> 0"
   479   shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> 0) n"
   480 proof-
   481   from fps_nonzero_nth_minimal[of a] a0
   482   obtain n where n: "a$n \<noteq> 0" "\<forall>m < n. a$m = 0" by blast
   483   from n have ln: "leastP (\<lambda>n. a$n \<noteq> 0) n" 
   484     by (auto simp add: leastP_def setge_def not_le[symmetric])
   485   moreover
   486   {fix m assume "leastP (\<lambda>n. a$n \<noteq> 0) m"
   487     then have "m = n" using ln
   488       apply (auto simp add: leastP_def setge_def)
   489       apply (erule allE[where x=n])
   490       apply (erule allE[where x=m])
   491       by simp}
   492   ultimately show ?thesis by blast
   493 qed
   494 
   495 lemma fps_eq_least_unique: assumes ab: "(a::('a::ab_group_add) fps) \<noteq> b"
   496   shows "\<exists>! n. leastP (\<lambda>n. a$n \<noteq> b$n) n"
   497 using fps_nonzero_least_unique[of "a - b"] ab
   498 by auto
   499 
   500 instantiation fps :: (comm_ring_1) metric_space
   501 begin
   502 
   503 definition open_fps_def: "open (S :: 'a fps set) = (\<forall>a \<in> S. \<exists>r. r >0 \<and> ball a r \<subseteq> S)"
   504 
   505 instance
   506 proof
   507   fix S :: "'a fps set" 
   508   show "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   509     by (auto simp add: open_fps_def ball_def subset_eq)
   510 next
   511 {  fix a b :: "'a fps"
   512   {assume ab: "a = b"
   513    then have "\<not> (\<exists>n. a$n \<noteq> b$n)" by simp
   514    then have "dist a b = 0" by (simp add: dist_fps_def)}
   515  moreover
   516  {assume d: "dist a b = 0"
   517    then have "\<forall>n. a$n = b$n" 
   518      by - (rule ccontr, simp add: dist_fps_def)
   519    then have "a = b" by (simp add: fps_eq_iff)}
   520  ultimately show "dist a b =0 \<longleftrightarrow> a = b" by blast}
   521 note th = this
   522 from th have th'[simp]: "\<And>a::'a fps. dist a a = 0" by simp
   523   fix a b c :: "'a fps"
   524   {assume ab: "a = b" then have d0: "dist a b = 0"  unfolding th .
   525     then have "dist a b \<le> dist a c + dist b c" 
   526       using dist_fps_ge0[of a c] dist_fps_ge0[of b c] by simp}
   527   moreover
   528   {assume c: "c = a \<or> c = b" then have "dist a b \<le> dist a c + dist b c"
   529       by (cases "c=a", simp_all add: th dist_fps_sym) }
   530   moreover
   531   {assume ab: "a \<noteq> b" and ac: "a \<noteq> c" and bc: "b \<noteq> c"
   532     let ?P = "\<lambda>a b n. a$n \<noteq> b$n"
   533     from fps_eq_least_unique[OF ab] fps_eq_least_unique[OF ac] 
   534       fps_eq_least_unique[OF bc]
   535     obtain nab nac nbc where nab: "leastP (?P a b) nab" 
   536       and nac: "leastP (?P a c) nac" 
   537       and nbc: "leastP (?P b c) nbc" by blast
   538     from nab have nab': "\<And>m. m < nab \<Longrightarrow> a$m = b$m" "a$nab \<noteq> b$nab"
   539       by (auto simp add: leastP_def setge_def)
   540     from nac have nac': "\<And>m. m < nac \<Longrightarrow> a$m = c$m" "a$nac \<noteq> c$nac"
   541       by (auto simp add: leastP_def setge_def)
   542     from nbc have nbc': "\<And>m. m < nbc \<Longrightarrow> b$m = c$m" "b$nbc \<noteq> c$nbc"
   543       by (auto simp add: leastP_def setge_def)
   544 
   545     have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a$n \<noteq> b$n)"
   546       by (simp add: fps_eq_iff)
   547     from ab ac bc nab nac nbc 
   548     have dab: "dist a b = inverse (2 ^ nab)" 
   549       and dac: "dist a c = inverse (2 ^ nac)" 
   550       and dbc: "dist b c = inverse (2 ^ nbc)"
   551       unfolding th0
   552       apply (simp_all add: dist_fps_def)
   553       apply (erule the1_equality[OF fps_eq_least_unique[OF ab]])
   554       apply (erule the1_equality[OF fps_eq_least_unique[OF ac]])
   555       by (erule the1_equality[OF fps_eq_least_unique[OF bc]])
   556     from ab ac bc have nz: "dist a b \<noteq> 0" "dist a c \<noteq> 0" "dist b c \<noteq> 0"
   557       unfolding th by simp_all
   558     from nz have pos: "dist a b > 0" "dist a c > 0" "dist b c > 0"
   559       using dist_fps_ge0[of a b] dist_fps_ge0[of a c] dist_fps_ge0[of b c] 
   560       by auto
   561     have th1: "\<And>n. (2::real)^n >0" by auto
   562     {assume h: "dist a b > dist a c + dist b c"
   563       then have gt: "dist a b > dist a c" "dist a b > dist b c"
   564         using pos by auto
   565       from gt have gtn: "nab < nbc" "nab < nac"
   566         unfolding dab dbc dac by (auto simp add: th1)
   567       from nac'(1)[OF gtn(2)] nbc'(1)[OF gtn(1)]
   568       have "a$nab = b$nab" by simp
   569       with nab'(2) have False  by simp}
   570     then have "dist a b \<le> dist a c + dist b c"
   571       by (auto simp add: not_le[symmetric]) }
   572   ultimately show "dist a b \<le> dist a c + dist b c" by blast
   573 qed
   574   
   575 end
   576 
   577 text{* The infinite sums and justification of the notation in textbooks*}
   578 
   579 lemma reals_power_lt_ex: assumes xp: "x > 0" and y1: "(y::real) > 1"
   580   shows "\<exists>k>0. (1/y)^k < x"
   581 proof-
   582   have yp: "y > 0" using y1 by simp
   583   from reals_Archimedean2[of "max 0 (- log y x) + 1"]
   584   obtain k::nat where k: "real k > max 0 (- log y x) + 1" by blast
   585   from k have kp: "k > 0" by simp
   586   from k have "real k > - log y x" by simp
   587   then have "ln y * real k > - ln x" unfolding log_def
   588     using ln_gt_zero_iff[OF yp] y1
   589     by (simp add: minus_divide_left field_simps del:minus_divide_left[symmetric] )
   590   then have "ln y * real k + ln x > 0" by simp
   591   then have "exp (real k * ln y + ln x) > exp 0"
   592     by (simp add: mult_ac)
   593   then have "y ^ k * x > 1"
   594     unfolding exp_zero exp_add exp_real_of_nat_mult
   595     exp_ln[OF xp] exp_ln[OF yp] by simp
   596   then have "x > (1/y)^k" using yp 
   597     by (simp add: field_simps nonzero_power_divide )
   598   then show ?thesis using kp by blast
   599 qed
   600 lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
   601 lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"
   602   by (simp add: X_power_iff)
   603  
   604 
   605 lemma fps_sum_rep_nth: "(setsum (%i. fps_const(a$i)*X^i) {0..m})$n = (if n \<le> m then a$n else (0::'a::comm_ring_1))"
   606   apply (auto simp add: fps_eq_iff fps_setsum_nth X_power_nth cond_application_beta cond_value_iff  cong del: if_weak_cong)
   607   by (simp add: setsum_delta')
   608   
   609 lemma fps_notation: 
   610   "(%n. setsum (%i. fps_const(a$i) * X^i) {0..n}) ----> a" (is "?s ----> a")
   611 proof-
   612     {fix r:: real
   613       assume rp: "r > 0"
   614       have th0: "(2::real) > 1" by simp
   615       from reals_power_lt_ex[OF rp th0] 
   616       obtain n0 where n0: "(1/2)^n0 < r" "n0 > 0" by blast
   617       {fix n::nat
   618         assume nn0: "n \<ge> n0"
   619         then have thnn0: "(1/2)^n <= (1/2 :: real)^n0"
   620           by (auto intro: power_decreasing)
   621         {assume "?s n = a" then have "dist (?s n) a < r" 
   622             unfolding dist_eq_0_iff[of "?s n" a, symmetric]
   623             using rp by (simp del: dist_eq_0_iff)}
   624         moreover
   625         {assume neq: "?s n \<noteq> a"
   626           from fps_eq_least_unique[OF neq] 
   627           obtain k where k: "leastP (\<lambda>i. ?s n $ i \<noteq> a$i) k" by blast
   628           have th0: "\<And>(a::'a fps) b. a \<noteq> b \<longleftrightarrow> (\<exists>n. a$n \<noteq> b$n)"
   629             by (simp add: fps_eq_iff)
   630           from neq have dth: "dist (?s n) a = (1/2)^k"
   631             unfolding th0 dist_fps_def
   632             unfolding the1_equality[OF fps_eq_least_unique[OF neq], OF k]
   633             by (auto simp add: inverse_eq_divide power_divide)
   634 
   635           from k have kn: "k > n"
   636             by (simp add: leastP_def setge_def fps_sum_rep_nth split:split_if_asm)
   637           then have "dist (?s n) a < (1/2)^n" unfolding dth
   638             by (auto intro: power_strict_decreasing)
   639           also have "\<dots> <= (1/2)^n0" using nn0
   640             by (auto intro: power_decreasing)
   641           also have "\<dots> < r" using n0 by simp
   642           finally have "dist (?s n) a < r" .}
   643         ultimately have "dist (?s n) a < r" by blast}
   644       then have "\<exists>n0. \<forall> n \<ge> n0. dist (?s n) a < r " by blast}
   645     then show ?thesis  unfolding  LIMSEQ_def by blast
   646   qed
   647 
   648 subsection{* Inverses of formal power series *}
   649 
   650 declare setsum_cong[fundef_cong]
   651 
   652 
   653 instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse
   654 begin
   655 
   656 fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where
   657   "natfun_inverse f 0 = inverse (f$0)"
   658 | "natfun_inverse f n = - inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}"
   659 
   660 definition fps_inverse_def:
   661   "inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))"
   662 definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
   663 instance ..
   664 end
   665 
   666 lemma fps_inverse_zero[simp]:
   667   "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
   668   by (simp add: fps_ext fps_inverse_def)
   669 
   670 lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
   671   apply (auto simp add: expand_fps_eq fps_inverse_def)
   672   by (case_tac n, auto)
   673 
   674 instance fps :: ("{comm_monoid_add,inverse, times, uminus}")  division_by_zero
   675   by default (rule fps_inverse_zero)
   676 
   677 lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
   678   shows "inverse f * f = 1"
   679 proof-
   680   have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
   681   from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n"
   682     by (simp add: fps_inverse_def)
   683   from f0 have th0: "(inverse f * f) $ 0 = 1"
   684     by (simp add: fps_mult_nth fps_inverse_def)
   685   {fix n::nat assume np: "n >0 "
   686     from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
   687     have d: "{0} \<inter> {1 .. n} = {}" by auto
   688     have f: "finite {0::nat}" "finite {1..n}" by auto
   689     from f0 np have th0: "- (inverse f$n) =
   690       (setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)"
   691       by (cases n, simp, simp add: divide_inverse fps_inverse_def)
   692     from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
   693     have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n} =
   694       - (f$0) * (inverse f)$n"
   695       by (simp add: ring_simps)
   696     have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n - i))"
   697       unfolding fps_mult_nth ifn ..
   698     also have "\<dots> = f$0 * natfun_inverse f n
   699       + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
   700       unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
   701       by simp
   702     also have "\<dots> = 0" unfolding th1 ifn by simp
   703     finally have "(inverse f * f)$n = 0" unfolding c . }
   704   with th0 show ?thesis by (simp add: fps_eq_iff)
   705 qed
   706 
   707 lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"
   708   by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)
   709 
   710 lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0"
   711 proof-
   712   {assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
   713   moreover
   714   {assume h: "inverse f = 0" and c: "f $0 \<noteq> 0"
   715     from inverse_mult_eq_1[OF c] h have False by simp}
   716   ultimately show ?thesis by blast
   717 qed
   718 
   719 lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
   720   shows "inverse (inverse f) = f"
   721 proof-
   722   from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp
   723   from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
   724   have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
   725   then show ?thesis using f0 unfolding mult_cancel_left by simp
   726 qed
   727 
   728 lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1"
   729   shows "inverse f = g"
   730 proof-
   731   from inverse_mult_eq_1[OF f0] fg
   732   have th0: "inverse f * f = g * f" by (simp add: mult_ac)
   733   then show ?thesis using f0  unfolding mult_cancel_right
   734     by (auto simp add: expand_fps_eq)
   735 qed
   736 
   737 lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
   738   = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
   739   apply (rule fps_inverse_unique)
   740   apply simp
   741   apply (simp add: fps_eq_iff fps_mult_nth)
   742 proof(clarsimp)
   743   fix n::nat assume n: "n > 0"
   744   let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
   745   let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
   746   let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
   747   have th1: "setsum ?f {0..n} = setsum ?g {0..n}"
   748     by (rule setsum_cong2) auto
   749   have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
   750     using n apply - by (rule setsum_cong2) auto
   751   have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
   752   from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto
   753   have f: "finite {0.. n - 1}" "finite {n}" by auto
   754   show "setsum ?f {0..n} = 0"
   755     unfolding th1
   756     apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
   757     unfolding th2
   758     by(simp add: setsum_delta)
   759 qed
   760 
   761 subsection{* Formal Derivatives, and the MacLaurin theorem around 0*}
   762 
   763 definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
   764 
   765 lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def)
   766 
   767 lemma fps_deriv_linear[simp]: "fps_deriv (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g"
   768   unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: ring_simps)
   769 
   770 lemma fps_deriv_mult[simp]:
   771   fixes f :: "('a :: comm_ring_1) fps"
   772   shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
   773 proof-
   774   let ?D = "fps_deriv"
   775   {fix n::nat
   776     let ?Zn = "{0 ..n}"
   777     let ?Zn1 = "{0 .. n + 1}"
   778     let ?f = "\<lambda>i. i + 1"
   779     have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
   780     have eq: "{1.. n+1} = ?f ` {0..n}" by auto
   781     let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n - i) +
   782         of_nat (i+1)* f $ (i+1) * g $ (n - i)"
   783     let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1) - i) +
   784         of_nat i* f $ i * g $ ((n + 1) - i)"
   785     {fix k assume k: "k \<in> {0..n}"
   786       have "?h (k + 1) = ?g k" using k by auto}
   787     note th0 = this
   788     have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
   789     have s0: "setsum (\<lambda>i. of_nat i * f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. of_nat (n + 1 - i) * f $ (n + 1 - i) * g $ i) ?Zn1"
   790       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
   791       apply (simp add: inj_on_def Ball_def)
   792       apply presburger
   793       apply (rule set_ext)
   794       apply (presburger add: image_iff)
   795       by simp
   796     have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
   797       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
   798       apply (simp add: inj_on_def Ball_def)
   799       apply presburger
   800       apply (rule set_ext)
   801       apply (presburger add: image_iff)
   802       by simp
   803     have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute)
   804     also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
   805       by (simp add: fps_mult_nth setsum_addf[symmetric])
   806     also have "\<dots> = setsum ?h {1..n+1}"
   807       using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
   808     also have "\<dots> = setsum ?h {0..n+1}"
   809       apply (rule setsum_mono_zero_left)
   810       apply simp
   811       apply (simp add: subset_eq)
   812       unfolding eq'
   813       by simp
   814     also have "\<dots> = (fps_deriv (f * g)) $ n"
   815       apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
   816       unfolding s0 s1
   817       unfolding setsum_addf[symmetric] setsum_right_distrib
   818       apply (rule setsum_cong2)
   819       by (auto simp add: of_nat_diff ring_simps)
   820     finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .}
   821   then show ?thesis unfolding fps_eq_iff by auto
   822 qed
   823 
   824 lemma fps_deriv_X[simp]: "fps_deriv X = 1"
   825   by (simp add: fps_deriv_def X_def fps_eq_iff)
   826 
   827 lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
   828   by (simp add: fps_eq_iff fps_deriv_def)
   829 lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
   830   using fps_deriv_linear[of 1 f 1 g] by simp
   831 
   832 lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
   833   unfolding diff_minus by simp
   834 
   835 lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
   836   by (simp add: fps_ext fps_deriv_def fps_const_def)
   837 
   838 lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
   839   by simp
   840 
   841 lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
   842   by (simp add: fps_deriv_def fps_eq_iff)
   843 
   844 lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
   845   by (simp add: fps_deriv_def fps_eq_iff )
   846 
   847 lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
   848   by simp
   849 
   850 lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
   851 proof-
   852   {assume "\<not> finite S" hence ?thesis by simp}
   853   moreover
   854   {assume fS: "finite S"
   855     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
   856   ultimately show ?thesis by blast
   857 qed
   858 
   859 lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))"
   860 proof-
   861   {assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
   862     hence "fps_deriv f = 0" by simp }
   863   moreover
   864   {assume z: "fps_deriv f = 0"
   865     hence "\<forall>n. (fps_deriv f)$n = 0" by simp
   866     hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
   867     hence "f = fps_const (f$0)"
   868       apply (clarsimp simp add: fps_eq_iff fps_const_def)
   869       apply (erule_tac x="n - 1" in allE)
   870       by simp}
   871   ultimately show ?thesis by blast
   872 qed
   873 
   874 lemma fps_deriv_eq_iff:
   875   fixes f:: "('a::{idom,semiring_char_0}) fps"
   876   shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
   877 proof-
   878   have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
   879   also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)$0)" unfolding fps_deriv_eq_0_iff ..
