src/HOL/Library/Formal_Power_Series.thy
author huffman
Thu, 12 Mar 2009 08:57:03 -0700
changeset 30488 5c4c3a9e9102
parent 30273 ecd6f0ca62ea
child 30661 54858c8ad226
child 30746 d6915b738bd9
permissions -rw-r--r--
remove trailing spaces
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     1
(*  Title:      Formal_Power_Series.thy
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
     2
    ID:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     3
    Author:     Amine Chaieb, University of Cambridge
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     4
*)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     5
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     6
header{* A formalization of formal power series *}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     7
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     8
theory Formal_Power_Series
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
     9
  imports Main Fact Parity
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    10
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    11
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
    12
subsection {* The type of formal power series*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    13
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    14
typedef (open) 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    15
  morphisms fps_nth Abs_fps
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    16
  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    17
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    18
notation fps_nth (infixl "$" 75)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    19
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    20
lemma expand_fps_eq: "p = q \<longleftrightarrow> (\<forall>n. p $ n = q $ n)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    21
  by (simp add: fps_nth_inject [symmetric] expand_fun_eq)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    22
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    23
lemma fps_ext: "(\<And>n. p $ n = q $ n) \<Longrightarrow> p = q"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    24
  by (simp add: expand_fps_eq)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    25
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    26
lemma fps_nth_Abs_fps [simp]: "Abs_fps f $ n = f n"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    27
  by (simp add: Abs_fps_inverse)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    28
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    29
text{* Definition of the basic elements 0 and 1 and the basic operations of addition, negation and multiplication *}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    30
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    31
instantiation fps :: (zero)  zero
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    32
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    33
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    34
definition fps_zero_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    35
  "0 = Abs_fps (\<lambda>n. 0)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    36
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    37
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    38
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    39
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    40
lemma fps_zero_nth [simp]: "0 $ n = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    41
  unfolding fps_zero_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    42
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    43
instantiation fps :: ("{one,zero}")  one
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    44
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    45
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    46
definition fps_one_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    47
  "1 = Abs_fps (\<lambda>n. if n = 0 then 1 else 0)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    48
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    49
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    50
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    51
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
    52
lemma fps_one_nth [simp]: "1 $ n = (if n = 0 then 1 else 0)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    53
  unfolding fps_one_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    54
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    55
instantiation fps :: (plus)  plus
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    56
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    57
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    58
definition fps_plus_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    59
  "op + = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n + g $ n))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    60
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    61
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    62
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    63
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    64
lemma fps_add_nth [simp]: "(f + g) $ n = f $ n + g $ n"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    65
  unfolding fps_plus_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    66
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    67
instantiation fps :: (minus) minus
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    68
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    69
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    70
definition fps_minus_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    71
  "op - = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    72
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    73
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    74
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    75
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    76
lemma fps_sub_nth [simp]: "(f - g) $ n = f $ n - g $ n"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    77
  unfolding fps_minus_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    78
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    79
instantiation fps :: (uminus) uminus
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    80
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    81
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    82
definition fps_uminus_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    83
  "uminus = (\<lambda>f. Abs_fps (\<lambda>n. - (f $ n)))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    84
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    85
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    86
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    87
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    88
lemma fps_neg_nth [simp]: "(- f) $ n = - (f $ n)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    89
  unfolding fps_uminus_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    90
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    91
instantiation fps :: ("{comm_monoid_add, times}")  times
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    92
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    93
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    94
definition fps_times_def:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    95
  "op * = (\<lambda>f g. Abs_fps (\<lambda>n. \<Sum>i=0..n. f $ i * g $ (n - i)))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
    96
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    97
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    98
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
    99
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   100
lemma fps_mult_nth: "(f * g) $ n = (\<Sum>i=0..n. f$i * g$(n - i))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   101
  unfolding fps_times_def by simp
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   102
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   103
declare atLeastAtMost_iff[presburger]
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   104
declare Bex_def[presburger]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   105
declare Ball_def[presburger]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   106
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   107
lemma mult_delta_left:
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   108
  fixes x y :: "'a::mult_zero"
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   109
  shows "(if b then x else 0) * y = (if b then x * y else 0)"
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   110
  by simp
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   111
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   112
lemma mult_delta_right:
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   113
  fixes x y :: "'a::mult_zero"
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   114
  shows "x * (if b then y else 0) = (if b then x * y else 0)"
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   115
  by simp
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   116
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   117
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   118
  by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   119
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   120
  by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   121
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   122
subsection{* Formal power series form a commutative ring with unity, if the range of sequences
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   123
  they represent is a commutative ring with unity*}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   124
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   125
instance fps :: (semigroup_add) semigroup_add
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   126
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   127
  fix a b c :: "'a fps" show "a + b + c = a + (b + c)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   128
    by (simp add: fps_ext add_assoc)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   129
qed
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   130
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   131
instance fps :: (ab_semigroup_add) ab_semigroup_add
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   132
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   133
  fix a b :: "'a fps" show "a + b = b + a"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   134
    by (simp add: fps_ext add_commute)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   135
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   136
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   137
lemma fps_mult_assoc_lemma:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   138
  fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   139
  shows "(\<Sum>j=0..k. \<Sum>i=0..j. f i (j - i) (n - j)) =
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   140
         (\<Sum>j=0..k. \<Sum>i=0..k - j. f j i (n - j - i))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   141
proof (induct k)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   142
  case 0 show ?case by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   143
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   144
  case (Suc k) thus ?case
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   145
    by (simp add: Suc_diff_le setsum_addf add_assoc
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   146
             cong: strong_setsum_cong)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   147
qed
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   148
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   149
instance fps :: (semiring_0) semigroup_mult
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   150
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   151
  fix a b c :: "'a fps"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   152
  show "(a * b) * c = a * (b * c)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   153
  proof (rule fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   154
    fix n :: nat
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   155
    have "(\<Sum>j=0..n. \<Sum>i=0..j. a$i * b$(j - i) * c$(n - j)) =
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   156
          (\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   157
      by (rule fps_mult_assoc_lemma)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   158
    thus "((a * b) * c) $ n = (a * (b * c)) $ n"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   159
      by (simp add: fps_mult_nth setsum_right_distrib
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   160
                    setsum_left_distrib mult_assoc)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   161
  qed
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   162
qed
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   163
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   164
lemma fps_mult_commute_lemma:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   165
  fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   166
  shows "(\<Sum>i=0..n. f i (n - i)) = (\<Sum>i=0..n. f (n - i) i)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   167
proof (rule setsum_reindex_cong)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   168
  show "inj_on (\<lambda>i. n - i) {0..n}"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   169
    by (rule inj_onI) simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   170
  show "{0..n} = (\<lambda>i. n - i) ` {0..n}"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   171
    by (auto, rule_tac x="n - x" in image_eqI, simp_all)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   172
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   173
  fix i assume "i \<in> {0..n}"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   174
  hence "n - (n - i) = i" by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   175
  thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   176
qed
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   177
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   178
instance fps :: (comm_semiring_0) ab_semigroup_mult
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   179
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   180
  fix a b :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   181
  show "a * b = b * a"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   182
  proof (rule fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   183
    fix n :: nat
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   184
    have "(\<Sum>i=0..n. a$i * b$(n - i)) = (\<Sum>i=0..n. a$(n - i) * b$i)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   185
      by (rule fps_mult_commute_lemma)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   186
    thus "(a * b) $ n = (b * a) $ n"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   187
      by (simp add: fps_mult_nth mult_commute)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   188
  qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   189
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   190
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   191
instance fps :: (monoid_add) monoid_add
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   192
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   193
  fix a :: "'a fps" show "0 + a = a "
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   194
    by (simp add: fps_ext)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   195
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   196
  fix a :: "'a fps" show "a + 0 = a "
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   197
    by (simp add: fps_ext)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   198
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   199
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   200
instance fps :: (comm_monoid_add) comm_monoid_add
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   201
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   202
  fix a :: "'a fps" show "0 + a = a "
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   203
    by (simp add: fps_ext)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   204
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   205
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   206
instance fps :: (semiring_1) monoid_mult
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   207
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   208
  fix a :: "'a fps" show "1 * a = a"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   209
    by (simp add: fps_ext fps_mult_nth mult_delta_left setsum_delta)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   210
next
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   211
  fix a :: "'a fps" show "a * 1 = a"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   212
    by (simp add: fps_ext fps_mult_nth mult_delta_right setsum_delta')
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   213
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   214
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   215
instance fps :: (cancel_semigroup_add) cancel_semigroup_add
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   216
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   217
  fix a b c :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   218
  assume "a + b = a + c" then show "b = c"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   219
    by (simp add: expand_fps_eq)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   220
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   221
  fix a b c :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   222
  assume "b + a = c + a" then show "b = c"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   223
    by (simp add: expand_fps_eq)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   224
qed
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   225
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   226
instance fps :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   227
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   228
  fix a b c :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   229
  assume "a + b = a + c" then show "b = c"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   230
    by (simp add: expand_fps_eq)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   231
qed
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   232
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   233
instance fps :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   234
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   235
instance fps :: (group_add) group_add
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   236
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   237
  fix a :: "'a fps" show "- a + a = 0"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   238
    by (simp add: fps_ext)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   239
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   240
  fix a b :: "'a fps" show "a - b = a + - b"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   241
    by (simp add: fps_ext diff_minus)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   242
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   243
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   244
instance fps :: (ab_group_add) ab_group_add
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   245
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   246
  fix a :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   247
  show "- a + a = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   248
    by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   249
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   250
  fix a b :: "'a fps"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   251
  show "a - b = a + - b"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   252
    by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   253
qed
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   254
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   255
instance fps :: (zero_neq_one) zero_neq_one
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   256
  by default (simp add: expand_fps_eq)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   257
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   258
instance fps :: (semiring_0) semiring
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   259
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   260
  fix a b c :: "'a fps"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   261
  show "(a + b) * c = a * c + b * c"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   262
    by (simp add: expand_fps_eq fps_mult_nth left_distrib setsum_addf)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   263
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   264
  fix a b c :: "'a fps"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   265
  show "a * (b + c) = a * b + a * c"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   266
    by (simp add: expand_fps_eq fps_mult_nth right_distrib setsum_addf)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   267
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   268
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   269
instance fps :: (semiring_0) semiring_0
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   270
proof
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   271
  fix a:: "'a fps" show "0 * a = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   272
    by (simp add: fps_ext fps_mult_nth)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   273
next
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   274
  fix a:: "'a fps" show "a * 0 = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   275
    by (simp add: fps_ext fps_mult_nth)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   276
qed
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   277
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   278
instance fps :: (semiring_0_cancel) semiring_0_cancel ..
