src/HOL/Library/Formal_Power_Series.thy
 author huffman Fri Feb 13 14:45:10 2009 -0800 (2009-02-13) changeset 29906 80369da39838 parent 29692 121289b1ae27 child 29911 c790a70a3d19 permissions -rw-r--r--
section -> subsection
```     1 (*  Title:      Formal_Power_Series.thy
```
```     2     ID:
```
```     3     Author:     Amine Chaieb, University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header{* A formalization of formal power series *}
```
```     7
```
```     8 theory Formal_Power_Series
```
```     9   imports Main Fact Parity
```
```    10 begin
```
```    11
```
```    12 subsection {* The type of formal power series*}
```
```    13
```
```    14 typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
```
```    15   by simp
```
```    16
```
```    17 text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *}
```
```    18
```
```    19 instantiation fps :: (zero)  zero
```
```    20 begin
```
```    21
```
```    22 definition fps_zero_def: "(0 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). 0)"
```
```    23 instance ..
```
```    24 end
```
```    25
```
```    26 instantiation fps :: ("{one,zero}")  one
```
```    27 begin
```
```    28
```
```    29 definition fps_one_def: "(1 :: 'a fps) \<equiv> Abs_fps (\<lambda>(n::nat). if n = 0 then 1 else 0)"
```
```    30 instance ..
```
```    31 end
```
```    32
```
```    33 instantiation fps :: (plus)  plus
```
```    34 begin
```
```    35
```
```    36 definition fps_plus_def: "op + \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). Rep_fps (f) n + Rep_fps (g) n))"
```
```    37 instance ..
```
```    38 end
```
```    39
```
```    40 instantiation fps :: (minus)  minus
```
```    41 begin
```
```    42
```
```    43 definition fps_minus_def: "op - \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). Rep_fps (f) n - Rep_fps (g) n))"
```
```    44 instance ..
```
```    45 end
```
```    46
```
```    47 instantiation fps :: (uminus)  uminus
```
```    48 begin
```
```    49
```
```    50 definition fps_uminus_def: "uminus \<equiv> (\<lambda>(f::'a fps). Abs_fps (\<lambda>(n::nat). - Rep_fps (f) n))"
```
```    51 instance ..
```
```    52 end
```
```    53
```
```    54 instantiation fps :: ("{comm_monoid_add, times}")  times
```
```    55 begin
```
```    56
```
```    57 definition fps_times_def:
```
```    58   "op * \<equiv> (\<lambda>(f::'a fps) (g:: 'a fps). Abs_fps (\<lambda>(n::nat). setsum (\<lambda>i. Rep_fps (f) i  * Rep_fps (g) (n - i)) {0.. n}))"
```
```    59 instance ..
```
```    60 end
```
```    61
```
```    62 text{* Some useful theorems to get rid of Abs and Rep *}
```
```    63
```
```    64 lemma mem_fps_set_trivial[intro, simp]: "f \<in> fps" unfolding fps_def by blast
```
```    65 lemma Rep_fps_Abs_fps[simp]: "Rep_fps (Abs_fps f) = f"
```
```    66   by (blast intro: Abs_fps_inverse)
```
```    67 lemma Abs_fps_Rep_fps[simp]: "Abs_fps (Rep_fps f) = f"
```
```    68   by (blast intro: Rep_fps_inverse)
```
```    69 lemma Abs_fps_eq[simp]: "Abs_fps f = Abs_fps g \<longleftrightarrow> f = g"
```
```    70 proof-
```
```    71   {assume "f = g" hence "Abs_fps f = Abs_fps g" by simp}
```
```    72   moreover
```
```    73   {assume a: "Abs_fps f = Abs_fps g"
```
```    74     from a have "Rep_fps (Abs_fps f) = Rep_fps (Abs_fps g)" by simp
```
```    75     hence "f = g" by simp}
```
```    76   ultimately show ?thesis by blast
```
```    77 qed
```
```    78
```
```    79 lemma Rep_fps_eq[simp]: "Rep_fps f = Rep_fps g \<longleftrightarrow> f = g"
```
```    80 proof-
```
```    81   {assume "Rep_fps f = Rep_fps g"
```
```    82     hence "Abs_fps (Rep_fps f) = Abs_fps (Rep_fps g)" by simp hence "f=g" by simp}
```
```    83   moreover
```
```    84   {assume "f = g" hence "Rep_fps f = Rep_fps g" by simp}
```
```    85   ultimately show ?thesis by blast
```
```    86 qed
```
```    87
```
```    88 declare atLeastAtMost_iff[presburger]
```
```    89 declare Bex_def[presburger]
```
```    90 declare Ball_def[presburger]
```
```    91
```
```    92 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
```
```    93   by auto
```
```    94 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
```
```    95   by auto
```
```    96
```
```    97 subsection{* Formal power series form a commutative ring with unity, if the range of sequences
```
```    98   they represent is a commutative ring with unity*}
```
```    99
```
```   100 instantiation fps :: (semigroup_add) semigroup_add
```
```   101 begin
```
```   102
```
```   103 instance
```
```   104 proof
```
```   105   fix a b c :: "'a fps" show "a + b + c = a + (b + c)"
```
```   106     by (auto simp add: fps_plus_def expand_fun_eq add_assoc)
```
```   107 qed
```
```   108
```
```   109 end
```
```   110
```
```   111 instantiation fps :: (ab_semigroup_add) ab_semigroup_add
```
```   112 begin
```
```   113
```
```   114 instance by (intro_classes, simp add: fps_plus_def expand_fun_eq add_commute)
```
```   115 end
```
```   116
```
```   117 instantiation fps :: (semiring_1) semigroup_mult
```
```   118 begin
```
```   119
```
```   120 instance
```
```   121 proof
```
```   122   fix a b c :: "'a fps"
```
```   123   let ?a = "Rep_fps a"
```
```   124   let ?b = "Rep_fps b"
```
```   125   let ?c = "Rep_fps c"
```
```   126   let ?x = "\<lambda> i k. if k \<le> i then (1::'a) else 0"
```
```   127   show "a*b*c = a* (b * c)"
```
```   128   proof(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
```
```   129     fix n::nat
```
```   130     let ?r = "\<lambda>i. n - i"
```
```   131     have i: "inj_on ?r {0..n}" by (auto simp add: inj_on_def)
```
```   132     have ri: "{0 .. n} = ?r ` {0..n}" apply (auto simp add: image_iff)
```
```   133       by presburger
```
```   134     let ?f = "\<lambda>i j. ?a j * ?b (i - j) * ?c (n -i)"
```
```   135     let ?g = "\<lambda>i j. ?a i * (?b j * ?c (n - (i + j)))"
```
```   136     have "setsum (\<lambda>i. setsum (?f i) {0..i}) {0..n}
```
```   137       = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..i}) {0..n}"
```
```   138       by (rule setsum_cong2)+ auto
```
```   139     also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f i j * ?x i j) {0..n}) {0..n}"
```
```   140     proof(rule setsum_cong2)
```
```   141       fix i assume i: "i \<in> {0..n}"
```
```   142       have eq: "{0 .. n} = {0 ..i} \<union> {i+1 .. n}" using i by auto
```
```   143       have d: "{0 ..i} \<inter> {i+1 .. n} = {}" using i by auto
```
```   144       have f: "finite {0..i}" "finite {i+1 ..n}" by auto
```
```   145       have s0: "setsum (\<lambda>j. ?f i j * ?x i j) {i+1 ..n} = 0" by simp
```
```   146       show "setsum (\<lambda>j. ?f i j * ?x i j) {0..i} = setsum (\<lambda>j. ?f i j * ?x i j) {0..n}"
```
```   147 	unfolding eq setsum_Un_disjoint[OF f d] s0
```
```   148 	by simp
```
```   149     qed
```
```   150     also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {0 .. n}) {0 .. n}"
```
```   151       by (rule setsum_commute)
```
```   152     also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. ?f j i * ?x j i) {i .. n}) {0 .. n}"
```
```   153       apply(rule setsum_cong2)
```
```   154       apply (rule setsum_mono_zero_right)
```
```   155       apply auto
```
```   156       done
```
```   157     also have "\<dots> = setsum (\<lambda>i. setsum (?g i) {0..n - i}) {0..n}"
```
```   158       apply (rule setsum_cong2)
```
```   159       apply (rule_tac f="\<lambda>i. i + x" in setsum_reindex_cong)
```
```   160       apply (simp add: inj_on_def)
```
```   161       apply (rule set_ext)
```
```   162       apply (presburger add: image_iff)
```
```   163       by (simp add: add_ac mult_assoc)
```
```   164     finally  show "setsum (\<lambda>i. setsum (\<lambda>j. ?a j * ?b (i - j) * ?c (n -i)) {0..i}) {0..n}
```
```   165       = setsum (\<lambda>i. setsum (\<lambda>j. ?a i * (?b j * ?c (n - (i + j)))) {0..n - i}) {0..n}".
```
```   166   qed
```
```   167 qed
```
```   168
```
```   169 end
```
```   170
```
```   171 instantiation fps :: (comm_semiring_1) ab_semigroup_mult
```
```   172 begin
```
```   173
```
```   174 instance
```
```   175 proof
```
```   176   fix a b :: "'a fps"
```
```   177   show "a*b = b*a"
```
```   178   apply(auto simp add: fps_times_def setsum_right_distrib setsum_left_distrib, rule ext)
```
```   179   apply (rule_tac f = "\<lambda>i. n - i" in setsum_reindex_cong)
```
```   180   apply (simp add: inj_on_def)
```
```   181   apply presburger
```
```   182   apply (rule set_ext)
```
```   183   apply (presburger add: image_iff)
```
```   184   by (simp add: mult_commute)
```
```   185 qed
```
```   186 end
```
```   187
```
```   188 instantiation fps :: (monoid_add) monoid_add
```
```   189 begin
```
```   190
```
```   191 instance
```
```   192 proof
```
```   193   fix a :: "'a fps" show "0 + a = a "
```
```   194     by (auto simp add: fps_plus_def fps_zero_def intro: ext)
```
```   195 next
```
```   196   fix a :: "'a fps" show "a + 0 = a "
```
```   197     by (auto simp add: fps_plus_def fps_zero_def intro: ext)
```
```   198 qed
```
```   199
```
```   200 end
```
```   201 instantiation fps :: (comm_monoid_add) comm_monoid_add
```
```   202 begin
```
```   203
```
```   204 instance
```
```   205 proof
```
```   206   fix a :: "'a fps" show "0 + a = a "
```
```   207     by (auto simp add: fps_plus_def fps_zero_def intro: ext)
```
```   208 qed
```
```   209
```
```   210 end
```
```   211
```
```   212 instantiation fps :: (semiring_1) monoid_mult
```
```   213 begin
```
```   214
```
```   215 instance
```
```   216 proof
```
```   217   fix a :: "'a fps" show "1 * a = a"
```
```   218     apply (auto simp add: fps_one_def fps_times_def)
```
```   219     apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
```
```   220     unfolding Abs_fps_eq
```
```   221     apply (rule ext)
```
```   222     by (simp add: cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
```
```   223 next
```
```   224   fix a :: "'a fps" show "a*1 = a"
```
```   225     apply (auto simp add: fps_one_def fps_times_def)
```
```   226     apply (subst (2) Abs_fps_Rep_fps[of a, symmetric])
```
```   227     unfolding Abs_fps_eq
```
```   228     apply (rule ext)
```
```   229     by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong)
```
```   230 qed
```
```   231 end
```
```   232
```
```   233 instantiation fps :: (cancel_semigroup_add) cancel_semigroup_add
```
```   234 begin
```
```   235
```
```   236 instance by (intro_classes) (auto simp add: fps_plus_def expand_fun_eq Rep_fps_eq[symmetric])
```
```   237 end
```
```   238
```
```   239 instantiation fps :: (group_add) group_add
```
```   240 begin
```
```   241
```
```   242 instance
```
```   243 proof
```
```   244   fix a :: "'a fps" show "- a + a = 0"
```
```   245     by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def intro: ext)
```
```   246 next
```
```   247   fix a b :: "'a fps" show "a - b = a + - b"
```
```   248     by (auto simp add: fps_plus_def fps_uminus_def fps_zero_def
```
```   249       fps_minus_def expand_fun_eq diff_minus)
```
```   250 qed
```
```   251 end
```
```   252
```
```   253 context comm_ring_1
```
```   254 begin
```
```   255 subclass group_add proof qed
```
```   256 end
```
```   257
```
```   258 instantiation fps :: (zero_neq_one) zero_neq_one
```
```   259 begin
```
```   260 instance by (intro_classes, auto simp add: zero_neq_one
```
```   261   fps_one_def fps_zero_def expand_fun_eq)
```
```   262 end
```
```   263
```
```   264 instantiation fps :: (semiring_1) semiring
```
```   265 begin
```
```   266
```
```   267 instance
```
```   268 proof
```
```   269   fix a b c :: "'a fps"
```
```   270   show "(a + b) * c = a * c + b*c"
```
```   271     apply (auto simp add: fps_plus_def fps_times_def, rule ext)
```
```   272     unfolding setsum_addf[symmetric]
```
```   273     apply (simp add: ring_simps)
```
```   274     done
```
```   275 next
```
```   276   fix a b c :: "'a fps"
```
```   277   show "a * (b + c) = a * b + a*c"
```
```   278     apply (auto simp add: fps_plus_def fps_times_def, rule ext)
```
```   279     unfolding setsum_addf[symmetric]
```
```   280     apply (simp add: ring_simps)
```
```   281     done
```
```   282 qed
```
```   283 end
```
```   284
```
```   285 instantiation fps :: (semiring_1) semiring_0
```
```   286 begin
```
```   287
```
```   288 instance
```
```   289 proof
```
```   290   fix a:: "'a fps" show "0 * a = 0" by (simp add: fps_zero_def fps_times_def)
```
```   291 next
```
```   292   fix a:: "'a fps" show "a*0 = 0" by (simp add: fps_zero_def fps_times_def)
```
```   293 qed
```
```   294 end
```
```   295
```
```   296 subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
```
```   297
```
```   298 definition fps_nth:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" (infixl "\$" 75)
```
```   299   where "f \$ n = Rep_fps f n"
```
```   300
```
```   301 lemma fps_nth_Abs_fps[simp]: "Abs_fps a \$ n = a n" by (simp add: fps_nth_def)
```
```   302 lemma fps_zero_nth[simp]: "0 \$ n = 0" by (simp add: fps_zero_def)
```
```   303 lemma fps_one_nth[simp]: "1 \$ n = (if n = 0 then 1 else 0)"
```
```   304   by (simp add: fps_one_def)
```
```   305 lemma fps_add_nth[simp]: "(f + g) \$ n = f\$n + g\$n" by (simp add: fps_plus_def fps_nth_def)
```
```   306 lemma fps_mult_nth: "(f * g) \$ n = setsum (\<lambda>i. f\$i * g\$(n - i)) {0..n}"
```
```   307   by (simp add: fps_times_def fps_nth_def)
```
```   308 lemma fps_neg_nth[simp]: "(- f) \$n = - (f \$n)" by (simp add: fps_nth_def fps_uminus_def)
```
```   309 lemma fps_sub_nth[simp]: "(f - g)\$n = f\$n - g\$n" by (simp add: fps_nth_def fps_minus_def)
```
```   310
```
```   311 lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f \$n \<noteq> 0)"
```
```   312 proof-
```
```   313   {assume "f \<noteq> 0"
```
```   314     hence "Rep_fps f \<noteq> Rep_fps 0" by simp
```
```   315     hence "\<exists>n. f \$n \<noteq> 0" by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
```
```   316   moreover
```
```   317   {assume "\<exists>n. f\$n \<noteq> 0" and "f = 0"
```
```   318     then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
```
```   319   ultimately show ?thesis by blast
```
```   320 qed
```
```   321
```
```   322 lemma fps_nonzero_nth_minimal: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f \$n \<noteq> 0 \<and> (\<forall>m <n. f \$m = 0))"
```
```   323 proof-
```
```   324   let ?S = "{n. f\$n \<noteq> 0}"
```
```   325   {assume "\<exists>n. f\$n \<noteq> 0 \<and> (\<forall>m <n. f \$m = 0)" and "f = 0"
```
```   326     then have False by (simp add: expand_fun_eq fps_zero_def fps_nth_def )}
```
```   327   moreover
```
```   328   {assume f0: "f \<noteq> 0"
```
```   329     from f0 fps_nonzero_nth have ex: "\<exists>n. f\$n \<noteq> 0" by blast
```
```   330     hence Se: "?S\<noteq> {}" by blast
```
```   331     from ex obtain n where n: "f\$n \<noteq> 0" by blast
```
```   332     from n have nS: "n \<in> ?S" by blast
```
```   333         let ?U = "?S \<inter> {0..n}"
```
```   334     have fU: "finite ?U" by auto
```
```   335     from n have Ue: "?U \<noteq> {}" by auto
```
```   336     let ?m = "Min ?U"
```
```   337     have mU: "?m \<in> ?U" using Min_in[OF fU Ue] .
