--- a/src/HOL/Library/Formal_Power_Series.thy Thu May 28 09:46:45 2009 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy Thu May 28 09:56:04 2009 +0200
@@ -5,7 +5,7 @@
header{* A formalization of formal power series *}
theory Formal_Power_Series
-imports Main Fact Parity
+imports Main Fact Parity Rational
begin
subsection {* The type of formal power series*}
@@ -392,8 +392,8 @@
begin
definition number_of_fps_def: "(number_of k::'a fps) = of_int k"
-instance
-by (intro_classes, rule number_of_fps_def)
+instance proof
+qed (rule number_of_fps_def)
end
subsection{* Inverses of formal power series *}
@@ -923,12 +923,19 @@
subsection{* Integration *}
-definition "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
+
+definition
+ fps_integral :: "'a::field_char_0 fps \<Rightarrow> 'a \<Rightarrow> 'a fps" where
+ "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else (a$(n - 1) / of_nat n))"
-lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a (a0 :: 'a :: {field, ring_char_0})) = a"
- by (simp add: fps_integral_def fps_deriv_def fps_eq_iff field_simps del: of_nat_Suc)
+lemma fps_deriv_fps_integral: "fps_deriv (fps_integral a a0) = a"
+ unfolding fps_integral_def fps_deriv_def
+ by (simp add: fps_eq_iff del: of_nat_Suc)
-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")
+lemma fps_integral_linear:
+ "fps_integral (fps_const a * 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")
proof-
have "fps_deriv ?l = fps_deriv ?r" by (simp add: fps_deriv_fps_integral)
moreover have "?l$0 = ?r$0" by (simp add: fps_integral_def)
@@ -1438,7 +1445,7 @@
qed
lemma power_radical:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a:: "'a::field_char_0 fps"
assumes a0: "a$0 \<noteq> 0"
shows "(r (Suc k) (a$0)) ^ Suc k = a$0 \<longleftrightarrow> (fps_radical r (Suc k) a) ^ (Suc k) = a"
proof-
@@ -1499,7 +1506,7 @@
(*
lemma power_radical:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a:: "'a::field_char_0 fps"
assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
shows "(fps_radical r (Suc k) a) ^ (Suc k) = a"
proof-
@@ -1561,7 +1568,7 @@
lemma radical_unique:
assumes r0: "(r (Suc k) (b$0)) ^ Suc k = b$0"
- and a0: "r (Suc k) (b$0 ::'a::{field, ring_char_0}) = a$0" and b0: "b$0 \<noteq> 0"
+ and a0: "r (Suc k) (b$0 ::'a::field_char_0) = a$0" and b0: "b$0 \<noteq> 0"
shows "a^(Suc k) = b \<longleftrightarrow> a = fps_radical r (Suc k) b"
proof-
let ?r = "fps_radical r (Suc k) b"
@@ -1655,7 +1662,7 @@
lemma radical_power:
assumes r0: "r (Suc k) ((a$0) ^ Suc k) = a$0"
- and a0: "(a$0 ::'a::{field, ring_char_0}) \<noteq> 0"
+ and a0: "(a$0 ::'a::field_char_0) \<noteq> 0"
shows "(fps_radical r (Suc k) (a ^ Suc k)) = a"
proof-
let ?ak = "a^ Suc k"
@@ -1667,7 +1674,7 @@
qed
lemma fps_deriv_radical:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a:: "'a::field_char_0 fps"
assumes r0: "(r (Suc k) (a$0)) ^ Suc k = a$0" and a0: "a$0 \<noteq> 0"
shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (fps_const (of_nat (Suc k)) * (fps_radical r (Suc k) a) ^ k)"
proof-
@@ -1688,7 +1695,7 @@
qed
lemma radical_mult_distrib:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a:: "'a::field_char_0 fps"
assumes
k: "k > 0"
and ra0: "r k (a $ 0) ^ k = a $ 0"
@@ -1722,7 +1729,7 @@
(*
lemma radical_mult_distrib:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a:: "'a::field_char_0 fps"
assumes
ra0: "r k (a $ 0) ^ k = a $ 0"
and rb0: "r k (b $ 0) ^ k = b $ 0"
@@ -1752,7 +1759,7 @@
by (simp add: fps_divide_def)
lemma radical_divide:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a :: "'a::field_char_0 fps"
assumes
kp: "k>0"
and ra0: "(r k (a $ 0)) ^ k = a $ 0"
@@ -1790,7 +1797,7 @@
qed
lemma radical_inverse:
- fixes a:: "'a ::{field, ring_char_0} fps"
+ fixes a :: "'a::field_char_0 fps"
assumes
k: "k>0"
and ra0: "r k (a $ 0) ^ k = a $ 0"
@@ -2159,7 +2166,7 @@
by (induct n, auto)
lemma fps_compose_radical:
- assumes b0: "b$0 = (0::'a::{field, ring_char_0})"
+ assumes b0: "b$0 = (0::'a::field_char_0)"
and ra0: "r (Suc k) (a$0) ^ Suc k = a$0"
and a0: "a$0 \<noteq> 0"
shows "fps_radical r (Suc k) a oo b = fps_radical r (Suc k) (a oo b)"
@@ -2260,7 +2267,7 @@
subsubsection{* Exponential series *}
definition "E x = Abs_fps (\<lambda>n. x^n / of_nat (fact n))"
-lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::{field, ring_char_0}) * E a" (is "?l = ?r")
+lemma E_deriv[simp]: "fps_deriv (E a) = fps_const (a::'a::field_char_0) * E a" (is "?l = ?r")
proof-
{fix n
have "?l$n = ?r $ n"
@@ -2270,7 +2277,7 @@
qed
lemma E_unique_ODE:
- "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * E (c :: 'a::{field, ring_char_0})"
+ "fps_deriv a = fps_const c * a \<longleftrightarrow> a = fps_const (a$0) * E (c :: 'a::field_char_0)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume d: ?lhs
@@ -2297,7 +2304,7 @@
ultimately show ?thesis by blast
qed
-lemma E_add_mult: "E (a + b) = E (a::'a::{ring_char_0, field}) * E b" (is "?l = ?r")
+lemma E_add_mult: "E (a + b) = E (a::'a::field_char_0) * E b" (is "?l = ?r")
proof-
have "fps_deriv (?r) = fps_const (a+b) * ?r"
by (simp add: fps_const_add[symmetric] ring_simps del: fps_const_add)
@@ -2312,7 +2319,7 @@
lemma E0[simp]: "E (0::'a::{field}) = 1"
by (simp add: fps_eq_iff power_0_left)
-lemma E_neg: "E (- a) = inverse (E (a::'a::{ring_char_0, field}))"
+lemma E_neg: "E (- a) = inverse (E (a::'a::field_char_0))"
proof-
from E_add_mult[of a "- a"] have th0: "E a * E (- a) = 1"
by (simp )
@@ -2320,7 +2327,7 @@
from fps_inverse_unique[OF th1 th0] show ?thesis by simp
qed
-lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::{field, ring_char_0})) = (fps_const a)^n * (E a)"
+lemma E_nth_deriv[simp]: "fps_nth_deriv n (E (a::'a::field_char_0)) = (fps_const a)^n * (E a)"
by (induct n, auto simp add: power_Suc)
@@ -2355,11 +2362,11 @@
from fps_inverse_unique[OF th' th] show ?thesis .