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   279
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   280
subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   281
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   282
lemma fps_nonzero_nth: "f \<noteq> 0 \<longleftrightarrow> (\<exists> n. f $n \<noteq> 0)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   283
  by (simp add: expand_fps_eq)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   284
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   285
lemma fps_nonzero_nth_minimal:
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   286
  "f \<noteq> 0 \<longleftrightarrow> (\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   287
proof
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   288
  let ?n = "LEAST n. f $ n \<noteq> 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   289
  assume "f \<noteq> 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   290
  then have "\<exists>n. f $ n \<noteq> 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   291
    by (simp add: fps_nonzero_nth)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   292
  then have "f $ ?n \<noteq> 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   293
    by (rule LeastI_ex)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   294
  moreover have "\<forall>m<?n. f $ m = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   295
    by (auto dest: not_less_Least)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   296
  ultimately have "f $ ?n \<noteq> 0 \<and> (\<forall>m<?n. f $ m = 0)" ..
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   297
  then show "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)" ..
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   298
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   299
  assume "\<exists>n. f $ n \<noteq> 0 \<and> (\<forall>m<n. f $ m = 0)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   300
  then show "f \<noteq> 0" by (auto simp add: expand_fps_eq)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   301
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   302
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   303
lemma fps_eq_iff: "f = g \<longleftrightarrow> (\<forall>n. f $ n = g $n)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   304
  by (rule expand_fps_eq)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   305
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   306
lemma fps_setsum_nth: "(setsum f S) $ n = setsum (\<lambda>k. (f k) $ n) S"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   307
proof (cases "finite S")
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   308
  assume "\<not> finite S" then show ?thesis by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   309
next
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   310
  assume "finite S"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   311
  then show ?thesis by (induct set: finite) auto
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   312
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   313
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   314
subsection{* Injection of the basic ring elements and multiplication by scalars *}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   315
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   316
definition
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   317
  "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   318
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   319
lemma fps_nth_fps_const [simp]: "fps_const c $ n = (if n = 0 then c else 0)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   320
  unfolding fps_const_def by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   321
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   322
lemma fps_const_0_eq_0 [simp]: "fps_const 0 = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   323
  by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   324
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   325
lemma fps_const_1_eq_1 [simp]: "fps_const 1 = 1"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   326
  by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   327
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   328
lemma fps_const_neg [simp]: "- (fps_const (c::'a::ring)) = fps_const (- c)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   329
  by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   330
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   331
lemma fps_const_add [simp]: "fps_const (c::'a\<Colon>monoid_add) + fps_const d = fps_const (c + d)"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   332
  by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   333
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   334
lemma fps_const_mult[simp]: "fps_const (c::'a\<Colon>ring) * fps_const d = fps_const (c * d)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   335
  by (simp add: fps_eq_iff fps_mult_nth setsum_0')
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   336
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   337
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)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   338
  by (simp add: fps_ext)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   339
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   340
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)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   341
  by (simp add: fps_ext)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   342
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   343
lemma fps_const_mult_left: "fps_const (c::'a\<Colon>semiring_0) * f = Abs_fps (\<lambda>n. c * f$n)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   344
  unfolding fps_eq_iff fps_mult_nth
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   345
  by (simp add: fps_const_def mult_delta_left setsum_delta)
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   346
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   347
lemma fps_const_mult_right: "f * fps_const (c::'a\<Colon>semiring_0) = Abs_fps (\<lambda>n. f$n * c)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   348
  unfolding fps_eq_iff fps_mult_nth
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   349
  by (simp add: fps_const_def mult_delta_right setsum_delta')
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   350
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   351
lemma fps_mult_left_const_nth [simp]: "(fps_const (c::'a::semiring_1) * f)$n = c* f$n"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   352
  by (simp add: fps_mult_nth mult_delta_left setsum_delta)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   353
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   354
lemma fps_mult_right_const_nth [simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   355
  by (simp add: fps_mult_nth mult_delta_right setsum_delta')
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   356
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   357
subsection {* Formal power series form an integral domain*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   358
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   359
instance fps :: (ring) ring ..
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   360
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   361
instance fps :: (ring_1) ring_1
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   362
  by (intro_classes, auto simp add: diff_minus left_distrib)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   363
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   364
instance fps :: (comm_ring_1) comm_ring_1
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   365
  by (intro_classes, auto simp add: diff_minus left_distrib)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   366
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   367
instance fps :: (ring_no_zero_divisors) ring_no_zero_divisors
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   368
proof
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   369
  fix a b :: "'a fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   370
  assume a0: "a \<noteq> 0" and b0: "b \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   371
  then obtain i j where i: "a$i\<noteq>0" "\<forall>k<i. a$k=0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   372
    and j: "b$j \<noteq>0" "\<forall>k<j. b$k =0" unfolding fps_nonzero_nth_minimal
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   373
    by blast+
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   374
  have "(a * b) $ (i+j) = (\<Sum>k=0..i+j. a$k * b$(i+j-k))"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   375
    by (rule fps_mult_nth)
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   376
  also have "\<dots> = (a$i * b$(i+j-i)) + (\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k))"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   377
    by (rule setsum_diff1') simp_all
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   378
  also have "(\<Sum>k\<in>{0..i+j}-{i}. a$k * b$(i+j-k)) = 0"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   379
    proof (rule setsum_0' [rule_format])
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   380
      fix k assume "k \<in> {0..i+j} - {i}"
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   381
      then have "k < i \<or> i+j-k < j" by auto
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   382
      then show "a$k * b$(i+j-k) = 0" using i j by auto
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   383
    qed
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   384
  also have "a$i * b$(i+j-i) + 0 = a$i * b$j" by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   385
  also have "a$i * b$j \<noteq> 0" using i j by simp
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   386
  finally have "(a*b) $ (i+j) \<noteq> 0" .
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   387
  then show "a*b \<noteq> 0" unfolding fps_nonzero_nth by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   388
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   389
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   390
instance fps :: (idom) idom ..