```
```   338     hence mn: "?m \<le> n" by simp
```
```   339     from mU have mf: "f \$ ?m \<noteq> 0" by blast
```
```   340     {fix m assume m: "m < ?m" and f: "f \$m \<noteq> 0"
```
```   341       from m mn have mn': "m < n" by arith
```
```   342       with f have mU': "m \<in> ?U" by simp
```
```   343       from Min_le[OF fU mU'] m have False by arith}
```
```   344     hence "\<forall>m <?m. f\$m = 0" by blast
```
```   345     with mf have "\<exists> n. f \$n \<noteq> 0 \<and> (\<forall>m <n. f \$m = 0)" by blast}
```
```   346   ultimately show ?thesis by blast
```
```   347 qed
```
```   348
```
```   349 lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f \$ n = g \$n)"
```
```   350   by (auto simp add: fps_nth_def Rep_fps_eq[unfolded expand_fun_eq])
```
```   351
```
```   352 lemma fps_setsum_nth: "(setsum f S) \$ n = setsum (\<lambda>k. (f k) \$ n) S"
```
```   353 proof-
```
```   354   {assume "\<not> finite S" hence ?thesis by simp}
```
```   355   moreover
```
```   356   {assume fS: "finite S"
```
```   357     have ?thesis by(induct rule: finite_induct[OF fS]) auto}
```
```   358   ultimately show ?thesis by blast
```
```   359 qed
```
```   360
```
```   361 subsection{* Injection of the basic ring elements and multiplication by scalars *}
```
```   362
```
```   363 definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
```
```   364 lemma fps_const_0_eq_0[simp]: "fps_const 0 = 0" by (simp add: fps_const_def fps_eq_iff)
```
```   365 lemma fps_const_1_eq_1[simp]: "fps_const 1 = 1" by (simp add: fps_const_def fps_eq_iff)
```
```   366 lemma fps_const_neg[simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
```
```   367   by (simp add: fps_uminus_def fps_const_def fps_eq_iff)
```
```   368 lemma fps_const_add[simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
```
```   369   by (simp add: fps_plus_def fps_const_def fps_eq_iff)
```
```   370 lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
```
```   371   by (auto simp add: fps_times_def fps_const_def fps_eq_iff intro: setsum_0')
```
```   372
```
```   373 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)"
```
```   374   unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)
```
```   375 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)"
```
```   376   unfolding fps_eq_iff fps_add_nth by (simp add: fps_const_def)
```
```   377
```
```   378 lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f\$n)"
```
```   379   unfolding fps_eq_iff fps_mult_nth
```
```   380   by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
```
```   381 lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f\$n * c)"
```
```   382   unfolding fps_eq_iff fps_mult_nth
```
```   383   by (simp add: fps_const_def cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
```
```   384
```
```   385 lemma fps_const_nth[simp]: "(fps_const c) \$n = (if n = 0 then c else 0)"
```
```   386   by (simp add: fps_const_def)
```
```   387
```
```   388 lemma fps_mult_left_const_nth[simp]: "(fps_const (c::'a::semiring_1) * f)\$n = c* f\$n"
```
```   389   by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
```
```   390
```
```   391 lemma fps_mult_right_const_nth[simp]: "(f * fps_const (c::'a::semiring_1))\$n = f\$n * c"
```
```   392   by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
```
```   393
```
```   394 subsection {* Formal power series form an integral domain*}
```
```   395
```
```   396 instantiation fps :: (ring_1) ring_1
```
```   397 begin
```
```   398
```
```   399 instance by (intro_classes, auto simp add: diff_minus left_distrib)
```
```   400 end
```
```   401
```
```   402 instantiation fps :: (comm_ring_1) comm_ring_1
```
```   403 begin
```
```   404
```
```   405 instance by (intro_classes, auto simp add: diff_minus left_distrib)
```
```   406 end
```
```   407 instantiation fps :: ("{ring_no_zero_divisors, comm_ring_1}") ring_no_zero_divisors
```
```   408 begin
```
```   409
```
```   410 instance
```
```   411 proof
```
```   412   fix a b :: "'a fps"
```
```   413   assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
```
```   414   then obtain i j where i: "a\$i\<noteq>0" "\<forall>k<i. a\$k=0"
```
```   415     and j: "b\$j \<noteq>0" "\<forall>k<j. b\$k =0" unfolding fps_nonzero_nth_minimal
```
```   416     by blast+
```
```   417   have eq: "({0..i+j} -{i}) \<union> {i} = {0..i+j}" by auto
```
```   418   have d: "({0..i+j} -{i}) \<inter> {i} = {}" by auto
```
```   419   have f: "finite ({0..i+j} -{i})" "finite {i}" by auto
```
```   420   have th0: "setsum (\<lambda>k. a\$k * b\$(i+j - k)) ({0..i+j} -{i}) = 0"
```
```   421     apply (rule setsum_0')
```
```   422     apply auto
```
```   423     apply (case_tac "aa < i")
```
```   424     using i
```
```   425     apply auto
```
```   426     apply (subgoal_tac "b \$ (i+j - aa) = 0")
```
```   427     apply blast
```
```   428     apply (rule j(2)[rule_format])
```
```   429     by arith
```
```   430   have "(a*b) \$ (i+j) =  setsum (\<lambda>k. a\$k * b\$(i+j - k)) {0..i+j}"
```
```   431     by (rule fps_mult_nth)
```
```   432   hence "(a*b) \$ (i+j) = a\$i * b\$j"
```
```   433     unfolding setsum_Un_disjoint[OF f d, unfolded eq] th0 by simp
```
```   434   with i j have "(a*b) \$ (i+j) \<noteq> 0" by simp
```
```   435   then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
```
```   436 qed
```
```   437 end
```
```   438
```
```   439 instantiation fps :: (idom) idom
```
```   440 begin
```
```   441
```
```   442 instance ..
```
```   443 end
```
```   444
```
```   445 subsection{* Inverses of formal power series *}
```
```   446
```
```   447 declare setsum_cong[fundef_cong]
```
```   448
```
```   449
```
```   450 instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse
```
```   451 begin
```
```   452
```
```   453 fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where
```
```   454   "natfun_inverse f 0 = inverse (f\$0)"
```
```   455 | "natfun_inverse f n = - inverse (f\$0) * setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}"
```
```   456
```
```   457 definition fps_inverse_def:
```
```   458   "inverse f = (if f\$0 = 0 then 0 else Abs_fps (natfun_inverse f))"
```
```   459 definition fps_divide_def: "divide \<equiv> (\<lambda>(f::'a fps) g. f * inverse g)"
```
```   460 instance ..
```
```   461 end
```
```   462
```
```   463 lemma fps_inverse_zero[simp]:
```
```   464   "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
```
```   465   by (simp add: fps_zero_def fps_inverse_def)
```
```   466
```
```   467 lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
```
```   468   apply (auto simp add: fps_one_def fps_inverse_def expand_fun_eq)
```
```   469   by (case_tac x, auto)
```
```   470
```
```   471 instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}")  division_by_zero
```
```   472 begin
```
```   473 instance
```
```   474   apply (intro_classes)
```
```   475   by (rule fps_inverse_zero)
```
```   476 end
```
```   477
```
```   478 lemma inverse_mult_eq_1[intro]: assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   479   shows "inverse f * f = 1"
```
```   480 proof-
```
```   481   have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
```
```   482   from f0 have ifn: "\<And>n. inverse f \$ n = natfun_inverse f n"
```
```   483     by (simp add: fps_inverse_def)
```
```   484   from f0 have th0: "(inverse f * f) \$ 0 = 1"
```
```   485     by (simp add: fps_inverse_def fps_one_def fps_mult_nth)
```
```   486   {fix n::nat assume np: "n >0 "
```
```   487     from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
```
```   488     have d: "{0} \<inter> {1 .. n} = {}" by auto
```
```   489     have f: "finite {0::nat}" "finite {1..n}" by auto
```
```   490     from f0 np have th0: "- (inverse f\$n) =
```
```   491       (setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n}) / (f\$0)"
```
```   492       by (cases n, simp_all add: divide_inverse fps_inverse_def fps_nth_def ring_simps)
```
```   493     from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
```
```   494     have th1: "setsum (\<lambda>i. f\$i * natfun_inverse f (n - i)) {1..n} =
```
```   495       - (f\$0) * (inverse f)\$n"
```
```   496       by (simp add: ring_simps)
```
```   497     have "(f * inverse f) \$ n = (\<Sum>i = 0..n. f \$i * natfun_inverse f (n - i))"
```
```   498       unfolding fps_mult_nth ifn ..
```
```   499     also have "\<dots> = f\$0 * natfun_inverse f n
```
```   500       + (\<Sum>i = 1..n. f\$i * natfun_inverse f (n-i))"
```
```   501       unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
```
```   502       by simp
```
```   503     also have "\<dots> = 0" unfolding th1 ifn by simp
```
```   504     finally have "(inverse f * f)\$n = 0" unfolding c . }
```
```   505   with th0 show ?thesis by (simp add: fps_eq_iff)
```
```   506 qed
```
```   507
```
```   508 lemma fps_inverse_0_iff[simp]: "(inverse f)\$0 = (0::'a::division_ring) \<longleftrightarrow> f\$0 = 0"
```
```   509   apply (simp add: fps_inverse_def)
```
```   510   by (metis fps_nth_def fps_nth_def inverse_zero_imp_zero)
```
```   511
```
```   512 lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f \$0 = 0"
```
```   513 proof-
```
```   514   {assume "f\$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
```
```   515   moreover
```
```   516   {assume h: "inverse f = 0" and c: "f \$0 \<noteq> 0"
```
```   517     from inverse_mult_eq_1[OF c] h have False by simp}
```
```   518   ultimately show ?thesis by blast
```
```   519 qed
```
```   520
```
```   521 lemma fps_inverse_idempotent[intro]: assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   522   shows "inverse (inverse f) = f"
```
```   523 proof-
```
```   524   from f0 have if0: "inverse f \$ 0 \<noteq> 0" by simp
```
```   525   from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
```
```   526   have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
```
```   527   then show ?thesis using f0 unfolding mult_cancel_left by simp
```
```   528 qed
```
```   529
```
```   530 lemma fps_inverse_unique: assumes f0: "f\$0 \<noteq> (0::'a::field)" and fg: "f*g = 1"
```
```   531   shows "inverse f = g"
```
```   532 proof-
```
```   533   from inverse_mult_eq_1[OF f0] fg
```
```   534   have th0: "inverse f * f = g * f" by (simp add: mult_ac)
```
```   535   then show ?thesis using f0  unfolding mult_cancel_right
```
```   536     unfolding Rep_fps_eq[of f 0, symmetric]
```
```   537     by (auto simp add: fps_zero_def expand_fun_eq fps_nth_def)
```
```   538 qed
```
```   539
```
```   540 lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
```
```   541   = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
```
```   542   apply (rule fps_inverse_unique)
```
```   543   apply simp
```
```   544   apply (simp add: fps_eq_iff fps_nth_def fps_times_def fps_one_def)
```
```   545 proof(clarsimp)
```
```   546   fix n::nat assume n: "n > 0"
```
```   547   let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
```
```   548   let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
```
```   549   let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
```
```   550   have th1: "setsum ?f {0..n} = setsum ?g {0..n}"
```
```   551     by (rule setsum_cong2) auto
```
```   552   have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
```
```   553     using n apply - by (rule setsum_cong2) auto
```
```   554   have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
```
```   555   from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto
```
```   556   have f: "finite {0.. n - 1}" "finite {n}" by auto
```
```   557   show "setsum ?f {0..n} = 0"
```
```   558     unfolding th1
```
```   559     apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
```
```   560     unfolding th2
```
```   561     by(simp add: setsum_delta)
```
```   562 qed
```
```   563
```
```   564 subsection{* Formal Derivatives, and the McLauren theorem around 0*}
```
```   565
```
```   566 definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f \$ (n + 1))"
```
```   567
```
```   568 lemma fps_deriv_nth[simp]: "fps_deriv f \$ n = of_nat (n +1) * f \$ (n+1)" by (simp add: fps_deriv_def)
```
```   569
```
```   570 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"
```
```   571   unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: ring_simps)
```
```   572
```
```   573 lemma fps_deriv_mult[simp]:
```
```   574   fixes f :: "('a :: comm_ring_1) fps"
```
```   575   shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
```
```   576 proof-
```
```   577   let ?D = "fps_deriv"
```
```   578   {fix n::nat
```
```   579     let ?Zn = "{0 ..n}"
```
```   580     let ?Zn1 = "{0 .. n + 1}"
```
```   581     let ?f = "\<lambda>i. i + 1"
```
```   582     have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
```
```   583     have eq: "{1.. n+1} = ?f ` {0..n}" by auto
```
```   584     let ?g = "\<lambda>i. of_nat (i+1) * g \$ (i+1) * f \$ (n - i) +
```
```   585         of_nat (i+1)* f \$ (i+1) * g \$ (n - i)"
```
```   586     let ?h = "\<lambda>i. of_nat i * g \$ i * f \$ ((n+1) - i) +
```
```   587         of_nat i* f \$ i * g \$ ((n + 1) - i)"
```
```   588     {fix k assume k: "k \<in> {0..n}"
```
```   589       have "?h (k + 1) = ?g k" using k by auto}
```
```   590     note th0 = this
```
```   591     have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
```
```   592     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"
```
```   593       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
```
```   594       apply (simp add: inj_on_def Ball_def)
```
```   595       apply presburger
```
```   596       apply (rule set_ext)
```
```   597       apply (presburger add: image_iff)
```
```   598       by simp
```
```   599     have s1: "setsum (\<lambda>i. f \$ i * g \$ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f \$ (n + 1 - i) * g \$ i) ?Zn1"
```
```   600       apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
```
```   601       apply (simp add: inj_on_def Ball_def)
```
```   602       apply presburger
```
```   603       apply (rule set_ext)
```
```   604       apply (presburger add: image_iff)
```
```   605       by simp
```
```   606     have "(f * ?D g + ?D f * g)\$n = (?D g * f + ?D f * g)\$n" by (simp only: mult_commute)
```
```   607     also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
```
```   608       by (simp add: fps_mult_nth setsum_addf[symmetric])
```
```   609     also have "\<dots> = setsum ?h {1..n+1}"
```
```   610       using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
```
```   611     also have "\<dots> = setsum ?h {0..n+1}"
```
```   612       apply (rule setsum_mono_zero_left)
```
```   613       apply simp
```
```   614       apply (simp add: subset_eq)
```
```   615       unfolding eq'
```
```   616       by simp
```
```   617     also have "\<dots> = (fps_deriv (f * g)) \$ n"
```
```   618       apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
```
```   619       unfolding s0 s1
```
```   620       unfolding setsum_addf[symmetric] setsum_right_distrib
```
```   621       apply (rule setsum_cong2)
```
```   622       by (auto simp add: of_nat_diff ring_simps)
```
```   623     finally have "(f * ?D g + ?D f * g) \$ n = ?D (f*g) \$ n" .}
```
```   624   then show ?thesis unfolding fps_eq_iff by auto
```
```   625 qed
```
```   626
```
```   627 lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
```
```   628   by (simp add: fps_uminus_def fps_eq_iff fps_deriv_def fps_nth_def)
```
```   629 lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
```
```   630   using fps_deriv_linear[of 1 f 1 g] by simp
```
```   631
```
```   632 lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
```
```   633   unfolding diff_minus by simp
```
```   634
```
```   635 lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
```
```   636   by (simp add: fps_deriv_def fps_const_def fps_zero_def)
```
```   637
```
```   638 lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
```
```   639   by simp
```
```   640
```
```   641 lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
```
```   642   by (simp add: fps_deriv_def fps_eq_iff)
```
```   643
```
```   644 lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
```
```   645   by (simp add: fps_deriv_def fps_eq_iff )
```
```   646
```
```   647 lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
```
```   648   by simp
```
```   649
```
```   650 lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
```
```   651 proof-
```
```   652   {assume "\<not> finite S" hence ?thesis by simp}
```
```   653   moreover
```
```   654   {assume fS: "finite S"
```
```   655     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
```
```   656   ultimately show ?thesis by blast
```
```   657 qed
```
```   658
```
```   659 lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f\$0 :: 'a::{idom,semiring_char_0}))"
```
```   660 proof-
```
```   661   {assume "f= fps_const (f\$0)" hence "fps_deriv f = fps_deriv (fps_const (f\$0))" by simp
```
```   662     hence "fps_deriv f = 0" by simp }
```
```   663   moreover
```
```   664   {assume z: "fps_deriv f = 0"
```
```   665     hence "\<forall>n. (fps_deriv f)\$n = 0" by simp
```
```   666     hence "\<forall>n. f\$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
```
```   667     hence "f = fps_const (f\$0)"
```
```   668       apply (clarsimp simp add: fps_eq_iff fps_const_def)
```
```   669       apply (erule_tac x="n - 1" in allE)
```
```   670       by simp}
```
```   671   ultimately show ?thesis by blast
```
```   672 qed
```
```   673
```
```   674 lemma fps_deriv_eq_iff:
```
```   675   fixes f:: "('a::{idom,semiring_char_0}) fps"
```
```   676   shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f\$0 - g\$0) + g)"
```
```   677 proof-
```
```   678   have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
```
```   679   also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)\$0)" unfolding fps_deriv_eq_0_iff ..
