--- a/src/HOL/Groups_Big.thy Tue Jan 21 13:05:22 2014 +0100
+++ b/src/HOL/Groups_Big.thy Tue Jan 21 13:21:55 2014 +0100
@@ -1332,38 +1332,6 @@
by induct (auto simp add: field_simps abs_mult)
qed auto
-lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
-apply (erule finite_induct)
-apply auto
-done
-
-lemma setprod_gen_delta:
- assumes fS: "finite S"
- shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
-proof-
- let ?f = "(\<lambda>k. if k=a then b k else c)"
- {assume a: "a \<notin> S"
- hence "\<forall> k\<in> S. ?f k = c" by simp
- hence ?thesis using a setprod_constant[OF fS, of c] by simp }
- moreover
- {assume a: "a \<in> S"
- let ?A = "S - {a}"
- let ?B = "{a}"
- have eq: "S = ?A \<union> ?B" using a by blast
- have dj: "?A \<inter> ?B = {}" by simp
- from fS have fAB: "finite ?A" "finite ?B" by auto
- have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
- apply (rule setprod_cong) by auto
- have cA: "card ?A = card S - 1" using fS a by auto
- have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto
- have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
- using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
- by simp
- then have ?thesis using a cA
- by (simp add: fA1 field_simps cong add: setprod_cong cong del: if_weak_cong)}
- ultimately show ?thesis by blast
-qed
-
lemma setprod_eq_1_iff [simp]:
"finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"
by (induct set: finite) auto