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   391
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   392
subsection{* Inverses of formal power series *}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   393
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   394
declare setsum_cong[fundef_cong]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   395
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   396
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   397
instantiation fps :: ("{comm_monoid_add,inverse, times, uminus}") inverse
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   398
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   399
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   400
fun natfun_inverse:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" where
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   401
  "natfun_inverse f 0 = inverse (f$0)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   402
| "natfun_inverse f n = - inverse (f$0) * setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   403
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   404
definition fps_inverse_def:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   405
  "inverse f = (if f$0 = 0 then 0 else Abs_fps (natfun_inverse f))"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   406
definition fps_divide_def: "divide = (\<lambda>(f::'a fps) g. f * inverse g)"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   407
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   408
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   409
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   410
lemma fps_inverse_zero[simp]:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   411
  "inverse (0 :: 'a::{comm_monoid_add,inverse, times, uminus} fps) = 0"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   412
  by (simp add: fps_ext fps_inverse_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   413
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   414
lemma fps_inverse_one[simp]: "inverse (1 :: 'a::{division_ring,zero_neq_one} fps) = 1"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   415
  apply (auto simp add: expand_fps_eq fps_inverse_def)
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   416
  by (case_tac n, auto)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   417
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   418
instance fps :: ("{comm_monoid_add,inverse, times, uminus}")  division_by_zero
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   419
  by default (rule fps_inverse_zero)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   420
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   421
lemma inverse_mult_eq_1[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   422
  shows "inverse f * f = 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   423
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   424
  have c: "inverse f * f = f * inverse f" by (simp add: mult_commute)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   425
  from f0 have ifn: "\<And>n. inverse f $ n = natfun_inverse f n"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   426
    by (simp add: fps_inverse_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   427
  from f0 have th0: "(inverse f * f) $ 0 = 1"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   428
    by (simp add: fps_mult_nth fps_inverse_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   429
  {fix n::nat assume np: "n >0 "
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   430
    from np have eq: "{0..n} = {0} \<union> {1 .. n}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   431
    have d: "{0} \<inter> {1 .. n} = {}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   432
    have f: "finite {0::nat}" "finite {1..n}" by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   433
    from f0 np have th0: "- (inverse f$n) =
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   434
      (setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n}) / (f$0)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   435
      by (cases n, simp, simp add: divide_inverse fps_inverse_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   436
    from th0[symmetric, unfolded nonzero_divide_eq_eq[OF f0]]
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   437
    have th1: "setsum (\<lambda>i. f$i * natfun_inverse f (n - i)) {1..n} =
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   438
      - (f$0) * (inverse f)$n"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   439
      by (simp add: ring_simps)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   440
    have "(f * inverse f) $ n = (\<Sum>i = 0..n. f $i * natfun_inverse f (n - i))"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   441
      unfolding fps_mult_nth ifn ..
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   442
    also have "\<dots> = f$0 * natfun_inverse f n
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   443
      + (\<Sum>i = 1..n. f$i * natfun_inverse f (n-i))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   444
      unfolding setsum_Un_disjoint[OF f d, unfolded eq[symmetric]]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   445
      by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   446
    also have "\<dots> = 0" unfolding th1 ifn by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   447
    finally have "(inverse f * f)$n = 0" unfolding c . }
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   448
  with th0 show ?thesis by (simp add: fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   449
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   450
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   451
lemma fps_inverse_0_iff[simp]: "(inverse f)$0 = (0::'a::division_ring) \<longleftrightarrow> f$0 = 0"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   452
  by (simp add: fps_inverse_def nonzero_imp_inverse_nonzero)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   453
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   454
lemma fps_inverse_eq_0_iff[simp]: "inverse f = (0:: ('a::field) fps) \<longleftrightarrow> f $0 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   455
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   456
  {assume "f$0 = 0" hence "inverse f = 0" by (simp add: fps_inverse_def)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   457
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   458
  {assume h: "inverse f = 0" and c: "f $0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   459
    from inverse_mult_eq_1[OF c] h have False by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   460
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   461
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   462
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   463
lemma fps_inverse_idempotent[intro]: assumes f0: "f$0 \<noteq> (0::'a::field)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   464
  shows "inverse (inverse f) = f"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   465
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   466
  from f0 have if0: "inverse f $ 0 \<noteq> 0" by simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   467
  from inverse_mult_eq_1[OF f0] inverse_mult_eq_1[OF if0]
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   468
  have th0: "inverse f * f = inverse f * inverse (inverse f)"   by (simp add: mult_ac)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   469
  then show ?thesis using f0 unfolding mult_cancel_left by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   470
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   471
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   472
lemma fps_inverse_unique: assumes f0: "f$0 \<noteq> (0::'a::field)" and fg: "f*g = 1"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   473
  shows "inverse f = g"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   474
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   475
  from inverse_mult_eq_1[OF f0] fg
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   476
  have th0: "inverse f * f = g * f" by (simp add: mult_ac)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   477
  then show ?thesis using f0  unfolding mult_cancel_right
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   478
    by (auto simp add: expand_fps_eq)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   479
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   480
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   481
lemma fps_inverse_gp: "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   482
  = Abs_fps (\<lambda>n. if n= 0 then 1 else if n=1 then - 1 else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   483
  apply (rule fps_inverse_unique)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   484
  apply simp
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   485
  apply (simp add: fps_eq_iff fps_mult_nth)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   486
proof(clarsimp)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   487
  fix n::nat assume n: "n > 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   488
  let ?f = "\<lambda>i. if n = i then (1\<Colon>'a) else if n - i = 1 then - 1 else 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   489
  let ?g = "\<lambda>i. if i = n then 1 else if i=n - 1 then - 1 else 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   490
  let ?h = "\<lambda>i. if i=n - 1 then - 1 else 0"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   491
  have th1: "setsum ?f {0..n} = setsum ?g {0..n}"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   492
    by (rule setsum_cong2) auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   493
  have th2: "setsum ?g {0..n - 1} = setsum ?h {0..n - 1}"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   494
    using n apply - by (rule setsum_cong2) auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   495
  have eq: "{0 .. n} = {0.. n - 1} \<union> {n}" by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   496
  from n have d: "{0.. n - 1} \<inter> {n} = {}" by auto
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   497
  have f: "finite {0.. n - 1}" "finite {n}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   498
  show "setsum ?f {0..n} = 0"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   499
    unfolding th1
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   500
    apply (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] del: One_nat_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   501
    unfolding th2
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   502
    by(simp add: setsum_delta)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   503
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   504
29912
f4ac160d2e77 fix spelling
huffman
parents: 29911
diff changeset
   505
subsection{* Formal Derivatives, and the MacLaurin theorem around 0*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   506
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   507
definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   508
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   509
lemma fps_deriv_nth[simp]: "fps_deriv f $ n = of_nat (n +1) * f $ (n+1)" by (simp add: fps_deriv_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   510
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   511
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   512
  unfolding fps_eq_iff fps_add_nth  fps_const_mult_left fps_deriv_nth by (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   513
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   514
lemma fps_deriv_mult[simp]:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   515
  fixes f :: "('a :: comm_ring_1) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   516
  shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   517
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   518
  let ?D = "fps_deriv"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   519
  {fix n::nat
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   520
    let ?Zn = "{0 ..n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   521
    let ?Zn1 = "{0 .. n + 1}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   522
    let ?f = "\<lambda>i. i + 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   523
    have fi: "inj_on ?f {0..n}" by (simp add: inj_on_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   524
    have eq: "{1.. n+1} = ?f ` {0..n}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   525
    let ?g = "\<lambda>i. of_nat (i+1) * g $ (i+1) * f $ (n - i) +
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   526
        of_nat (i+1)* f $ (i+1) * g $ (n - i)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   527
    let ?h = "\<lambda>i. of_nat i * g $ i * f $ ((n+1) - i) +
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   528
        of_nat i* f $ i * g $ ((n + 1) - i)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   529
    {fix k assume k: "k \<in> {0..n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   530
      have "?h (k + 1) = ?g k" using k by auto}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   531
    note th0 = this
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   532
    have eq': "{0..n +1}- {1 .. n+1} = {0}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   533
    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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   534
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   535
      apply (simp add: inj_on_def Ball_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   536
      apply presburger
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   537
      apply (rule set_ext)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   538
      apply (presburger add: image_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   539
      by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   540
    have s1: "setsum (\<lambda>i. f $ i * g $ (n + 1 - i)) ?Zn1 = setsum (\<lambda>i. f $ (n + 1 - i) * g $ i) ?Zn1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   541
      apply (rule setsum_reindex_cong[where f="\<lambda>i. n + 1 - i"])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   542
      apply (simp add: inj_on_def Ball_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   543