```
```   680   finally show ?thesis by (simp add: ring_simps)
```
```   681 qed
```
```   682
```
```   683 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)"
```
```   684   apply auto unfolding fps_deriv_eq_iff by blast
```
```   685
```
```   686
```
```   687 fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
```
```   688   "fps_nth_deriv 0 f = f"
```
```   689 | "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
```
```   690
```
```   691 lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
```
```   692   by (induct n arbitrary: f, auto)
```
```   693
```
```   694 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"
```
```   695   by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)
```
```   696
```
```   697 lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
```
```   698   by (induct n arbitrary: f, simp_all)
```
```   699
```
```   700 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"
```
```   701   using fps_nth_deriv_linear[of n 1 f 1 g] by simp
```
```   702
```
```   703 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"
```
```   704   unfolding diff_minus fps_nth_deriv_add by simp
```
```   705
```
```   706 lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
```
```   707   by (induct n, simp_all )
```
```   708
```
```   709 lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
```
```   710   by (induct n, simp_all )
```
```   711
```
```   712 lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
```
```   713   by (cases n, simp_all)
```
```   714
```
```   715 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"
```
```   716   using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
```
```   717
```
```   718 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"
```
```   719   using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
```
```   720
```
```   721 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"
```
```   722 proof-
```
```   723   {assume "\<not> finite S" hence ?thesis by simp}
```
```   724   moreover
```
```   725   {assume fS: "finite S"
```
```   726     have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
```
```   727   ultimately show ?thesis by blast
```
```   728 qed
```
```   729
```
```   730 lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) \$ 0 = of_nat (fact k) * f\$(k)"
```
```   731   by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)
```
```   732
```
```   733 subsection {* Powers*}
```
```   734
```
```   735 instantiation fps :: (semiring_1) power
```
```   736 begin
```
```   737
```
```   738 fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
```
```   739   "fps_pow 0 f = 1"
```
```   740 | "fps_pow (Suc n) f = f * fps_pow n f"
```
```   741
```
```   742 definition fps_power_def: "power (f::'a fps) n = fps_pow n f"
```
```   743 instance ..
```
```   744 end
```
```   745
```
```   746 instantiation fps :: (comm_ring_1) recpower
```
```   747 begin
```
```   748 instance
```
```   749   apply (intro_classes)
```
```   750   by (simp_all add: fps_power_def)
```
```   751 end
```
```   752
```
```   753 lemma eq_neg_iff_add_eq_0: "(a::'a::ring) = -b \<longleftrightarrow> a + b = 0"
```
```   754 proof-
```
```   755   {assume "a = -b" hence "b + a = b + -b" by simp
```
```   756     hence "a + b = 0" by (simp add: ring_simps)}
```
```   757   moreover
```
```   758   {assume "a + b = 0" hence "a + b - b = -b" by simp
```
```   759     hence "a = -b" by simp}
```
```   760   ultimately show ?thesis by blast
```
```   761 qed
```
```   762
```
```   763 lemma fps_sqrare_eq_iff: "(a:: 'a::idom fps)^ 2 = b^2  \<longleftrightarrow> (a = b \<or> a = -b)"
```
```   764 proof-
```
```   765   {assume "a = b \<or> a = -b" hence "a^2 = b^2" by auto}
```
```   766   moreover
```
```   767   {assume "a^2 = b^2 "
```
```   768     hence "a^2 - b^2 = 0" by simp
```
```   769     hence "(a-b) * (a+b) = 0" by (simp add: power2_eq_square ring_simps)
```
```   770     hence "a = b \<or> a = -b" by (simp add: eq_neg_iff_add_eq_0)}
```
```   771   ultimately show ?thesis by blast
```
```   772 qed
```
```   773
```
```   774 lemma fps_power_zeroth_eq_one: "a\$0 =1 \<Longrightarrow> a^n \$ 0 = (1::'a::semiring_1)"
```
```   775   by (induct n, auto simp add: fps_power_def fps_times_def fps_nth_def fps_one_def)
```
```   776
```
```   777 lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)\$0 =1 \<Longrightarrow> a^n \$ 1 = of_nat n * a\$1"
```
```   778 proof(induct n)
```
```   779   case 0 thus ?case by (simp add: fps_power_def)
```
```   780 next
```
```   781   case (Suc n)
```
```   782   note h = Suc.hyps[OF `a\$0 = 1`]
```
```   783   show ?case unfolding power_Suc fps_mult_nth
```
```   784     using h `a\$0 = 1`  fps_power_zeroth_eq_one[OF `a\$0=1`] by (simp add: ring_simps)
```
```   785 qed
```
```   786
```
```   787 lemma startsby_one_power:"a \$ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n \$ 0 = 1"
```
```   788   by (induct n, auto simp add: fps_power_def fps_mult_nth)
```
```   789
```
```   790 lemma startsby_zero_power:"a \$0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n \$0 = 0"
```
```   791   by (induct n, auto simp add: fps_power_def fps_mult_nth)
```
```   792
```
```   793 lemma startsby_power:"a \$0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n \$0 = v^n"
```
```   794   by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)
```
```   795
```
```   796 lemma startsby_zero_power_iff[simp]:
```
```   797   "a^n \$0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a\$0 = 0)"
```
```   798 apply (rule iffI)
```
```   799 apply (induct n, auto simp add: power_Suc fps_mult_nth)
```
```   800 by (rule startsby_zero_power, simp_all)
```
```   801
```
```   802 lemma startsby_zero_power_prefix:
```
```   803   assumes a0: "a \$0 = (0::'a::idom)"
```
```   804   shows "\<forall>n < k. a ^ k \$ n = 0"
```
```   805   using a0
```
```   806 proof(induct k rule: nat_less_induct)
```
```   807   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)"
```
```   808   let ?ths = "\<forall>m<k. a ^ k \$ m = 0"
```
```   809   {assume "k = 0" then have ?ths by simp}
```
```   810   moreover
```
```   811   {fix l assume k: "k = Suc l"
```
```   812     {fix m assume mk: "m < k"
```
```   813       {assume "m=0" hence "a^k \$ m = 0" using startsby_zero_power[of a k] k a0
```
```   814 	  by simp}
```
```   815       moreover
```
```   816       {assume m0: "m \<noteq> 0"
```
```   817 	have "a ^k \$ m = (a^l * a) \$m" by (simp add: k power_Suc mult_commute)
```
```   818 	also have "\<dots> = (\<Sum>i = 0..m. a ^ l \$ i * a \$ (m - i))" by (simp add: fps_mult_nth)
```
```   819 	also have "\<dots> = 0" apply (rule setsum_0')
```
```   820 	  apply auto
```
```   821 	  apply (case_tac "aa = m")
```
```   822 	  using a0
```
```   823 	  apply simp
```
```   824 	  apply (rule H[rule_format])
```
```   825 	  using a0 k mk by auto
```
```   826 	finally have "a^k \$ m = 0" .}
```
```   827     ultimately have "a^k \$ m = 0" by blast}
```
```   828     hence ?ths by blast}
```
```   829   ultimately show ?ths by (cases k, auto)
```
```   830 qed
```
```   831
```
```   832 lemma startsby_zero_setsum_depends:
```
```   833   assumes a0: "a \$0 = (0::'a::idom)" and kn: "n \<ge> k"
```
```   834   shows "setsum (\<lambda>i. (a ^ i)\$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)\$k) {0 .. k}"
```
```   835   apply (rule setsum_mono_zero_right)
```
```   836   using kn apply auto
```
```   837   apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```   838   by arith
```
```   839
```
```   840 lemma startsby_zero_power_nth_same: assumes a0: "a\$0 = (0::'a::{recpower, idom})"
```
```   841   shows "a^n \$ n = (a\$1) ^ n"
```
```   842 proof(induct n)
```
```   843   case 0 thus ?case by (simp add: power_0)
```
```   844 next
```
```   845   case (Suc n)
```
```   846   have "a ^ Suc n \$ (Suc n) = (a^n * a)\$(Suc n)" by (simp add: ring_simps power_Suc)
```
```   847   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
```
```   848   also have "\<dots> = setsum (\<lambda>i. a^n\$i * a \$ (Suc n - i)) {n .. Suc n}"
```
```   849     apply (rule setsum_mono_zero_right)
```
```   850     apply simp
```
```   851     apply clarsimp
```
```   852     apply clarsimp
```
```   853     apply (rule startsby_zero_power_prefix[rule_format, OF a0])
```
```   854     apply arith
```
```   855     done
```
```   856   also have "\<dots> = a^n \$ n * a\$1" using a0 by simp
```
```   857   finally show ?case using Suc.hyps by (simp add: power_Suc)
```
```   858 qed
```
```   859
```
```   860 lemma fps_inverse_power:
```
```   861   fixes a :: "('a::{field, recpower}) fps"
```
```   862   shows "inverse (a^n) = inverse a ^ n"
```
```   863 proof-
```
```   864   {assume a0: "a\$0 = 0"
```
```   865     hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
```
```   866     {assume "n = 0" hence ?thesis by simp}
```
```   867     moreover
```
```   868     {assume n: "n > 0"
```
```   869       from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
```
```   870 	by (simp add: fps_inverse_def)}
```
```   871     ultimately have ?thesis by blast}
```
```   872   moreover
```
```   873   {assume a0: "a\$0 \<noteq> 0"
```
```   874     have ?thesis
```
```   875       apply (rule fps_inverse_unique)
```
```   876       apply (simp add: a0)
```
```   877       unfolding power_mult_distrib[symmetric]
```
```   878       apply (rule ssubst[where t = "a * inverse a" and s= 1])
```
```   879       apply simp_all
```
```   880       apply (subst mult_commute)
```
```   881       by (rule inverse_mult_eq_1[OF a0])}
```
```   882   ultimately show ?thesis by blast
```
```   883 qed
```
```   884
```
```   885 lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
```
```   886   apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)
```
```   887   by (case_tac n, auto simp add: power_Suc ring_simps)
```
```   888
```
```   889 lemma fps_inverse_deriv:
```
```   890   fixes a:: "('a :: field) fps"
```
```   891   assumes a0: "a\$0 \<noteq> 0"
```
```   892   shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
```
```   893 proof-
```
```   894   from inverse_mult_eq_1[OF a0]
```
```   895   have "fps_deriv (inverse a * a) = 0" by simp
```
```   896   hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
```
```   897   hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
```
```   898   with inverse_mult_eq_1[OF a0]
```
```   899   have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
```
```   900     unfolding power2_eq_square
```
```   901     apply (simp add: ring_simps)
```
```   902     by (simp add: mult_assoc[symmetric])
```
```   903   hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
```
```   904     by simp
```
```   905   then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)
```
```   906 qed
```
```   907
```
```   908 lemma fps_inverse_mult:
```
```   909   fixes a::"('a :: field) fps"
```
```   910   shows "inverse (a * b) = inverse a * inverse b"
```
```   911 proof-
```
```   912   {assume a0: "a\$0 = 0" hence ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```   913     from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
```
```   914     have ?thesis unfolding th by simp}
```
```   915   moreover
```
```   916   {assume b0: "b\$0 = 0" hence ab0: "(a*b)\$0 = 0" by (simp add: fps_mult_nth)
```
```   917     from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
```
```   918     have ?thesis unfolding th by simp}
```
```   919   moreover
```
```   920   {assume a0: "a\$0 \<noteq> 0" and b0: "b\$0 \<noteq> 0"
```
```   921     from a0 b0 have ab0:"(a*b) \$ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
```
```   922     from inverse_mult_eq_1[OF ab0]
```
```   923     have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
```
```   924     then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
```
```   925       by (simp add: ring_simps)
```
```   926     then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
```
```   927 ultimately show ?thesis by blast
```
```   928 qed
```
```   929
```
```   930 lemma fps_inverse_deriv':
```
```   931   fixes a:: "('a :: field) fps"
```
```   932   assumes a0: "a\$0 \<noteq> 0"
```
```   933   shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
```
```   934   using fps_inverse_deriv[OF a0]
```
```   935   unfolding power2_eq_square fps_divide_def
```
```   936     fps_inverse_mult by simp
```
```   937
```
```   938 lemma inverse_mult_eq_1': assumes f0: "f\$0 \<noteq> (0::'a::field)"
```
```   939   shows "f * inverse f= 1"
```
```   940   by (metis mult_commute inverse_mult_eq_1 f0)
```
```   941
```
```   942 lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
```
```   943   assumes a0: "b\$0 \<noteq> 0"
```
```   944   shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
```
```   945   using fps_inverse_deriv[OF a0]
```
```   946   by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
```
```   947
```
```   948 subsection{* The eXtractor series X*}
```
```   949
```
```   950 lemma minus_one_power_iff: "(- (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else - 1)"
```
```   951   by (induct n, auto)
```
```   952
```
```   953 definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
```
```   954
```
```   955 lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
```
```   956   = 1 - X"
```
```   957   by (simp add: fps_inverse_gp fps_eq_iff X_def fps_minus_def fps_one_def)
```
```   958
```
```   959 lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   960 proof-
```
```   961   {assume n: "n \<noteq> 0"
```
```   962     have fN: "finite {0 .. n}" by simp
```
```   963     have "(X * f) \$n = (\<Sum>i = 0..n. X \$ i * f \$ (n - i))" by (simp add: fps_mult_nth)
```
```   964     also have "\<dots> = f \$ (n - 1)"
```
```   965       using n by (simp add: X_def cond_value_iff cond_application_beta setsum_delta[OF fN]
```
```   966 	del: One_nat_def cong del:  if_weak_cong)
```
```   967   finally have ?thesis using n by simp }
```
```   968   moreover
```
```   969   {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
```
```   970   ultimately show ?thesis by blast
```
```   971 qed
```
```   972
```
```   973 lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) \$n = (if n = 0 then 0 else f \$ (n - 1))"
```
```   974   by (metis X_mult_nth mult_commute)
```
```   975
```
```   976 lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
```
```   977 proof(induct k)
```
```   978   case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)
```
```   979 next
```
```   980   case (Suc k)
```
```   981   {fix m
```
```   982     have "(X^Suc k) \$ m = (if m = 0 then (0::'a) else (X^k) \$ (m - 1))"
```
```   983       by (simp add: power_Suc del: One_nat_def)
```
```   984     then     have "(X^Suc k) \$ m = (if m = Suc k then (1::'a) else 0)"
```
```   985       using Suc.hyps by (auto cong del: if_weak_cong)}
```
```   986   then show ?case by (simp add: fps_eq_iff)
```
```   987 qed
```
```   988
```
```   989 lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   990   apply (induct k arbitrary: n)
```
```   991   apply (simp)
```
```   992   unfolding power_Suc mult_assoc
```
```   993   by (case_tac n, auto)
```
```   994
```
```   995 lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) \$n = (if n < k then 0 else f \$ (n - k))"
```
```   996   by (metis X_power_mult_nth mult_commute)
```
```   997 lemma fps_deriv_X[simp]: "fps_deriv X = 1"
```
```   998   by (simp add: fps_deriv_def X_def fps_eq_iff)
```
```   999
```
```  1000 lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
```
```  1001   by (cases "n", simp_all)
```
```  1002
```
```  1003 lemma X_nth[simp]: "X\$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
```
```  1004 lemma X_power_nth[simp]: "(X^k) \$n = (if n = k then 1 else (0::'a::comm_ring_1))"
```
```  1005   by (simp add: X_power_iff)
```
```  1006
```
```  1007 lemma fps_inverse_X_plus1:
```
```  1008   "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{recpower, field})) ^ n)" (is "_ = ?r")
```
```  1009 proof-
```
```  1010   have eq: "(1 + X) * ?r = 1"
```
```  1011     unfolding minus_one_power_iff
```
```  1012     apply (auto simp add: ring_simps fps_eq_iff)
```
```  1013     by presburger+
```
```  1014   show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
```
```  1015 qed
```
```  1016
```
```  1017
```
```  1018 subsection{* Integration *}
```
```  1019 definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a\$(n - 1) / of_nat n))"
```
```  1020
```
```  1021 lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
```
```  1022   by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
```
```  1023
```
```  1024 lemma fps_integral_linear: "fps_integral (fps_const (a::'a::{field, ring_char_0}) * f + fps_const b * g) (a*a0 + b*b0) = fps_const a * fps_integral f a0 + fps_const b * fps_integral g b0" (is "?l = ?r")
```
```  1025 proof-
```
```  1026   have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
```
```  1027   moreover have "?l\$0 = ?r\$0" by (simp add: fps_integral_def)
```
```  1028   ultimately show ?thesis
```
```  1029     unfolding fps_deriv_eq_iff by auto
```
```  1030 qed
```
```  1031
```
```  1032 subsection {* Composition of FPSs *}
```
```  1033 definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
```
```  1034   fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n})"
```
```  1035
```
```  1036 lemma fps_compose_nth: "(a oo b)\$n = setsum (\<lambda>i. a\$i * (b^i\$n)) {0..n}" by (simp add: fps_compose_def)
```
```  1037
```
```  1038 lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
```
```  1039   by (auto simp add: fps_compose_def X_power_iff fps_eq_iff cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
