src/HOL/Library/Formal_Power_Series.thy
changeset 29906 80369da39838
parent 29692 121289b1ae27
child 29911 c790a70a3d19
     1.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Fri Feb 13 14:41:54 2009 -0800
     1.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Fri Feb 13 14:45:10 2009 -0800
     1.3 @@ -9,7 +9,7 @@
     1.4    imports Main Fact Parity
     1.5  begin
     1.6  
     1.7 -section {* The type of formal power series*}
     1.8 +subsection {* The type of formal power series*}
     1.9  
    1.10  typedef 'a fps = "{f :: nat \<Rightarrow> 'a. True}"
    1.11    by simp
    1.12 @@ -94,7 +94,7 @@
    1.13  lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
    1.14    by auto
    1.15  
    1.16 -section{* Formal power series form a commutative ring with unity, if the range of sequences 
    1.17 +subsection{* Formal power series form a commutative ring with unity, if the range of sequences 
    1.18    they represent is a commutative ring with unity*}
    1.19  
    1.20  instantiation fps :: (semigroup_add) semigroup_add
    1.21 @@ -293,7 +293,7 @@
    1.22  qed
    1.23  end
    1.24    
    1.25 -section {* Selection of the nth power of the implicit variable in the infinite sum*}
    1.26 +subsection {* Selection of the nth power of the implicit variable in the infinite sum*}
    1.27  
    1.28  definition fps_nth:: "'a fps \<Rightarrow> nat \<Rightarrow> 'a" (infixl "$" 75)
    1.29    where "f $ n = Rep_fps f n"
    1.30 @@ -358,7 +358,7 @@
    1.31    ultimately show ?thesis by blast
    1.32  qed
    1.33  
    1.34 -section{* Injection of the basic ring elements and multiplication by scalars *}
    1.35 +subsection{* Injection of the basic ring elements and multiplication by scalars *}
    1.36  
    1.37  definition "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else 0)"
    1.38  lemma fps_const_0_eq_0[simp]: "fps_const 0 = 0" by (simp add: fps_const_def fps_eq_iff)
    1.39 @@ -391,7 +391,7 @@
    1.40  lemma fps_mult_right_const_nth[simp]: "(f * fps_const (c::'a::semiring_1))$n = f$n * c"
    1.41    by (simp add: fps_mult_nth fps_const_nth cond_application_beta cond_value_iff setsum_delta' cong del: if_weak_cong)
    1.42  
    1.43 -section {* Formal power series form an integral domain*}
    1.44 +subsection {* Formal power series form an integral domain*}
    1.45  
    1.46  instantiation fps :: (ring_1) ring_1
    1.47  begin
    1.48 @@ -442,7 +442,7 @@
    1.49  instance ..
    1.50  end
    1.51  
    1.52 -section{* Inverses of formal power series *}
    1.53 +subsection{* Inverses of formal power series *}
    1.54  
    1.55  declare setsum_cong[fundef_cong]
    1.56  
    1.57 @@ -561,7 +561,7 @@
    1.58      by(simp add: setsum_delta)
    1.59  qed
    1.60  
    1.61 -section{* Formal Derivatives, and the McLauren theorem around 0*}
    1.62 +subsection{* Formal Derivatives, and the McLauren theorem around 0*}
    1.63  
    1.64  definition "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
    1.65  
    1.66 @@ -730,7 +730,7 @@
    1.67  lemma fps_deriv_maclauren_0: "(fps_nth_deriv k (f:: ('a::comm_semiring_1) fps)) $ 0 = of_nat (fact k) * f$(k)"
    1.68    by (induct k arbitrary: f) (auto simp add: ring_simps of_nat_mult)
    1.69  
    1.70 -section {* Powers*}
    1.71 +subsection {* Powers*}
    1.72  
    1.73  instantiation fps :: (semiring_1) power
    1.74  begin
    1.75 @@ -945,7 +945,7 @@
    1.76    using fps_inverse_deriv[OF a0]
    1.77    by (simp add: fps_divide_def ring_simps power2_eq_square fps_inverse_mult inverse_mult_eq_1'[OF a0])
    1.78    
    1.79 -section{* The eXtractor series X*}
    1.80 +subsection{* The eXtractor series X*}
    1.81  
    1.82  lemma minus_one_power_iff: "(- (1::'a :: {recpower, comm_ring_1})) ^ n = (if even n then 1 else - 1)"
    1.83    by (induct n, auto)
    1.84 @@ -1015,7 +1015,7 @@
    1.85  qed
    1.86  
    1.87    
    1.88 -section{* Integration *}
    1.89 +subsection{* Integration *}
    1.90  definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
    1.91  
    1.92  lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
    1.93 @@ -1029,7 +1029,7 @@
    1.94      unfolding fps_deriv_eq_iff by auto
    1.95  qed
    1.96    
    1.97 -section {* Composition of FPSs *}
    1.98 +subsection {* Composition of FPSs *}
    1.99  definition fps_compose :: "('a::semiring_1) fps \<Rightarrow> 'a fps \<Rightarrow> 'a fps" (infixl "oo" 55) where
   1.100    fps_compose_def: "a oo b = Abs_fps (\<lambda>n. setsum (\<lambda>i. a$i * (b^i$n)) {0..n})"
   1.101  