      apply presburger
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   544
      apply (rule set_ext)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   545
      apply (presburger add: image_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   546
      by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   547
    have "(f * ?D g + ?D f * g)$n = (?D g * f + ?D f * g)$n" by (simp only: mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   548
    also have "\<dots> = (\<Sum>i = 0..n. ?g i)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   549
      by (simp add: fps_mult_nth setsum_addf[symmetric])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   550
    also have "\<dots> = setsum ?h {1..n+1}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   551
      using th0 setsum_reindex_cong[OF fi eq, of "?g" "?h"] by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   552
    also have "\<dots> = setsum ?h {0..n+1}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   553
      apply (rule setsum_mono_zero_left)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   554
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   555
      apply (simp add: subset_eq)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   556
      unfolding eq'
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   557
      by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   558
    also have "\<dots> = (fps_deriv (f * g)) $ n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   559
      apply (simp only: fps_deriv_nth fps_mult_nth setsum_addf)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   560
      unfolding s0 s1
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   561
      unfolding setsum_addf[symmetric] setsum_right_distrib
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   562
      apply (rule setsum_cong2)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   563
      by (auto simp add: of_nat_diff ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   564
    finally have "(f * ?D g + ?D f * g) $ n = ?D (f*g) $ n" .}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   565
  then show ?thesis unfolding fps_eq_iff by auto
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   566
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   567
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   568
lemma fps_deriv_neg[simp]: "fps_deriv (- (f:: ('a:: comm_ring_1) fps)) = - (fps_deriv f)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   569
  by (simp add: fps_eq_iff fps_deriv_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   570
lemma fps_deriv_add[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) + g) = fps_deriv f + fps_deriv g"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   571
  using fps_deriv_linear[of 1 f 1 g] by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   572
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   573
lemma fps_deriv_sub[simp]: "fps_deriv ((f:: ('a::comm_ring_1) fps) - g) = fps_deriv f - fps_deriv g"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   574
  unfolding diff_minus by simp
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   575
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   576
lemma fps_deriv_const[simp]: "fps_deriv (fps_const c) = 0"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   577
  by (simp add: fps_ext fps_deriv_def fps_const_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   578
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   579
lemma fps_deriv_mult_const_left[simp]: "fps_deriv (fps_const (c::'a::comm_ring_1) * f) = fps_const c * fps_deriv f"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   580
  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   581
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   582
lemma fps_deriv_0[simp]: "fps_deriv 0 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   583
  by (simp add: fps_deriv_def fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   584
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   585
lemma fps_deriv_1[simp]: "fps_deriv 1 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   586
  by (simp add: fps_deriv_def fps_eq_iff )
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   587
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   588
lemma fps_deriv_mult_const_right[simp]: "fps_deriv (f * fps_const (c::'a::comm_ring_1)) = fps_deriv f * fps_const c"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   589
  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   590
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   591
lemma fps_deriv_setsum: "fps_deriv (setsum f S) = setsum (\<lambda>i. fps_deriv (f i :: ('a::comm_ring_1) fps)) S"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   592
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   593
  {assume "\<not> finite S" hence ?thesis by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   594
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   595
  {assume fS: "finite S"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   596
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   597
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   598
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   599
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   600
lemma fps_deriv_eq_0_iff[simp]: "fps_deriv f = 0 \<longleftrightarrow> (f = fps_const (f$0 :: 'a::{idom,semiring_char_0}))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   601
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   602
  {assume "f= fps_const (f$0)" hence "fps_deriv f = fps_deriv (fps_const (f$0))" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   603
    hence "fps_deriv f = 0" by simp }
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   604
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   605
  {assume z: "fps_deriv f = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   606
    hence "\<forall>n. (fps_deriv f)$n = 0" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   607
    hence "\<forall>n. f$(n+1) = 0" by (simp del: of_nat_Suc of_nat_add One_nat_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   608
    hence "f = fps_const (f$0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   609
      apply (clarsimp simp add: fps_eq_iff fps_const_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   610
      apply (erule_tac x="n - 1" in allE)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   611
      by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   612
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   613
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   614
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   615
lemma fps_deriv_eq_iff:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   616
  fixes f:: "('a::{idom,semiring_char_0}) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   617
  shows "fps_deriv f = fps_deriv g \<longleftrightarrow> (f = fps_const(f$0 - g$0) + g)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   618
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   619
  have "fps_deriv f = fps_deriv g \<longleftrightarrow> fps_deriv (f - g) = 0" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   620
  also have "\<dots> \<longleftrightarrow> f - g = fps_const ((f-g)$0)" unfolding fps_deriv_eq_0_iff ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   621
  finally show ?thesis by (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   622
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   623
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   624
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)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   625
  apply auto unfolding fps_deriv_eq_iff by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   626
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   627
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   628
fun fps_nth_deriv :: "nat \<Rightarrow> ('a::semiring_1) fps \<Rightarrow> 'a fps" where
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   629
  "fps_nth_deriv 0 f = f"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   630
| "fps_nth_deriv (Suc n) f = fps_nth_deriv n (fps_deriv f)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   631
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   632
lemma fps_nth_deriv_commute: "fps_nth_deriv (Suc n) f = fps_deriv (fps_nth_deriv n f)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   633
  by (induct n arbitrary: f, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   634
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   635
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   636
  by (induct n arbitrary: f g, auto simp add: fps_nth_deriv_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   637
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   638
lemma fps_nth_deriv_neg[simp]: "fps_nth_deriv n (- (f:: ('a:: comm_ring_1) fps)) = - (fps_nth_deriv n f)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   639
  by (induct n arbitrary: f, simp_all)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   640
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   641
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   642
  using fps_nth_deriv_linear[of n 1 f 1 g] by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   643
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   644
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"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   645
  unfolding diff_minus fps_nth_deriv_add by simp
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   646
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   647
lemma fps_nth_deriv_0[simp]: "fps_nth_deriv n 0 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   648
  by (induct n, simp_all )
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   649
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   650
lemma fps_nth_deriv_1[simp]: "fps_nth_deriv n 1 = (if n = 0 then 1 else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   651
  by (induct n, simp_all )
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   652
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   653
lemma fps_nth_deriv_const[simp]: "fps_nth_deriv n (fps_const c) = (if n = 0 then fps_const c else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   654
  by (cases n, simp_all)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   655
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   656
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   657
  using fps_nth_deriv_linear[of n "c" f 0 0 ] by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   658
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   659
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   660
  using fps_nth_deriv_linear[of n "c" f 0 0] by (simp add: mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   661
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   662
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"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   663
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   664
  {assume "\<not> finite S" hence ?thesis by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   665
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   666
  {assume fS: "finite S"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   667
    have ?thesis  by (induct rule: finite_induct[OF fS], simp_all)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   668
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   669
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   670
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   671
lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   672
  by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   673
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   674
subsection {* Powers*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   675
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   676
instantiation fps :: (semiring_1) power
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   677
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   678
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   679
fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   680
  "fps_pow 0 f = 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   681
| "fps_pow (Suc n) f = f * fps_pow n f"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   682
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   683
definition fps_power_def: "power (f::'a fps) n = fps_pow n f"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   684
instance ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   685
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   686
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   687
instantiation fps :: (comm_ring_1) recpower
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   688
begin
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   689
instance
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   690
  apply (intro_classes)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   691
  by (simp_all add: fps_power_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   692
end
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   693
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   694
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   695
  by (induct n, auto simp add: fps_power_def expand_fps_eq fps_mult_nth)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   696
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   697
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   698
proof(induct n)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   699
  case 0 thus ?case by (simp add: fps_power_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   700
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   701
  case (Suc n)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   702
  note h = Suc.hyps[OF `a$0 = 1`]
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   703
  show ?case unfolding power_Suc fps_mult_nth
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   704
    using h `a$0 = 1`  fps_power_zeroth_eq_one[OF `a$0=1`] by (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   705
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   706
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   707
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   708
