--- a/src/HOL/Multivariate_Analysis/Integration.thy Fri Sep 07 15:00:03 2012 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Fri Sep 07 15:15:07 2012 +0200
@@ -1771,31 +1771,33 @@
lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
by (metis option.nchotomy)
-lemma exists_option:
- "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
+lemma exists_option: "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
by (metis option.nchotomy)
-fun lifted where
- "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
- "lifted opp None _ = (None::'b option)" |
- "lifted opp _ None = None"
+fun lifted
+where
+ "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some (opp x y)"
+| "lifted opp None _ = (None::'b option)"
+| "lifted opp _ None = None"
lemma lifted_simp_1[simp]: "lifted opp v None = None"
- apply(induct v) by auto
+ by (induct v) auto
definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and>
(\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
(\<forall>x. opp (neutral opp) x = x)"
-lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
+lemma monoidalI:
+ assumes "\<And>x y. opp x y = opp y x"
"\<And>x y z. opp x (opp y z) = opp (opp x y) z"
"\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
unfolding monoidal_def using assms by fastforce
-lemma monoidal_ac: assumes "monoidal opp"
+lemma monoidal_ac:
+ assumes "monoidal opp"
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
"opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)"
- using assms unfolding monoidal_def apply- by metis+
+ using assms unfolding monoidal_def by metis+
lemma monoidal_simps[simp]: assumes "monoidal opp"
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
@@ -1804,10 +1806,14 @@
lemma neutral_lifted[cong]: assumes "monoidal opp"
shows "neutral (lifted opp) = Some(neutral opp)"
apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
-proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
- thus "x = Some (neutral opp)" apply(induct x) defer
+proof -
+ fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
+ thus "x = Some (neutral opp)"
+ apply(induct x) defer
apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
- apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
+ apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE)
+ apply auto
+ done
qed(auto simp add:monoidal_ac[OF assms])
lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
@@ -1825,7 +1831,8 @@
lemma support_clauses:
"\<And>f g s. support opp f {} = {}"
- "\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
+ "\<And>f g s. support opp f (insert x s) =
+ (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
"\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
"\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
"\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"