   880   finally show ?thesis by (simp add: ring_simps)
   881 qed
   882 
   883 lemma fps_deriv_eq_iff_ex: "(fps_deriv f = fps_deriv g) \<longleftrightarrow> (\<exists>(c::'a::{idom,semiring_char_0}). f = fps_const c + g)"
   884   apply auto unfolding fps_deriv_eq_iff by blast
   885 
   886 
   887 fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
   888   "fps_nth_deriv 0 f = f"
   889 | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
   890 
   891 lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
   892   by (induct n arbitrary: f, auto)
   893 
   894 lemma fps_nth_deriv_linear[simp]: "fps_nth_deriv n (fps_const (a::'a::comm_semiring_1) * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
   895   by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)
   896 
   897 lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
   898   by (induct n arbitrary: f, simp_all)
   899 
   900 lemma fps_nth_deriv_add[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) + g) = fps_nth_deriv n f + fps_nth_deriv n g"
   901   using fps_nth_deriv_linear[of n 1 f 1 g] by simp
   902 
   903 lemma fps_nth_deriv_sub[simp]: "fps_nth_deriv n ((f:: ('a::comm_ring_1) fps) - g) = fps_nth_deriv n f - fps_nth_deriv n g"
   904   unfolding diff_minus fps_nth_deriv_add by simp
   905 
   906 lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
   907   by (induct n, simp_all )
   908 
   909 lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
   910   by (induct n, simp_all )
   911 
   912 lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
   913   by (cases n, simp_all)
   914 
   915 lemma fps_nth_deriv_mult_const_left[simp]: "fps_nth_deriv n (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_nth_deriv n f"
   916   using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
   917 
   918 lemma fps_nth_deriv_mult_const_right[simp]: "fps_nth_deriv n (f * fps_const (c::'a::comm_ring_1)) = fps_nth_deriv n f * fps_const c"
   919   using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
   920 
   921 lemma fps_nth_deriv_setsum: "fps_nth_deriv n (setsum f S) = setsum (\<lambda>i. fps_nth_deriv n (f i :: ('a::comm_ring_1) fps)) S"
   922 proof-
   923   {assume "\<not> finite S" hence ?thesis by simp}
   924   moreover
   925   {assume fS: "finite S"
   926     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
   927   ultimately show ?thesis by blast
   928 qed
   929 
   930 lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"
   931   by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)
   932 
   933 subsection {* Powers*}
   934 
   935 lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
   936   by (induct n, auto simp add: expand_fps_eq fps_mult_nth)
   937 
   938 lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
   939 proof(induct n)
   940   case 0 thus ?case by simp
   941 next
   942   case (Suc n)
   943   note h = Suc.hyps[OF `a$0 = 1`]
   944   show ?case unfolding power_Suc fps_mult_nth
   945     using h `a$0 = 1`  fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps)
   946 qed
   947 
   948 lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
   949   by (induct n, auto simp add: fps_mult_nth)
   950 
   951 lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
   952   by (induct n, auto simp add: fps_mult_nth)
   953 
   954 lemma startsby_power:"a $0 = (v::'a::{comm_ring_1}) \<Longrightarrow> a^n $0 = v^n"
   955   by (induct n, auto simp add: fps_mult_nth power_Suc)
   956 
   957 lemma startsby_zero_power_iff[simp]:
   958   "a^n $0 = (0::'a::{idom}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"
   959 apply (rule iffI)
   960 apply (induct n, auto simp add: power_Suc fps_mult_nth)
   961 by (rule startsby_zero_power, simp_all)
   962 
   963 lemma startsby_zero_power_prefix:
   964   assumes a0: "a $0 = (0::'a::idom)"
   965   shows "\<forall>n < k. a ^ k $ n = 0"
   966   using a0
   967 proof(induct k rule: nat_less_induct)
   968   fix k assume H: "\<forall>m<k. a $0 =  0 \<longrightarrow> (\<forall>n<m. a ^ m $ n = 0)" and a0: "a $0 = (0\<Colon>'a)"
   969   let ?ths = "\<forall>m<k. a ^ k $ m = 0"
   970   {assume "k = 0" then have ?ths by simp}
   971   moreover
   972   {fix l assume k: "k = Suc l"
   973     {fix m assume mk: "m < k"
   974       {assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0
   975           by simp}
   976       moreover
   977       {assume m0: "m \<noteq> 0"
   978         have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute)
   979         also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
   980         also have "\<dots> = 0" apply (rule setsum_0')
   981           apply auto
   982           apply (case_tac "aa = m")
   983           using a0
   984           apply simp
   985           apply (rule H[rule_format])
   986           using a0 k mk by auto
   987         finally have "a^k $ m = 0" .}
   988     ultimately have "a^k $ m = 0" by blast}
   989     hence ?ths by blast}
   990   ultimately show ?ths by (cases k, auto)
   991 qed
   992 
   993 lemma startsby_zero_setsum_depends:
   994   assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k"
   995   shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}"
   996   apply (rule setsum_mono_zero_right)
   997   using kn apply auto
   998   apply (rule startsby_zero_power_prefix[rule_format, OF a0])
   999   by arith
  1000 
  1001 lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{idom})"
  1002   shows "a^n $ n = (a$1) ^ n"
  1003 proof(induct n)
  1004   case 0 thus ?case by (simp add: power_0)
  1005 next
  1006   case (Suc n)
  1007   have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc)
  1008   also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
  1009   also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
  1010     apply (rule setsum_mono_zero_right)
  1011     apply simp
  1012     apply clarsimp
  1013     apply clarsimp
  1014     apply (rule startsby_zero_power_prefix[rule_format, OF a0])
  1015     apply arith
  1016     done
  1017   also have "\<dots> = a^n $ n * a$1" using a0 by simp
  1018   finally show ?case using Suc.hyps by (simp add: power_Suc)
  1019 qed
  1020 
  1021 lemma fps_inverse_power:
  1022   fixes a :: "('a::{field}) fps"
  1023   shows "inverse (a^n) = inverse a ^ n"
  1024 proof-
  1025   {assume a0: "a$0 = 0"
  1026     hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
  1027     {assume "n = 0" hence ?thesis by simp}
  1028     moreover
  1029     {assume n: "n > 0"
  1030       from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
  1031         by (simp add: fps_inverse_def)}
  1032     ultimately have ?thesis by blast}
  1033   moreover
  1034   {assume a0: "a$0 \<noteq> 0"
  1035     have ?thesis
  1036       apply (rule fps_inverse_unique)
  1037       apply (simp add: a0)
  1038       unfolding power_mult_distrib[symmetric]
  1039       apply (rule ssubst[where t = "a * inverse a" and s= 1])
  1040       apply simp_all
  1041       apply (subst mult_commute)
  1042       by (rule inverse_mult_eq_1[OF a0])}
  1043   ultimately show ?thesis by blast
  1044 qed
  1045 
  1046 lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
  1047   apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)
  1048   by (case_tac n, auto simp add: power_Suc ring_simps)
  1049 
  1050 lemma fps_inverse_deriv:
  1051   fixes a:: "('a :: field) fps"
  1052   assumes a0: "a$0 \<noteq> 0"
  1053   shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
  1054 proof-
  1055   from inverse_mult_eq_1[OF a0]
  1056   have "fps_deriv (inverse a * a) = 0" by simp
  1057   hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
  1058   hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
  1059   with inverse_mult_eq_1[OF a0]
  1060   have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
  1061     unfolding power2_eq_square
  1062     apply (simp add: ring_simps)
  1063     by (simp add: mult_assoc[symmetric])
  1064   hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
  1065     by simp
  1066   then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)
  1067 qed
  1068 
  1069 lemma fps_inverse_mult:
  1070   fixes a::"('a :: field) fps"
  1071   shows "inverse (a * b) = inverse a * inverse b"
  1072 proof-
  1073   {assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
  1074     from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
  1075     have ?thesis unfolding th by simp}
  1076   moreover
  1077   {assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
  1078     from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
  1079     have ?thesis unfolding th by simp}
  1080   moreover
  1081   {assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"
  1082     from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
  1083     from inverse_mult_eq_1[OF ab0]
  1084     have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
  1085     then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
  1086       by (simp add: ring_simps)
  1087     then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
  1088 ultimately show ?thesis by blast
  1089 qed
  1090 
  1091 lemma fps_inverse_deriv':
  1092   fixes a:: "('a :: field) fps"
  1093   assumes a0: "a$0 \<noteq> 0"
  1094   shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
  1095   using fps_inverse_deriv[OF a0]
  1096   unfolding power2_eq_square fps_divide_def
  1097     fps_inverse_mult by simp
  1098 
  1099 lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)"
  1100   shows "f * inverse f= 1"
  1101   by (metis mult_commute inverse_mult_eq_1 f0)
  1102 
  1103 lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
  1104   assumes a0: "b$0 \<noteq> 0"
  1105   shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
  1106   using fps_inverse_deriv[OF a0]
  1107   by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
  1108 
  1109 
  1110 lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
  1111   = 1 - X"
  1112   by (simp add: fps_inverse_gp fps_eq_iff X_def)
  1113 
  1114 lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
  1115   by (cases "n", simp_all)
  1116 
  1117 
  1118 lemma fps_inverse_X_plus1:
  1119   "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{field})) ^ n)" (is "_ = ?r")
  1120 proof-
  1121   have eq: "(1 + X) * ?r = 1"
  1122     unfolding minus_one_power_iff
  1123     by (auto simp add: ring_simps fps_eq_iff)
  1124   show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
  1125 qed
  1126 
  1127 
  1128 subsection{* Integration *}
  1129 
  1130 definition
  1131   fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps" where
  1132   "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
  1133 
  1134 lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
  1135   unfolding fps_integral_def fps_deriv_def
  1136   by (simp add: fps_eq_iff del: of_nat_Suc)
  1137 
  1138 lemma fps_integral_linear:
  1139   "fps_integral (fps_const a * f + fps_const b * g) (a*a0 + b*b0) =
  1140     fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0"
  1141   (is "?l = ?r")
  1142 proof-
  1143   have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
  1144   moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
  1145   ultimately show ?thesis
  1146     unfolding fps_deriv_eq_iff by auto
  1147 qed
  1148 
  1149 subsection {* Composition of FPSs *}
  1150 definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
  1151   fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"
  1152 
  1153 lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def)
  1154 
  1155 lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
  1156   by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta')
  1157 
  1158 lemma fps_const_compose[simp]:
  1159   "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
  1160   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
  1161 
  1162 lemma number_of_compose[simp]: "(number_of k::('a::{comm_ring_1}) fps) oo b = number_of k"
  1163   unfolding number_of_fps_const by simp
  1164 
  1165 lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
  1166   by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta
  1167                 power_Suc not_le)
  1168 
  1169 
  1170 subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
  1171 
  1172 subsubsection {* Rule 1 *}
  1173   (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
  1174 
  1175 lemma fps_power_mult_eq_shift:
  1176   "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: comm_ring_1) * X^i) {0 .. k}" (is "?lhs = ?rhs")
  1177 proof-
  1178   {fix n:: nat
  1179     have "?lhs $ n = (if n < Suc k then 0 else a n)"
  1180       unfolding X_power_mult_nth by auto
  1181     also have "\<dots> = ?rhs $ n"
  1182     proof(induct k)
  1183       case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
  1184     next
  1185       case (Suc k)
  1186       note th = Suc.hyps[symmetric]
  1187       have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
  1188       also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
  1189         using th
  1190         unfolding fps_sub_nth by simp
  1191       also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
  1192         unfolding X_power_mult_right_nth
  1193         apply (auto simp add: not_less fps_const_def)
  1194         apply (rule cong[of a a, OF refl])
  1195         by arith
  1196       finally show ?case by simp
  1197     qed
  1198     finally have "?lhs $ n = ?rhs $ n"  .}
  1199   then show ?thesis by (simp add: fps_eq_iff)
  1200 qed
  1201 
  1202 subsubsection{* Rule 2*}
  1203 
  1204   (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
  1205   (* If f reprents {a_n} and P is a polynomial, then
  1206         P(xD) f represents {P(n) a_n}*)
  1207 
  1208 definition "XD = op * X o fps_deriv"
  1209 
  1210 lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
  1211   by (simp add: XD_def ring_simps)
  1212 
  1213 lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
  1214   by (simp add: XD_def ring_simps)
  1215 
  1216 lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) = fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)"
  1217   by simp
  1218 
  1219 lemma XDN_linear:
  1220   "(XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)"
  1221   by (induct n, simp_all)
  1222 
  1223 lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
  1224 
  1225 
  1226 lemma fps_mult_XD_shift:
  1227   "(XD ^^ k) (a:: ('a::{comm_ring_1}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
  1228   by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
  1229 
  1230 subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
  1231 subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
  1232 
  1233 lemma fps_divide_X_minus1_setsum_lemma:
  1234   "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
  1235 proof-
  1236   let ?X = "X::('a::comm_ring_1) fps"
  1237   let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
  1238   have th0: "\<And>i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
  1239   {fix n:: nat
  1240     {assume "n=0" hence "a$n = ((1 - ?X) * ?sa) $ n"
  1241         by (simp add: fps_mult_nth)}
  1242     moreover
  1243     {assume n0: "n \<noteq> 0"
  1244       then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
  1245         "{0..n - 1}\<union>{n} = {0..n}"
  1246         by (auto simp: expand_set_eq)
  1247       have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}"
  1248         "{0..n - 1}\<inter>{n} ={}" using n0 by simp_all
  1249       have f: "finite {0}" "finite {1}" "finite {2 .. n}"
  1250         "finite {0 .. n - 1}" "finite {n}" by simp_all
  1251     have "((1 - ?X) * ?sa) $ n = setsum (\<lambda>i. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}"
  1252       by (simp add: fps_mult_nth)
  1253     also have "\<dots> = a$n" unfolding th0
  1254       unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
  1255       unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
  1256       apply (simp)
  1257       unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
  1258       by simp
  1259     finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp}
  1260   ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast}
  1261 then show ?thesis
  1262   unfolding fps_eq_iff by blast
  1263 qed
  1264 
  1265 lemma fps_divide_X_minus1_setsum:
  1266   "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
  1267 proof-
  1268   let ?X = "1 - (X::('a::field) fps)"
  1269   have th0: "?X $ 0 \<noteq> 0" by simp
  1270   have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X"
  1271     using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
  1272     by (simp add: fps_divide_def mult_assoc)
  1273   also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) "
  1274     by (simp add: mult_ac)
  1275   finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
  1276 qed
  1277 
  1278 subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
  1279   finite product of FPS, also the relvant instance of powers of a FPS*}
  1280 
  1281 definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
  1282 
  1283 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
  1284   apply (auto simp add: natpermute_def)
  1285   apply (case_tac x, auto)
  1286   done
  1287 
  1288 lemma foldl_add_start0:
  1289   "foldl op + x xs = x + foldl op + (0::nat) xs"
  1290   apply (induct xs arbitrary: x)
  1291   apply simp
  1292   unfolding foldl.simps
  1293   apply atomize
  1294   apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
  1295   apply (erule_tac x="x + a" in allE)
  1296   apply (erule_tac x="a" in allE)
  1297   apply simp
  1298   apply assumption
  1299   done
  1300 
  1301 lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
  1302   apply (induct ys arbitrary: x xs)
  1303   apply auto
  1304   apply (subst (2) foldl_add_start0)
  1305   apply simp
  1306   apply (subst (2) foldl_add_start0)
  1307   by simp
  1308 
  1309 lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
  1310 proof(induct xs arbitrary: x)
  1311   case Nil thus ?case by simp
  1312 next
  1313   case (Cons a as x)
  1314   have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
  1315     apply (rule setsum_reindex_cong [where f=Suc])
  1316     by (simp_all add: inj_on_def)
  1317   have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
  1318   have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
  1319   have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
  1320   have "foldl op + x (a#as) = x + foldl op + a as "
  1321     apply (subst foldl_add_start0)    by simp
  1322   also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
  1323   also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
  1324     unfolding eq[symmetric]
  1325     unfolding setsum_Un_disjoint[OF f d, unfolded seq]
  1326     by simp
  1327   finally show ?case  .