  by (induct n, auto simp add: fps_power_def fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   709
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   710
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   711
  by (induct n, auto simp add: fps_power_def fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   712
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   713
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n $0 = v^n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   714
  by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   715
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   716
lemma startsby_zero_power_iff[simp]:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   717
  "a^n $0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   718
apply (rule iffI)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   719
apply (induct n, auto simp add: power_Suc fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   720
by (rule startsby_zero_power, simp_all)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   721
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   722
lemma startsby_zero_power_prefix:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   723
  assumes a0: "a $0 = (0::'a::idom)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   724
  shows "\<forall>n < k. a ^ k $ n = 0"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   725
  using a0
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   726
proof(induct k rule: nat_less_induct)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   727
  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)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   728
  let ?ths = "\<forall>m<k. a ^ k $ m = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   729
  {assume "k = 0" then have ?ths by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   730
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   731
  {fix l assume k: "k = Suc l"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   732
    {fix m assume mk: "m < k"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   733
      {assume "m=0" hence "a^k $ m = 0" using startsby_zero_power[of a k] k a0
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   734
	  by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   735
      moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   736
      {assume m0: "m \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   737
	have "a ^k $ m = (a^l * a) $m" by (simp add: k power_Suc mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   738
	also have "\<dots> = (\<Sum>i = 0..m. a ^ l $ i * a $ (m - i))" by (simp add: fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   739
	also have "\<dots> = 0" apply (rule setsum_0')
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   740
	  apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   741
	  apply (case_tac "aa = m")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   742
	  using a0
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   743
	  apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   744
	  apply (rule H[rule_format])
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   745
	  using a0 k mk by auto
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   746
	finally have "a^k $ m = 0" .}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   747
    ultimately have "a^k $ m = 0" by blast}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   748
    hence ?ths by blast}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   749
  ultimately show ?ths by (cases k, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   750
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   751
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   752
lemma startsby_zero_setsum_depends:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   753
  assumes a0: "a $0 = (0::'a::idom)" and kn: "n \<ge> k"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   754
  shows "setsum (\<lambda>i. (a ^ i)$k) {0 .. n} = setsum (\<lambda>i. (a ^ i)$k) {0 .. k}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   755
  apply (rule setsum_mono_zero_right)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   756
  using kn apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   757
  apply (rule startsby_zero_power_prefix[rule_format, OF a0])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   758
  by arith
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   759
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   760
lemma startsby_zero_power_nth_same: assumes a0: "a$0 = (0::'a::{recpower, idom})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   761
  shows "a^n $ n = (a$1) ^ n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   762
proof(induct n)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   763
  case 0 thus ?case by (simp add: power_0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   764
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   765
  case (Suc n)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   766
  have "a ^ Suc n $ (Suc n) = (a^n * a)$(Suc n)" by (simp add: ring_simps power_Suc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   767
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {0.. Suc n}" by (simp add: fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   768
  also have "\<dots> = setsum (\<lambda>i. a^n$i * a $ (Suc n - i)) {n .. Suc n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   769
    apply (rule setsum_mono_zero_right)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   770
    apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   771
    apply clarsimp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   772
    apply clarsimp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   773
    apply (rule startsby_zero_power_prefix[rule_format, OF a0])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   774
    apply arith
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   775
    done
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   776
  also have "\<dots> = a^n $ n * a$1" using a0 by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   777
  finally show ?case using Suc.hyps by (simp add: power_Suc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   778
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   779
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   780
lemma fps_inverse_power:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   781
  fixes a :: "('a::{field, recpower}) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   782
  shows "inverse (a^n) = inverse a ^ n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   783
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   784
  {assume a0: "a$0 = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   785
    hence eq: "inverse a = 0" by (simp add: fps_inverse_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   786
    {assume "n = 0" hence ?thesis by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   787
    moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   788
    {assume n: "n > 0"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   789
      from startsby_zero_power[OF a0 n] eq a0 n have ?thesis
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   790
	by (simp add: fps_inverse_def)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   791
    ultimately have ?thesis by blast}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   792
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   793
  {assume a0: "a$0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   794
    have ?thesis
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   795
      apply (rule fps_inverse_unique)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   796
      apply (simp add: a0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   797
      unfolding power_mult_distrib[symmetric]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   798
      apply (rule ssubst[where t = "a * inverse a" and s= 1])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   799
      apply simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   800
      apply (subst mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   801
      by (rule inverse_mult_eq_1[OF a0])}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   802
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   803
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   804
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   805
lemma fps_deriv_power: "fps_deriv (a ^ n) = fps_const (of_nat n :: 'a:: comm_ring_1) * fps_deriv a * a ^ (n - 1)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   806
  apply (induct n, auto simp add: power_Suc ring_simps fps_const_add[symmetric] simp del: fps_const_add)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   807
  by (case_tac n, auto simp add: power_Suc ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   808
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   809
lemma fps_inverse_deriv:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   810
  fixes a:: "('a :: field) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   811
  assumes a0: "a$0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   812
  shows "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   813
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   814
  from inverse_mult_eq_1[OF a0]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   815
  have "fps_deriv (inverse a * a) = 0" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   816
  hence "inverse a * fps_deriv a + fps_deriv (inverse a) * a = 0" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   817
  hence "inverse a * (inverse a * fps_deriv a + fps_deriv (inverse a) * a) = 0"  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   818
  with inverse_mult_eq_1[OF a0]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   819
  have "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   820
    unfolding power2_eq_square
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   821
    apply (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   822
    by (simp add: mult_assoc[symmetric])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   823
  hence "inverse a ^ 2 * fps_deriv a + fps_deriv (inverse a) - fps_deriv a * inverse a ^ 2 = 0 - fps_deriv a * inverse a ^ 2"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   824
    by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   825
  then show "fps_deriv (inverse a) = - fps_deriv a * inverse a ^ 2" by (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   826
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   827
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   828
lemma fps_inverse_mult:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   829
  fixes a::"('a :: field) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   830
  shows "inverse (a * b) = inverse a * inverse b"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   831
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   832
  {assume a0: "a$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   833
    from a0 ab0 have th: "inverse a = 0" "inverse (a*b) = 0" by simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   834
    have ?thesis unfolding th by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   835
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   836
  {assume b0: "b$0 = 0" hence ab0: "(a*b)$0 = 0" by (simp add: fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   837
    from b0 ab0 have th: "inverse b = 0" "inverse (a*b) = 0" by simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   838
    have ?thesis unfolding th by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   839
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   840
  {assume a0: "a$0 \<noteq> 0" and b0: "b$0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   841
    from a0 b0 have ab0:"(a*b) $ 0 \<noteq> 0" by (simp  add: fps_mult_nth)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   842
    from inverse_mult_eq_1[OF ab0]
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   843
    have "inverse (a*b) * (a*b) * inverse a * inverse b = 1 * inverse a * inverse b" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   844
    then have "inverse (a*b) * (inverse a * a) * (inverse b * b) = inverse a * inverse b"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   845
      by (simp add: ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   846
    then have ?thesis using inverse_mult_eq_1[OF a0] inverse_mult_eq_1[OF b0] by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   847
ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   848
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   849
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   850
lemma fps_inverse_deriv':
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   851
  fixes a:: "('a :: field) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   852
  assumes a0: "a$0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   853
  shows "fps_deriv (inverse a) = - fps_deriv a / a ^ 2"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   854
  using fps_inverse_deriv[OF a0]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   855
  unfolding power2_eq_square fps_divide_def
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   856
    fps_inverse_mult by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   857
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   858
lemma inverse_mult_eq_1': assumes f0: "f$0 \<noteq> (0::'a::field)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   859
  shows "f * inverse f= 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   860
  by (metis mult_commute inverse_mult_eq_1 f0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   861
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   862
lemma fps_divide_deriv:   fixes a:: "('a :: field) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   863
  assumes a0: "b$0 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   864
  shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b ^ 2"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   865
  using fps_inverse_deriv[OF a0]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   866
  by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   867
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   868
subsection{* The eXtractor series X*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   869
