author huffman Thu, 28 May 2009 13:52:13 -0700 changeset 31287 6c593b431f04 parent 31286 424874813840 child 31288 67dff9c5b2bd
use class field_char_0
```--- a/src/HOL/Library/Binomial.thy	Thu May 28 13:43:45 2009 -0700
+++ b/src/HOL/Library/Binomial.thy	Thu May 28 13:52:13 2009 -0700
@@ -6,7 +6,7 @@

theory Binomial
-imports Fact SetInterval Presburger Main
+imports Fact SetInterval Presburger Main Rational
begin

text {* This development is based on the work of Andy Gordon and
@@ -290,7 +290,7 @@

subsection{* Generalized binomial coefficients *}

-definition gbinomial :: "'a::{field, ring_char_0} \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
+definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
where "a gchoose n = (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"

lemma gbinomial_0[simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
@@ -345,7 +345,7 @@

lemma binomial_fact:
assumes kn: "k \<le> n"
-  shows "(of_nat (n choose k) :: 'a::{field, ring_char_0}) = of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
+  shows "(of_nat (n choose k) :: 'a::field_char_0) = of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
using binomial_fact_lemma[OF kn]
by (simp add: field_simps fact_not_eq_zero of_nat_mult[symmetric])

@@ -418,16 +418,16 @@

lemma gbinomial_mult_fact:
-  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::{field, ring_char_0}) gchoose (Suc k)) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
+  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
unfolding gbinomial_Suc
by (simp_all add: field_simps del: fact_Suc)

lemma gbinomial_mult_fact':
-  "((a::'a::{field, ring_char_0}) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
+  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
using gbinomial_mult_fact[of k a]
apply (subst mult_commute) .

-lemma gbinomial_Suc_Suc: "((a::'a::{field, ring_char_0}) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
+lemma gbinomial_Suc_Suc: "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
proof-
{assume "k = 0" then have ?thesis by simp}
moreover```
```--- a/src/HOL/ex/Formal_Power_Series_Examples.thy	Thu May 28 13:43:45 2009 -0700
+++ b/src/HOL/ex/Formal_Power_Series_Examples.thy	Thu May 28 13:52:13 2009 -0700
@@ -11,7 +11,7 @@

section{* The generalized binomial theorem *}
lemma gbinomial_theorem:
-  "((a::'a::{ring_char_0, field, division_by_zero})+b) ^ n = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
+  "((a::'a::{field_char_0, division_by_zero})+b) ^ n = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
proof-
have "(E (a + b)) \$ n = (E a * E b)\$n" by simp
@@ -38,7 +38,7 @@

lemma fps_binomial_ODE_unique:
-  fixes c :: "'a::{field, ring_char_0}"
+  fixes c :: "'a::field_char_0"
shows "fps_deriv a = (fps_const c * a) / (1 + X) \<longleftrightarrow> a = fps_const (a\$0) * fps_binomial c"
(is "?lhs \<longleftrightarrow> ?rhs")
proof-
@@ -302,4 +302,4 @@
finally show ?thesis .
qed

-end
\ No newline at end of file
+end```