  1328 qed
  1329 
  1330 
  1331 lemma append_natpermute_less_eq:
  1332   assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
  1333 proof-
  1334   {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
  1335     hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
  1336   note th = this
  1337   {from th show "foldl op + 0 xs \<le> n" by simp}
  1338   {from th show "foldl op + 0 ys \<le> n" by simp}
  1339 qed
  1340 
  1341 lemma natpermute_split:
  1342   assumes mn: "h \<le> k"
  1343   shows "natpermute n k = (\<Union>m \<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})" (is "?L = ?R" is "?L = (\<Union>m \<in>{0..n}. ?S m)")
  1344 proof-
  1345   {fix l assume l: "l \<in> ?R"
  1346     from l obtain m xs ys where h: "m \<in> {0..n}" and xs: "xs \<in> natpermute m h" and ys: "ys \<in> natpermute (n - m) (k - h)"  and leq: "l = xs@ys" by blast
  1347     from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
  1348     from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
  1349     have "l \<in> ?L" using leq xs ys h
  1350       apply simp
  1351       apply (clarsimp simp add: natpermute_def simp del: foldl_append)
  1352       apply (simp add: foldl_add_append[unfolded foldl_append])
  1353       unfolding xs' ys'
  1354       using mn xs ys
  1355       unfolding natpermute_def by simp}
  1356   moreover
  1357   {fix l assume l: "l \<in> natpermute n k"
  1358     let ?xs = "take h l"
  1359     let ?ys = "drop h l"
  1360     let ?m = "foldl op + 0 ?xs"
  1361     from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
  1362     have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)
  1363     have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
  1364       by (simp add: natpermute_def)
  1365     from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
  1366     from xs ys ls have "l \<in> ?R"
  1367       apply auto
  1368       apply (rule bexI[where x = "?m"])
  1369       apply (rule exI[where x = "?xs"])
  1370       apply (rule exI[where x = "?ys"])
  1371       using ls l unfolding foldl_add_append
  1372       by (auto simp add: natpermute_def)}
  1373   ultimately show ?thesis by blast
  1374 qed
  1375 
  1376 lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
  1377   by (auto simp add: natpermute_def)
  1378 lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
  1379   apply (auto simp add: set_replicate_conv_if natpermute_def)
  1380   apply (rule nth_equalityI)
  1381   by simp_all
  1382 
  1383 lemma natpermute_finite: "finite (natpermute n k)"
  1384 proof(induct k arbitrary: n)
  1385   case 0 thus ?case
  1386     apply (subst natpermute_split[of 0 0, simplified])
  1387     by (simp add: natpermute_0)
  1388 next
  1389   case (Suc k)
  1390   then show ?case unfolding natpermute_split[of k "Suc k", simplified]
  1391     apply -
  1392     apply (rule finite_UN_I)
  1393     apply simp
  1394     unfolding One_nat_def[symmetric] natlist_trivial_1
  1395     apply simp
  1396     done
  1397 qed
  1398 
  1399 lemma natpermute_contain_maximal:
  1400   "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
  1401   (is "?A = ?B")
  1402 proof-
  1403   {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
  1404     from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
  1405       unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
  1406     have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
  1407     have f: "finite({0..k} - {i})" "finite {i}" by auto
  1408     have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
  1409     from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
  1410       unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
  1411     also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
  1412       unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
  1413     finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
  1414     from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
  1415     from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
  1416       unfolding length_replicate  by arith+
  1417     have "xs = replicate (k+1) 0 [i := n]"
  1418       apply (rule nth_equalityI)
  1419       unfolding xsl length_list_update length_replicate
  1420       apply simp
  1421       apply clarify
  1422       unfolding nth_list_update[OF i'(1)]
  1423       using i zxs
  1424       by (case_tac "ia=i", auto simp del: replicate.simps)
  1425     then have "xs \<in> ?B" using i by blast}
  1426   moreover
  1427   {fix i assume i: "i \<in> {0..k}"
  1428     let ?xs = "replicate (k+1) 0 [i:=n]"
  1429     have nxs: "n \<in> set ?xs"
  1430       apply (rule set_update_memI) using i by simp
  1431     have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
  1432     have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
  1433       unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
  1434     also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
  1435       apply (rule setsum_cong2) by (simp del: replicate.simps)
  1436     also have "\<dots> = n" using i by (simp add: setsum_delta)
  1437     finally
  1438     have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def
  1439       by blast
  1440     then have "?xs \<in> ?A"  using nxs  by blast}
  1441   ultimately show ?thesis by auto
  1442 qed
  1443 
  1444     (* The general form *)
  1445 lemma fps_setprod_nth:
  1446   fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
  1447   shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
  1448   (is "?P m n")
  1449 proof(induct m arbitrary: n rule: nat_less_induct)
  1450   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
  1451   {assume m0: "m = 0"
  1452     hence "?P m n" apply simp
  1453       unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
  1454   moreover
  1455   {fix k assume k: "m = Suc k"
  1456     have km: "k < m" using k by arith
  1457     have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger
  1458     have f0: "finite {0 .. k}" "finite {m}" by auto
  1459     have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
  1460     have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
  1461       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
  1462     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))"
  1463       unfolding fps_mult_nth H[rule_format, OF km] ..
  1464     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
  1465       apply (simp add: k)
  1466       unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
  1467       apply (subst setsum_UN_disjoint)
  1468       apply simp
  1469       apply simp
  1470       unfolding image_Collect[symmetric]
  1471       apply clarsimp
  1472       apply (rule finite_imageI)
  1473       apply (rule natpermute_finite)
  1474       apply (clarsimp simp add: expand_set_eq)
  1475       apply auto
  1476       apply (rule setsum_cong2)
  1477       unfolding setsum_left_distrib
  1478       apply (rule sym)
  1479       apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
  1480       apply (simp add: inj_on_def)
  1481       apply auto
  1482       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
  1483       apply (clarsimp simp add: natpermute_def nth_append)
  1484       done
  1485     finally have "?P m n" .}
  1486   ultimately show "?P m n " by (cases m, auto)
  1487 qed
  1488 
  1489 text{* The special form for powers *}
  1490 lemma fps_power_nth_Suc:
  1491   fixes m :: nat and a :: "('a::comm_ring_1) fps"
  1492   shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
  1493 proof-
  1494   have f: "finite {0 ..m}" by simp
  1495   have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
  1496   show ?thesis unfolding th0 fps_setprod_nth ..
  1497 qed
  1498 lemma fps_power_nth:
  1499   fixes m :: nat and a :: "('a::comm_ring_1) fps"
  1500   shows "(a ^m)$n = (if m=0 then 1$n else setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m - 1}) (natpermute n m))"
  1501   by (cases m, simp_all add: fps_power_nth_Suc del: power_Suc)
  1502 
  1503 lemma fps_nth_power_0:
  1504   fixes m :: nat and a :: "('a::{comm_ring_1}) fps"
  1505   shows "(a ^m)$0 = (a$0) ^ m"
  1506 proof-
  1507   {assume "m=0" hence ?thesis by simp}
  1508   moreover
  1509   {fix n assume m: "m = Suc n"
  1510     have c: "m = card {0..n}" using m by simp
  1511    have "(a ^m)$0 = setprod (\<lambda>i. a$0) {0..n}"
  1512      by (simp add: m fps_power_nth del: replicate.simps power_Suc)
  1513    also have "\<dots> = (a$0) ^ m"
  1514      unfolding c by (rule setprod_constant, simp)
  1515    finally have ?thesis .}
  1516  ultimately show ?thesis by (cases m, auto)
  1517 qed
  1518 
  1519 lemma fps_compose_inj_right:
  1520   assumes a0: "a$0 = (0::'a::{idom})"
  1521   and a1: "a$1 \<noteq> 0"
  1522   shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
  1523 proof-
  1524   {assume ?rhs then have "?lhs" by simp}
  1525   moreover
  1526   {assume h: ?lhs
  1527     {fix n have "b$n = c$n"
  1528       proof(induct n rule: nat_less_induct)
  1529         fix n assume H: "\<forall>m<n. b$m = c$m"
  1530         {assume n0: "n=0"
  1531           from h have "(b oo a)$n = (c oo a)$n" by simp
  1532           hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)}
  1533         moreover
  1534         {fix n1 assume n1: "n = Suc n1"
  1535           have f: "finite {0 .. n1}" "finite {n}" by simp_all
  1536           have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
  1537           have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
  1538           have seq: "(\<Sum>i = 0..n1. b $ i * a ^ i $ n) = (\<Sum>i = 0..n1. c $ i * a ^ i $ n)"
  1539             apply (rule setsum_cong2)
  1540             using H n1 by auto
  1541           have th0: "(b oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + b$n * (a$1)^n"
  1542             unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
  1543             using startsby_zero_power_nth_same[OF a0]
  1544             by simp
  1545           have th1: "(c oo a) $n = (\<Sum>i = 0..n1. c $ i * a ^ i $ n) + c$n * (a$1)^n"
  1546             unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
  1547             using startsby_zero_power_nth_same[OF a0]
  1548             by simp
  1549           from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
  1550           have "b$n = c$n" by auto}
  1551         ultimately show "b$n = c$n" by (cases n, auto)
  1552       qed}
  1553     then have ?rhs by (simp add: fps_eq_iff)}
  1554   ultimately show ?thesis by blast
  1555 qed
  1556 
  1557 
  1558 subsection {* Radicals *}
  1559 
  1560 declare setprod_cong[fundef_cong]
  1561 function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a::{field}) fps \<Rightarrow> nat \<Rightarrow> 'a" where
  1562   "radical r 0 a 0 = 1"
  1563 | "radical r 0 a (Suc n) = 0"
  1564 | "radical r (Suc k) a 0 = r (Suc k) (a$0)"
  1565 | "radical r (Suc k) a (Suc n) = (a$ Suc n - setsum (\<lambda>xs. setprod (\<lambda>j. radical r (Suc k) a (xs ! j)) {0..k}) {xs. xs \<in> natpermute (Suc n) (Suc k) \<and> Suc n \<notin> set xs}) / (of_nat (Suc k) * (radical r (Suc k) a 0)^k)"
  1566 by pat_completeness auto
  1567 
  1568 termination radical
  1569 proof
  1570   let ?R = "measure (\<lambda>(r, k, a, n). n)"
  1571   {
  1572     show "wf ?R" by auto}
  1573   {fix r k a n xs i
  1574     assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
  1575     {assume c: "Suc n \<le> xs ! i"
  1576       from xs i have "xs !i \<noteq> Suc n" by (auto simp add: in_set_conv_nth natpermute_def)
  1577       with c have c': "Suc n < xs!i" by arith
  1578       have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
  1579       have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
  1580       have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
  1581       from xs have "Suc n = foldl op + 0 xs" by (simp add: natpermute_def)
  1582       also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
  1583         by (simp add: natpermute_def)
  1584       also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
  1585         unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
  1586         unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
  1587         by simp
  1588       finally have False using c' by simp}
  1589     then show "((r,Suc k,a,xs!i), r, Suc k, a, Suc n) \<in> ?R"
  1590       apply auto by (metis not_less)}
  1591   {fix r k a n
  1592     show "((r,Suc k, a, 0),r, Suc k, a, Suc n) \<in> ?R" by simp}
  1593 qed
  1594 
  1595 definition "fps_radical r n a = Abs_fps (radical r n a)"
  1596 
  1597 lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
  1598   apply (auto simp add: fps_eq_iff fps_radical_def)  by (case_tac n, auto)
  1599 
  1600 lemma fps_radical_nth_0[simp]: "fps_radical r n a $ 0 = (if n=0 then 1 else r n (a$0))"
  1601   by (cases n, simp_all add: fps_radical_def)
  1602 
  1603 lemma fps_radical_power_nth[simp]:
  1604   assumes r: "(r k (a$0)) ^ k = a$0"
  1605   shows "fps_radical r k a ^ k $ 0 = (if k = 0 then 1 else a$0)"
  1606 proof-
  1607   {assume "k=0" hence ?thesis by simp }
  1608   moreover
  1609   {fix h assume h: "k = Suc h"
  1610     have fh: "finite {0..h}" by simp
  1611     have eq1: "fps_radical r k a ^ k $ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a $ (replicate k 0) ! j)"
  1612       unfolding fps_power_nth h by simp
  1613     also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a$0))"
  1614       apply (rule setprod_cong)
  1615       apply simp
  1616       using h
  1617       apply (subgoal_tac "replicate k (0::nat) ! x = 0")
  1618       by (auto intro: nth_replicate simp del: replicate.simps)
  1619     also have "\<dots> = a$0"
  1620       unfolding setprod_constant[OF fh] using r by (simp add: h)
  1621     finally have ?thesis using h by simp}
  1622   ultimately show ?thesis by (cases k, auto)
  1623 qed
  1624 
  1625 lemma natpermute_max_card: assumes n0: "n\<noteq>0"
  1626   shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k+1"
  1627   unfolding natpermute_contain_maximal
  1628 proof-
  1629   let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
  1630   let ?K = "{0 ..k}"
  1631   have fK: "finite ?K" by simp
  1632   have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
  1633   have d: "\<forall>i\<in> ?K. \<forall>j\<in> ?K. i \<noteq> j \<longrightarrow> {replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
  1634   proof(clarify)
  1635     fix i j assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
  1636     {assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
  1637       have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps)
  1638       moreover
  1639       have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps)
  1640       ultimately have False using eq n0 by (simp del: replicate.simps)}
  1641     then show "{replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
  1642       by auto
  1643   qed
  1644   from card_UN_disjoint[OF fK fAK d]
  1645   show "card (\<Union>i\<in>{0..k}. {replicate (k + 1) 0[i := n]}) = k+1" by simp
  1646 qed
  1647 
  1648 lemma power_radical:
  1649   fixes a:: "'a::field_char_0 fps"
  1650   assumes a0: "a$0 \<noteq> 0"
  1651   shows "(r (Suc k) (a$0)) ^ Suc k = a$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a"
  1652 proof-
  1653   let ?r = "fps_radical r (Suc k) a"
  1654   {assume r0: "(r (Suc k) (a$0)) ^ Suc k = a$0"
  1655     from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto
  1656     {fix z have "?r ^ Suc k $ z = a$z"
  1657       proof(induct z rule: nat_less_induct)
  1658         fix n assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
  1659         {assume "n = 0" hence "?r ^ Suc k $ n = a $n"
  1660             using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
  1661         moreover
  1662         {fix n1 assume n1: "n = Suc n1"
  1663           have fK: "finite {0..k}" by simp
  1664           have nz: "n \<noteq> 0" using n1 by arith
  1665           let ?Pnk = "natpermute n (k + 1)"
  1666           let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
  1667           let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
  1668           have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
  1669           have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
  1670           have f: "finite ?Pnkn" "finite ?Pnknn"
  1671             using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
  1672             by (metis natpermute_finite)+
  1673           let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
  1674           have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
  1675           proof(rule setsum_cong2)
  1676             fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
  1677             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"
  1678           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
  1679             unfolding natpermute_contain_maximal by auto
  1680           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))"
  1681             apply (rule setprod_cong, simp)
  1682             using i r0 by (simp del: replicate.simps)
  1683           also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
  1684             unfolding setprod_gen_delta[OF fK] using i r0 by simp
  1685           finally show ?ths .
  1686         qed
  1687         then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
  1688           by (simp add: natpermute_max_card[OF nz, simplified])
  1689         also have "\<dots> = a$n - setsum ?f ?Pnknn"
  1690           unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
  1691         finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
  1692         have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
  1693           unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
  1694         also have "\<dots> = a$n" unfolding fn by simp
  1695         finally have "?r ^ Suc k $ n = a $n" .}
  1696       ultimately  show "?r ^ Suc k $ n = a $n" by (cases n, auto)
  1697     qed }
  1698   then have ?thesis using r0 by (simp add: fps_eq_iff)}
  1699 moreover 
  1700 { assume h: "(fps_radical r (Suc k) a) ^ (Suc k) = a"
  1701   hence "((fps_radical r (Suc k) a) ^ (Suc k))$0 = a$0" by simp
  1702   then have "(r (Suc k) (a$0)) ^ Suc k = a$0"
  1703     unfolding fps_power_nth_Suc
  1704     by (simp add: setprod_constant del: replicate.simps)}
  1705 ultimately show ?thesis by blast
  1706 qed
  1707 
  1708 (*
  1709 lemma power_radical:
  1710   fixes a:: "'a::field_char_0 fps"
  1711   assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
  1712   shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
  1713 proof-
  1714   let ?r = "fps_radical r (Suc k) a"
  1715   from a0 r0 have r00: "r (Suc k) (a$0) \<noteq> 0" by auto
  1716   {fix z have "?r ^ Suc k $ z = a$z"
  1717     proof(induct z rule: nat_less_induct)
  1718       fix n assume H: "\<forall>m<n. ?r ^ Suc k $ m = a$m"
  1719       {assume "n = 0" hence "?r ^ Suc k $ n = a $n"
  1720           using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
  1721       moreover
  1722       {fix n1 assume n1: "n = Suc n1"
  1723         have fK: "finite {0..k}" by simp
  1724         have nz: "n \<noteq> 0" using n1 by arith
  1725         let ?Pnk = "natpermute n (k + 1)"
  1726         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
  1727         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
  1728         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
  1729         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
  1730         have f: "finite ?Pnkn" "finite ?Pnknn"
  1731           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
  1732           by (metis natpermute_finite)+
  1733         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
  1734         have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r $ n * r (Suc k) (a $ 0) ^ k) ?Pnkn"
  1735         proof(rule setsum_cong2)
  1736           fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
  1737           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"
  1738           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
  1739             unfolding natpermute_contain_maximal by auto
  1740           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))"
  1741             apply (rule setprod_cong, simp)
  1742             using i r0 by (simp del: replicate.simps)
  1743           also have "\<dots> = (fps_radical r (Suc k) a $ n) * r (Suc k) (a$0) ^ k"
  1744             unfolding setprod_gen_delta[OF fK] using i r0 by simp
  1745           finally show ?ths .