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   870
lemma minus_one_power_iff: "(- (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else - 1)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   871
  by (induct n, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   872
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   873
definition "X = Abs_fps (\<lambda>n. if n = 1 then 1 else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   874
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   875
lemma fps_inverse_gp': "inverse (Abs_fps(\<lambda>n. (1::'a::field)))
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   876
  = 1 - X"
29911
c790a70a3d19 declare fps_nth as a typedef morphism; clean up instance proofs
huffman
parents: 29906
diff changeset
   877
  by (simp add: fps_inverse_gp fps_eq_iff X_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   878
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   879
lemma X_mult_nth[simp]: "(X * (f :: ('a::semiring_1) fps)) $n = (if n = 0 then 0 else f $ (n - 1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   880
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   881
  {assume n: "n \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   882
    have fN: "finite {0 .. n}" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   883
    have "(X * f) $n = (\<Sum>i = 0..n. X $ i * f $ (n - i))" by (simp add: fps_mult_nth)
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   884
    also have "\<dots> = f $ (n - 1)"
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   885
      using n by (simp add: X_def mult_delta_left setsum_delta [OF fN])
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   886
  finally have ?thesis using n by simp }
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   887
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   888
  {assume n: "n=0" hence ?thesis by (simp add: fps_mult_nth X_def)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   889
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   890
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   891
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   892
lemma X_mult_right_nth[simp]: "((f :: ('a::comm_semiring_1) fps) * X) $n = (if n = 0 then 0 else f $ (n - 1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   893
  by (metis X_mult_nth mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   894
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   895
lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   896
proof(induct k)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   897
  case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   898
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   899
  case (Suc k)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   900
  {fix m
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   901
    have "(X^Suc k) $ m = (if m = 0 then (0::'a) else (X^k) $ (m - 1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   902
      by (simp add: power_Suc del: One_nat_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   903
    then     have "(X^Suc k) $ m = (if m = Suc k then (1::'a) else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   904
      using Suc.hyps by (auto cong del: if_weak_cong)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   905
  then show ?case by (simp add: fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   906
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   907
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   908
lemma X_power_mult_nth: "(X^k * (f :: ('a::comm_ring_1) fps)) $n = (if n < k then 0 else f $ (n - k))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   909
  apply (induct k arbitrary: n)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   910
  apply (simp)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   911
  unfolding power_Suc mult_assoc
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   912
  by (case_tac n, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   913
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   914
lemma X_power_mult_right_nth: "((f :: ('a::comm_ring_1) fps) * X^k) $n = (if n < k then 0 else f $ (n - k))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   915
  by (metis X_power_mult_nth mult_commute)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   916
lemma fps_deriv_X[simp]: "fps_deriv X = 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   917
  by (simp add: fps_deriv_def X_def fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   918
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   919
lemma fps_nth_deriv_X[simp]: "fps_nth_deriv n X = (if n = 0 then X else if n=1 then 1 else 0)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   920
  by (cases "n", simp_all)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   921
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   922
lemma X_nth[simp]: "X$n = (if n = 1 then 1 else 0)" by (simp add: X_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   923
lemma X_power_nth[simp]: "(X^k) $n = (if n = k then 1 else (0::'a::comm_ring_1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   924
  by (simp add: X_power_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   925
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   926
lemma fps_inverse_X_plus1:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   927
  "inverse (1 + X) = Abs_fps (\<lambda>n. (- (1::'a::{recpower, field})) ^ n)" (is "_ = ?r")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   928
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   929
  have eq: "(1 + X) * ?r = 1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   930
    unfolding minus_one_power_iff
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   931
    apply (auto simp add: ring_simps fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   932
    by presburger+
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   933
  show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   934
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   935
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   936
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   937
subsection{* Integration *}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   938
definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   939
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   940
lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   941
  by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   942
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   943
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")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   944
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   945
  have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   946
  moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   947
  ultimately show ?thesis
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   948
    unfolding fps_deriv_eq_iff by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   949
qed
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   950
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   951
subsection {* Composition of FPSs *}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   952
definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   953
  fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   954
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   955
lemma fps_compose_nth: "(a oo b)$n = setsum (\<lambda>i. a$i * (b^i$n)) {0..n}" by (simp add: fps_compose_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   956
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   957
lemma fps_compose_X[simp]: "a oo X = (a :: ('a :: comm_ring_1) fps)"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   958
  by (simp add: fps_ext fps_compose_def mult_delta_right setsum_delta')
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   959
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   960
lemma fps_const_compose[simp]:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   961
  "fps_const (a::'a::{comm_ring_1}) oo b = fps_const (a)"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   962
  by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   963
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   964
lemma X_fps_compose_startby0[simp]: "a$0 = 0 \<Longrightarrow> X oo a = (a :: ('a :: comm_ring_1) fps)"
29913
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   965
  by (simp add: fps_eq_iff fps_compose_def mult_delta_left setsum_delta
89eadbe71e97 add mult_delta lemmas; simplify some proofs
huffman
parents: 29912
diff changeset
   966
                power_Suc not_le)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   967
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   968
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   969
subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   970
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
   971
subsubsection {* Rule 1 *}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   972
  (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   973
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   974
lemma fps_power_mult_eq_shift:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   975
  "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")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   976
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   977
  {fix n:: nat
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   978
    have "?lhs $ n = (if n < Suc k then 0 else a n)"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   979
      unfolding X_power_mult_nth by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   980
    also have "\<dots> = ?rhs $ n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   981
    proof(induct k)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   982
      case 0 thus ?case by (simp add: fps_setsum_nth power_Suc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   983
    next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   984
      case (Suc k)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   985
      note th = Suc.hyps[symmetric]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   986
      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)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   987
      also  have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
   988
	using th
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   989
	unfolding fps_sub_nth by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   990
      also have "\<dots> = (if n < Suc (Suc k) then 0 else a n)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   991
	unfolding X_power_mult_right_nth
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   992
	apply (auto simp add: not_less fps_const_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   993
	apply (rule cong[of a a, OF refl])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   994
	by arith
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   995
      finally show ?case by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   996
    qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   997
    finally have "?lhs $ n = ?rhs $ n"  .}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   998
  then show ?thesis by (simp add: fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
   999
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1000
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
  1001
subsubsection{* Rule 2*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1002
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1003
  (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1004
  (* If f reprents {a_n} and P is a polynomial, then
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1005
        P(xD) f represents {P(n) a_n}*)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1006
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1007
definition "XD = op * X o fps_deriv"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1008
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1009
lemma XD_add[simp]:"XD (a + b) = XD a + XD (b :: ('a::comm_ring_1) fps)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1010
  by (simp add: XD_def ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1011
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1012
lemma XD_mult_const[simp]:"XD (fps_const (c::'a::comm_ring_1) * a) = fps_const c * XD a"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1013
  by (simp add: XD_def ring_simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1014
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1015
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)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1016
  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1017
29689
dd086f26ee4f removed definition of funpow , reusing that of Relation_Power
chaieb
parents: 29687
diff changeset
  1018
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)"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1019
  by (induct n, simp_all)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1020
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1021
lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1022
29689
dd086f26ee4f removed definition of funpow , reusing that of Relation_Power
chaieb
parents: 29687
diff changeset
  1023
lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1024
by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1025
29906
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
  1026
subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
80369da39838 section -> subsection
huffman
parents: 29692
diff changeset
  1027
subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1028
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1029
lemma fps_divide_X_minus1_setsum_lemma:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1030
  "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1031
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1032
  let ?X = "X::('a::comm_ring_1) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1033
  let ?sa = "Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1034
  have th0: "\<And>i. (1 - (X::'a fps)) $ i = (if i = 0 then 1 else if i = 1 then - 1 else 0)" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1035
  {fix n:: nat
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1036