qed
-lemma E_power_mult: "(E (c::'a::{field,ring_char_0}))^n = E (of_nat n * c)"
+lemma E_power_mult: "(E (c::'a::field_char_0))^n = E (of_nat n * c)"
by (induct n, auto simp add: ring_simps E_add_mult power_Suc)
subsubsection{* Logarithmic series *}
-definition "(L::'a::{field, ring_char_0} fps)
+definition "(L::'a::field_char_0 fps)
= Abs_fps (\<lambda>n. (- 1) ^ Suc n / of_nat n)"
lemma fps_deriv_L: "fps_deriv L = inverse (1 + X)"
@@ -2371,7 +2378,7 @@
by (simp add: L_def)
lemma L_E_inv:
- assumes a: "a\<noteq> (0::'a::{field,division_by_zero,ring_char_0})"
+ assumes a: "a \<noteq> (0::'a::{field_char_0,division_by_zero})"
shows "L = fps_const a * fps_inv (E a - 1)" (is "?l = ?r")
proof-
let ?b = "E a - 1"
@@ -2395,16 +2402,17 @@
subsubsection{* Formal trigonometric functions *}
-definition "fps_sin (c::'a::{field, ring_char_0}) =
+definition "fps_sin (c::'a::field_char_0) =
Abs_fps (\<lambda>n. if even n then 0 else (- 1) ^((n - 1) div 2) * c^n /(of_nat (fact n)))"
-definition "fps_cos (c::'a::{field, ring_char_0}) = Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
+definition "fps_cos (c::'a::field_char_0) =
+ Abs_fps (\<lambda>n. if even n then (- 1) ^ (n div 2) * c^n / (of_nat (fact n)) else 0)"
lemma fps_sin_deriv:
"fps_deriv (fps_sin c) = fps_const c * fps_cos c"
(is "?lhs = ?rhs")
-proof-
- {fix n::nat
+proof (rule fps_ext)
+ fix n::nat
{assume en: "even n"
have "?lhs$n = of_nat (n+1) * (fps_sin c $ (n+1))" by simp
also have "\<dots> = of_nat (n+1) * ((- 1)^(n div 2) * c^Suc n / of_nat (fact (Suc n)))"
@@ -2416,18 +2424,18 @@
by (simp add: field_simps del: of_nat_add of_nat_Suc)
finally have "?lhs $n = ?rhs$n" using en
by (simp add: fps_cos_def ring_simps power_Suc )}
- then have "?lhs $ n = ?rhs $ n"
- by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def) }
- then show ?thesis by (auto simp add: fps_eq_iff)
+ then show "?lhs $ n = ?rhs $ n"
+ by (cases "even n", simp_all add: fps_deriv_def fps_sin_def fps_cos_def)
qed
lemma fps_cos_deriv:
"fps_deriv (fps_cos c) = fps_const (- c)* (fps_sin c)"
(is "?lhs = ?rhs")
-proof-
+proof (rule fps_ext)
have th0: "\<And>n. - ((- 1::'a) ^ n) = (- 1)^Suc n" by (simp add: power_Suc)
- have th1: "\<And>n. odd n\<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2" by presburger (* FIXME: VERY slow! *)
- {fix n::nat
+ have th1: "\<And>n. odd n \<Longrightarrow> Suc ((n - 1) div 2) = Suc n div 2"
+ by (case_tac n, simp_all)
+ fix n::nat
{assume en: "odd n"
from en have n0: "n \<noteq>0 " by presburger
have "?lhs$n = of_nat (n+1) * (fps_cos c $ (n+1))" by simp
@@ -2442,10 +2450,9 @@
unfolding th0 unfolding th1[OF en] by simp
finally have "?lhs $n = ?rhs$n" using en
by (simp add: fps_sin_def ring_simps power_Suc)}
- then have "?lhs $ n = ?rhs $ n"
+ then show "?lhs $ n = ?rhs $ n"
by (cases "even n", simp_all add: fps_deriv_def fps_sin_def
- fps_cos_def) }
- then show ?thesis by (auto simp add: fps_eq_iff)
+ fps_cos_def)
qed
lemma fps_sin_cos_sum_of_squares:
@@ -2461,6 +2468,110 @@
finally show ?thesis .