   1.102 @@ -1051,9 +1051,9 @@
   1.103    by simp_all
   1.104  
   1.105  
   1.106 -section {* Rules from Herbert Wilf's Generatingfunctionology*}
   1.107 +subsection {* Rules from Herbert Wilf's Generatingfunctionology*}
   1.108  
   1.109 -subsection {* Rule 1 *}
   1.110 +subsubsection {* Rule 1 *}
   1.111    (* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
   1.112  
   1.113  lemma fps_power_mult_eq_shift: 
   1.114 @@ -1083,7 +1083,7 @@
   1.115    then show ?thesis by (simp add: fps_eq_iff)
   1.116  qed
   1.117  
   1.118 -subsection{* Rule 2*}
   1.119 +subsubsection{* Rule 2*}
   1.120  
   1.121    (* We can not reach the form of Wilf, but still near to it using rewrite rules*)
   1.122    (* If f reprents {a_n} and P is a polynomial, then 
   1.123 @@ -1108,8 +1108,8 @@
   1.124  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)"
   1.125  by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
   1.126  
   1.127 -subsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
   1.128 -subsection{* Rule 5 --- summation and "division" by (1 - X)*}
   1.129 +subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
   1.130 +subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
   1.131  
   1.132  lemma fps_divide_X_minus1_setsum_lemma:
   1.133    "a = ((1::('a::comm_ring_1) fps) - X) * Abs_fps (\<lambda>n. setsum (\<lambda>i. a $ i) {0..n})"
   1.134 @@ -1157,7 +1157,7 @@
   1.135    finally show ?thesis by (simp add: inverse_mult_eq_1[OF th0])
   1.136  qed
   1.137  
   1.138 -subsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary 
   1.139 +subsubsection{* Rule 4 in its more general form: generalizes Rule 3 for an arbitrary 
   1.140    finite product of FPS, also the relvant instance of powers of a FPS*}
   1.141  
   1.142  definition "natpermute n k = {l:: nat list. length l = k \<and> foldl op + 0 l = n}"
   1.143 @@ -1447,7 +1447,7 @@
   1.144  qed
   1.145  
   1.146  
   1.147 -section {* Radicals *}
   1.148 +subsection {* Radicals *}
   1.149  
   1.150  declare setprod_cong[fundef_cong]
   1.151  function radical :: "(nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> ('a::{field, recpower}) fps \<Rightarrow> nat \<Rightarrow> 'a" where
   1.152 @@ -1832,7 +1832,7 @@
   1.153  ultimately show ?thesis by blast
   1.154  qed
   1.155  
   1.156 -section{* Derivative of composition *}
   1.157 +subsection{* Derivative of composition *}
   1.158  
   1.159  lemma fps_compose_deriv: 
   1.160    fixes a:: "('a::idom) fps"
   1.161 @@ -1898,7 +1898,7 @@
   1.162    ultimately show ?thesis by (cases n, auto)
   1.163  qed
   1.164  
   1.165 -section{* Finite FPS (i.e. polynomials) and X *}
   1.166 +subsection{* Finite FPS (i.e. polynomials) and X *}
   1.167  lemma fps_poly_sum_X:
   1.168    assumes z: "\<forall>i > n. a$i = (0::'a::comm_ring_1)" 
   1.169    shows "a = setsum (\<lambda>i. fps_const (a$i) * X^i) {0..n}" (is "a = ?r")
   1.170 @@ -1911,7 +1911,7 @@
   1.171    then show ?thesis unfolding fps_eq_iff by blast
   1.172  qed
   1.173  
   1.174 -section{* Compositional inverses *}
   1.175 +subsection{* Compositional inverses *}
   1.176  
   1.177  
   1.178  fun compinv :: "'a fps \<Rightarrow> nat \<Rightarrow> 'a::{recpower,field}" where
   1.179 @@ -2217,9 +2217,9 @@
   1.180    show "?dia = inverse ?d" by simp
   1.181  qed
   1.182  
   1.183 -section{* Elementary series *}
   1.184 +subsection{* Elementary series *}
   1.185  
   1.186 -subsection{* Exponential series *}
   1.187 +subsubsection{* Exponential series *}
   1.188  definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"   
   1.189  
   1.190  lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::{field, recpower, ring_char_0}) * E a" (is "?l = ?r")
   1.191 @@ -2332,7 +2332,7 @@
   1.192  lemma E_power_mult: "(E (c::'a::{field,recpower,ring_char_0}))^n = E (of_nat n * c)"
   1.193    by (induct n, auto simp add: ring_simps E_add_mult power_Suc)
   1.194  
   1.195 -subsection{* Logarithmic series *}  
   1.196 +subsubsection{* Logarithmic series *}  
   1.197  definition "(L::'a::{field, ring_char_0,recpower} fps) 
   1.198    = Abs_fps (\<lambda>n. (- 1) ^ Suc n / of_nat n)"
   1.199  
   1.200 @@ -2366,7 +2366,7 @@
   1.201      by (simp add: L_nth fps_inv_def)
   1.202  qed
   1.203  
   1.204 -subsection{* Formal trigonometric functions  *}
   1.205 +subsubsection{* Formal trigonometric functions  *}
   1.206  
   1.207  definition "fps_sin (c::'a::{field, recpower, ring_char_0}) = 
   1.208    Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"