    {assume "n=0" hence "a$n = ((1 - ?X) * ?sa) $ n"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1037
	by (simp add: fps_mult_nth)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1038
    moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1039
    {assume n0: "n \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1040
      then have u: "{0} \<union> ({1} \<union> {2..n}) = {0..n}" "{1}\<union>{2..n} = {1..n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1041
	"{0..n - 1}\<union>{n} = {0..n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1042
	apply (simp_all add: expand_set_eq) by presburger+
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1043
      have d: "{0} \<inter> ({1} \<union> {2..n}) = {}" "{1} \<inter> {2..n} = {}"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1044
	"{0..n - 1}\<inter>{n} ={}" using n0
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1045
	by (simp_all add: expand_set_eq, presburger+)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1046
      have f: "finite {0}" "finite {1}" "finite {2 .. n}"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1047
	"finite {0 .. n - 1}" "finite {n}" by simp_all
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1048
    have "((1 - ?X) * ?sa) $ n = setsum (\<lambda>i. (1 - ?X)$ i * ?sa $ (n - i)) {0 .. n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1049
      by (simp add: fps_mult_nth)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1050
    also have "\<dots> = a$n" unfolding th0
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1051
      unfolding setsum_Un_disjoint[OF f(1) finite_UnI[OF f(2,3)] d(1), unfolded u(1)]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1052
      unfolding setsum_Un_disjoint[OF f(2) f(3) d(2)]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1053
      apply (simp)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1054
      unfolding setsum_Un_disjoint[OF f(4,5) d(3), unfolded u(3)]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1055
      by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1056
    finally have "a$n = ((1 - ?X) * ?sa) $ n" by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1057
  ultimately have "a$n = ((1 - ?X) * ?sa) $ n" by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1058
then show ?thesis
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1059
  unfolding fps_eq_iff by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1060
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1061
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1062
lemma fps_divide_X_minus1_setsum:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1063
  "a /((1::('a::field) fps) - X)  = Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1064
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1065
  let ?X = "1 - (X::('a::field) fps)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1066
  have th0: "?X $ 0 \<noteq> 0" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1067
  have "a /?X = ?X *  Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) * inverse ?X"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1068
    using fps_divide_X_minus1_setsum_lemma[of a, symmetric] th0
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1069
    by (simp add: fps_divide_def mult_assoc)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1070
  also have "\<dots> = (inverse ?X * ?X) * Abs_fps (\<lambda>n\<Colon>nat. setsum (op $ a) {0..n}) "
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1071
    by (simp add: mult_ac)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1072
  finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1073
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1074
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1075
subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1076
  finite product of FPS, also the relvant instance of powers of a FPS*}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1077
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1078
definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1079
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1080
lemma natlist_trivial_1: "natpermute n 1 = {[n]}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1081
  apply (auto simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1082
  apply (case_tac x, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1083
  done
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1084
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1085
lemma foldl_add_start0:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1086
  "foldl op + x xs = x + foldl op + (0::nat) xs"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1087
  apply (induct xs arbitrary: x)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1088
  apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1089
  unfolding foldl.simps
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1090
  apply atomize
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1091
  apply (subgoal_tac "\<forall>x\<Colon>nat. foldl op + x xs = x + foldl op + (0\<Colon>nat) xs")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1092
  apply (erule_tac x="x + a" in allE)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1093
  apply (erule_tac x="a" in allE)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1094
  apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1095
  apply assumption
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1096
  done
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1097
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1098
lemma foldl_add_append: "foldl op + (x::nat) (xs@ys) = foldl op + x xs + foldl op + 0 ys"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1099
  apply (induct ys arbitrary: x xs)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1100
  apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1101
  apply (subst (2) foldl_add_start0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1102
  apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1103
  apply (subst (2) foldl_add_start0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1104
  by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1105
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1106
lemma foldl_add_setsum: "foldl op + (x::nat) xs = x + setsum (nth xs) {0..<length xs}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1107
proof(induct xs arbitrary: x)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1108
  case Nil thus ?case by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1109
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1110
  case (Cons a as x)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1111
  have eq: "setsum (op ! (a#as)) {1..<length (a#as)} = setsum (op ! as) {0..<length as}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1112
    apply (rule setsum_reindex_cong [where f=Suc])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1113
    by (simp_all add: inj_on_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1114
  have f: "finite {0}" "finite {1 ..< length (a#as)}" by simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1115
  have d: "{0} \<inter> {1..<length (a#as)} = {}" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1116
  have seq: "{0} \<union> {1..<length(a#as)} = {0 ..<length (a#as)}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1117
  have "foldl op + x (a#as) = x + foldl op + a as "
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1118
    apply (subst foldl_add_start0)    by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1119
  also have "\<dots> = x + a + setsum (op ! as) {0..<length as}" unfolding Cons.hyps by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1120
  also have "\<dots> = x + setsum (op ! (a#as)) {0..<length (a#as)}"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1121
    unfolding eq[symmetric]
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1122
    unfolding setsum_Un_disjoint[OF f d, unfolded seq]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1123
    by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1124
  finally show ?case  .
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1125
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1126
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1127
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1128
lemma append_natpermute_less_eq:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1129
  assumes h: "xs@ys \<in> natpermute n k" shows "foldl op + 0 xs \<le> n" and "foldl op + 0 ys \<le> n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1130
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1131
  {from h have "foldl op + 0 (xs@ ys) = n" by (simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1132
    hence "foldl op + 0 xs + foldl op + 0 ys = n" unfolding foldl_add_append .}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1133
  note th = this
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1134
  {from th show "foldl op + 0 xs \<le> n" by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1135
  {from th show "foldl op + 0 ys \<le> n" by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1136
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1137
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1138
lemma natpermute_split:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1139
  assumes mn: "h \<le> k"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1140
  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)")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1141
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1142
  {fix l assume l: "l \<in> ?R"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1143
    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
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1144
    from xs have xs': "foldl op + 0 xs = m" by (simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1145
    from ys have ys': "foldl op + 0 ys = n - m" by (simp add: natpermute_def)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1146
    have "l \<in> ?L" using leq xs ys h
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1147
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1148
      apply (clarsimp simp add: natpermute_def simp del: foldl_append)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1149
      apply (simp add: foldl_add_append[unfolded foldl_append])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1150
      unfolding xs' ys'
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1151
      using mn xs ys
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1152
      unfolding natpermute_def by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1153
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1154
  {fix l assume l: "l \<in> natpermute n k"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1155
    let ?xs = "take h l"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1156
    let ?ys = "drop h l"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1157
    let ?m = "foldl op + 0 ?xs"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1158
    from l have ls: "foldl op + 0 (?xs @ ?ys) = n" by (simp add: natpermute_def)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1159
    have xs: "?xs \<in> natpermute ?m h" using l mn by (simp add: natpermute_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1160
    have ys: "?ys \<in> natpermute (n - ?m) (k - h)" using l mn ls[unfolded foldl_add_append]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1161
      by (simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1162
    from ls have m: "?m \<in> {0..n}"  unfolding foldl_add_append by simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1163
    from xs ys ls have "l \<in> ?R"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1164
      apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1165
      apply (rule bexI[where x = "?m"])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1166
      apply (rule exI[where x = "?xs"])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1167
      apply (rule exI[where x = "?ys"])
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1168
      using ls l unfolding foldl_add_append
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1169
      by (auto simp add: natpermute_def)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1170
  ultimately show ?thesis by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1171
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1172
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1173
lemma natpermute_0: "natpermute n 0 = (if n = 0 then {[]} else {})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1174
  by (auto simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1175
lemma natpermute_0'[simp]: "natpermute 0 k = (if k = 0 then {[]} else {replicate k 0})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1176
  apply (auto simp add: set_replicate_conv_if natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1177
  apply (rule nth_equalityI)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1178
  by simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1179
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1180
lemma natpermute_finite: "finite (natpermute n k)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1181
proof(induct k arbitrary: n)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1182
  case 0 thus ?case
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1183
    apply (subst natpermute_split[of 0 0, simplified])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1184
    by (simp add: natpermute_0)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1185
next
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1186
  case (Suc k)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1187
  then show ?case unfolding natpermute_split[of k "Suc k", simplified]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1188
    apply -
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1189
    apply (rule finite_UN_I)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1190
    apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1191
    unfolding One_nat_def[symmetric] natlist_trivial_1
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1192
    apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1193
    unfolding image_Collect[symmetric]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1194
    unfolding Collect_def mem_def