  1746         qed
  1747         then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r $ n * r (Suc k) (a $ 0) ^ k"
  1748           by (simp add: natpermute_max_card[OF nz, simplified])
  1749         also have "\<dots> = a$n - setsum ?f ?Pnknn"
  1750           unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
  1751         finally have fn: "setsum ?f ?Pnkn = a$n - setsum ?f ?Pnknn" .
  1752         have "(?r ^ Suc k)$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
  1753           unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
  1754         also have "\<dots> = a$n" unfolding fn by simp
  1755         finally have "?r ^ Suc k $ n = a $n" .}
  1756       ultimately  show "?r ^ Suc k $ n = a $n" by (cases n, auto)
  1757   qed }
  1758   then show ?thesis by (simp add: fps_eq_iff)
  1759 qed
  1760 
  1761 *)
  1762 lemma eq_divide_imp': assumes c0: "(c::'a::field) ~= 0" and eq: "a * c = b"
  1763   shows "a = b / c"
  1764 proof-
  1765   from eq have "a * c * inverse c = b * inverse c" by simp
  1766   hence "a * (inverse c * c) = b/c" by (simp only: field_simps divide_inverse)
  1767   then show "a = b/c" unfolding  field_inverse[OF c0] by simp
  1768 qed
  1769 
  1770 lemma radical_unique:
  1771   assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0"
  1772   and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0" and b0: "b$0 \<noteq> 0"
  1773   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
  1774 proof-
  1775   let ?r = "fps_radical r (Suc k) b"
  1776   have r00: "r (Suc k) (b$0) \<noteq> 0" using b0 r0 by auto
  1777   {assume H: "a = ?r"
  1778     from H have "a^Suc k = b" using power_radical[OF b0, of r k, unfolded r0] by simp}
  1779   moreover
  1780   {assume H: "a^Suc k = b"
  1781     have ceq: "card {0..k} = Suc k" by simp
  1782     have fk: "finite {0..k}" by simp
  1783     from a0 have a0r0: "a$0 = ?r$0" by simp
  1784     {fix n have "a $ n = ?r $ n"
  1785       proof(induct n rule: nat_less_induct)
  1786         fix n assume h: "\<forall>m<n. a$m = ?r $m"
  1787         {assume "n = 0" hence "a$n = ?r $n" using a0 by simp }
  1788         moreover
  1789         {fix n1 assume n1: "n = Suc n1"
  1790           have fK: "finite {0..k}" by simp
  1791         have nz: "n \<noteq> 0" using n1 by arith
  1792         let ?Pnk = "natpermute n (Suc k)"
  1793         let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
  1794         let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
  1795         have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
  1796         have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
  1797         have f: "finite ?Pnkn" "finite ?Pnknn"
  1798           using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
  1799           by (metis natpermute_finite)+
  1800         let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r $ v ! j"
  1801         let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a $ v ! j"
  1802         have "setsum ?g ?Pnkn = setsum (\<lambda>v. a $ n * (?r$0)^k) ?Pnkn"
  1803         proof(rule setsum_cong2)
  1804           fix v assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
  1805           let ?ths = "(\<Prod>j\<in>{0..k}. a $ v ! j) = a $ n * (?r$0)^k"
  1806           from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
  1807             unfolding Suc_eq_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps)
  1808           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))"
  1809             apply (rule setprod_cong, simp)
  1810             using i a0 by (simp del: replicate.simps)
  1811           also have "\<dots> = a $ n * (?r $ 0)^k"
  1812             unfolding  setprod_gen_delta[OF fK] using i by simp
  1813           finally show ?ths .
  1814         qed
  1815         then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a $ n * (?r $ 0)^k"
  1816           by (simp add: natpermute_max_card[OF nz, simplified])
  1817         have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
  1818         proof (rule setsum_cong2, rule setprod_cong, simp)
  1819           fix xs i assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
  1820           {assume c: "n \<le> xs ! i"
  1821             from xs i have "xs !i \<noteq> n" by (auto simp add: in_set_conv_nth natpermute_def)
  1822             with c have c': "n < xs!i" by arith
  1823             have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
  1824             have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
  1825             have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
  1826             from xs have "n = foldl op + 0 xs" by (simp add: natpermute_def)
  1827             also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
  1828               by (simp add: natpermute_def)
  1829             also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
  1830               unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
  1831               unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
  1832               by simp
  1833             finally have False using c' by simp}
  1834           then have thn: "xs!i < n" by arith
  1835           from h[rule_format, OF thn]
  1836           show "a$(xs !i) = ?r$(xs!i)" .
  1837         qed
  1838         have th00: "\<And>(x::'a). of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
  1839           by (simp add: field_simps del: of_nat_Suc)
  1840         from H have "b$n = a^Suc k $ n" by (simp add: fps_eq_iff)
  1841         also have "a ^ Suc k$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
  1842           unfolding fps_power_nth_Suc
  1843           using setsum_Un_disjoint[OF f d, unfolded Suc_eq_plus1[symmetric],
  1844             unfolded eq, of ?g] by simp
  1845         also have "\<dots> = of_nat (k+1) * a $ n * (?r $ 0)^k + setsum ?f ?Pnknn" unfolding th0 th1 ..
  1846         finally have "of_nat (k+1) * a $ n * (?r $ 0)^k = b$n - setsum ?f ?Pnknn" by simp
  1847         then have "a$n = (b$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r $ 0)^k)"
  1848           apply -
  1849           apply (rule eq_divide_imp')
  1850           using r00
  1851           apply (simp del: of_nat_Suc)
  1852           by (simp add: mult_ac)
  1853         then have "a$n = ?r $n"
  1854           apply (simp del: of_nat_Suc)
  1855           unfolding fps_radical_def n1
  1856           by (simp add: field_simps n1 th00 del: of_nat_Suc)}
  1857         ultimately show "a$n = ?r $ n" by (cases n, auto)
  1858       qed}
  1859     then have "a = ?r" by (simp add: fps_eq_iff)}
  1860   ultimately show ?thesis by blast
  1861 qed
  1862 
  1863 
  1864 lemma radical_power:
  1865   assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
  1866   and a0: "(a$0 ::'a::field_char_0) \<noteq> 0"
  1867   shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
  1868 proof-
  1869   let ?ak = "a^ Suc k"
  1870   have ak0: "?ak $ 0 = (a$0) ^ Suc k" by (simp add: fps_nth_power_0 del: power_Suc)
  1871   from r0 have th0: "r (Suc k) (a ^ Suc k $ 0) ^ Suc k = a ^ Suc k $ 0" using ak0 by auto
  1872   from r0 ak0 have th1: "r (Suc k) (a ^ Suc k $ 0) = a $ 0" by auto
  1873   from ak0 a0 have ak00: "?ak $ 0 \<noteq>0 " by auto
  1874   from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis by metis
  1875 qed
  1876 
  1877 lemma fps_deriv_radical:
  1878   fixes a:: "'a::field_char_0 fps"
  1879   assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
  1880   shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
  1881 proof-
  1882   let ?r= "fps_radical r (Suc k) a"
  1883   let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
  1884   from a0 r0 have r0': "r (Suc k) (a$0) \<noteq> 0" by auto
  1885   from r0' have w0: "?w $ 0 \<noteq> 0" by (simp del: of_nat_Suc)
  1886   note th0 = inverse_mult_eq_1[OF w0]
  1887   let ?iw = "inverse ?w"
  1888   from iffD1[OF power_radical[of a r], OF a0 r0]
  1889   have "fps_deriv (?r ^ Suc k) = fps_deriv a" by simp
  1890   hence "fps_deriv ?r * ?w = fps_deriv a"
  1891     by (simp add: fps_deriv_power mult_ac del: power_Suc)
  1892   hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp
  1893   hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
  1894     by (simp add: fps_divide_def)
  1895   then show ?thesis unfolding th0 by simp
  1896 qed
  1897 
  1898 lemma radical_mult_distrib:
  1899   fixes a:: "'a::field_char_0 fps"
  1900   assumes
  1901   k: "k > 0"
  1902   and ra0: "r k (a $ 0) ^ k = a $ 0"
  1903   and rb0: "r k (b $ 0) ^ k = b $ 0"
  1904   and a0: "a$0 \<noteq> 0"
  1905   and b0: "b$0 \<noteq> 0"
  1906   shows "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0) \<longleftrightarrow> fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
  1907 proof-
  1908   {assume  r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
  1909   from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
  1910     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
  1911   {assume "k=0" hence ?thesis using r0' by simp}
  1912   moreover
  1913   {fix h assume k: "k = Suc h"
  1914   let ?ra = "fps_radical r (Suc h) a"
  1915   let ?rb = "fps_radical r (Suc h) b"
  1916   have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
  1917     using r0' k by (simp add: fps_mult_nth)
  1918   have ab0: "(a*b) $ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
  1919   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]
  1920     iffD1[OF power_radical[of _ r], OF a0 ra0[unfolded k]] iffD1[OF power_radical[of _ r], OF b0 rb0[unfolded k]] k r0'
  1921   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
  1922 ultimately have ?thesis by (cases k, auto)}
  1923 moreover
  1924 {assume h: "fps_radical r k (a*b) = fps_radical r k a * fps_radical r k b"
  1925   hence "(fps_radical r k (a*b))$0 = (fps_radical r k a * fps_radical r k b)$0" by simp
  1926   then have "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
  1927     using k by (simp add: fps_mult_nth)}
  1928 ultimately show ?thesis by blast
  1929 qed
  1930 
  1931 (*
  1932 lemma radical_mult_distrib:
  1933   fixes a:: "'a::field_char_0 fps"
  1934   assumes
  1935   ra0: "r k (a $ 0) ^ k = a $ 0"
  1936   and rb0: "r k (b $ 0) ^ k = b $ 0"
  1937   and r0': "r k ((a * b) $ 0) = r k (a $ 0) * r k (b $ 0)"
  1938   and a0: "a$0 \<noteq> 0"
  1939   and b0: "b$0 \<noteq> 0"
  1940   shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
  1941 proof-
  1942   from r0' have r0: "(r (k) ((a*b)$0)) ^ k = (a*b)$0"
  1943     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
  1944   {assume "k=0" hence ?thesis by simp}
  1945   moreover
  1946   {fix h assume k: "k = Suc h"
  1947   let ?ra = "fps_radical r (Suc h) a"
  1948   let ?rb = "fps_radical r (Suc h) b"
  1949   have th0: "r (Suc h) ((a * b) $ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) $ 0"
  1950     using r0' k by (simp add: fps_mult_nth)
  1951   have ab0: "(a*b) $ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
  1952   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]
  1953     power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
  1954   have ?thesis by (auto simp add: power_mult_distrib simp del: power_Suc)}
  1955 ultimately show ?thesis by (cases k, auto)
  1956 qed
  1957 *)
  1958 
  1959 lemma fps_divide_1[simp]: "(a:: ('a::field) fps) / 1 = a"
  1960   by (simp add: fps_divide_def)
  1961 
  1962 lemma radical_divide:
  1963   fixes a :: "'a::field_char_0 fps"
  1964   assumes
  1965   kp: "k>0"
  1966   and ra0: "(r k (a $ 0)) ^ k = a $ 0"
  1967   and rb0: "(r k (b $ 0)) ^ k = b $ 0"
  1968   and a0: "a$0 \<noteq> 0"
  1969   and b0: "b$0 \<noteq> 0"
  1970   shows "r k ((a $ 0) / (b$0)) = r k (a$0) / r k (b $ 0) \<longleftrightarrow> fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b" (is "?lhs = ?rhs")
  1971 proof-
  1972   let ?r = "fps_radical r k"
  1973   from kp obtain h where k: "k = Suc h" by (cases k, auto)
  1974   have ra0': "r k (a$0) \<noteq> 0" using a0 ra0 k by auto
  1975   have rb0': "r k (b$0) \<noteq> 0" using b0 rb0 k by auto
  1976 
  1977   {assume ?rhs
  1978     then have "?r (a/b) $ 0 = (?r a / ?r b)$0" by simp
  1979     then have ?lhs using k a0 b0 rb0' 
  1980       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse) }
  1981   moreover
  1982   {assume h: ?lhs
  1983     from a0 b0 have ab0[simp]: "(a/b)$0 = a$0 / b$0" 
  1984       by (simp add: fps_divide_def fps_mult_nth divide_inverse fps_inverse_def)
  1985     have th0: "r k ((a/b)$0) ^ k = (a/b)$0"
  1986       by (simp add: h nonzero_power_divide[OF rb0'] ra0 rb0 del: k)
  1987     from a0 b0 ra0' rb0' kp h 
  1988     have th1: "r k ((a / b) $ 0) = (fps_radical r k a / fps_radical r k b) $ 0"
  1989       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def divide_inverse del: k)
  1990     from a0 b0 ra0' rb0' kp have ab0': "(a / b) $ 0 \<noteq> 0"
  1991       by (simp add: fps_divide_def fps_mult_nth fps_inverse_def nonzero_imp_inverse_nonzero)
  1992     note tha[simp] = iffD1[OF power_radical[where r=r and k=h], OF a0 ra0[unfolded k], unfolded k[symmetric]]
  1993     note thb[simp] = iffD1[OF power_radical[where r=r and k=h], OF b0 rb0[unfolded k], unfolded k[symmetric]]
  1994     have th2: "(?r a / ?r b)^k = a/b"
  1995       by (simp add: fps_divide_def power_mult_distrib fps_inverse_power[symmetric])
  1996     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] have ?rhs .}
  1997   ultimately show ?thesis by blast
  1998 qed
  1999 
  2000 lemma radical_inverse:
  2001   fixes a :: "'a::field_char_0 fps"
  2002   assumes
  2003   k: "k>0"
  2004   and ra0: "r k (a $ 0) ^ k = a $ 0"
  2005   and r1: "(r k 1)^k = 1"
  2006   and a0: "a$0 \<noteq> 0"
  2007   shows "r k (inverse (a $ 0)) = r k 1 / (r k (a $ 0)) \<longleftrightarrow> fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a"
  2008   using radical_divide[where k=k and r=r and a=1 and b=a, OF k ] ra0 r1 a0
  2009   by (simp add: divide_inverse fps_divide_def)
  2010 
  2011 subsection{* Derivative of composition *}
  2012 
  2013 lemma fps_compose_deriv:
  2014   fixes a:: "('a::idom) fps"
  2015   assumes b0: "b$0 = 0"
  2016   shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)"
  2017 proof-
  2018   {fix n
  2019     have "(fps_deriv (a oo b))$n = setsum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
  2020       by (simp add: fps_compose_def ring_simps setsum_right_distrib del: of_nat_Suc)
  2021     also have "\<dots> = setsum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
  2022       by (simp add: ring_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
  2023   also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
  2024     unfolding fps_mult_left_const_nth  by (simp add: ring_simps)
  2025   also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (setsum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {0.. Suc n}"
  2026     unfolding fps_mult_nth ..
  2027   also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (setsum (\<lambda>j. (b^ (i - 1))$j * (fps_deriv b)$(n - j)) {0..n})) {1.. Suc n}"
  2028     apply (rule setsum_mono_zero_right)
  2029     apply (auto simp add: mult_delta_left setsum_delta not_le)
  2030     done
  2031   also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
  2032     unfolding fps_deriv_nth
  2033     apply (rule setsum_reindex_cong[where f="Suc"])
  2034     by (auto simp add: mult_assoc)
  2035   finally have th0: "(fps_deriv (a oo b))$n = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}" .