```
```  1040
```
```  1041 lemma fps_const_compose[simp]:
```
```  1042   "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
```
```  1043   apply (auto simp add: fps_eq_iff fps_compose_nth fps_mult_nth
```
```  1044   cond_application_beta cond_value_iff cong del: if_weak_cong)
```
```  1045   by (simp add: setsum_delta )
```
```  1046
```
```  1047 lemma X_fps_compose_startby0[simp]: "a\$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
```
```  1048   apply (auto simp add: fps_compose_def fps_eq_iff cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
```
```  1049   apply (simp add: power_Suc)
```
```  1050   apply (subgoal_tac "n = 0")
```
```  1051   by simp_all
```
```  1052
```
```  1053
```
```  1054 subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
```
```  1055
```
```  1056 subsubsection {* Rule 1 *}
```
```  1057   (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
```
```  1058
```
```  1059 lemma fps_power_mult_eq_shift:
```
```  1060   "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k}" (is "?lhs = ?rhs")
```
```  1061 proof-
```
```  1062   {fix n:: nat
```
```  1063     have "?lhs \$ n = (if n < Suc k then 0 else a n)"
```
```  1064       unfolding X_power_mult_nth by auto
```
```  1065     also have "\<dots> = ?rhs \$ n"
```
```  1066     proof(induct k)
```
```  1067       case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
```
```  1068     next
```
```  1069       case (Suc k)
```
```  1070       note th = Suc.hyps[symmetric]
```
```  1071       have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. Suc k})\$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) \$ n" by (simp add: ring_simps)
```
```  1072       also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)\$n"
```
```  1073 	using th
```
```  1074 	unfolding fps_sub_nth by simp
```
```  1075       also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
```
```  1076 	unfolding X_power_mult_right_nth
```
```  1077 	apply (auto simp add: not_less fps_const_def)
```
```  1078 	apply (rule cong[of a a, OF refl])
```
```  1079 	by arith
```
```  1080       finally show ?case by simp
```
```  1081     qed
```
```  1082     finally have "?lhs \$ n = ?rhs \$ n"  .}
```
```  1083   then show ?thesis by (simp add: fps_eq_iff)
```
```  1084 qed
```
```  1085
```
```  1086 subsubsection{* Rule 2*}
```
```  1087
```
```  1088   (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
```
```  1089   (* If f reprents {a_n} and P is a polynomial, then
```
```  1090         P(xD) f represents {P(n) a_n}*)
```
```  1091
```
```  1092 definition "XD = op * X o fps_deriv"
```
```  1093
```
```  1094 lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
```
```  1095   by (simp add: XD_def ring_simps)
```
```  1096
```
```  1097 lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
```
```  1098   by (simp add: XD_def ring_simps)
```
```  1099
```
```  1100 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)"
```
```  1101   by simp
```
```  1102
```
```  1103 lemma XDN_linear: "(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)"
```
```  1104   by (induct n, simp_all)
```
```  1105
```
```  1106 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)
```
```  1107
```
```  1108 lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a\$n)"
```
```  1109 by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
```
```  1110
```
```  1111 subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
```
```  1112 subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
```
```  1113
```
```  1114 lemma fps_divide_X_minus1_setsum_lemma:
```
```  1115   "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1116 proof-
```
```  1117   let ?X = "X::('a::comm_ring_1) fps"
```
```  1118   let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1119   have th0: "\<And>i. (1 - (X::'a fps)) \$ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
```
```  1120   {fix n:: nat
```
```  1121     {assume "n=0" hence "a\$n = ((1 - ?X) * ?sa) \$ n"
```
```  1122 	by (simp add: fps_mult_nth)}
```
```  1123     moreover
```
```  1124     {assume n0: "n \<noteq> 0"
```
```  1125       then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
```
```  1126 	"{0..n - 1}\<union>{n} = {0..n}"
```
```  1127 	apply (simp_all add: expand_set_eq) by presburger+
```
```  1128       have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}"
```
```  1129 	"{0..n - 1}\<inter>{n} ={}" using n0
```
```  1130 	by (simp_all add: expand_set_eq, presburger+)
```
```  1131       have f: "finite {0}" "finite {1}" "finite {2 .. n}"
```
```  1132 	"finite {0 .. n - 1}" "finite {n}" by simp_all
```
```  1133     have "((1 - ?X) * ?sa) \$ n = setsum (\<lambda>i. (1 - ?X)\$ i * ?sa \$ (n - i)) {0 .. n}"
```
```  1134       by (simp add: fps_mult_nth)
```
```  1135     also have "\<dots> = a\$n" unfolding th0
```
```  1136       unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
```
```  1137       unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
```
```  1138       apply (simp)
```
```  1139       unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
```
```  1140       by simp
```
```  1141     finally have "a\$n = ((1 - ?X) * ?sa) \$ n" by simp}
```
```  1142   ultimately have "a\$n = ((1 - ?X) * ?sa) \$ n" by blast}
```
```  1143 then show ?thesis
```
```  1144   unfolding fps_eq_iff by blast
```
```  1145 qed
```
```  1146
```
```  1147 lemma fps_divide_X_minus1_setsum:
```
```  1148   "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a \$ i) {0..n})"
```
```  1149 proof-
```
```  1150   let ?X = "1 - (X::('a::field) fps)"
```
```  1151   have th0: "?X \$ 0 \<noteq> 0" by simp
```
```  1152   have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op \$ a) {0..n}) * inverse ?X"
```
```  1153     using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
```
```  1154     by (simp add: fps_divide_def mult_assoc)
```
```  1155   also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op \$ a) {0..n}) "
```
```  1156     by (simp add: mult_ac)
```
```  1157   finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
```
```  1158 qed
```
```  1159
```
```  1160 subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
```
```  1161   finite product of FPS, also the relvant instance of powers of a FPS*}
```
```  1162
```
```  1163 definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
```
```  1164
```
```  1165 lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
```
```  1166   apply (auto simp add: natpermute_def)
```
```  1167   apply (case_tac x, auto)
```
```  1168   done
```
```  1169
```
```  1170 lemma foldl_add_start0:
```
```  1171   "foldl op + x xs = x + foldl op + (0::nat) xs"
```
```  1172   apply (induct xs arbitrary: x)
```
```  1173   apply simp
```
```  1174   unfolding foldl.simps
```
```  1175   apply atomize
```
```  1176   apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
```
```  1177   apply (erule_tac x="x + a" in allE)
```
```  1178   apply (erule_tac x="a" in allE)
```
```  1179   apply simp
```
```  1180   apply assumption
```
```  1181   done
```
```  1182
```
```  1183 lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
```
```  1184   apply (induct ys arbitrary: x xs)
```
```  1185   apply auto
```
```  1186   apply (subst (2) foldl_add_start0)
```
```  1187   apply simp
```
```  1188   apply (subst (2) foldl_add_start0)
```
```  1189   by simp
```
```  1190
```
```  1191 lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
```
```  1192 proof(induct xs arbitrary: x)
```
```  1193   case Nil thus ?case by simp
```
```  1194 next
```
```  1195   case (Cons a as x)
```
```  1196   have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
```
```  1197     apply (rule setsum_reindex_cong [where f=Suc])
```
```  1198     by (simp_all add: inj_on_def)
```
```  1199   have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
```
```  1200   have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
```
```  1201   have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
```
```  1202   have "foldl op + x (a#as) = x + foldl op + a as "
```
```  1203     apply (subst foldl_add_start0)    by simp
```
```  1204   also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
```
```  1205   also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
```
```  1206     unfolding eq[symmetric]
```
```  1207     unfolding setsum_Un_disjoint[OF f d, unfolded seq]
```
```  1208     by simp
```
```  1209   finally show ?case  .
```
```  1210 qed
```
```  1211
```
```  1212
```
```  1213 lemma append_natpermute_less_eq:
```
```  1214   assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
```
```  1215 proof-
```
```  1216   {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
```
```  1217     hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
```
```  1218   note th = this
```
```  1219   {from th show "foldl op + 0 xs \<le> n" by simp}
```
```  1220   {from th show "foldl op + 0 ys \<le> n" by simp}
```
```  1221 qed
```
```  1222
```
```  1223 lemma natpermute_split:
```
```  1224   assumes mn: "h \<le> k"
```
```  1225   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)")
```
```  1226 proof-
```
```  1227   {fix l assume l: "l \<in> ?R"
```
```  1228     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
```
```  1229     from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
```
```  1230     from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
```
```  1231     have "l \<in> ?L" using leq xs ys h
```
```  1232       apply simp
```
```  1233       apply (clarsimp simp add: natpermute_def simp del: foldl_append)
```
```  1234       apply (simp add: foldl_add_append[unfolded foldl_append])
```
```  1235       unfolding xs' ys'
```
```  1236       using mn xs ys
```
```  1237       unfolding natpermute_def by simp}
```
```  1238   moreover
```
```  1239   {fix l assume l: "l \<in> natpermute n k"
```
```  1240     let ?xs = "take h l"
```
```  1241     let ?ys = "drop h l"
```
```  1242     let ?m = "foldl op + 0 ?xs"
```
```  1243     from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
```
```  1244     have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)
```
```  1245     have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
```
```  1246       by (simp add: natpermute_def)
```
```  1247     from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
```
```  1248     from xs ys ls have "l \<in> ?R"
```
```  1249       apply auto
```
```  1250       apply (rule bexI[where x = "?m"])
```
```  1251       apply (rule exI[where x = "?xs"])
```
```  1252       apply (rule exI[where x = "?ys"])
```
```  1253       using ls l unfolding foldl_add_append
```
```  1254       by (auto simp add: natpermute_def)}
```
```  1255   ultimately show ?thesis by blast
```
```  1256 qed
```
```  1257
```
```  1258 lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
```
```  1259   by (auto simp add: natpermute_def)
```
```  1260 lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
```
```  1261   apply (auto simp add: set_replicate_conv_if natpermute_def)
```
```  1262   apply (rule nth_equalityI)
```
```  1263   by simp_all
```
```  1264
```
```  1265 lemma natpermute_finite: "finite (natpermute n k)"
```
```  1266 proof(induct k arbitrary: n)
```
```  1267   case 0 thus ?case
```
```  1268     apply (subst natpermute_split[of 0 0, simplified])
```
```  1269     by (simp add: natpermute_0)
```
```  1270 next
```
```  1271   case (Suc k)
```
```  1272   then show ?case unfolding natpermute_split[of k "Suc k", simplified]
```
```  1273     apply -
```
```  1274     apply (rule finite_UN_I)
```
```  1275     apply simp
```
```  1276     unfolding One_nat_def[symmetric] natlist_trivial_1
```
```  1277     apply simp
```
```  1278     unfolding image_Collect[symmetric]
```
```  1279     unfolding Collect_def mem_def
```
```  1280     apply (rule finite_imageI)
```
```  1281     apply blast
```
```  1282     done
```
```  1283 qed
```
```  1284
```
```  1285 lemma natpermute_contain_maximal:
```
```  1286   "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
```
```  1287   (is "?A = ?B")
```
```  1288 proof-
```
```  1289   {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
```
```  1290     from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
```
```  1291       unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
```
```  1292     have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
```
```  1293     have f: "finite({0..k} - {i})" "finite {i}" by auto
```
```  1294     have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
```
```  1295     from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
```
```  1296       unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
```
```  1297     also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
```
```  1298       unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
```
```  1299     finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
```
```  1300     from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
```
```  1301     from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
```
```  1302       unfolding length_replicate  by arith+
```
```  1303     have "xs = replicate (k+1) 0 [i := n]"
```
```  1304       apply (rule nth_equalityI)
```
```  1305       unfolding xsl length_list_update length_replicate
```
```  1306       apply simp
```
```  1307       apply clarify
```
```  1308       unfolding nth_list_update[OF i'(1)]
```
```  1309       using i zxs
```
```  1310       by (case_tac "ia=i", auto simp del: replicate.simps)
```
```  1311     then have "xs \<in> ?B" using i by blast}
```
```  1312   moreover
```
```  1313   {fix i assume i: "i \<in> {0..k}"
```
```  1314     let ?xs = "replicate (k+1) 0 [i:=n]"
```
```  1315     have nxs: "n \<in> set ?xs"
```
```  1316       apply (rule set_update_memI) using i by simp
```
```  1317     have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
```
```  1318     have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
```
```  1319       unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
```
```  1320     also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
```
```  1321       apply (rule setsum_cong2) by (simp del: replicate.simps)
```
```  1322     also have "\<dots> = n" using i by (simp add: setsum_delta)
```
```  1323     finally
```
```  1324     have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def
```
```  1325       by blast
```
```  1326     then have "?xs \<in> ?A"  using nxs  by blast}
```
```  1327   ultimately show ?thesis by auto
```
```  1328 qed
```
```  1329
```
```  1330     (* The general form *)
```
```  1331 lemma fps_setprod_nth:
```
```  1332   fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
```
```  1333   shows "(setprod a {0 .. m})\$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1334   (is "?P m n")
```
```  1335 proof(induct m arbitrary: n rule: nat_less_induct)
```
```  1336   fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
```
```  1337   {assume m0: "m = 0"
```
```  1338     hence "?P m n" apply simp
```
```  1339       unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
```
```  1340   moreover
```
```  1341   {fix k assume k: "m = Suc k"
```
```  1342     have km: "k < m" using k by arith
```
```  1343     have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger
```
```  1344     have f0: "finite {0 .. k}" "finite {m}" by auto
```
```  1345     have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
```
```  1346     have "(setprod a {0 .. m}) \$ n = (setprod a {0 .. k} * a m) \$ n"
```
```  1347       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
```
```  1348     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))"
```
```  1349       unfolding fps_mult_nth H[rule_format, OF km] ..
```
```  1350     also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j \$ v ! j)"
```
```  1351       apply (simp add: k)
```
```  1352       unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
```
```  1353       apply (subst setsum_UN_disjoint)
```
```  1354       apply simp
```
```  1355       apply simp
```
```  1356       unfolding image_Collect[symmetric]
```
```  1357       apply clarsimp
```
```  1358       apply (rule finite_imageI)
```
```  1359       apply (rule natpermute_finite)
```
```  1360       apply (clarsimp simp add: expand_set_eq)
```
```  1361       apply auto
```
```  1362       apply (rule setsum_cong2)
```
```  1363       unfolding setsum_left_distrib
```
```  1364       apply (rule sym)
```
```  1365       apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
```
```  1366       apply (simp add: inj_on_def)
```
```  1367       apply auto
```
```  1368       unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
```
```  1369       apply (clarsimp simp add: natpermute_def nth_append)
```
```  1370       apply (rule_tac f="\<lambda>x. x * a (Suc k) \$ (n - foldl op + 0 aa)" in cong[OF refl])
```
```  1371       apply (rule setprod_cong)
```
```  1372       apply simp
```
```  1373       apply simp
```
```  1374       done
```
```  1375     finally have "?P m n" .}
```
```  1376   ultimately show "?P m n " by (cases m, auto)
```
```  1377 qed
```
```  1378
```
```  1379 text{* The special form for powers *}
```
```  1380 lemma fps_power_nth_Suc:
```
```  1381   fixes m :: nat and a :: "('a::comm_ring_1) fps"
```
```  1382   shows "(a ^ Suc m)\$n = setsum (\<lambda>v. setprod (\<lambda>j. a \$ (v!j)) {0..m}) (natpermute n (m+1))"
```
```  1383 proof-
```
```  1384   have f: "finite {0 ..m}" by simp
```
```  1385   have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
```
```  1386   show ?thesis unfolding th0 fps_setprod_nth ..