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1195
    apply (rule finite_imageI)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1196
    apply blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1197
    done
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1198
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1199
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1200
lemma natpermute_contain_maximal:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1201
  "{xs \<in> natpermute n (k+1). n \<in> set xs} = UNION {0 .. k} (\<lambda>i. {(replicate (k+1) 0) [i:=n]})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1202
  (is "?A = ?B")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1203
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1204
  {fix xs assume H: "xs \<in> natpermute n (k+1)" and n: "n \<in> set xs"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1205
    from n obtain i where i: "i \<in> {0.. k}" "xs!i = n" using H
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1206
      unfolding in_set_conv_nth by (auto simp add: less_Suc_eq_le natpermute_def)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1207
    have eqs: "({0..k} - {i}) \<union> {i} = {0..k}" using i by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1208
    have f: "finite({0..k} - {i})" "finite {i}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1209
    have d: "({0..k} - {i}) \<inter> {i} = {}" using i by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1210
    from H have "n = setsum (nth xs) {0..k}" apply (simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1211
      unfolding foldl_add_setsum by (auto simp add: atLeastLessThanSuc_atLeastAtMost)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1212
    also have "\<dots> = n + setsum (nth xs) ({0..k} - {i})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1213
      unfolding setsum_Un_disjoint[OF f d, unfolded eqs] using i by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1214
    finally have zxs: "\<forall> j\<in> {0..k} - {i}. xs!j = 0" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1215
    from H have xsl: "length xs = k+1" by (simp add: natpermute_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1216
    from i have i': "i < length (replicate (k+1) 0)"   "i < k+1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1217
      unfolding length_replicate  by arith+
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1218
    have "xs = replicate (k+1) 0 [i := n]"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1219
      apply (rule nth_equalityI)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1220
      unfolding xsl length_list_update length_replicate
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1221
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1222
      apply clarify
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1223
      unfolding nth_list_update[OF i'(1)]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1224
      using i zxs
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1225
      by (case_tac "ia=i", auto simp del: replicate.simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1226
    then have "xs \<in> ?B" using i by blast}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1227
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1228
  {fix i assume i: "i \<in> {0..k}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1229
    let ?xs = "replicate (k+1) 0 [i:=n]"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1230
    have nxs: "n \<in> set ?xs"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1231
      apply (rule set_update_memI) using i by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1232
    have xsl: "length ?xs = k+1" by (simp only: length_replicate length_list_update)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1233
    have "foldl op + 0 ?xs = setsum (nth ?xs) {0..<k+1}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1234
      unfolding foldl_add_setsum add_0 length_replicate length_list_update ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1235
    also have "\<dots> = setsum (\<lambda>j. if j = i then n else 0) {0..< k+1}"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1236
      apply (rule setsum_cong2) by (simp del: replicate.simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1237
    also have "\<dots> = n" using i by (simp add: setsum_delta)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1238
    finally
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1239
    have "?xs \<in> natpermute n (k+1)" using xsl unfolding natpermute_def Collect_def mem_def
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1240
      by blast
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1241
    then have "?xs \<in> ?A"  using nxs  by blast}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1242
  ultimately show ?thesis by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1243
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1244
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1245
    (* The general form *)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1246
lemma fps_setprod_nth:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1247
  fixes m :: nat and a :: "nat \<Rightarrow> ('a::comm_ring_1) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1248
  shows "(setprod a {0 .. m})$n = setsum (\<lambda>v. setprod (\<lambda>j. (a j) $ (v!j)) {0..m}) (natpermute n (m+1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1249
  (is "?P m n")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1250
proof(induct m arbitrary: n rule: nat_less_induct)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1251
  fix m n assume H: "\<forall>m' < m. \<forall>n. ?P m' n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1252
  {assume m0: "m = 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1253
    hence "?P m n" apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1254
      unfolding natlist_trivial_1[where n = n, unfolded One_nat_def] by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1255
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1256
  {fix k assume k: "m = Suc k"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1257
    have km: "k < m" using k by arith
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1258
    have u0: "{0 .. k} \<union> {m} = {0..m}" using k apply (simp add: expand_set_eq) by presburger
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1259
    have f0: "finite {0 .. k}" "finite {m}" by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1260
    have d0: "{0 .. k} \<inter> {m} = {}" using k by auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1261
    have "(setprod a {0 .. m}) $ n = (setprod a {0 .. k} * a m) $ n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1262
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0] by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1263
    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))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1264
      unfolding fps_mult_nth H[rule_format, OF km] ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1265
    also have "\<dots> = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j\<in>{0..m}. a j $ v ! j)"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1266
      apply (simp add: k)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1267
      unfolding natpermute_split[of m "m + 1", simplified, of n, unfolded natlist_trivial_1[unfolded One_nat_def] k]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1268
      apply (subst setsum_UN_disjoint)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1269
      apply simp
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1270
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1271
      unfolding image_Collect[symmetric]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1272
      apply clarsimp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1273
      apply (rule finite_imageI)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1274
      apply (rule natpermute_finite)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1275
      apply (clarsimp simp add: expand_set_eq)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1276
      apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1277
      apply (rule setsum_cong2)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1278
      unfolding setsum_left_distrib
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1279
      apply (rule sym)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1280
      apply (rule_tac f="\<lambda>xs. xs @[n - x]" in  setsum_reindex_cong)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1281
      apply (simp add: inj_on_def)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1282
      apply auto
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1283
      unfolding setprod_Un_disjoint[OF f0 d0, unfolded u0, unfolded k]
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1284
      apply (clarsimp simp add: natpermute_def nth_append)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1285
      apply (rule_tac f="\<lambda>x. x * a (Suc k) $ (n - foldl op + 0 aa)" in cong[OF refl])
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1286
      apply (rule setprod_cong)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1287
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1288
      apply simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1289
      done
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1290
    finally have "?P m n" .}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1291
  ultimately show "?P m n " by (cases m, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1292
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1293
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1294
text{* The special form for powers *}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1295
lemma fps_power_nth_Suc:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1296
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1297
  shows "(a ^ Suc m)$n = setsum (\<lambda>v. setprod (\<lambda>j. a $ (v!j)) {0..m}) (natpermute n (m+1))"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1298
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1299
  have f: "finite {0 ..m}" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1300
  have th0: "a^Suc m = setprod (\<lambda>i. a) {0..m}" unfolding setprod_constant[OF f, of a] by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1301
  show ?thesis unfolding th0 fps_setprod_nth ..
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1302
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1303
lemma fps_power_nth:
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1304
  fixes m :: nat and a :: "('a::comm_ring_1) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1305
  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))"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 29915
diff changeset
  1306
  by (cases m, simp_all add: fps_power_nth_Suc del: power_Suc)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1307
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1308
lemma fps_nth_power_0:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1309
  fixes m :: nat and a :: "('a::{comm_ring_1, recpower}) fps"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1310
  shows "(a ^m)$0 = (a$0) ^ m"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1311
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1312
  {assume "m=0" hence ?thesis by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1313
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1314
  {fix n assume m: "m = Suc n"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1315
    have c: "m = card {0..n}" using m by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1316
   have "(a ^m)$0 = setprod (\<lambda>i. a$0) {0..n}"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 29915
diff changeset
  1317
     apply (simp add: m fps_power_nth del: replicate.simps power_Suc)
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1318
     apply (rule setprod_cong)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1319
     by (simp_all del: replicate.simps)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1320
   also have "\<dots> = (a$0) ^ m"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1321
     unfolding c by (rule setprod_constant, simp)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1322
   finally have ?thesis .}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1323
 ultimately show ?thesis by (cases m, auto)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1324
qed
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1325
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1326
lemma fps_compose_inj_right:
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1327
  assumes a0: "a$0 = (0::'a::{recpower,idom})"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1328
  and a1: "a$1 \<noteq> 0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1329
  shows "(b oo a = c oo a) \<longleftrightarrow> b = c" (is "?lhs \<longleftrightarrow>?rhs")
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1330
proof-
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1331
  {assume ?rhs then have "?lhs" by simp}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1332
  moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1333
  {assume h: ?lhs
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30273
diff changeset
  1334
    {fix n have "b$n = c$n"
29687
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1335
      proof(induct n rule: nat_less_induct)
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1336
	fix n assume H: "\<forall>m<n. b$m = c$m"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1337
	{assume n0: "n=0"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1338
	  from h have "(b oo a)$n = (c oo a)$n" by simp
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1339
	  hence "b$n = c$n" using n0 by (simp add: fps_compose_nth)}
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1340
	moreover
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1341
	{fix n1 assume n1: "n = Suc n1"
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1342
	  have f: "finite {0 .. n1}" "finite {n}" by simp_all
4d934a895d11 A formalization of formal power series
chaieb
parents:
diff changeset
  1343
	  have eq: "{0 .. n1} \<union> {n} = {0 .. n}" using n1 by auto