--- a/src/HOL/Finite_Set.thy Thu Jun 04 12:09:07 2009 +0200
+++ b/src/HOL/Finite_Set.thy Thu Jun 04 13:26:51 2009 +0200
@@ -1333,6 +1333,10 @@
apply (auto simp add: insert_Diff_if add_ac)
done
+lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
+ shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
+unfolding setsum_diff1'[OF assms] by auto
+
(* By Jeremy Siek: *)
lemma setsum_diff_nat:
--- a/src/HOL/Fun.thy Thu Jun 04 12:09:07 2009 +0200
+++ b/src/HOL/Fun.thy Thu Jun 04 13:26:51 2009 +0200
@@ -250,6 +250,10 @@
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
by (simp add: bij_betw_def)
+lemma bij_betw_trans:
+ "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
+by(auto simp add:bij_betw_def comp_inj_on)
+
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
proof -
have i: "inj_on f A" and s: "f ` A = B"
@@ -300,6 +304,9 @@
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
done
+lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
+by(blast dest: inj_onD)
+
lemma inj_on_image_Int:
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B"
apply (simp add: inj_on_def, blast)
--- a/src/HOL/SetInterval.thy Thu Jun 04 12:09:07 2009 +0200
+++ b/src/HOL/SetInterval.thy Thu Jun 04 13:26:51 2009 +0200
@@ -468,7 +468,6 @@
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
-
lemma ex_bij_betw_nat_finite:
"finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
apply(drule finite_imp_nat_seg_image_inj_on)
@@ -479,6 +478,17 @@
"finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
+lemma finite_same_card_bij:
+ "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
+apply(drule ex_bij_betw_finite_nat)
+apply(drule ex_bij_betw_nat_finite)
+apply(auto intro!:bij_betw_trans)
+done
+
+lemma ex_bij_betw_nat_finite_1:
+ "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
+by (rule finite_same_card_bij) auto
+
subsection {* Intervals of integers *}