```
```  1387 qed
```
```  1388 lemma fps_power_nth:
```
```  1389   fixes m :: nat and a :: "('a::comm_ring_1) fps"
```
```  1390   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))"
```
```  1391   by (cases m, simp_all add: fps_power_nth_Suc)
```
```  1392
```
```  1393 lemma fps_nth_power_0:
```
```  1394   fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps"
```
```  1395   shows "(a ^m)\$0 = (a\$0) ^ m"
```
```  1396 proof-
```
```  1397   {assume "m=0" hence ?thesis by simp}
```
```  1398   moreover
```
```  1399   {fix n assume m: "m = Suc n"
```
```  1400     have c: "m = card {0..n}" using m by simp
```
```  1401    have "(a ^m)\$0 = setprod (\<lambda>i. a\$0) {0..n}"
```
```  1402      apply (simp add: m fps_power_nth del: replicate.simps)
```
```  1403      apply (rule setprod_cong)
```
```  1404      by (simp_all del: replicate.simps)
```
```  1405    also have "\<dots> = (a\$0) ^ m"
```
```  1406      unfolding c by (rule setprod_constant, simp)
```
```  1407    finally have ?thesis .}
```
```  1408  ultimately show ?thesis by (cases m, auto)
```
```  1409 qed
```
```  1410
```
```  1411 lemma fps_compose_inj_right:
```
```  1412   assumes a0: "a\$0 = (0::'a::{recpower,idom})"
```
```  1413   and a1: "a\$1 \<noteq> 0"
```
```  1414   shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
```
```  1415 proof-
```
```  1416   {assume ?rhs then have "?lhs" by simp}
```
```  1417   moreover
```
```  1418   {assume h: ?lhs
```
```  1419     {fix n have "b\$n = c\$n"
```
```  1420       proof(induct n rule: nat_less_induct)
```
```  1421 	fix n assume H: "\<forall>m<n. b\$m = c\$m"
```
```  1422 	{assume n0: "n=0"
```
```  1423 	  from h have "(b oo a)\$n = (c oo a)\$n" by simp
```
```  1424 	  hence "b\$n = c\$n" using n0 by (simp add: fps_compose_nth)}
```
```  1425 	moreover
```
```  1426 	{fix n1 assume n1: "n = Suc n1"
```
```  1427 	  have f: "finite {0 .. n1}" "finite {n}" by simp_all
```
```  1428 	  have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto
```
```  1429 	  have d: "{0 .. n1} \<inter> {n} = {}" using n1 by auto
```
```  1430 	  have seq: "(\<Sum>i = 0..n1. b \$ i * a ^ i \$ n) = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n)"
```
```  1431 	    apply (rule setsum_cong2)
```
```  1432 	    using H n1 by auto
```
```  1433 	  have th0: "(b oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + b\$n * (a\$1)^n"
```
```  1434 	    unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq] seq
```
```  1435 	    using startsby_zero_power_nth_same[OF a0]
```
```  1436 	    by simp
```
```  1437 	  have th1: "(c oo a) \$n = (\<Sum>i = 0..n1. c \$ i * a ^ i \$ n) + c\$n * (a\$1)^n"
```
```  1438 	    unfolding fps_compose_nth setsum_Un_disjoint[OF f d, unfolded eq]
```
```  1439 	    using startsby_zero_power_nth_same[OF a0]
```
```  1440 	    by simp
```
```  1441 	  from h[unfolded fps_eq_iff, rule_format, of n] th0 th1 a1
```
```  1442 	  have "b\$n = c\$n" by auto}
```
```  1443 	ultimately show "b\$n = c\$n" by (cases n, auto)
```
```  1444       qed}
```
```  1445     then have ?rhs by (simp add: fps_eq_iff)}
```
```  1446   ultimately show ?thesis by blast
```
```  1447 qed
```
```  1448
```
```  1449
```
```  1450 subsection {* Radicals *}
```
```  1451
```
```  1452 declare setprod_cong[fundef_cong]
```
```  1453 function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a::{field, recpower}) fps \<Rightarrow> nat \<Rightarrow> 'a" where
```
```  1454   "radical r 0 a 0 = 1"
```
```  1455 | "radical r 0 a (Suc n) = 0"
```
```  1456 | "radical r (Suc k) a 0 = r (Suc k) (a\$0)"
```
```  1457 | "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)"
```
```  1458 by pat_completeness auto
```
```  1459
```
```  1460 termination radical
```
```  1461 proof
```
```  1462   let ?R = "measure (\<lambda>(r, k, a, n). n)"
```
```  1463   {
```
```  1464     show "wf ?R" by auto}
```
```  1465   {fix r k a n xs i
```
```  1466     assume xs: "xs \<in> {xs \<in> natpermute (Suc n) (Suc k). Suc n \<notin> set xs}" and i: "i \<in> {0..k}"
```
```  1467     {assume c: "Suc n \<le> xs ! i"
```
```  1468       from xs i have "xs !i \<noteq> Suc n" by (auto simp add: in_set_conv_nth natpermute_def)
```
```  1469       with c have c': "Suc n < xs!i" by arith
```
```  1470       have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
```
```  1471       have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
```
```  1472       have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
```
```  1473       from xs have "Suc n = foldl op + 0 xs" by (simp add: natpermute_def)
```
```  1474       also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
```
```  1475 	by (simp add: natpermute_def)
```
```  1476       also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  1477 	unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  1478 	unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
```
```  1479 	by simp
```
```  1480       finally have False using c' by simp}
```
```  1481     then show "((r,Suc k,a,xs!i), r, Suc k, a, Suc n) \<in> ?R"
```
```  1482       apply auto by (metis not_less)}
```
```  1483   {fix r k a n
```
```  1484     show "((r,Suc k, a, 0),r, Suc k, a, Suc n) \<in> ?R" by simp}
```
```  1485 qed
```
```  1486
```
```  1487 definition "fps_radical r n a = Abs_fps (radical r n a)"
```
```  1488
```
```  1489 lemma fps_radical0[simp]: "fps_radical r 0 a = 1"
```
```  1490   apply (auto simp add: fps_eq_iff fps_radical_def)  by (case_tac n, auto)
```
```  1491
```
```  1492 lemma fps_radical_nth_0[simp]: "fps_radical r n a \$ 0 = (if n=0 then 1 else r n (a\$0))"
```
```  1493   by (cases n, simp_all add: fps_radical_def)
```
```  1494
```
```  1495 lemma fps_radical_power_nth[simp]:
```
```  1496   assumes r: "(r k (a\$0)) ^ k = a\$0"
```
```  1497   shows "fps_radical r k a ^ k \$ 0 = (if k = 0 then 1 else a\$0)"
```
```  1498 proof-
```
```  1499   {assume "k=0" hence ?thesis by simp }
```
```  1500   moreover
```
```  1501   {fix h assume h: "k = Suc h"
```
```  1502     have fh: "finite {0..h}" by simp
```
```  1503     have eq1: "fps_radical r k a ^ k \$ 0 = (\<Prod>j\<in>{0..h}. fps_radical r k a \$ (replicate k 0) ! j)"
```
```  1504       unfolding fps_power_nth h by simp
```
```  1505     also have "\<dots> = (\<Prod>j\<in>{0..h}. r k (a\$0))"
```
```  1506       apply (rule setprod_cong)
```
```  1507       apply simp
```
```  1508       using h
```
```  1509       apply (subgoal_tac "replicate k (0::nat) ! x = 0")
```
```  1510       by (auto intro: nth_replicate simp del: replicate.simps)
```
```  1511     also have "\<dots> = a\$0"
```
```  1512       unfolding setprod_constant[OF fh] using r by (simp add: h)
```
```  1513     finally have ?thesis using h by simp}
```
```  1514   ultimately show ?thesis by (cases k, auto)
```
```  1515 qed
```
```  1516
```
```  1517 lemma natpermute_max_card: assumes n0: "n\<noteq>0"
```
```  1518   shows "card {xs \<in> natpermute n (k+1). n \<in> set xs} = k+1"
```
```  1519   unfolding natpermute_contain_maximal
```
```  1520 proof-
```
```  1521   let ?A= "\<lambda>i. {replicate (k + 1) 0[i := n]}"
```
```  1522   let ?K = "{0 ..k}"
```
```  1523   have fK: "finite ?K" by simp
```
```  1524   have fAK: "\<forall>i\<in>?K. finite (?A i)" by auto
```
```  1525   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]} = {}"
```
```  1526   proof(clarify)
```
```  1527     fix i j assume i: "i \<in> ?K" and j: "j\<in> ?K" and ij: "i\<noteq>j"
```
```  1528     {assume eq: "replicate (k+1) 0 [i:=n] = replicate (k+1) 0 [j:= n]"
```
```  1529       have "(replicate (k+1) 0 [i:=n] ! i) = n" using i by (simp del: replicate.simps)
```
```  1530       moreover
```
```  1531       have "(replicate (k+1) 0 [j:=n] ! i) = 0" using i ij by (simp del: replicate.simps)
```
```  1532       ultimately have False using eq n0 by (simp del: replicate.simps)}
```
```  1533     then show "{replicate (k + 1) 0[i := n]} \<inter> {replicate (k + 1) 0[j := n]} = {}"
```
```  1534       by auto
```
```  1535   qed
```
```  1536   from card_UN_disjoint[OF fK fAK d]
```
```  1537   show "card (\<Union>i\<in>{0..k}. {replicate (k + 1) 0[i := n]}) = k+1" by simp
```
```  1538 qed
```
```  1539
```
```  1540 lemma power_radical:
```
```  1541   fixes a:: "'a ::{field, ring_char_0, recpower} fps"
```
```  1542   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 \<noteq> 0"
```
```  1543   shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
```
```  1544 proof-
```
```  1545   let ?r = "fps_radical r (Suc k) a"
```
```  1546   from a0 r0 have r00: "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1547   {fix z have "?r ^ Suc k \$ z = a\$z"
```
```  1548     proof(induct z rule: nat_less_induct)
```
```  1549       fix n assume H: "\<forall>m<n. ?r ^ Suc k \$ m = a\$m"
```
```  1550       {assume "n = 0" hence "?r ^ Suc k \$ n = a \$n"
```
```  1551 	  using fps_radical_power_nth[of r "Suc k" a, OF r0] by simp}
```
```  1552       moreover
```
```  1553       {fix n1 assume n1: "n = Suc n1"
```
```  1554 	have fK: "finite {0..k}" by simp
```
```  1555 	have nz: "n \<noteq> 0" using n1 by arith
```
```  1556 	let ?Pnk = "natpermute n (k + 1)"
```
```  1557 	let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1558 	let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1559 	have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1560 	have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1561 	have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1562 	  using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1563 	  by (metis natpermute_finite)+
```
```  1564 	let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1565 	have "setsum ?f ?Pnkn = setsum (\<lambda>v. ?r \$ n * r (Suc k) (a \$ 0) ^ k) ?Pnkn"
```
```  1566 	proof(rule setsum_cong2)
```
```  1567 	  fix v assume v: "v \<in> {xs \<in> natpermute n (k + 1). n \<in> set xs}"
```
```  1568 	  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"
```
```  1569 	  from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1570 	    unfolding natpermute_contain_maximal by auto
```
```  1571 	  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))"
```
```  1572 	    apply (rule setprod_cong, simp)
```
```  1573 	    using i r0 by (simp del: replicate.simps)
```
```  1574 	  also have "\<dots> = (fps_radical r (Suc k) a \$ n) * r (Suc k) (a\$0) ^ k"
```
```  1575 	    unfolding setprod_gen_delta[OF fK] using i r0 by simp
```
```  1576 	  finally show ?ths .
```
```  1577 	qed
```
```  1578 	then have "setsum ?f ?Pnkn = of_nat (k+1) * ?r \$ n * r (Suc k) (a \$ 0) ^ k"
```
```  1579 	  by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1580 	also have "\<dots> = a\$n - setsum ?f ?Pnknn"
```
```  1581 	  unfolding n1 using r00 a0 by (simp add: field_simps fps_radical_def del: of_nat_Suc )
```
```  1582 	finally have fn: "setsum ?f ?Pnkn = a\$n - setsum ?f ?Pnknn" .
```
```  1583 	have "(?r ^ Suc k)\$n = setsum ?f ?Pnkn + setsum ?f ?Pnknn"
```
```  1584 	  unfolding fps_power_nth_Suc setsum_Un_disjoint[OF f d, unfolded eq] ..
```
```  1585 	also have "\<dots> = a\$n" unfolding fn by simp
```
```  1586 	finally have "?r ^ Suc k \$ n = a \$n" .}
```
```  1587       ultimately  show "?r ^ Suc k \$ n = a \$n" by (cases n, auto)
```
```  1588   qed }
```
```  1589   then show ?thesis by (simp add: fps_eq_iff)
```
```  1590 qed
```
```  1591
```
```  1592 lemma eq_divide_imp': assumes c0: "(c::'a::field) ~= 0" and eq: "a * c = b"
```
```  1593   shows "a = b / c"
```
```  1594 proof-
```
```  1595   from eq have "a * c * inverse c = b * inverse c" by simp
```
```  1596   hence "a * (inverse c * c) = b/c" by (simp only: field_simps divide_inverse)
```
```  1597   then show "a = b/c" unfolding  field_inverse[OF c0] by simp
```
```  1598 qed
```
```  1599
```
```  1600 lemma radical_unique:
```
```  1601   assumes r0: "(r (Suc k) (b\$0)) ^ Suc k = b\$0"
```
```  1602   and a0: "r (Suc k) (b\$0 ::'a::{field, ring_char_0, recpower}) = a\$0" and b0: "b\$0 \<noteq> 0"
```
```  1603   shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
```
```  1604 proof-
```
```  1605   let ?r = "fps_radical r (Suc k) b"
```
```  1606   have r00: "r (Suc k) (b\$0) \<noteq> 0" using b0 r0 by auto
```
```  1607   {assume H: "a = ?r"
```
```  1608     from H have "a^Suc k = b" using power_radical[of r k, OF r0 b0] by simp}
```
```  1609   moreover
```
```  1610   {assume H: "a^Suc k = b"
```
```  1611     (* Generally a\$0 would need to be the k+1 st root of b\$0 *)
```
```  1612     have ceq: "card {0..k} = Suc k" by simp
```
```  1613     have fk: "finite {0..k}" by simp
```
```  1614     from a0 have a0r0: "a\$0 = ?r\$0" by simp
```
```  1615     {fix n have "a \$ n = ?r \$ n"
```
```  1616       proof(induct n rule: nat_less_induct)
```
```  1617 	fix n assume h: "\<forall>m<n. a\$m = ?r \$m"
```
```  1618 	{assume "n = 0" hence "a\$n = ?r \$n" using a0 by simp }
```
```  1619 	moreover
```
```  1620 	{fix n1 assume n1: "n = Suc n1"
```
```  1621 	  have fK: "finite {0..k}" by simp
```
```  1622 	have nz: "n \<noteq> 0" using n1 by arith
```
```  1623 	let ?Pnk = "natpermute n (Suc k)"
```
```  1624 	let ?Pnkn = "{xs \<in> ?Pnk. n \<in> set xs}"
```
```  1625 	let ?Pnknn = "{xs \<in> ?Pnk. n \<notin> set xs}"
```
```  1626 	have eq: "?Pnkn \<union> ?Pnknn = ?Pnk" by blast
```
```  1627 	have d: "?Pnkn \<inter> ?Pnknn = {}" by blast
```
```  1628 	have f: "finite ?Pnkn" "finite ?Pnknn"
```
```  1629 	  using finite_Un[of ?Pnkn ?Pnknn, unfolded eq]
```
```  1630 	  by (metis natpermute_finite)+
```
```  1631 	let ?f = "\<lambda>v. \<Prod>j\<in>{0..k}. ?r \$ v ! j"
```
```  1632 	let ?g = "\<lambda>v. \<Prod>j\<in>{0..k}. a \$ v ! j"
```
```  1633 	have "setsum ?g ?Pnkn = setsum (\<lambda>v. a \$ n * (?r\$0)^k) ?Pnkn"
```
```  1634 	proof(rule setsum_cong2)
```
```  1635 	  fix v assume v: "v \<in> {xs \<in> natpermute n (Suc k). n \<in> set xs}"
```
```  1636 	  let ?ths = "(\<Prod>j\<in>{0..k}. a \$ v ! j) = a \$ n * (?r\$0)^k"
```
```  1637 	  from v obtain i where i: "i \<in> {0..k}" "v = replicate (k+1) 0 [i:= n]"
```
```  1638 	    unfolding Suc_plus1 natpermute_contain_maximal by (auto simp del: replicate.simps)
```
```  1639 	  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))"
```
```  1640 	    apply (rule setprod_cong, simp)
```
```  1641 	    using i a0 by (simp del: replicate.simps)
```
```  1642 	  also have "\<dots> = a \$ n * (?r \$ 0)^k"
```
```  1643 	    unfolding  setprod_gen_delta[OF fK] using i by simp
```
```  1644 	  finally show ?ths .