qed
+lemma fact_1 [simp]: "fact 1 = 1"
+unfolding One_nat_def fact_Suc by simp
+
+lemma divide_eq_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x / a = y \<longleftrightarrow> x = y * a"
+by auto
+
+lemma eq_divide_iff: "a \<noteq> (0::'a::field) \<Longrightarrow> x = y / a \<longleftrightarrow> x * a = y"
+by auto
+
+lemma fps_sin_nth_0 [simp]: "fps_sin c $ 0 = 0"
+unfolding fps_sin_def by simp
+
+lemma fps_sin_nth_1 [simp]: "fps_sin c $ 1 = c"
+unfolding fps_sin_def by simp
+
+lemma fps_sin_nth_add_2:
+ "fps_sin c $ (n + 2) = - (c * c * fps_sin c $ n / (of_nat(n+1) * of_nat(n+2)))"
+unfolding fps_sin_def
+apply (cases n, simp)
+apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
+apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
+done
+
+lemma fps_cos_nth_0 [simp]: "fps_cos c $ 0 = 1"
+unfolding fps_cos_def by simp
+
+lemma fps_cos_nth_1 [simp]: "fps_cos c $ 1 = 0"
+unfolding fps_cos_def by simp
+
+lemma fps_cos_nth_add_2:
+ "fps_cos c $ (n + 2) = - (c * c * fps_cos c $ n / (of_nat(n+1) * of_nat(n+2)))"
+unfolding fps_cos_def
+apply (simp add: divide_eq_iff eq_divide_iff del: of_nat_Suc fact_Suc)
+apply (simp add: of_nat_mult del: of_nat_Suc mult_Suc)
+done
+
+lemma nat_induct2:
+ "\<lbrakk>P 0; P 1; \<And>n. P n \<Longrightarrow> P (n + 2)\<rbrakk> \<Longrightarrow> P (n::nat)"
+unfolding One_nat_def numeral_2_eq_2
+apply (induct n rule: nat_less_induct)
+apply (case_tac n, simp)
+apply (rename_tac m, case_tac m, simp)
+apply (rename_tac k, case_tac k, simp_all)
+done
+
+lemma nat_add_1_add_1: "(n::nat) + 1 + 1 = n + 2"
+by simp
+
+lemma eq_fps_sin:
+ assumes 0: "a $ 0 = 0" and 1: "a $ 1 = c"
+ and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
+ shows "a = fps_sin c"
+apply (rule fps_ext)
+apply (induct_tac n rule: nat_induct2)
+apply (simp add: fps_sin_nth_0 0)
+apply (simp add: fps_sin_nth_1 1 del: One_nat_def)
+apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
+apply (simp add: nat_add_1_add_1 fps_sin_nth_add_2
+ del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
+apply (subst minus_divide_left)
+apply (subst eq_divide_iff)
+apply (simp del: of_nat_add of_nat_Suc)
+apply (simp only: mult_ac)
+done
+
+lemma eq_fps_cos:
+ assumes 0: "a $ 0 = 1" and 1: "a $ 1 = 0"
+ and 2: "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
+ shows "a = fps_cos c"
+apply (rule fps_ext)
+apply (induct_tac n rule: nat_induct2)
+apply (simp add: fps_cos_nth_0 0)
+apply (simp add: fps_cos_nth_1 1 del: One_nat_def)
+apply (rename_tac m, cut_tac f="\<lambda>a. a $ m" in arg_cong [OF 2])
+apply (simp add: nat_add_1_add_1 fps_cos_nth_add_2
+ del: One_nat_def of_nat_Suc of_nat_add add_2_eq_Suc')
+apply (subst minus_divide_left)
+apply (subst eq_divide_iff)
+apply (simp del: of_nat_add of_nat_Suc)
+apply (simp only: mult_ac)
+done
+
+lemma mult_nth_0 [simp]: "(a * b) $ 0 = a $ 0 * b $ 0"
+by (simp add: fps_mult_nth)
+
+lemma mult_nth_1 [simp]: "(a * b) $ 1 = a $ 0 * b $ 1 + a $ 1 * b $ 0"
+by (simp add: fps_mult_nth)
+
+lemma fps_sin_add:
+ "fps_sin (a + b) = fps_sin a * fps_cos b + fps_cos a * fps_sin b"
+apply (rule eq_fps_sin [symmetric], simp, simp del: One_nat_def)
+apply (simp del: fps_const_neg fps_const_add fps_const_mult
+ add: fps_const_add [symmetric] fps_const_neg [symmetric]
+ fps_sin_deriv fps_cos_deriv algebra_simps)
+done
+
+lemma fps_cos_add:
+ "fps_cos (a + b) = fps_cos a * fps_cos b - fps_sin a * fps_sin b"
+apply (rule eq_fps_cos [symmetric], simp, simp del: One_nat_def)
+apply (simp del: fps_const_neg fps_const_add fps_const_mult
+ add: fps_const_add [symmetric] fps_const_neg [symmetric]
+ fps_sin_deriv fps_cos_deriv algebra_simps)
+done
+
definition "fps_tan c = fps_sin c / fps_cos c"
lemma fps_tan_deriv: "fps_deriv(fps_tan c) = fps_const c/ (fps_cos c ^ 2)"