  2036 
  2037   have "(((fps_deriv a) oo b) * (fps_deriv b))$n = setsum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
  2038     unfolding fps_mult_nth by (simp add: mult_ac)
  2039   also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
  2040     unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
  2041     apply (rule setsum_cong2)
  2042     apply (rule setsum_mono_zero_left)
  2043     apply (simp_all add: subset_eq)
  2044     apply clarify
  2045     apply (subgoal_tac "b^i$x = 0")
  2046     apply simp
  2047     apply (rule startsby_zero_power_prefix[OF b0, rule_format])
  2048     by simp
  2049   also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
  2050     unfolding setsum_right_distrib
  2051     apply (subst setsum_commute)
  2052     by ((rule setsum_cong2)+) simp
  2053   finally have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n"
  2054     unfolding th0 by simp}
  2055 then show ?thesis by (simp add: fps_eq_iff)
  2056 qed
  2057 
  2058 lemma fps_mult_X_plus_1_nth:
  2059   "((1+X)*a) $n = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
  2060 proof-
  2061   {assume "n = 0" hence ?thesis by (simp add: fps_mult_nth )}
  2062   moreover
  2063   {fix m assume m: "n = Suc m"
  2064     have "((1+X)*a) $n = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0..n}"
  2065       by (simp add: fps_mult_nth)
  2066     also have "\<dots> = setsum (\<lambda>i. (1+X)$i * a$(n-i)) {0.. 1}"
  2067       unfolding m
  2068       apply (rule setsum_mono_zero_right)
  2069       by (auto simp add: )
  2070     also have "\<dots> = (if n = 0 then (a$n :: 'a::comm_ring_1) else a$n + a$(n - 1))"
  2071       unfolding m
  2072       by (simp add: )
  2073     finally have ?thesis .}
  2074   ultimately show ?thesis by (cases n, auto)
  2075 qed
  2076 
  2077 subsection{* Finite FPS (i.e. polynomials) and X *}
  2078 lemma fps_poly_sum_X:
  2079   assumes z: "\<forall>i > n. a$i = (0::'a::comm_ring_1)"
  2080   shows "a = setsum (\<lambda>i. fps_const (a$i) * X^i) {0..n}" (is "a = ?r")
  2081 proof-
  2082   {fix i
  2083     have "a$i = ?r$i"
  2084       unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
  2085       by (simp add: mult_delta_right setsum_delta' z)
  2086   }
  2087   then show ?thesis unfolding fps_eq_iff by blast
  2088 qed
  2089 
  2090 subsection{* Compositional inverses *}
  2091 
  2092 
  2093 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::{field}" where
  2094   "compinv a 0 = X$0"
  2095 | "compinv a (Suc n) = (X$ Suc n - setsum (\<lambda>i. (compinv a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
  2096 
  2097 definition "fps_inv a = Abs_fps (compinv a)"
  2098 
  2099 lemma fps_inv: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
  2100   shows "fps_inv a oo a = X"
  2101 proof-
  2102   let ?i = "fps_inv a oo a"
  2103   {fix n
  2104     have "?i $n = X$n"
  2105     proof(induct n rule: nat_less_induct)
  2106       fix n assume h: "\<forall>m<n. ?i$m = X$m"
  2107       {assume "n=0" hence "?i $n = X$n" using a0
  2108           by (simp add: fps_compose_nth fps_inv_def)}
  2109       moreover
  2110       {fix n1 assume n1: "n = Suc n1"
  2111         have "?i $ n = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + fps_inv a $ Suc n1 * (a $ 1)^ Suc n1"
  2112           by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0]
  2113                    del: power_Suc)
  2114         also have "\<dots> = setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1} + (X$ Suc n1 - setsum (\<lambda>i. (fps_inv a $ i) * (a^i)$n) {0 .. n1})"
  2115           using a0 a1 n1 by (simp add: fps_inv_def)
  2116         also have "\<dots> = X$n" using n1 by simp
  2117         finally have "?i $ n = X$n" .}
  2118       ultimately show "?i $ n = X$n" by (cases n, auto)
  2119     qed}
  2120   then show ?thesis by (simp add: fps_eq_iff)
  2121 qed
  2122 
  2123 
  2124 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::{field}" where
  2125   "gcompinv b a 0 = b$0"
  2126 | "gcompinv b a (Suc n) = (b$ Suc n - setsum (\<lambda>i. (gcompinv b a i) * (a^i)$Suc n) {0 .. n}) / (a$1) ^ Suc n"
  2127 
  2128 definition "fps_ginv b a = Abs_fps (gcompinv b a)"
  2129 
  2130 lemma fps_ginv: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
  2131   shows "fps_ginv b a oo a = b"
  2132 proof-
  2133   let ?i = "fps_ginv b a oo a"
  2134   {fix n
  2135     have "?i $n = b$n"
  2136     proof(induct n rule: nat_less_induct)
  2137       fix n assume h: "\<forall>m<n. ?i$m = b$m"
  2138       {assume "n=0" hence "?i $n = b$n" using a0
  2139           by (simp add: fps_compose_nth fps_ginv_def)}
  2140       moreover
  2141       {fix n1 assume n1: "n = Suc n1"
  2142         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"
  2143           by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0]
  2144                    del: power_Suc)
  2145         also have "\<dots> = setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1} + (b$ Suc n1 - setsum (\<lambda>i. (fps_ginv b a $ i) * (a^i)$n) {0 .. n1})"
  2146           using a0 a1 n1 by (simp add: fps_ginv_def)
  2147         also have "\<dots> = b$n" using n1 by simp
  2148         finally have "?i $ n = b$n" .}
  2149       ultimately show "?i $ n = b$n" by (cases n, auto)
  2150     qed}
  2151   then show ?thesis by (simp add: fps_eq_iff)
  2152 qed
  2153 
  2154 lemma fps_inv_ginv: "fps_inv = fps_ginv X"
  2155   apply (auto simp add: expand_fun_eq fps_eq_iff fps_inv_def fps_ginv_def)
  2156   apply (induct_tac n rule: nat_less_induct, auto)
  2157   apply (case_tac na)
  2158   apply simp
  2159   apply simp
  2160   done
  2161 
  2162 lemma fps_compose_1[simp]: "1 oo a = 1"
  2163   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
  2164 
  2165 lemma fps_compose_0[simp]: "0 oo a = 0"
  2166   by (simp add: fps_eq_iff fps_compose_nth)
  2167 
  2168 lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a$0)"
  2169   by (auto simp add: fps_eq_iff fps_compose_nth power_0_left setsum_0')
  2170 
  2171 lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
  2172   by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_addf)
  2173 
  2174 lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
  2175 proof-
  2176   {assume "\<not> finite S" hence ?thesis by simp}
  2177   moreover
  2178   {assume fS: "finite S"
  2179     have ?thesis
  2180     proof(rule finite_induct[OF fS])
  2181       show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
  2182     next
  2183       fix x F assume fF: "finite F" and xF: "x \<notin> F" and h: "setsum f F oo a = setsum (\<lambda>i. f i oo a) F"
  2184       show "setsum f (insert x F) oo a  = setsum (\<lambda>i. f i oo a) (insert x F)"
  2185         using fF xF h by (simp add: fps_compose_add_distrib)
  2186     qed}
  2187   ultimately show ?thesis by blast
  2188 qed
  2189 
  2190 lemma convolution_eq:
  2191   "setsum (%i. a (i :: nat) * b (n - i)) {0 .. n} = setsum (%(i,j). a i * b j) {(i,j). i <= n \<and> j \<le> n \<and> i + j = n}"
  2192   apply (rule setsum_reindex_cong[where f=fst])
  2193   apply (clarsimp simp add: inj_on_def)
  2194   apply (auto simp add: expand_set_eq image_iff)
  2195   apply (rule_tac x= "x" in exI)
  2196   apply clarsimp
  2197   apply (rule_tac x="n - x" in exI)
  2198   apply arith
  2199   done
  2200 
  2201 lemma product_composition_lemma:
  2202   assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0"
  2203   shows "((a oo c) * (b oo d))$n = setsum (%(k,m). a$k * b$m * (c^k * d^m) $ n) {(k,m). k + m \<le> n}" (is "?l = ?r")
  2204 proof-
  2205   let ?S = "{(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
  2206   have s: "?S \<subseteq> {0..n} <*> {0..n}" by (auto simp add: subset_eq)
  2207   have f: "finite {(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
  2208     apply (rule finite_subset[OF s])
  2209     by auto
  2210   have "?r =  setsum (%i. setsum (%(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
  2211     apply (simp add: fps_mult_nth setsum_right_distrib)
  2212     apply (subst setsum_commute)
  2213     apply (rule setsum_cong2)
  2214     by (auto simp add: ring_simps)
  2215   also have "\<dots> = ?l"
  2216     apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
  2217     apply (rule setsum_cong2)
  2218     apply (simp add: setsum_cartesian_product mult_assoc)
  2219     apply (rule setsum_mono_zero_right[OF f])
  2220     apply (simp add: subset_eq) apply presburger
  2221     apply clarsimp
  2222     apply (rule ccontr)
  2223     apply (clarsimp simp add: not_le)
  2224     apply (case_tac "x < aa")
  2225     apply simp
  2226     apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
  2227     apply blast
  2228     apply simp
  2229     apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
  2230     apply blast
  2231     done
  2232   finally show ?thesis by simp
  2233 qed
  2234 
  2235 lemma product_composition_lemma':
  2236   assumes c0: "c$0 = (0::'a::idom)" and d0: "d$0 = 0"
  2237   shows "((a oo c) * (b oo d))$n = setsum (%k. setsum (%m. a$k * b$m * (c^k * d^m) $ n) {0..n}) {0..n}" (is "?l = ?r")
  2238   unfolding product_composition_lemma[OF c0 d0]
  2239   unfolding setsum_cartesian_product
  2240   apply (rule setsum_mono_zero_left)
  2241   apply simp
  2242   apply (clarsimp simp add: subset_eq)
  2243   apply clarsimp
  2244   apply (rule ccontr)
  2245   apply (subgoal_tac "(c^aa * d^ba) $ n = 0")
  2246   apply simp
  2247   unfolding fps_mult_nth
  2248   apply (rule setsum_0')
  2249   apply (clarsimp simp add: not_le)
  2250   apply (case_tac "aaa < aa")
  2251   apply (rule startsby_zero_power_prefix[OF c0, rule_format])
  2252   apply simp
  2253   apply (subgoal_tac "n - aaa < ba")
  2254   apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
  2255   apply simp
  2256   apply arith
  2257   done
  2258 
  2259 
  2260 lemma setsum_pair_less_iff:
  2261   "setsum (%((k::nat),m). a k * b m * c (k + m)) {(k,m). k + m \<le> n} = setsum (%s. setsum (%i. a i * b (s - i) * c s) {0..s}) {0..n}" (is "?l = ?r")
  2262 proof-
  2263   let ?KM=  "{(k,m). k + m \<le> n}"
  2264   let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})"
  2265   have th0: "?KM = UNION {0..n} ?f"
  2266     apply (simp add: expand_set_eq)
  2267     apply arith (* FIXME: VERY slow! *)
  2268     done
  2269   show "?l = ?r "
  2270     unfolding th0
  2271     apply (subst setsum_UN_disjoint)
  2272     apply auto
  2273     apply (subst setsum_UN_disjoint)
  2274     apply auto
  2275     done
  2276 qed
  2277 
  2278 lemma fps_compose_mult_distrib_lemma:
  2279   assumes c0: "c$0 = (0::'a::idom)"
  2280   shows "((a oo c) * (b oo c))$n = setsum (%s. setsum (%i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}" (is "?l = ?r")
  2281   unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
  2282   unfolding setsum_pair_less_iff[where a = "%k. a$k" and b="%m. b$m" and c="%s. (c ^ s)$n" and n = n] ..
  2283 
  2284 
  2285 lemma fps_compose_mult_distrib:
  2286   assumes c0: "c$0 = (0::'a::idom)"
  2287   shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r")
  2288   apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
  2289   by (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
  2290 lemma fps_compose_setprod_distrib:
  2291   assumes c0: "c$0 = (0::'a::idom)"
  2292   shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r")
  2293   apply (cases "finite S")
  2294   apply simp_all
  2295   apply (induct S rule: finite_induct)
  2296   apply simp
  2297   apply (simp add: fps_compose_mult_distrib[OF c0])
  2298   done
  2299 
  2300 lemma fps_compose_power:   assumes c0: "c$0 = (0::'a::idom)"
  2301   shows "(a oo c)^n = a^n oo c" (is "?l = ?r")
  2302 proof-
  2303   {assume "n=0" then have ?thesis by simp}
  2304   moreover
  2305   {fix m assume m: "n = Suc m"
  2306     have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}"
  2307       by (simp_all add: setprod_constant m)
  2308     then have ?thesis
  2309       by (simp add: fps_compose_setprod_distrib[OF c0])}
  2310   ultimately show ?thesis by (cases n, auto)
  2311 qed
  2312 
  2313 lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
  2314   by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_negf[symmetric])
  2315 
  2316 lemma fps_compose_sub_distrib:
  2317   shows "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
  2318   unfolding diff_minus fps_compose_uminus fps_compose_add_distrib ..
  2319 
  2320 lemma X_fps_compose:"X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a$n)"
  2321   by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta power_Suc)
  2322 
  2323 lemma fps_inverse_compose:
  2324   assumes b0: "(b$0 :: 'a::field) = 0" and a0: "a$0 \<noteq> 0"
  2325   shows "inverse a oo b = inverse (a oo b)"
  2326 proof-
  2327   let ?ia = "inverse a"
  2328   let ?ab = "a oo b"
  2329   let ?iab = "inverse ?ab"
  2330 
  2331 from a0 have ia0: "?ia $ 0 \<noteq> 0" by (simp )
  2332 from a0 have ab0: "?ab $ 0 \<noteq> 0" by (simp add: fps_compose_def)
  2333 have "(?ia oo b) *  (a oo b) = 1"
  2334 unfolding fps_compose_mult_distrib[OF b0, symmetric]
  2335 unfolding inverse_mult_eq_1[OF a0]
  2336 fps_compose_1 ..
  2337 
  2338 then have "(?ia oo b) *  (a oo b) * ?iab  = 1 * ?iab" by simp
  2339 then have "(?ia oo b) *  (?iab * (a oo b))  = ?iab" by simp
  2340 then show ?thesis 
  2341   unfolding inverse_mult_eq_1[OF ab0] by simp
  2342 qed
  2343 
  2344 lemma fps_divide_compose:
  2345   assumes c0: "(c$0 :: 'a::field) = 0" and b0: "b$0 \<noteq> 0"
  2346   shows "(a/b) oo c = (a oo c) / (b oo c)"
  2347     unfolding fps_divide_def fps_compose_mult_distrib[OF c0]
  2348     fps_inverse_compose[OF c0 b0] ..
  2349 
  2350 lemma gp: assumes a0: "a$0 = (0::'a::field)"
  2351   shows "(Abs_fps (\<lambda>n. 1)) oo a = 1/(1 - a)" (is "?one oo a = _")
  2352 proof-
  2353   have o0: "?one $ 0 \<noteq> 0" by simp
  2354   have th0: "(1 - X) $ 0 \<noteq> (0::'a)" by simp  
  2355   from fps_inverse_gp[where ?'a = 'a]
  2356   have "inverse ?one = 1 - X" by (simp add: fps_eq_iff)
  2357   hence "inverse (inverse ?one) = inverse (1 - X)" by simp
  2358   hence th: "?one = 1/(1 - X)" unfolding fps_inverse_idempotent[OF o0] 
  2359     by (simp add: fps_divide_def)
  2360   show ?thesis unfolding th
  2361     unfolding fps_divide_compose[OF a0 th0]
  2362     fps_compose_1 fps_compose_sub_distrib X_fps_compose_startby0[OF a0] ..
  2363 qed
  2364 
  2365 lemma fps_const_power[simp]: "fps_const (c::'a::ring_1) ^ n = fps_const (c^n)"
  2366 by (induct n, auto)
  2367 
  2368 lemma fps_compose_radical:
  2369   assumes b0: "b$0 = (0::'a::field_char_0)"
  2370   and ra0: "r (Suc k) (a$0) ^ Suc k = a$0"
  2371   and a0: "a$0 \<noteq> 0"
  2372   shows "fps_radical r (Suc k)  a oo b = fps_radical r (Suc k) (a oo b)"
  2373 proof-
  2374   let ?r = "fps_radical r (Suc k)"
  2375   let ?ab = "a oo b"
  2376   have ab0: "?ab $ 0 = a$0" by (simp add: fps_compose_def)
  2377   from ab0 a0 ra0 have rab0: "?ab $ 0 \<noteq> 0" "r (Suc k) (?ab $ 0) ^ Suc k = ?ab $ 0" by simp_all
  2378   have th00: "r (Suc k) ((a oo b) $ 0) = (fps_radical r (Suc k) a oo b) $ 0"
  2379     by (simp add: ab0 fps_compose_def)
  2380   have th0: "(?r a oo b) ^ (Suc k) = a  oo b"
  2381     unfolding fps_compose_power[OF b0]
  2382     unfolding iffD1[OF power_radical[of a r k], OF a0 ra0]  .. 
  2383   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] show ?thesis  . 