```
```  1645 	qed
```
```  1646 	then have th0: "setsum ?g ?Pnkn = of_nat (k+1) * a \$ n * (?r \$ 0)^k"
```
```  1647 	  by (simp add: natpermute_max_card[OF nz, simplified])
```
```  1648 	have th1: "setsum ?g ?Pnknn = setsum ?f ?Pnknn"
```
```  1649 	proof (rule setsum_cong2, rule setprod_cong, simp)
```
```  1650 	  fix xs i assume xs: "xs \<in> ?Pnknn" and i: "i \<in> {0..k}"
```
```  1651 	  {assume c: "n \<le> xs ! i"
```
```  1652 	    from xs i have "xs !i \<noteq> n" by (auto simp add: in_set_conv_nth natpermute_def)
```
```  1653 	    with c have c': "n < xs!i" by arith
```
```  1654 	    have fths: "finite {0 ..< i}" "finite {i}" "finite {i+1..<Suc k}" by simp_all
```
```  1655 	    have d: "{0 ..< i} \<inter> ({i} \<union> {i+1 ..< Suc k}) = {}" "{i} \<inter> {i+1..< Suc k} = {}" by auto
```
```  1656 	    have eqs: "{0..<Suc k} = {0 ..< i} \<union> ({i} \<union> {i+1 ..< Suc k})" using i by auto
```
```  1657 	    from xs have "n = foldl op + 0 xs" by (simp add: natpermute_def)
```
```  1658 	    also have "\<dots> = setsum (nth xs) {0..<Suc k}" unfolding foldl_add_setsum using xs
```
```  1659 	      by (simp add: natpermute_def)
```
```  1660 	    also have "\<dots> = xs!i + setsum (nth xs) {0..<i} + setsum (nth xs) {i+1..<Suc k}"
```
```  1661 	      unfolding eqs  setsum_Un_disjoint[OF fths(1) finite_UnI[OF fths(2,3)] d(1)]
```
```  1662 	      unfolding setsum_Un_disjoint[OF fths(2) fths(3) d(2)]
```
```  1663 	      by simp
```
```  1664 	    finally have False using c' by simp}
```
```  1665 	  then have thn: "xs!i < n" by arith
```
```  1666 	  from h[rule_format, OF thn]
```
```  1667 	  show "a\$(xs !i) = ?r\$(xs!i)" .
```
```  1668 	qed
```
```  1669 	have th00: "\<And>(x::'a). of_nat (Suc k) * (x * inverse (of_nat (Suc k))) = x"
```
```  1670 	  by (simp add: field_simps del: of_nat_Suc)
```
```  1671 	from H have "b\$n = a^Suc k \$ n" by (simp add: fps_eq_iff)
```
```  1672 	also have "a ^ Suc k\$n = setsum ?g ?Pnkn + setsum ?g ?Pnknn"
```
```  1673 	  unfolding fps_power_nth_Suc
```
```  1674 	  using setsum_Un_disjoint[OF f d, unfolded Suc_plus1[symmetric],
```
```  1675 	    unfolded eq, of ?g] by simp
```
```  1676 	also have "\<dots> = of_nat (k+1) * a \$ n * (?r \$ 0)^k + setsum ?f ?Pnknn" unfolding th0 th1 ..
```
```  1677 	finally have "of_nat (k+1) * a \$ n * (?r \$ 0)^k = b\$n - setsum ?f ?Pnknn" by simp
```
```  1678 	then have "a\$n = (b\$n - setsum ?f ?Pnknn) / (of_nat (k+1) * (?r \$ 0)^k)"
```
```  1679 	  apply -
```
```  1680 	  apply (rule eq_divide_imp')
```
```  1681 	  using r00
```
```  1682 	  apply (simp del: of_nat_Suc)
```
```  1683 	  by (simp add: mult_ac)
```
```  1684 	then have "a\$n = ?r \$n"
```
```  1685 	  apply (simp del: of_nat_Suc)
```
```  1686 	  unfolding fps_radical_def n1
```
```  1687 	  by (simp add: field_simps n1 fps_nth_def th00 del: of_nat_Suc)}
```
```  1688 	ultimately show "a\$n = ?r \$ n" by (cases n, auto)
```
```  1689       qed}
```
```  1690     then have "a = ?r" by (simp add: fps_eq_iff)}
```
```  1691   ultimately show ?thesis by blast
```
```  1692 qed
```
```  1693
```
```  1694
```
```  1695 lemma radical_power:
```
```  1696   assumes r0: "r (Suc k) ((a\$0) ^ Suc k) = a\$0"
```
```  1697   and a0: "(a\$0 ::'a::{field, ring_char_0, recpower}) \<noteq> 0"
```
```  1698   shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
```
```  1699 proof-
```
```  1700   let ?ak = "a^ Suc k"
```
```  1701   have ak0: "?ak \$ 0 = (a\$0) ^ Suc k" by (simp add: fps_nth_power_0)
```
```  1702   from r0 have th0: "r (Suc k) (a ^ Suc k \$ 0) ^ Suc k = a ^ Suc k \$ 0" using ak0 by auto
```
```  1703   from r0 ak0 have th1: "r (Suc k) (a ^ Suc k \$ 0) = a \$ 0" by auto
```
```  1704   from ak0 a0 have ak00: "?ak \$ 0 \<noteq>0 " by auto
```
```  1705   from radical_unique[of r k ?ak a, OF th0 th1 ak00] show ?thesis by metis
```
```  1706 qed
```
```  1707
```
```  1708 lemma fps_deriv_radical:
```
```  1709   fixes a:: "'a ::{field, ring_char_0, recpower} fps"
```
```  1710   assumes r0: "(r (Suc k) (a\$0)) ^ Suc k = a\$0" and a0: "a\$0 \<noteq> 0"
```
```  1711   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)"
```
```  1712 proof-
```
```  1713   let ?r= "fps_radical r (Suc k) a"
```
```  1714   let ?w = "(fps_const (of_nat (Suc k)) * ?r ^ k)"
```
```  1715   from a0 r0 have r0': "r (Suc k) (a\$0) \<noteq> 0" by auto
```
```  1716   from r0' have w0: "?w \$ 0 \<noteq> 0" by (simp del: of_nat_Suc)
```
```  1717   note th0 = inverse_mult_eq_1[OF w0]
```
```  1718   let ?iw = "inverse ?w"
```
```  1719   from power_radical[of r, OF r0 a0]
```
```  1720   have "fps_deriv (?r ^ Suc k) = fps_deriv a" by simp
```
```  1721   hence "fps_deriv ?r * ?w = fps_deriv a"
```
```  1722     by (simp add: fps_deriv_power mult_ac)
```
```  1723   hence "?iw * fps_deriv ?r * ?w = ?iw * fps_deriv a" by simp
```
```  1724   hence "fps_deriv ?r * (?iw * ?w) = fps_deriv a / ?w"
```
```  1725     by (simp add: fps_divide_def)
```
```  1726   then show ?thesis unfolding th0 by simp
```
```  1727 qed
```
```  1728
```
```  1729 lemma radical_mult_distrib:
```
```  1730   fixes a:: "'a ::{field, ring_char_0, recpower} fps"
```
```  1731   assumes
```
```  1732   ra0: "r (k) (a \$ 0) ^ k = a \$ 0"
```
```  1733   and rb0: "r (k) (b \$ 0) ^ k = b \$ 0"
```
```  1734   and r0': "r (k) ((a * b) \$ 0) = r (k) (a \$ 0) * r (k) (b \$ 0)"
```
```  1735   and a0: "a\$0 \<noteq> 0"
```
```  1736   and b0: "b\$0 \<noteq> 0"
```
```  1737   shows "fps_radical r (k) (a*b) = fps_radical r (k) a * fps_radical r (k) (b)"
```
```  1738 proof-
```
```  1739   from r0' have r0: "(r (k) ((a*b)\$0)) ^ k = (a*b)\$0"
```
```  1740     by (simp add: fps_mult_nth ra0 rb0 power_mult_distrib)
```
```  1741   {assume "k=0" hence ?thesis by simp}
```
```  1742   moreover
```
```  1743   {fix h assume k: "k = Suc h"
```
```  1744   let ?ra = "fps_radical r (Suc h) a"
```
```  1745   let ?rb = "fps_radical r (Suc h) b"
```
```  1746   have th0: "r (Suc h) ((a * b) \$ 0) = (fps_radical r (Suc h) a * fps_radical r (Suc h) b) \$ 0"
```
```  1747     using r0' k by (simp add: fps_mult_nth)
```
```  1748   have ab0: "(a*b) \$ 0 \<noteq> 0" using a0 b0 by (simp add: fps_mult_nth)
```
```  1749   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]
```
```  1750     power_radical[of r, OF ra0[unfolded k] a0] power_radical[of r, OF rb0[unfolded k] b0] k
```
```  1751   have ?thesis by (auto simp add: power_mult_distrib)}
```
```  1752 ultimately show ?thesis by (cases k, auto)
```
```  1753 qed
```
```  1754
```
```  1755 lemma radical_inverse:
```
```  1756   fixes a:: "'a ::{field, ring_char_0, recpower} fps"
```
```  1757   assumes
```
```  1758   ra0: "r (k) (a \$ 0) ^ k = a \$ 0"
```
```  1759   and ria0: "r (k) (inverse (a \$ 0)) = inverse (r (k) (a \$ 0))"
```
```  1760   and r1: "(r (k) 1) = 1"
```
```  1761   and a0: "a\$0 \<noteq> 0"
```
```  1762   shows "fps_radical r (k) (inverse a) = inverse (fps_radical r (k) a)"
```
```  1763 proof-
```
```  1764   {assume "k=0" then have ?thesis by simp}
```
```  1765   moreover
```
```  1766   {fix h assume k[simp]: "k = Suc h"
```
```  1767     let ?ra = "fps_radical r (Suc h) a"
```
```  1768     let ?ria = "fps_radical r (Suc h) (inverse a)"
```
```  1769     from ra0 a0 have th00: "r (Suc h) (a\$0) \<noteq> 0" by auto
```
```  1770     have ria0': "r (Suc h) (inverse a \$ 0) ^ Suc h = inverse a\$0"
```
```  1771     using ria0 ra0 a0
```
```  1772     by (simp add: fps_inverse_def  nonzero_power_inverse[OF th00, symmetric])
```
```  1773   from inverse_mult_eq_1[OF a0] have th0: "a * inverse a = 1"
```
```  1774     by (simp add: mult_commute)
```
```  1775   from radical_unique[where a=1 and b=1 and r=r and k=h, simplified, OF r1[unfolded k]]
```
```  1776   have th01: "fps_radical r (Suc h) 1 = 1" .
```
```  1777   have th1: "r (Suc h) ((a * inverse a) \$ 0) ^ Suc h = (a * inverse a) \$ 0"
```
```  1778     "r (Suc h) ((a * inverse a) \$ 0) =
```
```  1779 r (Suc h) (a \$ 0) * r (Suc h) (inverse a \$ 0)"
```
```  1780     using r1 unfolding th0  apply (simp_all add: ria0[symmetric])
```
```  1781     apply (simp add: fps_inverse_def a0)
```
```  1782     unfolding ria0[unfolded k]
```
```  1783     using th00 by simp
```
```  1784   from nonzero_imp_inverse_nonzero[OF a0] a0
```
```  1785   have th2: "inverse a \$ 0 \<noteq> 0" by (simp add: fps_inverse_def)
```
```  1786   from radical_mult_distrib[of r "Suc h" a "inverse a", OF ra0[unfolded k] ria0' th1(2) a0 th2]
```
```  1787   have th3: "?ra * ?ria = 1" unfolding th0 th01 by simp
```
```  1788   from th00 have ra0: "?ra \$ 0 \<noteq> 0" by simp
```
```  1789   from fps_inverse_unique[OF ra0 th3] have ?thesis by simp}
```
```  1790 ultimately show ?thesis by (cases k, auto)
```
```  1791 qed
```
```  1792
```
```  1793 lemma fps_divide_inverse: "(a::('a::field) fps) / b = a * inverse b"
```
```  1794   by (simp add: fps_divide_def)
```
```  1795
```
```  1796 lemma radical_divide:
```
```  1797   fixes a:: "'a ::{field, ring_char_0, recpower} fps"
```
```  1798   assumes
```
```  1799       ra0: "r k (a \$ 0) ^ k = a \$ 0"
```
```  1800   and rb0: "r k (b \$ 0) ^ k = b \$ 0"
```
```  1801   and r1: "r k 1 = 1"
```
```  1802   and rb0': "r k (inverse (b \$ 0)) = inverse (r k (b \$ 0))"
```
```  1803   and raib': "r k (a\$0 / (b\$0)) = r k (a\$0) / r k (b\$0)"
```
```  1804   and a0: "a\$0 \<noteq> 0"
```
```  1805   and b0: "b\$0 \<noteq> 0"
```
```  1806   shows "fps_radical r k (a/b) = fps_radical r k a / fps_radical r k b"
```
```  1807 proof-
```
```  1808   from raib'
```
```  1809   have raib: "r k (a\$0 / (b\$0)) = r k (a\$0) * r k (inverse (b\$0))"
```
```  1810     by (simp add: divide_inverse rb0'[symmetric])
```
```  1811
```
```  1812   {assume "k=0" hence ?thesis by (simp add: fps_divide_def)}
```
```  1813   moreover
```
```  1814   {assume k0: "k\<noteq> 0"
```
```  1815     from b0 k0 rb0 have rbn0: "r k (b \$0) \<noteq> 0"
```
```  1816       by (auto simp add: power_0_left)
```
```  1817
```
```  1818     from rb0 rb0' have rib0: "(r k (inverse (b \$ 0)))^k = inverse (b\$0)"
```
```  1819     by (simp add: nonzero_power_inverse[OF rbn0, symmetric])
```
```  1820   from rib0 have th0: "r k (inverse b \$ 0) ^ k = inverse b \$ 0"
```
```  1821     by (simp add:fps_inverse_def b0)
```
```  1822   from raib
```
```  1823   have th1: "r k ((a * inverse b) \$ 0) = r k (a \$ 0) * r k (inverse b \$ 0)"
```
```  1824     by (simp add: divide_inverse fps_inverse_def  b0 fps_mult_nth)
```
```  1825   from nonzero_imp_inverse_nonzero[OF b0] b0 have th2: "inverse b \$ 0 \<noteq> 0"
```
```  1826     by (simp add: fps_inverse_def)
```
```  1827   from radical_mult_distrib[of r k a "inverse b", OF ra0 th0 th1 a0 th2]
```
```  1828   have th: "fps_radical r k (a/b) = fps_radical r k a * fps_radical r k (inverse b)"
```
```  1829     by (simp add: fps_divide_def)
```
```  1830   with radical_inverse[of r k b, OF rb0 rb0' r1 b0]
```
```  1831   have ?thesis by (simp add: fps_divide_def)}
```
```  1832 ultimately show ?thesis by blast
```
```  1833 qed
```
```  1834
```
```  1835 subsection{* Derivative of composition *}
```
```  1836
```
```  1837 lemma fps_compose_deriv:
```
```  1838   fixes a:: "('a::idom) fps"
```
```  1839   assumes b0: "b\$0 = 0"
```
```  1840   shows "fps_deriv (a oo b) = ((fps_deriv a) oo b) * (fps_deriv b)"
```
```  1841 proof-
```
```  1842   {fix n
```
```  1843     have "(fps_deriv (a oo b))\$n = setsum (\<lambda>i. a \$ i * (fps_deriv (b^i))\$n) {0.. Suc n}"
```
```  1844       by (simp add: fps_compose_def ring_simps setsum_right_distrib del: of_nat_Suc)
```
```  1845     also have "\<dots> = setsum (\<lambda>i. a\$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))\$n) {0.. Suc n}"
```
```  1846       by (simp add: ring_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
```
```  1847   also have "\<dots> = setsum (\<lambda>i. of_nat i * a\$i * (((b^(i - 1)) * fps_deriv b))\$n) {0.. Suc n}"
```
```  1848     unfolding fps_mult_left_const_nth  by (simp add: ring_simps)
```
```  1849   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}"
```
```  1850     unfolding fps_mult_nth ..
```
```  1851   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}"
```
```  1852     apply (rule setsum_mono_zero_right)
```
```  1853     by (auto simp add: cond_value_iff cond_application_beta setsum_delta
```
```  1854       not_le cong del: if_weak_cong)
```
```  1855   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}"
```
```  1856     unfolding fps_deriv_nth
```
```  1857     apply (rule setsum_reindex_cong[where f="Suc"])
```
```  1858     by (auto simp add: mult_assoc)
```
```  1859   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}" .
