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src/HOL/List.thy

changeset 17629 | f8ea8068c6d9 |

parent 17589 | 58eeffd73be1 |

child 17724 | e969fc0a4925 |

--- a/src/HOL/List.thy Sat Sep 24 16:43:41 2005 +0200 +++ b/src/HOL/List.thy Sat Sep 24 21:13:15 2005 +0200 @@ -778,6 +778,48 @@ qed qed +lemma Cons_eq_filterD: + "x#xs = filter P ys \<Longrightarrow> + \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" + (concl is "\<exists>us vs. ?P ys us vs") +proof(induct ys) + case Nil thus ?case by simp +next + case (Cons y ys) + show ?case (is "\<exists>x. ?Q x") + proof cases + assume Py: "P y" + show ?thesis + proof cases + assume xy: "x = y" + show ?thesis + proof from Py xy Cons(2) show "?Q []" by simp qed + next + assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp + qed + next + assume Py: "\<not> P y" + with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp + show ?thesis (is "? us. ?Q us") + proof show "?Q (y#us)" using 1 by simp qed + qed +qed + +lemma filter_eq_ConsD: + "filter P ys = x#xs \<Longrightarrow> + \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" +by(rule Cons_eq_filterD) simp + +lemma filter_eq_Cons_iff: + "(filter P ys = x#xs) = + (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" +by(auto dest:filter_eq_ConsD) + +lemma Cons_eq_filter_iff: + "(x#xs = filter P ys) = + (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" +by(auto dest:Cons_eq_filterD) + lemma filter_cong: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys" apply simp