  2384 qed
  2385 
  2386 lemma fps_const_mult_apply_left:
  2387   "fps_const c * (a oo b) = (fps_const c * a) oo b"
  2388   by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc)
  2389 
  2390 lemma fps_const_mult_apply_right:
  2391   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
  2392   by (auto simp add: fps_const_mult_apply_left mult_commute)
  2393 
  2394 lemma fps_compose_assoc:
  2395   assumes c0: "c$0 = (0::'a::idom)" and b0: "b$0 = 0"
  2396   shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
  2397 proof-
  2398   {fix n
  2399     have "?l$n = (setsum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n"
  2400       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left setsum_right_distrib mult_assoc fps_setsum_nth)
  2401     also have "\<dots> = ((setsum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
  2402       by (simp add: fps_compose_setsum_distrib)
  2403     also have "\<dots> = ?r$n"
  2404       apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
  2405       apply (rule setsum_cong2)
  2406       apply (rule setsum_mono_zero_right)
  2407       apply (auto simp add: not_le)
  2408       by (erule startsby_zero_power_prefix[OF b0, rule_format])
  2409     finally have "?l$n = ?r$n" .}
  2410   then show ?thesis by (simp add: fps_eq_iff)
  2411 qed
  2412 
  2413 
  2414 lemma fps_X_power_compose:
  2415   assumes a0: "a$0=0" shows "X^k oo a = (a::('a::idom fps))^k" (is "?l = ?r")
  2416 proof-
  2417   {assume "k=0" hence ?thesis by simp}
  2418   moreover
  2419   {fix h assume h: "k = Suc h"
  2420     {fix n
  2421       {assume kn: "k>n" hence "?l $ n = ?r $n" using a0 startsby_zero_power_prefix[OF a0] h
  2422           by (simp add: fps_compose_nth del: power_Suc)}
  2423       moreover
  2424       {assume kn: "k \<le> n"
  2425         hence "?l$n = ?r$n"
  2426           by (simp add: fps_compose_nth mult_delta_left setsum_delta)}
  2427       moreover have "k >n \<or> k\<le> n"  by arith
  2428       ultimately have "?l$n = ?r$n"  by blast}
  2429     then have ?thesis unfolding fps_eq_iff by blast}
  2430   ultimately show ?thesis by (cases k, auto)
  2431 qed
  2432 
  2433 lemma fps_inv_right: assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
  2434   shows "a oo fps_inv a = X"
  2435 proof-
  2436   let ?ia = "fps_inv a"
  2437   let ?iaa = "a oo fps_inv a"
  2438   have th0: "?ia $ 0 = 0" by (simp add: fps_inv_def)
  2439   have th1: "?iaa $ 0 = 0" using a0 a1
  2440     by (simp add: fps_inv_def fps_compose_nth)
  2441   have th2: "X$0 = 0" by simp
  2442   from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
  2443   then have "(a oo fps_inv a) oo a = X oo a"
  2444     by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0])
  2445   with fps_compose_inj_right[OF a0 a1]
  2446   show ?thesis by simp
  2447 qed
  2448 
  2449 lemma fps_inv_deriv:
  2450   assumes a0:"a$0 = (0::'a::{field})" and a1: "a$1 \<noteq> 0"
  2451   shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
  2452 proof-
  2453   let ?ia = "fps_inv a"
  2454   let ?d = "fps_deriv a oo ?ia"
  2455   let ?dia = "fps_deriv ?ia"
  2456   have ia0: "?ia$0 = 0" by (simp add: fps_inv_def)
  2457   have th0: "?d$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth fps_deriv_nth)
  2458   from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
  2459     by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
  2460   hence "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
  2461   with inverse_mult_eq_1[OF th0]
  2462   show "?dia = inverse ?d" by simp
  2463 qed
  2464 
  2465 lemma fps_inv_idempotent: 
  2466   assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
  2467   shows "fps_inv (fps_inv a) = a"
  2468 proof-
  2469   let ?r = "fps_inv"
  2470   have ra0: "?r a $ 0 = 0" by (simp add: fps_inv_def)
  2471   from a1 have ra1: "?r a $ 1 \<noteq> 0" by (simp add: fps_inv_def field_simps)
  2472   have X0: "X$0 = 0" by simp
  2473   from fps_inv[OF ra0 ra1] have "?r (?r a) oo ?r a = X" .
  2474   then have "?r (?r a) oo ?r a oo a = X oo a" by simp
  2475   then have "?r (?r a) oo (?r a oo a) = a" 
  2476     unfolding X_fps_compose_startby0[OF a0]
  2477     unfolding fps_compose_assoc[OF a0 ra0, symmetric] .
  2478   then show ?thesis unfolding fps_inv[OF a0 a1] by simp
  2479 qed
  2480 
  2481 lemma fps_ginv_ginv:
  2482   assumes a0: "a$0 = 0" and a1: "a$1 \<noteq> 0"
  2483   and c0: "c$0 = 0" and  c1: "c$1 \<noteq> 0"
  2484   shows "fps_ginv b (fps_ginv c a) = b oo a oo fps_inv c"
  2485 proof-
  2486   let ?r = "fps_ginv"
  2487   from c0 have rca0: "?r c a $0 = 0" by (simp add: fps_ginv_def)
  2488   from a1 c1 have rca1: "?r c a $ 1 \<noteq> 0" by (simp add: fps_ginv_def field_simps)
  2489   from fps_ginv[OF rca0 rca1] 
  2490   have "?r b (?r c a) oo ?r c a = b" .
  2491   then have "?r b (?r c a) oo ?r c a oo a = b oo a" by simp
  2492   then have "?r b (?r c a) oo (?r c a oo a) = b oo a"
  2493     apply (subst fps_compose_assoc)
  2494     using a0 c0 by (auto simp add: fps_ginv_def)
  2495   then have "?r b (?r c a) oo c = b oo a"
  2496     unfolding fps_ginv[OF a0 a1] .
  2497   then have "?r b (?r c a) oo c oo fps_inv c= b oo a oo fps_inv c" by simp
  2498   then have "?r b (?r c a) oo (c oo fps_inv c) = b oo a oo fps_inv c"
  2499     apply (subst fps_compose_assoc)
  2500     using a0 c0 by (auto simp add: fps_inv_def)
  2501   then show ?thesis unfolding fps_inv_right[OF c0 c1] by simp
  2502 qed
  2503 
  2504 lemma fps_ginv_deriv:
  2505   assumes a0:"a$0 = (0::'a::{field})" and a1: "a$1 \<noteq> 0"
  2506   shows "fps_deriv (fps_ginv b a) = (fps_deriv b / fps_deriv a) oo fps_ginv X a"
  2507 proof-
  2508   let ?ia = "fps_ginv b a"
  2509   let ?iXa = "fps_ginv X a"
  2510   let ?d = "fps_deriv"
  2511   let ?dia = "?d ?ia"
  2512   have iXa0: "?iXa $ 0 = 0" by (simp add: fps_ginv_def)
  2513   have da0: "?d a $ 0 \<noteq> 0" using a1 by simp
  2514   from fps_ginv[OF a0 a1, of b] have "?d (?ia oo a) = fps_deriv b" by simp
  2515   then have "(?d ?ia oo a) * ?d a = ?d b" unfolding fps_compose_deriv[OF a0] .
  2516   then have "(?d ?ia oo a) * ?d a * inverse (?d a) = ?d b * inverse (?d a)" by simp
  2517   then have "(?d ?ia oo a) * (inverse (?d a) * ?d a) = ?d b / ?d a" 
  2518     by (simp add: fps_divide_def)
  2519   then have "(?d ?ia oo a) oo ?iXa =  (?d b / ?d a) oo ?iXa "
  2520     unfolding inverse_mult_eq_1[OF da0] by simp
  2521   then have "?d ?ia oo (a oo ?iXa) =  (?d b / ?d a) oo ?iXa"
  2522     unfolding fps_compose_assoc[OF iXa0 a0] .
  2523   then show ?thesis unfolding fps_inv_ginv[symmetric]
  2524     unfolding fps_inv_right[OF a0 a1] by simp
  2525 qed
  2526 
  2527 subsection{* Elementary series *}
  2528 
  2529 subsubsection{* Exponential series *}
  2530 definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
  2531 
  2532 lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::field_char_0) * E a" (is "?l = ?r")
  2533 proof-
  2534   {fix n
  2535     have "?l$n = ?r $ n"
  2536   apply (auto simp add: E_def field_simps power_Suc[symmetric]simp del: fact_Suc of_nat_Suc power_Suc)
  2537   by (simp add: of_nat_mult ring_simps)}
  2538 then show ?thesis by (simp add: fps_eq_iff)
  2539 qed
  2540 
  2541 lemma E_unique_ODE:
  2542   "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * E (c :: 'a::field_char_0)"
  2543   (is "?lhs \<longleftrightarrow> ?rhs")
  2544 proof-
  2545   {assume d: ?lhs
  2546   from d have th: "\<And>n. a $ Suc n = c * a$n / of_nat (Suc n)"
  2547     by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
  2548   {fix n have "a$n = a$0 * c ^ n/ (of_nat (fact n))"
  2549       apply (induct n)
  2550       apply simp
  2551       unfolding th
  2552       using fact_gt_zero_nat
  2553       apply (simp add: field_simps del: of_nat_Suc fact_Suc)
  2554       apply (drule sym)
  2555       by (simp add: ring_simps of_nat_mult power_Suc)}
  2556   note th' = this
  2557   have ?rhs
  2558     by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro : th')}
  2559 moreover
  2560 {assume h: ?rhs
  2561   have ?lhs
  2562     apply (subst h)
  2563     apply simp
  2564     apply (simp only: h[symmetric])
  2565   by simp}
  2566 ultimately show ?thesis by blast
  2567 qed
  2568 
  2569 lemma E_add_mult: "E (a + b) = E (a::'a::field_char_0) * E b" (is "?l = ?r")
  2570 proof-
  2571   have "fps_deriv (?r) = fps_const (a+b) * ?r"
  2572     by (simp add: fps_const_add[symmetric] ring_simps del: fps_const_add)
  2573   then have "?r = ?l" apply (simp only: E_unique_ODE)
  2574     by (simp add: fps_mult_nth E_def)
  2575   then show ?thesis ..
  2576 qed
  2577 
  2578 lemma E_nth[simp]: "E a $ n = a^n / of_nat (fact n)"
  2579   by (simp add: E_def)
  2580 
  2581 lemma E0[simp]: "E (0::'a::{field}) = 1"
  2582   by (simp add: fps_eq_iff power_0_left)
  2583 
  2584 lemma E_neg: "E (- a) = inverse (E (a::'a::field_char_0))"
  2585 proof-
  2586   from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
  2587     by (simp )
  2588   have th1: "E a $ 0 \<noteq> 0" by simp
  2589   from fps_inverse_unique[OF th1 th0] show ?thesis by simp
  2590 qed
  2591 
  2592 lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::field_char_0)) = (fps_const a)^n * (E a)"
  2593   by (induct n, auto simp add: power_Suc)
  2594 
  2595 lemma X_compose_E[simp]: "X oo E (a::'a::{field}) = E a - 1"
  2596   by (simp add: fps_eq_iff X_fps_compose)
  2597 
  2598 lemma LE_compose:
  2599   assumes a: "a\<noteq>0"
  2600   shows "fps_inv (E a - 1) oo (E a - 1) = X"
  2601   and "(E a - 1) oo fps_inv (E a - 1) = X"
  2602 proof-
  2603   let ?b = "E a - 1"
  2604   have b0: "?b $ 0 = 0" by simp
  2605   have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
  2606   from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" .
  2607   from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" .
  2608 qed
  2609 
  2610 
  2611 lemma fps_const_inverse:
  2612   "a \<noteq> 0 \<Longrightarrow> inverse (fps_const (a::'a::field)) = fps_const (inverse a)"
  2613   apply (auto simp add: fps_eq_iff fps_inverse_def) by (case_tac "n", auto)
  2614 
  2615 lemma inverse_one_plus_X:
  2616   "inverse (1 + X) = Abs_fps (\<lambda>n. (- 1 ::'a::{field})^n)"
  2617   (is "inverse ?l = ?r")
  2618 proof-
  2619   have th: "?l * ?r = 1"
  2620     by (auto simp add: ring_simps fps_eq_iff minus_one_power_iff)
  2621   have th': "?l $ 0 \<noteq> 0" by (simp add: )
  2622   from fps_inverse_unique[OF th' th] show ?thesis .
  2623 qed
  2624 
  2625 lemma E_power_mult: "(E (c::'a::field_char_0))^n = E (of_nat n * c)"
  2626   by (induct n, auto simp add: ring_simps E_add_mult power_Suc)
  2627 
  2628 lemma radical_E:
  2629   assumes r: "r (Suc k) 1 = 1" 
  2630   shows "fps_radical r (Suc k) (E (c::'a::{field_char_0})) = E (c / of_nat (Suc k))"
  2631 proof-
  2632   let ?ck = "(c / of_nat (Suc k))"
  2633   let ?r = "fps_radical r (Suc k)"
  2634   have eq0[simp]: "?ck * of_nat (Suc k) = c" "of_nat (Suc k) * ?ck = c"
  2635     by (simp_all del: of_nat_Suc)
  2636   have th0: "E ?ck ^ (Suc k) = E c" unfolding E_power_mult eq0 ..
  2637   have th: "r (Suc k) (E c $0) ^ Suc k = E c $ 0"
  2638     "r (Suc k) (E c $ 0) = E ?ck $ 0" "E c $ 0 \<noteq> 0" using r by simp_all
  2639   from th0 radical_unique[where r=r and k=k, OF th]
  2640   show ?thesis by auto 
  2641 qed
  2642 
  2643 lemma Ec_E1_eq: 
  2644   "E (1::'a::{field_char_0}) oo (fps_const c * X) = E c"
  2645   apply (auto simp add: fps_eq_iff E_def fps_compose_def power_mult_distrib)
  2646   by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
  2647 
  2648 text{* The generalized binomial theorem as a  consequence of @{thm E_add_mult} *}
  2649 
  2650 lemma gbinomial_theorem: 
  2651   "((a::'a::{field_char_0, division_by_zero})+b) ^ n = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
  2652 proof-
  2653   from E_add_mult[of a b] 
  2654   have "(E (a + b)) $ n = (E a * E b)$n" by simp
  2655   then have "(a + b) ^ n = (\<Sum>i\<Colon>nat = 0\<Colon>nat..n. a ^ i * b ^ (n - i)  * (of_nat (fact n) / of_nat (fact i * fact (n - i))))"
  2656     by (simp add: field_simps fps_mult_nth of_nat_mult[symmetric] setsum_right_distrib)
  2657   then show ?thesis 
  2658     apply simp
  2659     apply (rule setsum_cong2)
  2660     apply simp
  2661     apply (frule binomial_fact[where ?'a = 'a, symmetric])
  2662     by (simp add: field_simps of_nat_mult)
  2663 qed
  2664 
  2665 text{* And the nat-form -- also available from Binomial.thy *}
  2666 lemma binomial_theorem: "(a+b) ^ n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
  2667   using gbinomial_theorem[of "of_nat a" "of_nat b" n]
  2668   unfolding of_nat_add[symmetric] of_nat_power[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric]
  2669   by simp
  2670 
  2671 subsubsection{* Logarithmic series *}
  2672 
  2673 lemma Abs_fps_if_0: 
  2674   "Abs_fps(%n. if n=0 then (v::'a::ring_1) else f n) = fps_const v + X * Abs_fps (%n. f (Suc n))"
  2675   by (auto simp add: fps_eq_iff)
  2676 
  2677 definition L:: "'a::{field_char_0} \<Rightarrow> 'a fps" where 
  2678   "L c \<equiv> fps_const (1/c) * Abs_fps (\<lambda>n. if n = 0 then 0 else (- 1) ^ (n - 1) / of_nat n)"
  2679 
  2680 lemma fps_deriv_L: "fps_deriv (L c) = fps_const (1/c) * inverse (1 + X)"
  2681   unfolding inverse_one_plus_X
  2682   by (simp add: L_def fps_eq_iff del: of_nat_Suc)
  2683 
  2684 lemma L_nth: "L c $ n = (if n=0 then 0 else 1/c * ((- 1) ^ (n - 1) / of_nat n))"
  2685   by (simp add: L_def field_simps)
  2686 
  2687 lemma L_0[simp]: "L c $ 0 = 0" by (simp add: L_def)
  2688 lemma L_E_inv:
  2689   assumes a: "a\<noteq> (0::'a::{field_char_0})"
  2690   shows "L a = fps_inv (E a - 1)" (is "?l = ?r")
  2691 proof-
  2692   let ?b = "E a - 1"
  2693   have b0: "?b $ 0 = 0" by simp
  2694   have b1: "?b $ 1 \<noteq> 0" by (simp add: a)
  2695   have "fps_deriv (E a - 1) oo fps_inv (E a - 1) = (fps_const a * (E a - 1) + fps_const a) oo fps_inv (E a - 1)"
  2696     by (simp add: ring_simps)
  2697   also have "\<dots> = fps_const a * (X + 1)" apply (simp add: fps_compose_add_distrib fps_const_mult_apply_left[symmetric] fps_inv_right[OF b0 b1])
  2698     by (simp add: ring_simps)
  2699   finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
  2700   from fps_inv_deriv[OF b0 b1, unfolded eq]
  2701   have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
  2702     using a 
  2703     by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
  2704   hence "fps_deriv ?l = fps_deriv ?r"
  2705     by (simp add: fps_deriv_L add_commute fps_divide_def divide_inverse)
  2706   then show ?thesis unfolding fps_deriv_eq_iff
  2707     by (simp add: L_nth fps_inv_def)
  2708 qed
  2709 
  2710 lemma L_mult_add: 
  2711   assumes c0: "c\<noteq>0" and d0: "d\<noteq>0"
  2712   shows "L c + L d = fps_const (c+d) * L (c*d)"
  2713   (is "?r = ?l")
  2714 proof-
  2715   from c0 d0 have eq: "1/c + 1/d = (c+d)/(c*d)" by (simp add: field_simps)
  2716   have "fps_deriv ?r = fps_const (1/c + 1/d) * inverse (1 + X)"
  2717     by (simp add: fps_deriv_L fps_const_add[symmetric] algebra_simps del: fps_const_add)
  2718   also have "\<dots> = fps_deriv ?l"
  2719     apply (simp add: fps_deriv_L)
  2720     by (simp add: fps_eq_iff eq)
  2721   finally show ?thesis
  2722     unfolding fps_deriv_eq_iff by simp
  2723 qed
  2724 
  2725 subsubsection{* Binomial series *}
  2726 
  2727 definition "fps_binomial a = Abs_fps (\<lambda>n. a gchoose n)"
  2728 
  2729 lemma fps_binomial_nth[simp]: "fps_binomial a $ n = a gchoose n"
  2730   by (simp add: fps_binomial_def)
  2731 
  2732 lemma fps_binomial_ODE_unique:
  2733   fixes c :: "'a::field_char_0"
  2734   shows "fps_deriv a = (fps_const c * a) / (1 + X) \<longleftrightarrow> a = fps_const (a$0) * fps_binomial c"
  2735   (is "?lhs \<longleftrightarrow> ?rhs")
  2736 proof-
  2737   let ?da = "fps_deriv a"
  2738   let ?x1 = "(1 + X):: 'a fps"
  2739   let ?l = "?x1 * ?da"
  2740   let ?r = "fps_const c * a"
  2741   have x10: "?x1 $ 0 \<noteq> 0" by simp
  2742   have "?l = ?r \<longleftrightarrow> inverse ?x1 * ?l = inverse ?x1 * ?r" by simp
  2743   also have "\<dots> \<longleftrightarrow> ?da = (fps_const c * a) / ?x1"
  2744     apply (simp only: fps_divide_def  mult_assoc[symmetric] inverse_mult_eq_1[OF x10])
  2745     by (simp add: ring_simps)
  2746   finally have eq: "?l = ?r \<longleftrightarrow> ?lhs" by simp
  2747   moreover
  2748   {assume h: "?l = ?r" 
  2749     {fix n
  2750       from h have lrn: "?l $ n = ?r$n" by simp
  2751       
  2752       from lrn 
  2753       have "a$ Suc n = ((c - of_nat n) / of_nat (Suc n)) * a $n" 
  2754         apply (simp add: ring_simps del: of_nat_Suc)
  2755         by (cases n, simp_all add: field_simps del: of_nat_Suc)
  2756     }
  2757     note th0 = this
  2758     {fix n have "a$n = (c gchoose n) * a$0"
  2759       proof(induct n)
  2760         case 0 thus ?case by simp
  2761       next
  2762         case (Suc m)
  2763         thus ?case unfolding th0
  2764           apply (simp add: field_simps del: of_nat_Suc)
  2765           unfolding mult_assoc[symmetric] gbinomial_mult_1
  2766           by (simp add: ring_simps)
  2767       qed}
  2768     note th1 = this
  2769     have ?rhs
  2770       apply (simp add: fps_eq_iff)
  2771       apply (subst th1)
  2772       by (simp add: ring_simps)}
  2773   moreover
  2774   {assume h: ?rhs
  2775   have th00:"\<And>x y. x * (a$0 * y) = a$0 * (x*y)" by (simp add: mult_commute)
  2776     have "?l = ?r" 
  2777       apply (subst h)
  2778       apply (subst (2) h)
  2779       apply (clarsimp simp add: fps_eq_iff ring_simps)
  2780       unfolding mult_assoc[symmetric] th00 gbinomial_mult_1
  2781       by (simp add: ring_simps gbinomial_mult_1)}
  2782   ultimately show ?thesis by blast
  2783 qed
  2784 
  2785 lemma fps_binomial_deriv: "fps_deriv (fps_binomial c) = fps_const c * fps_binomial c / (1 + X)"
  2786 proof-
  2787   let ?a = "fps_binomial c"
  2788   have th0: "?a = fps_const (?a$0) * ?a" by (simp)
  2789   from iffD2[OF fps_binomial_ODE_unique, OF th0] show ?thesis .