```
```  1860
```
```  1861   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}"
```
```  1862     unfolding fps_mult_nth by (simp add: mult_ac)
```
```  1863   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}"
```
```  1864     unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult_assoc
```
```  1865     apply (rule setsum_cong2)
```
```  1866     apply (rule setsum_mono_zero_left)
```
```  1867     apply (simp_all add: subset_eq)
```
```  1868     apply clarify
```
```  1869     apply (subgoal_tac "b^i\$x = 0")
```
```  1870     apply simp
```
```  1871     apply (rule startsby_zero_power_prefix[OF b0, rule_format])
```
```  1872     by simp
```
```  1873   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}"
```
```  1874     unfolding setsum_right_distrib
```
```  1875     apply (subst setsum_commute)
```
```  1876     by ((rule setsum_cong2)+) simp
```
```  1877   finally have "(fps_deriv (a oo b))\$n = (((fps_deriv a) oo b) * (fps_deriv b)) \$n"
```
```  1878     unfolding th0 by simp}
```
```  1879 then show ?thesis by (simp add: fps_eq_iff)
```
```  1880 qed
```
```  1881
```
```  1882 lemma fps_mult_X_plus_1_nth:
```
```  1883   "((1+X)*a) \$n = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  1884 proof-
```
```  1885   {assume "n = 0" hence ?thesis by (simp add: fps_mult_nth )}
```
```  1886   moreover
```
```  1887   {fix m assume m: "n = Suc m"
```
```  1888     have "((1+X)*a) \$n = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0..n}"
```
```  1889       by (simp add: fps_mult_nth)
```
```  1890     also have "\<dots> = setsum (\<lambda>i. (1+X)\$i * a\$(n-i)) {0.. 1}"
```
```  1891       unfolding m
```
```  1892       apply (rule setsum_mono_zero_right)
```
```  1893       by (auto simp add: )
```
```  1894     also have "\<dots> = (if n = 0 then (a\$n :: 'a::comm_ring_1) else a\$n + a\$(n - 1))"
```
```  1895       unfolding m
```
```  1896       by (simp add: )
```
```  1897     finally have ?thesis .}
```
```  1898   ultimately show ?thesis by (cases n, auto)
```
```  1899 qed
```
```  1900
```
```  1901 subsection{* Finite FPS (i.e. polynomials) and X *}
```
```  1902 lemma fps_poly_sum_X:
```
```  1903   assumes z: "\<forall>i > n. a\$i = (0::'a::comm_ring_1)"
```
```  1904   shows "a = setsum (\<lambda>i. fps_const (a\$i) * X^i) {0..n}" (is "a = ?r")
```
```  1905 proof-
```
```  1906   {fix i
```
```  1907     have "a\$i = ?r\$i"
```
```  1908       unfolding fps_setsum_nth fps_mult_left_const_nth X_power_nth
```
```  1909       apply (simp add: cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
```
```  1910       using z by auto}
```
```  1911   then show ?thesis unfolding fps_eq_iff by blast
```
```  1912 qed
```
```  1913
```
```  1914 subsection{* Compositional inverses *}
```
```  1915
```
```  1916
```
```  1917 fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::{recpower,field}" where
```
```  1918   "compinv a 0 = X\$0"
```
```  1919 | "compinv a (Suc n) = (X\$ Suc n - setsum (\<lambda>i. (compinv a i) * (a^i)\$Suc n) {0 .. n}) / (a\$1) ^ Suc n"
```
```  1920
```
```  1921 definition "fps_inv a = Abs_fps (compinv a)"
```
```  1922
```
```  1923 lemma fps_inv: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  1924   shows "fps_inv a oo a = X"
```
```  1925 proof-
```
```  1926   let ?i = "fps_inv a oo a"
```
```  1927   {fix n
```
```  1928     have "?i \$n = X\$n"
```
```  1929     proof(induct n rule: nat_less_induct)
```
```  1930       fix n assume h: "\<forall>m<n. ?i\$m = X\$m"
```
```  1931       {assume "n=0" hence "?i \$n = X\$n" using a0
```
```  1932 	  by (simp add: fps_compose_nth fps_inv_def)}
```
```  1933       moreover
```
```  1934       {fix n1 assume n1: "n = Suc n1"
```
```  1935 	have "?i \$ n = setsum (\<lambda>i. (fps_inv a \$ i) * (a^i)\$n) {0 .. n1} + fps_inv a \$ Suc n1 * (a \$ 1)^ Suc n1"
```
```  1936 	  by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0])
```
```  1937 	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})"
```
```  1938 	  using a0 a1 n1 by (simp add: fps_inv_def fps_nth_def)
```
```  1939 	also have "\<dots> = X\$n" using n1 by simp
```
```  1940 	finally have "?i \$ n = X\$n" .}
```
```  1941       ultimately show "?i \$ n = X\$n" by (cases n, auto)
```
```  1942     qed}
```
```  1943   then show ?thesis by (simp add: fps_eq_iff)
```
```  1944 qed
```
```  1945
```
```  1946
```
```  1947 fun gcompinv :: "'a fps \<Rightarrow> 'a fps \<Rightarrow> nat \<Rightarrow> 'a::{recpower,field}" where
```
```  1948   "gcompinv b a 0 = b\$0"
```
```  1949 | "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"
```
```  1950
```
```  1951 definition "fps_ginv b a = Abs_fps (gcompinv b a)"
```
```  1952
```
```  1953 lemma fps_ginv: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  1954   shows "fps_ginv b a oo a = b"
```
```  1955 proof-
```
```  1956   let ?i = "fps_ginv b a oo a"
```
```  1957   {fix n
```
```  1958     have "?i \$n = b\$n"
```
```  1959     proof(induct n rule: nat_less_induct)
```
```  1960       fix n assume h: "\<forall>m<n. ?i\$m = b\$m"
```
```  1961       {assume "n=0" hence "?i \$n = b\$n" using a0
```
```  1962 	  by (simp add: fps_compose_nth fps_ginv_def)}
```
```  1963       moreover
```
```  1964       {fix n1 assume n1: "n = Suc n1"
```
```  1965 	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"
```
```  1966 	  by (simp add: fps_compose_nth n1 startsby_zero_power_nth_same[OF a0])
```
```  1967 	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})"
```
```  1968 	  using a0 a1 n1 by (simp add: fps_ginv_def fps_nth_def)
```
```  1969 	also have "\<dots> = b\$n" using n1 by simp
```
```  1970 	finally have "?i \$ n = b\$n" .}
```
```  1971       ultimately show "?i \$ n = b\$n" by (cases n, auto)
```
```  1972     qed}
```
```  1973   then show ?thesis by (simp add: fps_eq_iff)
```
```  1974 qed
```
```  1975
```
```  1976 lemma fps_inv_ginv: "fps_inv = fps_ginv X"
```
```  1977   apply (auto simp add: expand_fun_eq fps_eq_iff fps_inv_def fps_ginv_def)
```
```  1978   apply (induct_tac n rule: nat_less_induct, auto)
```
```  1979   apply (case_tac na)
```
```  1980   apply simp
```
```  1981   apply simp
```
```  1982   done
```
```  1983
```
```  1984 lemma fps_compose_1[simp]: "1 oo a = 1"
```
```  1985   apply (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
```
```  1986   apply (simp add: setsum_delta)
```
```  1987   done
```
```  1988
```
```  1989 lemma fps_compose_0[simp]: "0 oo a = 0"
```
```  1990   by (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
```
```  1991
```
```  1992 lemma fps_pow_0: "fps_pow n 0 = (if n = 0 then 1 else 0)"
```
```  1993   by (induct n, simp_all)
```
```  1994
```
```  1995 lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a\$0)"
```
```  1996   apply (auto simp add: fps_eq_iff fps_compose_nth fps_power_def cond_value_iff cond_application_beta cong del: if_weak_cong)
```
```  1997   by (case_tac n, auto simp add: fps_pow_0 intro: setsum_0')
```
```  1998
```
```  1999 lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
```
```  2000   by (simp add: fps_eq_iff fps_compose_nth  ring_simps setsum_addf)
```
```  2001
```
```  2002 lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
```
```  2003 proof-
```
```  2004   {assume "\<not> finite S" hence ?thesis by simp}
```
```  2005   moreover
```
```  2006   {assume fS: "finite S"
```
```  2007     have ?thesis
```
```  2008     proof(rule finite_induct[OF fS])
```
```  2009       show "setsum f {} oo a = (\<Sum>i\<in>{}. f i oo a)" by simp
```
```  2010     next
```
```  2011       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"
```
```  2012       show "setsum f (insert x F) oo a  = setsum (\<lambda>i. f i oo a) (insert x F)"
```
```  2013 	using fF xF h by (simp add: fps_compose_add_distrib)
```
```  2014     qed}
```
```  2015   ultimately show ?thesis by blast
```
```  2016 qed
```
```  2017
```
```  2018 lemma convolution_eq:
```
```  2019   "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}"
```
```  2020   apply (rule setsum_reindex_cong[where f=fst])
```
```  2021   apply (clarsimp simp add: inj_on_def)
```
```  2022   apply (auto simp add: expand_set_eq image_iff)
```
```  2023   apply (rule_tac x= "x" in exI)
```
```  2024   apply clarsimp
```
```  2025   apply (rule_tac x="n - x" in exI)
```
```  2026   apply arith
```
```  2027   done
```
```  2028
```
```  2029 lemma product_composition_lemma:
```
```  2030   assumes c0: "c\$0 = (0::'a::idom)" and d0: "d\$0 = 0"
```
```  2031   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")
```
```  2032 proof-
```
```  2033   let ?S = "{(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
```
```  2034   have s: "?S \<subseteq> {0..n} <*> {0..n}" by (auto simp add: subset_eq)
```
```  2035   have f: "finite {(k\<Colon>nat, m\<Colon>nat). k + m \<le> n}"
```
```  2036     apply (rule finite_subset[OF s])
```
```  2037     by auto
```
```  2038   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}"
```
```  2039     apply (simp add: fps_mult_nth setsum_right_distrib)
```
```  2040     apply (subst setsum_commute)
```
```  2041     apply (rule setsum_cong2)
```
```  2042     by (auto simp add: ring_simps)
```
```  2043   also have "\<dots> = ?l"
```
```  2044     apply (simp add: fps_mult_nth fps_compose_nth setsum_product)
```
```  2045     apply (rule setsum_cong2)
```
```  2046     apply (simp add: setsum_cartesian_product mult_assoc)
```
```  2047     apply (rule setsum_mono_zero_right[OF f])
```
```  2048     apply (simp add: subset_eq) apply presburger
```
```  2049     apply clarsimp
```
```  2050     apply (rule ccontr)
```
```  2051     apply (clarsimp simp add: not_le)
```
```  2052     apply (case_tac "x < aa")
```
```  2053     apply simp
```
```  2054     apply (frule_tac startsby_zero_power_prefix[rule_format, OF c0])
```
```  2055     apply blast
```
```  2056     apply simp
```
```  2057     apply (frule_tac startsby_zero_power_prefix[rule_format, OF d0])
```
```  2058     apply blast
```
```  2059     done
```
```  2060   finally show ?thesis by simp
```
```  2061 qed
```
```  2062
```
```  2063 lemma product_composition_lemma':
```
```  2064   assumes c0: "c\$0 = (0::'a::idom)" and d0: "d\$0 = 0"
```
```  2065   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")
```
```  2066   unfolding product_composition_lemma[OF c0 d0]
```
```  2067   unfolding setsum_cartesian_product
```
```  2068   apply (rule setsum_mono_zero_left)
```
```  2069   apply simp
```
```  2070   apply (clarsimp simp add: subset_eq)
```
```  2071   apply clarsimp
```
```  2072   apply (rule ccontr)
```
```  2073   apply (subgoal_tac "(c^aa * d^ba) \$ n = 0")
```
```  2074   apply simp
```
```  2075   unfolding fps_mult_nth
```
```  2076   apply (rule setsum_0')
```
```  2077   apply (clarsimp simp add: not_le)
```
```  2078   apply (case_tac "aaa < aa")
```
```  2079   apply (rule startsby_zero_power_prefix[OF c0, rule_format])
```
```  2080   apply simp
```
```  2081   apply (subgoal_tac "n - aaa < ba")
```
```  2082   apply (frule_tac k = "ba" in startsby_zero_power_prefix[OF d0, rule_format])
```
```  2083   apply simp
```
```  2084   apply arith
```
```  2085   done
```
```  2086
```
```  2087
```
```  2088 lemma setsum_pair_less_iff:
```
```  2089   "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")
```
```  2090 proof-
```
```  2091   let ?KM=  "{(k,m). k + m \<le> n}"
```
```  2092   let ?f = "%s. UNION {(0::nat)..s} (%i. {(i,s - i)})"
```
```  2093   have th0: "?KM = UNION {0..n} ?f"
```
```  2094     apply (simp add: expand_set_eq)
```
```  2095     apply arith
```
```  2096     done
```
```  2097   show "?l = ?r "
```
```  2098     unfolding th0
```
```  2099     apply (subst setsum_UN_disjoint)
```
```  2100     apply auto
```
```  2101     apply (subst setsum_UN_disjoint)
```
```  2102     apply auto
```
```  2103     done
```
```  2104 qed
```
```  2105
```
```  2106 lemma fps_compose_mult_distrib_lemma:
```
```  2107   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2108   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")
```
```  2109   unfolding product_composition_lemma[OF c0 c0] power_add[symmetric]
```
```  2110   unfolding setsum_pair_less_iff[where a = "%k. a\$k" and b="%m. b\$m" and c="%s. (c ^ s)\$n" and n = n] ..