  2790 qed
  2791 
  2792 lemma fps_binomial_add_mult: "fps_binomial (c+d) = fps_binomial c * fps_binomial d" (is "?l = ?r")
  2793 proof-
  2794   let ?P = "?r - ?l"
  2795   let ?b = "fps_binomial"
  2796   let ?db = "\<lambda>x. fps_deriv (?b x)"
  2797   have "fps_deriv ?P = ?db c * ?b d + ?b c * ?db d - ?db (c + d)"  by simp
  2798   also have "\<dots> = inverse (1 + X) * (fps_const c * ?b c * ?b d + fps_const d * ?b c * ?b d - fps_const (c+d) * ?b (c + d))"
  2799     unfolding fps_binomial_deriv
  2800     by (simp add: fps_divide_def ring_simps)
  2801   also have "\<dots> = (fps_const (c + d)/ (1 + X)) * ?P"
  2802     by (simp add: ring_simps fps_divide_def fps_const_add[symmetric] del: fps_const_add)
  2803   finally have th0: "fps_deriv ?P = fps_const (c+d) * ?P / (1 + X)"
  2804     by (simp add: fps_divide_def)
  2805   have "?P = fps_const (?P$0) * ?b (c + d)"
  2806     unfolding fps_binomial_ODE_unique[symmetric]
  2807     using th0 by simp
  2808   hence "?P = 0" by (simp add: fps_mult_nth)
  2809   then show ?thesis by simp
  2810 qed
  2811 
  2812 lemma fps_minomial_minus_one: "fps_binomial (- 1) = inverse (1 + X)"
  2813   (is "?l = inverse ?r")
  2814 proof-
  2815   have th: "?r$0 \<noteq> 0" by simp
  2816   have th': "fps_deriv (inverse ?r) = fps_const (- 1) * inverse ?r / (1 + X)"
  2817     by (simp add: fps_inverse_deriv[OF th] fps_divide_def power2_eq_square mult_commute fps_const_neg[symmetric] del: fps_const_neg)
  2818   have eq: "inverse ?r $ 0 = 1"
  2819     by (simp add: fps_inverse_def)
  2820   from iffD1[OF fps_binomial_ODE_unique[of "inverse (1 + X)" "- 1"] th'] eq
  2821   show ?thesis by (simp add: fps_inverse_def)
  2822 qed
  2823 
  2824 text{* Vandermonde's Identity as a consequence *}
  2825 lemma gbinomial_Vandermonde: "setsum (\<lambda>k. (a gchoose k) * (b gchoose (n - k))) {0..n} = (a + b) gchoose n"
  2826 proof-
  2827   let ?ba = "fps_binomial a"
  2828   let ?bb = "fps_binomial b"
  2829   let ?bab = "fps_binomial (a + b)"
  2830   from fps_binomial_add_mult[of a b] have "?bab $ n = (?ba * ?bb)$n" by simp
  2831   then show ?thesis by (simp add: fps_mult_nth)
  2832 qed
  2833 
  2834 lemma binomial_Vandermonde: "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
  2835   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
  2836   
  2837   apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric] of_nat_setsum[symmetric] of_nat_add[symmetric])
  2838   by simp
  2839   
  2840 lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)^2) {0..n} = (2*n) choose n"
  2841   using binomial_Vandermonde[of n n n,symmetric]
  2842   unfolding nat_mult_2 apply (simp add: power2_eq_square)
  2843   apply (rule setsum_cong2)
  2844   by (auto intro:  binomial_symmetric)
  2845 
  2846 lemma Vandermonde_pochhammer_lemma:
  2847   fixes a :: "'a::field_char_0"
  2848   assumes b: "\<forall> j\<in>{0 ..<n}. b \<noteq> of_nat j"
  2849   shows "setsum (%k. (pochhammer (- a) k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer (b - of_nat n + 1) k)) {0..n} = pochhammer (- (a+ b)) n / pochhammer (- b) n" (is "?l = ?r")
  2850 proof-
  2851   let ?m1 = "%m. (- 1 :: 'a) ^ m"
  2852   let ?f = "%m. of_nat (fact m)"
  2853   let ?p = "%(x::'a). pochhammer (- x)"
  2854   from b have bn0: "?p b n \<noteq> 0" unfolding pochhammer_eq_0_iff by simp
  2855   {fix k assume kn: "k \<in> {0..n}"
  2856     {assume c:"pochhammer (b - of_nat n + 1) n = 0"
  2857       then obtain j where j: "j < n" "b - of_nat n + 1 = - of_nat j"
  2858         unfolding pochhammer_eq_0_iff by blast
  2859       from j have "b = of_nat n - of_nat j - of_nat 1" 
  2860         by (simp add: algebra_simps)
  2861       then have "b = of_nat (n - j - 1)" 
  2862         using j kn by (simp add: of_nat_diff)
  2863       with b have False using j by auto}
  2864     then have nz: "pochhammer (1 + b - of_nat n) n \<noteq> 0" 
  2865       by (auto simp add: algebra_simps)
  2866     
  2867     from nz kn have nz': "pochhammer (1 + b - of_nat n) k \<noteq> 0" 
  2868       by (simp add: pochhammer_neq_0_mono)
  2869     {assume k0: "k = 0 \<or> n =0" 
  2870       then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" 
  2871         using kn
  2872         by (cases "k=0", simp_all add: gbinomial_pochhammer)}
  2873     moreover
  2874     {assume n0: "n \<noteq> 0" and k0: "k \<noteq> 0" 
  2875       then obtain m where m: "n = Suc m" by (cases n, auto)
  2876       from k0 obtain h where h: "k = Suc h" by (cases k, auto)
  2877       {assume kn: "k = n"
  2878         then have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
  2879           using kn pochhammer_minus'[where k=k and n=n and b=b]
  2880           apply (simp add:  pochhammer_same)
  2881           using bn0
  2882           by (simp add: field_simps power_add[symmetric])}
  2883       moreover
  2884       {assume nk: "k \<noteq> n"
  2885         have m1nk: "?m1 n = setprod (%i. - 1) {0..m}" 
  2886           "?m1 k = setprod (%i. - 1) {0..h}"
  2887           by (simp_all add: setprod_constant m h)
  2888         from kn nk have kn': "k < n" by simp
  2889         have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
  2890           using bn0 kn 
  2891           unfolding pochhammer_eq_0_iff
  2892           apply auto
  2893           apply (erule_tac x= "n - ka - 1" in allE)
  2894           by (auto simp add: algebra_simps of_nat_diff)
  2895         have eq1: "setprod (%k. (1::'a) + of_nat m - of_nat k) {0 .. h} = setprod of_nat {Suc (m - h) .. Suc m}"        
  2896           apply (rule strong_setprod_reindex_cong[where f="%k. Suc m - k "])
  2897           using kn' h m
  2898           apply (auto simp add: inj_on_def image_def)
  2899           apply (rule_tac x="Suc m - x" in bexI)
  2900           apply (simp_all add: of_nat_diff)
  2901           done
  2902         
  2903         have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
  2904           unfolding m1nk 
  2905           
  2906           unfolding m h pochhammer_Suc_setprod
  2907           apply (simp add: field_simps del: fact_Suc id_def)
  2908           unfolding fact_altdef_nat id_def
  2909           unfolding of_nat_setprod
  2910           unfolding setprod_timesf[symmetric]
  2911           apply auto
  2912           unfolding eq1
  2913           apply (subst setprod_Un_disjoint[symmetric])
  2914           apply (auto)
  2915           apply (rule setprod_cong)
  2916           apply auto
  2917           done
  2918         have th20: "?m1 n * ?p b n = setprod (%i. b - of_nat i) {0..m}"
  2919           unfolding m1nk 
  2920           unfolding m h pochhammer_Suc_setprod
  2921           unfolding setprod_timesf[symmetric]
  2922           apply (rule setprod_cong)
  2923           apply auto
  2924           done
  2925         have th21:"pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {n - k .. n - 1}"
  2926           unfolding h m 
  2927           unfolding pochhammer_Suc_setprod
  2928           apply (rule strong_setprod_reindex_cong[where f="%k. n - 1 - k"])
  2929           using kn
  2930           apply (auto simp add: inj_on_def m h image_def)
  2931           apply (rule_tac x= "m - x" in bexI)
  2932           by (auto simp add: of_nat_diff)
  2933         
  2934         have "?m1 n * ?p b n = pochhammer (b - of_nat n + 1) k * setprod (%i. b - of_nat i) {0.. n - k - 1}"
  2935           unfolding th20 th21
  2936           unfolding h m
  2937           apply (subst setprod_Un_disjoint[symmetric])
  2938           using kn' h m
  2939           apply auto
  2940           apply (rule setprod_cong)
  2941           apply auto
  2942           done
  2943         then have th2: "(?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k = setprod (%i. b - of_nat i) {0.. n - k - 1}" 
  2944           using nz' by (simp add: field_simps)
  2945         have "(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k) = ((?m1 k * ?p (of_nat n) k) / ?f n) * ((?m1 n * ?p b n)/pochhammer (b - of_nat n + 1) k)"
  2946           using bnz0
  2947           by (simp add: field_simps)
  2948         also have "\<dots> = b gchoose (n - k)" 
  2949           unfolding th1 th2
  2950           using kn' by (simp add: gbinomial_def)
  2951         finally have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" by simp}
  2952       ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
  2953         by (cases "k =n", auto)}
  2954     ultimately have "b gchoose (n - k) = (?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)" "pochhammer (1 + b - of_nat n) k \<noteq> 0 "
  2955       using nz' 
  2956       apply (cases "n=0", auto)
  2957       by (cases "k", auto)}
  2958   note th00 = this
  2959   have "?r = ((a + b) gchoose n) * (of_nat (fact n)/ (?m1 n * pochhammer (- b) n))"
  2960     unfolding gbinomial_pochhammer 
  2961     using bn0 by (auto simp add: field_simps)
  2962   also have "\<dots> = ?l"
  2963     unfolding gbinomial_Vandermonde[symmetric]
  2964     apply (simp add: th00)
  2965     unfolding gbinomial_pochhammer
  2966     using bn0 apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
  2967     apply (rule setsum_cong2)
  2968     apply (drule th00(2))
  2969     by (simp add: field_simps power_add[symmetric])
  2970   finally show ?thesis by simp
  2971 qed 
  2972 
  2973     
  2974 lemma Vandermonde_pochhammer:
  2975    fixes a :: "'a::field_char_0"
  2976   assumes c: "ALL i : {0..< n}. c \<noteq> - of_nat i"
  2977   shows "setsum (%k. (pochhammer a k * pochhammer (- (of_nat n)) k) / (of_nat (fact k) * pochhammer c k)) {0..n} = pochhammer (c - a) n / pochhammer c n"
  2978 proof-
  2979   let ?a = "- a"
  2980   let ?b = "c + of_nat n - 1"
  2981   have h: "\<forall> j \<in>{0..< n}. ?b \<noteq> of_nat j" using c
  2982     apply (auto simp add: algebra_simps of_nat_diff)
  2983     apply (erule_tac x= "n - j - 1" in ballE)
  2984     by (auto simp add: of_nat_diff algebra_simps)
  2985   have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
  2986     unfolding pochhammer_minus[OF le_refl]
  2987     by (simp add: algebra_simps)
  2988   have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
  2989     unfolding pochhammer_minus[OF le_refl]
  2990     by simp
  2991   have nz: "pochhammer c n \<noteq> 0" using c
  2992     by (simp add: pochhammer_eq_0_iff)
  2993   from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
  2994   show ?thesis using nz by (simp add: field_simps setsum_right_distrib)
  2995 qed
  2996 
  2997 subsubsection{* Formal trigonometric functions  *}
  2998 
  2999 definition "fps_sin (c::'a::field_char_0) =
  3000   Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
  3001 
  3002 definition "fps_cos (c::'a::field_char_0) =
  3003   Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
  3004 
  3005 lemma fps_sin_deriv:
  3006   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
  3007   (is "?lhs = ?rhs")
  3008 proof (rule fps_ext)
  3009   fix n::nat
  3010     {assume en: "even n"
  3011       have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
  3012       also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
  3013         using en by (simp add: fps_sin_def)
  3014       also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
  3015         unfolding fact_Suc of_nat_mult
  3016         by (simp add: field_simps del: of_nat_add of_nat_Suc)
  3017       also have "\<dots> = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
  3018         by (simp add: field_simps del: of_nat_add of_nat_Suc)
  3019       finally have "?lhs $n = ?rhs$n" using en
  3020         by (simp add: fps_cos_def ring_simps power_Suc )}
  3021     then show "?lhs $ n = ?rhs $ n"
  3022       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
  3023 qed
  3024 
  3025 lemma fps_cos_deriv:
  3026   "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
  3027   (is "?lhs = ?rhs")
  3028 proof (rule fps_ext)
  3029   have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc)
  3030   have th1: "\<And>n. odd n \<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2"
  3031     by (case_tac n, simp_all)
  3032   fix n::nat
  3033     {assume en: "odd n"
  3034       from en have n0: "n \<noteq>0 " by presburger
  3035       have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
  3036       also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
  3037         using en by (simp add: fps_cos_def)
  3038       also have "\<dots> = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
  3039         unfolding fact_Suc of_nat_mult
  3040         by (simp add: field_simps del: of_nat_add of_nat_Suc)
  3041       also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
  3042         by (simp add: field_simps del: of_nat_add of_nat_Suc)
  3043       also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
  3044         unfolding th0 unfolding th1[OF en] by simp
  3045       finally have "?lhs $n = ?rhs$n" using en
  3046         by (simp add: fps_sin_def ring_simps power_Suc)}
  3047     then show "?lhs $ n = ?rhs $ n"
  3048       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def
  3049         fps_cos_def)
  3050 qed
  3051 
  3052 lemma fps_sin_cos_sum_of_squares:
  3053   "fps_cos c ^ 2 + fps_sin c ^ 2 = 1" (is "?lhs = 1")
  3054 proof-
  3055   have "fps_deriv ?lhs = 0"
  3056     apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc)
  3057     by (simp add: ring_simps fps_const_neg[symmetric] del: fps_const_neg)
  3058   then have "?lhs = fps_const (?lhs $ 0)"
  3059     unfolding fps_deriv_eq_0_iff .
  3060   also have "\<dots> = 1"
  3061     by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
  3062   finally show ?thesis .
  3063 qed
  3064 
  3065 lemma divide_eq_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x / a = y \<longleftrightarrow> x = y * a"
  3066 by auto
  3067 
  3068 lemma eq_divide_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x = y / a \<longleftrightarrow> x * a = y"
  3069 by auto
  3070 
  3071 lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0"
  3072 unfolding fps_sin_def by simp
  3073 
  3074 lemma fps_sin_nth_1 [simp]: "fps_sin c $ 1 = c"
  3075 unfolding fps_sin_def by simp
  3076 
  3077 lemma fps_sin_nth_add_2:
  3078   "fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat(n+1) * of_nat(n+2)))"
  3079 unfolding fps_sin_def
  3080 apply (cases n, simp)
  3081 apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
  3082 apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
  3083 done
  3084 
  3085 lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1"
  3086 unfolding fps_cos_def by simp
  3087 
  3088 lemma fps_cos_nth_1 [simp]: "fps_cos c $ 1 = 0"
  3089 unfolding fps_cos_def by simp
  3090 
  3091 lemma fps_cos_nth_add_2:
  3092   "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat(n+1) * of_nat(n+2)))"
  3093 unfolding fps_cos_def
  3094 apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
  3095 apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
  3096 done
  3097 
  3098 lemma nat_induct2:
  3099   "\<lbrakk>P 0; P 1; \<And>n. P n \<Longrightarrow> P (n + 2)\<rbrakk> \<Longrightarrow> P (n::nat)"
  3100 unfolding One_nat_def numeral_2_eq_2
  3101 apply (induct n rule: nat_less_induct)
  3102 apply (case_tac n, simp)
  3103 apply (rename_tac m, case_tac m, simp)
  3104 apply (rename_tac k, case_tac k, simp_all)
  3105 done
  3106 
  3107 lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
  3108 by simp
  3109 
  3110 lemma eq_fps_sin:
  3111   assumes 0: "a $ 0 = 0" and 1: "a $ 1 = c"
  3112   and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
  3113   shows "a = fps_sin c"
  3114 apply (rule fps_ext)
  3115 apply (induct_tac n rule: nat_induct2)
  3116 apply (simp add: fps_sin_nth_0 0)
  3117 apply (simp add: fps_sin_nth_1 1 del: One_nat_def)
  3118 apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
  3119 apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
  3120             del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
  3121 apply (subst minus_divide_left)
  3122 apply (subst eq_divide_iff)
  3123 apply (simp del: of_nat_add of_nat_Suc)
  3124 apply (simp only: mult_ac)
  3125 done
  3126 
  3127 lemma eq_fps_cos:
  3128   assumes 0: "a $ 0 = 1" and 1: "a $ 1 = 0"
  3129   and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
  3130   shows "a = fps_cos c"
  3131 apply (rule fps_ext)
  3132 apply (induct_tac n rule: nat_induct2)
  3133 apply (simp add: fps_cos_nth_0 0)
  3134 apply (simp add: fps_cos_nth_1 1 del: One_nat_def)
  3135 apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
  3136 apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
  3137             del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
  3138 apply (subst minus_divide_left)
  3139 apply (subst eq_divide_iff)
  3140 apply (simp del: of_nat_add of_nat_Suc)
  3141 apply (simp only: mult_ac)
  3142 done
  3143 
  3144 lemma mult_nth_0 [simp]: "(a * b) $ 0 = a $ 0 * b $ 0"
  3145 by (simp add: fps_mult_nth)
  3146 
  3147 lemma mult_nth_1 [simp]: "(a * b) $ 1 = a $ 0 * b $ 1 + a $ 1 * b $ 0"
  3148 by (simp add: fps_mult_nth)
  3149 
  3150 lemma fps_sin_add:
  3151   "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
  3152 apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
  3153 apply (simp del: fps_const_neg fps_const_add fps_const_mult
  3154             add: fps_const_add [symmetric] fps_const_neg [symmetric]
  3155                  fps_sin_deriv fps_cos_deriv algebra_simps)
  3156 done
  3157 
  3158 lemma fps_cos_add:
  3159   "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
  3160 apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
  3161 apply (simp del: fps_const_neg fps_const_add fps_const_mult
  3162             add: fps_const_add [symmetric] fps_const_neg [symmetric]
  3163                  fps_sin_deriv fps_cos_deriv algebra_simps)
  3164 done
  3165 
  3166 lemma fps_sin_even: "fps_sin (- c) = - fps_sin c"
  3167   by (auto simp add: fps_eq_iff fps_sin_def)
  3168 
  3169 lemma fps_cos_odd: "fps_cos (- c) = fps_cos c"
  3170   by (auto simp add: fps_eq_iff fps_cos_def)
  3171 
  3172 definition "fps_tan c = fps_sin c / fps_cos c"
  3173 
  3174 lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)"
  3175 proof-
  3176   have th0: "fps_cos c $ 0 \<noteq> 0" by (simp add: fps_cos_def)
  3177   show ?thesis
  3178     using fps_sin_cos_sum_of_squares[of c]
  3179     apply (simp add: fps_tan_def fps_divide_deriv[OF th0] fps_sin_deriv fps_cos_deriv add: fps_const_neg[symmetric] ring_simps power2_eq_square del: fps_const_neg)
  3180     unfolding right_distrib[symmetric]
  3181     by simp
  3182 qed
  3183 
  3184 text {* Connection to E c over the complex numbers --- Euler and De Moivre*}
  3185 lemma Eii_sin_cos:
  3186   "E (ii * c) = fps_cos c + fps_const ii * fps_sin c "
  3187   (is "?l = ?r")
  3188 proof-
  3189   {fix n::nat
  3190     {assume en: "even n"
  3191       from en obtain m where m: "n = 2*m" 
  3192         unfolding even_mult_two_ex by blast
  3193       
  3194       have "?l $n = ?r$n" 
  3195         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
  3196           power_mult power_minus)}
  3197     moreover
  3198     {assume on: "odd n"
  3199       from on obtain m where m: "n = 2*m + 1" 
  3200         unfolding odd_nat_equiv_def2 by (auto simp add: nat_mult_2)  
  3201       have "?l $n = ?r$n" 
  3202         by (simp add: m fps_sin_def fps_cos_def power_mult_distrib
  3203           power_mult power_minus)}
  3204     ultimately have "?l $n = ?r$n"  by blast}
  3205   then show ?thesis by (simp add: fps_eq_iff)
  3206 qed
  3207 
  3208 lemma E_minus_ii_sin_cos: "E (- (ii * c)) = fps_cos c - fps_const ii * fps_sin c "
  3209   unfolding minus_mult_right Eii_sin_cos by (simp add: fps_sin_even fps_cos_odd)
  3210 
  3211 lemma fps_const_minus: "fps_const (c::'a::group_add) - fps_const d = fps_const (c - d)"
  3212   by (simp add: fps_eq_iff fps_const_def)
  3213 
  3214 lemma fps_number_of_fps_const: "number_of i = fps_const (number_of i :: 'a:: {comm_ring_1, number_ring})"
  3215   apply (subst (2) number_of_eq)
  3216 apply(rule int_induct[of _ 0])
  3217 apply (simp_all add: number_of_fps_def)
  3218 by (simp_all add: fps_const_add[symmetric] fps_const_minus[symmetric])
  3219 
  3220 lemma fps_cos_Eii:
  3221   "fps_cos c = (E (ii * c) + E (- ii * c)) / fps_const 2"
  3222 proof-
  3223   have th: "fps_cos c + fps_cos c = fps_cos c * fps_const 2" 
  3224     by (simp add: fps_eq_iff fps_number_of_fps_const complex_number_of_def[symmetric])
  3225   show ?thesis
  3226   unfolding Eii_sin_cos minus_mult_commute
  3227   by (simp add: fps_sin_even fps_cos_odd fps_number_of_fps_const
  3228     fps_divide_def fps_const_inverse th complex_number_of_def[symmetric])
  3229 qed
  3230 
  3231 lemma fps_sin_Eii:
  3232   "fps_sin c = (E (ii * c) - E (- ii * c)) / fps_const (2*ii)"
  3233 proof-
  3234   have th: "fps_const \<i> * fps_sin c + fps_const \<i> * fps_sin c = fps_sin c * fps_const (2 * ii)" 
  3235     by (simp add: fps_eq_iff fps_number_of_fps_const complex_number_of_def[symmetric])
  3236   show ?thesis
  3237   unfolding Eii_sin_cos minus_mult_commute
  3238   by (simp add: fps_sin_even fps_cos_odd fps_divide_def fps_const_inverse th)
  3239 qed
  3240 
  3241 lemma fps_tan_Eii:
  3242   "fps_tan c = (E (ii * c) - E (- ii * c)) / (fps_const ii * (E (ii * c) + E (- ii * c)))"
  3243   unfolding fps_tan_def fps_sin_Eii fps_cos_Eii mult_minus_left E_neg
  3244   apply (simp add: fps_divide_def fps_inverse_mult fps_const_mult[symmetric] fps_const_inverse del: fps_const_mult)
  3245   by simp
  3246 
  3247 lemma fps_demoivre: "(fps_cos a + fps_const ii * fps_sin a)^n = fps_cos (of_nat n * a) + fps_const ii * fps_sin (of_nat n * a)"
  3248   unfolding Eii_sin_cos[symmetric] E_power_mult
  3249   by (simp add: mult_ac)
  3250 
  3251 subsection {* Hypergeometric series *}
  3252 
  3253 
  3254 definition "F as bs (c::'a::{field_char_0, division_by_zero}) = Abs_fps (%n. (foldl (%r a. r* pochhammer a n) 1 as * c^n)/ (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n)))"
  3255 
  3256 lemma F_nth[simp]: "F as bs c $ n =  (foldl (%r a. r* pochhammer a n) 1 as * c^n)/ (foldl (%r b. r * pochhammer b n) 1 bs * of_nat (fact n))"
  3257   by (simp add: F_def)
  3258 
  3259 lemma foldl_mult_start:
  3260   "foldl (%r x. r * f x) (v::'a::comm_ring_1) as * x = foldl (%r x. r * f x) (v * x) as "
  3261   by (induct as arbitrary: x v, auto simp add: algebra_simps)
  3262 
  3263 lemma foldr_mult_foldl: "foldr (%x r. r * f x) as v = foldl (%r x. r * f x) (v :: 'a::comm_ring_1) as"
  3264   by (induct as arbitrary: v, auto simp add: foldl_mult_start)
  3265 
  3266 lemma F_nth_alt: "F as bs c $ n = foldr (\<lambda>a r. r * pochhammer a n) as (c ^ n) /
  3267     foldr (\<lambda>b r. r * pochhammer b n) bs (of_nat (fact n))"
  3268   by (simp add: foldl_mult_start foldr_mult_foldl)
  3269 
  3270 lemma F_E[simp]: "F [] [] c = E c" 
  3271   by (simp add: fps_eq_iff)
  3272 
  3273 lemma F_1_0[simp]: "F [1] [] c = 1/(1 - fps_const c * X)"
  3274 proof-
  3275   let ?a = "(Abs_fps (\<lambda>n. 1)) oo (fps_const c * X)"
  3276   have th0: "(fps_const c * X) $ 0 = 0" by simp
  3277   show ?thesis unfolding gp[OF th0, symmetric]
  3278     by (auto simp add: fps_eq_iff pochhammer_fact[symmetric] fps_compose_nth power_mult_distrib cond_value_iff setsum_delta' cong del: if_weak_cong)
  3279 qed
  3280 
  3281 lemma F_B[simp]: "F [-a] [] (- 1) = fps_binomial a"
  3282   by (simp add: fps_eq_iff gbinomial_pochhammer algebra_simps)
  3283 
  3284 lemma F_0[simp]: "F as bs c $0 = 1"
  3285   apply simp
  3286   apply (subgoal_tac "ALL as. foldl (%(r::'a) (a::'a). r) 1 as = 1")
  3287   apply auto
  3288   apply (induct_tac as, auto)
  3289   done
  3290 
  3291 lemma foldl_prod_prod: "foldl (%(r::'b::comm_ring_1) (x::'a::comm_ring_1). r * f x) v as * foldl (%r x. r * g x) w as = foldl (%r x. r * f x * g x) (v*w) as"
  3292   by (induct as arbitrary: v w, auto simp add: algebra_simps)
  3293 
  3294 
  3295 lemma F_rec: "F as bs c $ Suc n = ((foldl (%r a. r* (a + of_nat n)) c as)/ (foldl (%r b. r * (b + of_nat n)) (of_nat (Suc n)) bs )) * F as bs c $ n"
  3296   apply (simp del: of_nat_Suc of_nat_add fact_Suc)
  3297   apply (simp add: foldl_mult_start del: fact_Suc of_nat_Suc)
  3298   unfolding foldl_prod_prod[unfolded foldl_mult_start] pochhammer_Suc
  3299   by (simp add: algebra_simps of_nat_mult)
  3300 
  3301 lemma XD_nth[simp]: "XD a $ n = (if n=0 then 0 else of_nat n * a$n)"
  3302   by (simp add: XD_def)
  3303 
  3304 lemma XD_0th[simp]: "XD a $ 0 = 0" by simp
  3305 lemma XD_Suc[simp]:" XD a $ Suc n = of_nat (Suc n) * a $ Suc n" by simp
  3306 
  3307 definition "XDp c a = XD a + fps_const c * a"
  3308 
  3309 lemma XDp_nth[simp]: "XDp c a $ n = (c + of_nat n) * a$n"
  3310   by (simp add: XDp_def algebra_simps)
  3311 
  3312 lemma XDp_commute:
  3313   shows "XDp b o XDp (c::'a::comm_ring_1) = XDp c o XDp b"
  3314   by (auto simp add: XDp_def expand_fun_eq fps_eq_iff algebra_simps)
  3315 
  3316 lemma XDp0[simp]: "XDp 0 = XD"
  3317   by (simp add: expand_fun_eq fps_eq_iff)
  3318 
  3319 lemma XDp_fps_integral[simp]:"XDp 0 (fps_integral a c) = X * a"
  3320   by (simp add: fps_eq_iff fps_integral_def)
  3321 
  3322 lemma F_minus_nat: 
  3323   "F [- of_nat n] [- of_nat (n + m)] (c::'a::{field_char_0, division_by_zero}) $ k = (if k <= n then pochhammer (- of_nat n) k * c ^ k /
  3324     (pochhammer (- of_nat (n + m)) k * of_nat (fact k)) else 0)"
  3325   "F [- of_nat m] [- of_nat (m + n)] (c::'a::{field_char_0, division_by_zero}) $ k = (if k <= m then pochhammer (- of_nat m) k * c ^ k /
  3326     (pochhammer (- of_nat (m + n)) k * of_nat (fact k)) else 0)"
  3327   by (auto simp add: pochhammer_eq_0_iff)
  3328 
  3329 lemma setsum_eq_if: "setsum f {(n::nat) .. m} = (if m < n then 0 else f n + setsum f {n+1 .. m})"
  3330   apply simp
  3331   apply (subst setsum_insert[symmetric])
  3332   by (auto simp add: not_less setsum_head_Suc)
  3333 
  3334 lemma pochhammer_rec_if: 
  3335   "pochhammer a n = (if n = 0 then 1 else a * pochhammer (a + 1) (n - 1))"
  3336   by (cases n, simp_all add: pochhammer_rec)
  3337 
  3338 lemma XDp_foldr_nth[simp]: "foldr (%c r. XDp c o r) cs (%c. XDp c a) c0 $ n = 
  3339   foldr (%c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n"
  3340   by (induct cs arbitrary: c0, auto simp add: algebra_simps)
  3341 
  3342 lemma genric_XDp_foldr_nth:
  3343   assumes 
  3344   f: "ALL n c a. f c a $ n = (of_nat n + k c) * a$n"
  3345 
  3346   shows "foldr (%c r. f c o r) cs (%c. g c a) c0 $ n = 
  3347   foldr (%c r. (k c + of_nat n) * r) cs (g c0 a $ n)"
  3348   by (induct cs arbitrary: c0, auto simp add: algebra_simps f )
  3349 
  3350 end