```
```  2111
```
```  2112
```
```  2113 lemma fps_compose_mult_distrib:
```
```  2114   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2115   shows "(a * b) oo c = (a oo c) * (b oo c)" (is "?l = ?r")
```
```  2116   apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma[OF c0])
```
```  2117   by (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
```
```  2118 lemma fps_compose_setprod_distrib:
```
```  2119   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2120   shows "(setprod a S) oo c = setprod (%k. a k oo c) S" (is "?l = ?r")
```
```  2121   apply (cases "finite S")
```
```  2122   apply simp_all
```
```  2123   apply (induct S rule: finite_induct)
```
```  2124   apply simp
```
```  2125   apply (simp add: fps_compose_mult_distrib[OF c0])
```
```  2126   done
```
```  2127
```
```  2128 lemma fps_compose_power:   assumes c0: "c\$0 = (0::'a::idom)"
```
```  2129   shows "(a oo c)^n = a^n oo c" (is "?l = ?r")
```
```  2130 proof-
```
```  2131   {assume "n=0" then have ?thesis by simp}
```
```  2132   moreover
```
```  2133   {fix m assume m: "n = Suc m"
```
```  2134     have th0: "a^n = setprod (%k. a) {0..m}" "(a oo c) ^ n = setprod (%k. a oo c) {0..m}"
```
```  2135       by (simp_all add: setprod_constant m)
```
```  2136     then have ?thesis
```
```  2137       by (simp add: fps_compose_setprod_distrib[OF c0])}
```
```  2138   ultimately show ?thesis by (cases n, auto)
```
```  2139 qed
```
```  2140
```
```  2141 lemma fps_const_mult_apply_left:
```
```  2142   "fps_const c * (a oo b) = (fps_const c * a) oo b"
```
```  2143   by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult_assoc)
```
```  2144
```
```  2145 lemma fps_const_mult_apply_right:
```
```  2146   "(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
```
```  2147   by (auto simp add: fps_const_mult_apply_left mult_commute)
```
```  2148
```
```  2149 lemma fps_compose_assoc:
```
```  2150   assumes c0: "c\$0 = (0::'a::idom)" and b0: "b\$0 = 0"
```
```  2151   shows "a oo (b oo c) = a oo b oo c" (is "?l = ?r")
```
```  2152 proof-
```
```  2153   {fix n
```
```  2154     have "?l\$n = (setsum (\<lambda>i. (fps_const (a\$i) * b^i) oo c) {0..n})\$n"
```
```  2155       by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left setsum_right_distrib mult_assoc fps_setsum_nth)
```
```  2156     also have "\<dots> = ((setsum (\<lambda>i. fps_const (a\$i) * b^i) {0..n}) oo c)\$n"
```
```  2157       by (simp add: fps_compose_setsum_distrib)
```
```  2158     also have "\<dots> = ?r\$n"
```
```  2159       apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult_assoc)
```
```  2160       apply (rule setsum_cong2)
```
```  2161       apply (rule setsum_mono_zero_right)
```
```  2162       apply (auto simp add: not_le)
```
```  2163       by (erule startsby_zero_power_prefix[OF b0, rule_format])
```
```  2164     finally have "?l\$n = ?r\$n" .}
```
```  2165   then show ?thesis by (simp add: fps_eq_iff)
```
```  2166 qed
```
```  2167
```
```  2168
```
```  2169 lemma fps_X_power_compose:
```
```  2170   assumes a0: "a\$0=0" shows "X^k oo a = (a::('a::idom fps))^k" (is "?l = ?r")
```
```  2171 proof-
```
```  2172   {assume "k=0" hence ?thesis by simp}
```
```  2173   moreover
```
```  2174   {fix h assume h: "k = Suc h"
```
```  2175     {fix n
```
```  2176       {assume kn: "k>n" hence "?l \$ n = ?r \$n" using a0 startsby_zero_power_prefix[OF a0] h
```
```  2177 	  by (simp add: fps_compose_nth)}
```
```  2178       moreover
```
```  2179       {assume kn: "k \<le> n"
```
```  2180 	hence "?l\$n = ?r\$n" apply (simp only: fps_compose_nth X_power_nth)
```
```  2181 	  by (simp add: cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)}
```
```  2182       moreover have "k >n \<or> k\<le> n"  by arith
```
```  2183       ultimately have "?l\$n = ?r\$n"  by blast}
```
```  2184     then have ?thesis unfolding fps_eq_iff by blast}
```
```  2185   ultimately show ?thesis by (cases k, auto)
```
```  2186 qed
```
```  2187
```
```  2188 lemma fps_inv_right: assumes a0: "a\$0 = 0" and a1: "a\$1 \<noteq> 0"
```
```  2189   shows "a oo fps_inv a = X"
```
```  2190 proof-
```
```  2191   let ?ia = "fps_inv a"
```
```  2192   let ?iaa = "a oo fps_inv a"
```
```  2193   have th0: "?ia \$ 0 = 0" by (simp add: fps_inv_def)
```
```  2194   have th1: "?iaa \$ 0 = 0" using a0 a1
```
```  2195     by (simp add: fps_inv_def fps_compose_nth)
```
```  2196   have th2: "X\$0 = 0" by simp
```
```  2197   from fps_inv[OF a0 a1] have "a oo (fps_inv a oo a) = a oo X" by simp
```
```  2198   then have "(a oo fps_inv a) oo a = X oo a"
```
```  2199     by (simp add: fps_compose_assoc[OF a0 th0] X_fps_compose_startby0[OF a0])
```
```  2200   with fps_compose_inj_right[OF a0 a1]
```
```  2201   show ?thesis by simp
```
```  2202 qed
```
```  2203
```
```  2204 lemma fps_inv_deriv:
```
```  2205   assumes a0:"a\$0 = (0::'a::{recpower,field})" and a1: "a\$1 \<noteq> 0"
```
```  2206   shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
```
```  2207 proof-
```
```  2208   let ?ia = "fps_inv a"
```
```  2209   let ?d = "fps_deriv a oo ?ia"
```
```  2210   let ?dia = "fps_deriv ?ia"
```
```  2211   have ia0: "?ia\$0 = 0" by (simp add: fps_inv_def)
```
```  2212   have th0: "?d\$0 \<noteq> 0" using a1 by (simp add: fps_compose_nth fps_deriv_nth)
```
```  2213   from fps_inv_right[OF a0 a1] have "?d * ?dia = 1"
```
```  2214     by (simp add: fps_compose_deriv[OF ia0, of a, symmetric] )
```
```  2215   hence "inverse ?d * ?d * ?dia = inverse ?d * 1" by simp
```
```  2216   with inverse_mult_eq_1[OF th0]
```
```  2217   show "?dia = inverse ?d" by simp
```
```  2218 qed
```
```  2219
```
```  2220 subsection{* Elementary series *}
```
```  2221
```
```  2222 subsubsection{* Exponential series *}
```
```  2223 definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
```
```  2224
```
```  2225 lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::{field, recpower, ring_char_0}) * E a" (is "?l = ?r")
```
```  2226 proof-
```
```  2227   {fix n
```
```  2228     have "?l\$n = ?r \$ n"
```
```  2229   apply (auto simp add: E_def field_simps power_Suc[symmetric]simp del: fact_Suc of_nat_Suc)
```
```  2230   by (simp add: of_nat_mult ring_simps)}
```
```  2231 then show ?thesis by (simp add: fps_eq_iff)
```
```  2232 qed
```
```  2233
```
```  2234 lemma E_unique_ODE:
```
```  2235   "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a\$0) * E (c :: 'a::{field, ring_char_0, recpower})"
```
```  2236   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  2237 proof-
```
```  2238   {assume d: ?lhs
```
```  2239   from d have th: "\<And>n. a \$ Suc n = c * a\$n / of_nat (Suc n)"
```
```  2240     by (simp add: fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
```
```  2241   {fix n have "a\$n = a\$0 * c ^ n/ (of_nat (fact n))"
```
```  2242       apply (induct n)
```
```  2243       apply simp
```
```  2244       unfolding th
```
```  2245       using fact_gt_zero
```
```  2246       apply (simp add: field_simps del: of_nat_Suc fact.simps)
```
```  2247       apply (drule sym)
```
```  2248       by (simp add: ring_simps of_nat_mult power_Suc)}
```
```  2249   note th' = this
```
```  2250   have ?rhs
```
```  2251     by (auto simp add: fps_eq_iff fps_const_mult_left E_def intro : th')}
```
```  2252 moreover
```
```  2253 {assume h: ?rhs
```
```  2254   have ?lhs
```
```  2255     apply (subst h)
```
```  2256     apply simp
```
```  2257     apply (simp only: h[symmetric])
```
```  2258   by simp}
```
```  2259 ultimately show ?thesis by blast
```
```  2260 qed
```
```  2261
```
```  2262 lemma E_add_mult: "E (a + b) = E (a::'a::{ring_char_0, field, recpower}) * E b" (is "?l = ?r")
```
```  2263 proof-
```
```  2264   have "fps_deriv (?r) = fps_const (a+b) * ?r"
```
```  2265     by (simp add: fps_const_add[symmetric] ring_simps del: fps_const_add)
```
```  2266   then have "?r = ?l" apply (simp only: E_unique_ODE)
```
```  2267     by (simp add: fps_mult_nth E_def)
```
```  2268   then show ?thesis ..
```
```  2269 qed
```
```  2270
```
```  2271 lemma E_nth[simp]: "E a \$ n = a^n / of_nat (fact n)"
```
```  2272   by (simp add: E_def)
```
```  2273
```
```  2274 lemma E0[simp]: "E (0::'a::{field, recpower}) = 1"
```
```  2275   by (simp add: fps_eq_iff power_0_left)
```
```  2276
```
```  2277 lemma E_neg: "E (- a) = inverse (E (a::'a::{ring_char_0, field, recpower}))"
```
```  2278 proof-
```
```  2279   from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
```
```  2280     by (simp )
```
```  2281   have th1: "E a \$ 0 \<noteq> 0" by simp
```
```  2282   from fps_inverse_unique[OF th1 th0] show ?thesis by simp
```
```  2283 qed
```
```  2284
```
```  2285 lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::{field, recpower, ring_char_0})) = (fps_const a)^n * (E a)"
```
```  2286   by (induct n, auto simp add: power_Suc)
```
```  2287
```
```  2288 lemma fps_compose_uminus: "- (a::'a::ring_1 fps) oo c = - (a oo c)"
```
```  2289   by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_negf[symmetric])
```
```  2290
```
```  2291 lemma fps_compose_sub_distrib:
```
```  2292   shows "(a - b) oo (c::'a::ring_1 fps) = (a oo c) - (b oo c)"
```
```  2293   unfolding diff_minus fps_compose_uminus fps_compose_add_distrib ..
```
```  2294
```
```  2295 lemma X_fps_compose:"X oo a = Abs_fps (\<lambda>n. if n = 0 then (0::'a::comm_ring_1) else a\$n)"
```
```  2296   apply (simp add: fps_eq_iff fps_compose_nth)
```
```  2297   by (simp add: cond_value_iff cond_application_beta setsum_delta power_Suc cong del: if_weak_cong)
```
```  2298
```
```  2299 lemma X_compose_E[simp]: "X oo E (a::'a::{field, recpower}) = E a - 1"
```
```  2300   by (simp add: fps_eq_iff X_fps_compose)
```
```  2301
```
```  2302 lemma LE_compose:
```
```  2303   assumes a: "a\<noteq>0"
```
```  2304   shows "fps_inv (E a - 1) oo (E a - 1) = X"
```
```  2305   and "(E a - 1) oo fps_inv (E a - 1) = X"
```
```  2306 proof-
```
```  2307   let ?b = "E a - 1"
```
```  2308   have b0: "?b \$ 0 = 0" by simp
```
```  2309   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  2310   from fps_inv[OF b0 b1] show "fps_inv (E a - 1) oo (E a - 1) = X" .
```
```  2311   from fps_inv_right[OF b0 b1] show "(E a - 1) oo fps_inv (E a - 1) = X" .
```
```  2312 qed
```
```  2313
```
```  2314
```
```  2315 lemma fps_const_inverse:
```
```  2316   "inverse (fps_const (a::'a::{field, division_by_zero})) = fps_const (inverse a)"
```
```  2317   apply (auto simp add: fps_eq_iff fps_inverse_def) by (case_tac "n", auto)
```
```  2318
```
```  2319
```
```  2320 lemma inverse_one_plus_X:
```
```  2321   "inverse (1 + X) = Abs_fps (\<lambda>n. (- 1 ::'a::{field, recpower})^n)"
```
```  2322   (is "inverse ?l = ?r")
```
```  2323 proof-
```
```  2324   have th: "?l * ?r = 1"
```
```  2325     apply (auto simp add: ring_simps fps_eq_iff X_mult_nth  minus_one_power_iff)
```
```  2326     apply presburger+
```
```  2327     done
```
```  2328   have th': "?l \$ 0 \<noteq> 0" by (simp add: )
```
```  2329   from fps_inverse_unique[OF th' th] show ?thesis .
```
```  2330 qed
```
```  2331
```
```  2332 lemma E_power_mult: "(E (c::'a::{field,recpower,ring_char_0}))^n = E (of_nat n * c)"
```
```  2333   by (induct n, auto simp add: ring_simps E_add_mult power_Suc)
```
```  2334
```
```  2335 subsubsection{* Logarithmic series *}
```
```  2336 definition "(L::'a::{field, ring_char_0,recpower} fps)
```
```  2337   = Abs_fps (\<lambda>n. (- 1) ^ Suc n / of_nat n)"
```
```  2338
```
```  2339 lemma fps_deriv_L: "fps_deriv L = inverse (1 + X)"
```
```  2340   unfolding inverse_one_plus_X
```
```  2341   by (simp add: L_def fps_eq_iff power_Suc del: of_nat_Suc)
```
```  2342
```
```  2343 lemma L_nth: "L \$ n = (- 1) ^ Suc n / of_nat n"
```
```  2344   by (simp add: L_def)
```
```  2345
```
```  2346 lemma L_E_inv:
```
```  2347   assumes a: "a\<noteq> (0::'a::{field,division_by_zero,ring_char_0,recpower})"
```
```  2348   shows "L = fps_const a * fps_inv (E a - 1)" (is "?l = ?r")
```
```  2349 proof-
```
```  2350   let ?b = "E a - 1"
```
```  2351   have b0: "?b \$ 0 = 0" by simp
```
```  2352   have b1: "?b \$ 1 \<noteq> 0" by (simp add: a)
```
```  2353   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)"
```
```  2354     by (simp add: ring_simps)
```
```  2355   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])
```
```  2356     by (simp add: ring_simps)
```
```  2357   finally have eq: "fps_deriv (E a - 1) oo fps_inv (E a - 1) = fps_const a * (X + 1)" .
```
```  2358   from fps_inv_deriv[OF b0 b1, unfolded eq]
```
```  2359   have "fps_deriv (fps_inv ?b) = fps_const (inverse a) / (X + 1)"
```
```  2360     by (simp add: fps_const_inverse eq fps_divide_def fps_inverse_mult)
```
```  2361   hence "fps_deriv (fps_const a * fps_inv ?b) = inverse (X + 1)"
```
```  2362     using a by (simp add: fps_divide_def field_simps)
```
```  2363   hence "fps_deriv ?l = fps_deriv ?r"
```
```  2364     by (simp add: fps_deriv_L add_commute)
```
```  2365   then show ?thesis unfolding fps_deriv_eq_iff
```
```  2366     by (simp add: L_nth fps_inv_def)
```
```  2367 qed
```
```  2368
```
```  2369 subsubsection{* Formal trigonometric functions  *}
```
```  2370
```
```  2371 definition "fps_sin (c::'a::{field, recpower, ring_char_0}) =
```
```  2372   Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
```
```  2373
```
```  2374 definition "fps_cos (c::'a::{field, recpower, ring_char_0}) = Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
```
```  2375
```
```  2376 lemma fps_sin_deriv:
```
```  2377   "fps_deriv (fps_sin c) = fps_const c * fps_cos c"
```
```  2378   (is "?lhs = ?rhs")
```
```  2379 proof-
```
```  2380   {fix n::nat
```
```  2381     {assume en: "even n"
```
```  2382       have "?lhs\$n = of_nat (n+1) * (fps_sin c \$ (n+1))" by simp
```
```  2383       also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  2384 	using en by (simp add: fps_sin_def)
```
```  2385       also have "\<dots> = (- 1)^(n div 2) * c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  2386 	unfolding fact_Suc of_nat_mult
```
```  2387 	by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  2388       also have "\<dots> = (- 1)^(n div 2) *c^Suc n / of_nat (fact n)"
```
```  2389 	by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  2390       finally have "?lhs \$n = ?rhs\$n" using en
```
```  2391 	by (simp add: fps_cos_def ring_simps power_Suc )}
```
```  2392     then have "?lhs \$ n = ?rhs \$ n"
```
```  2393       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def) }
```
```  2394   then show ?thesis by (auto simp add: fps_eq_iff)
```
```  2395 qed
```
```  2396
```
```  2397 lemma fps_cos_deriv:
```
```  2398   "fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
```
```  2399   (is "?lhs = ?rhs")
```
```  2400 proof-
```
```  2401   have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc)
```
```  2402   have th1: "\<And>n. odd n\<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2" by presburger
```
```  2403   {fix n::nat
```
```  2404     {assume en: "odd n"
```
```  2405       from en have n0: "n \<noteq>0 " by presburger
```
```  2406       have "?lhs\$n = of_nat (n+1) * (fps_cos c \$ (n+1))" by simp
```
```  2407       also have "\<dots> = of_nat (n+1) * ((- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact (Suc n)))"
```
```  2408 	using en by (simp add: fps_cos_def)
```
```  2409       also have "\<dots> = (- 1)^((n + 1) div 2)*c^Suc n * (of_nat (n+1) / (of_nat (Suc n) * of_nat (fact n)))"
```
```  2410 	unfolding fact_Suc of_nat_mult
```
```  2411 	by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  2412       also have "\<dots> = (- 1)^((n + 1) div 2) * c^Suc n / of_nat (fact n)"
```
```  2413 	by (simp add: field_simps del: of_nat_add of_nat_Suc)
```
```  2414       also have "\<dots> = (- ((- 1)^((n - 1) div 2))) * c^Suc n / of_nat (fact n)"
```
```  2415 	unfolding th0 unfolding th1[OF en] by simp
```
```  2416       finally have "?lhs \$n = ?rhs\$n" using en
```
```  2417 	by (simp add: fps_sin_def fps_uminus_def ring_simps power_Suc)}
```
```  2418     then have "?lhs \$ n = ?rhs \$ n"
```
```  2419       by (cases "even n", simp_all add: fps_deriv_def fps_sin_def
```
```  2420 	fps_cos_def fps_uminus_def) }
```
```  2421   then show ?thesis by (auto simp add: fps_eq_iff)
```
```  2422 qed
```
```  2423
```
```  2424 lemma fps_sin_cos_sum_of_squares:
```
```  2425   "fps_cos c ^ 2 + fps_sin c ^ 2 = 1" (is "?lhs = 1")
```
```  2426 proof-
```
```  2427   have "fps_deriv ?lhs = 0"
```
```  2428     apply (simp add:  fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc)
```
```  2429     by (simp add: fps_power_def ring_simps fps_const_neg[symmetric] del: fps_const_neg)
```
```  2430   then have "?lhs = fps_const (?lhs \$ 0)"
```
```  2431     unfolding fps_deriv_eq_0_iff .
```
```  2432   also have "\<dots> = 1"
```
```  2433     by (auto simp add: fps_eq_iff fps_power_def nat_number fps_mult_nth fps_cos_def fps_sin_def)
```
```  2434   finally show ?thesis .
```
```  2435 qed
```
```  2436
```
```  2437 definition "fps_tan c = fps_sin c / fps_cos c"
```
```  2438
```
```  2439 lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)"
```
```  2440 proof-
```
```  2441   have th0: "fps_cos c \$ 0 \<noteq> 0" by (simp add: fps_cos_def)
```
```  2442   show ?thesis
```
```  2443     using fps_sin_cos_sum_of_squares[of c]
```
```  2444     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)
```
```  2445     unfolding right_distrib[symmetric]
```
```  2446     by simp
```
```  2447 qed
```
`  2448 end`