--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Moment.thy~ Wed Jan 27 19:28:42 2016 +0800
@@ -0,0 +1,225 @@
+theory Moment
+imports Main
+begin
+
+definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where "moment n s = rev (take n (rev s))"
+
+value "moment 3 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9::int]"
+value "moment 2 [5, 4, 3, 2, 1, 0::int]"
+
+(*
+lemma length_moment_le:
+ assumes le_k: "k \<le> length s"
+ shows "length (moment k s) = k"
+using le_k unfolding moment_def by auto
+*)
+
+(*
+lemma length_moment_ge:
+ assumes le_k: "length s \<le> k"
+ shows "length (moment k s) = (length s)"
+using assms unfolding moment_def by simp
+*)
+
+lemma moment_app [simp]:
+ assumes ile: "i \<le> length s"
+ shows "moment i (s' @ s) = moment i s"
+using assms unfolding moment_def by simp
+
+lemma moment_eq [simp]: "moment (length s) (s' @ s) = s"
+ unfolding moment_def by simp
+
+lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
+ by (unfold moment_def, simp)
+
+lemma moment_zero [simp]: "moment 0 s = []"
+ by (simp add:moment_def)
+
+lemma p_split_gen:
+ "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>
+ (\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof (induct s, simp)
+ fix a s
+ assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>
+ \<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"
+ and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"
+ have le_k: "k \<le> length s"
+ proof -
+ { assume "length s < k"
+ hence "length (a#s) \<le> k" by simp
+ from moment_ge [OF this] and nq and qa
+ have "False" by auto
+ } thus ?thesis by arith
+ qed
+ have nq_k: "\<not> Q (moment k s)"
+ proof -
+ have "moment k (a#s) = moment k s"
+ proof -
+ from moment_app [OF le_k, of "[a]"] show ?thesis by simp
+ qed
+ with nq show ?thesis by simp
+ qed
+ show "\<exists>i<length (a # s). k \<le> i \<and> \<not> Q (moment i (a # s)) \<and> (\<forall>i'>i. Q (moment i' (a # s)))"
+ proof -
+ { assume "Q s"
+ from ih [OF this nq_k]
+ obtain i where lti: "i < length s"
+ and nq: "\<not> Q (moment i s)"
+ and rst: "\<forall>i'>i. Q (moment i' s)"
+ and lki: "k \<le> i" by auto
+ have ?thesis
+ proof -
+ from lti have "i < length (a # s)" by auto
+ moreover have " \<not> Q (moment i (a # s))"
+ proof -
+ from lti have "i \<le> (length s)" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment i (a # s) = moment i s" by simp
+ with nq show ?thesis by auto
+ qed
+ moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"
+ proof -
+ {
+ fix i'
+ assume lti': "i < i'"
+ have "Q (moment i' (a # s))"
+ proof(cases "length (a#s) \<le> i'")
+ case True
+ from True have "moment i' (a#s) = a#s" by simp
+ with qa show ?thesis by simp
+ next
+ case False
+ from False have "i' \<le> length s" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment i' (a#s) = moment i' s" by simp
+ with rst lti' show ?thesis by auto
+ qed
+ } thus ?thesis by auto
+ qed
+ moreover note lki
+ ultimately show ?thesis by auto
+ qed
+ } moreover {
+ assume ns: "\<not> Q s"
+ have ?thesis
+ proof -
+ let ?i = "length s"
+ have "\<not> Q (moment ?i (a#s))"
+ proof -
+ have "?i \<le> length s" by simp
+ from moment_app [OF this, of "[a]"]
+ have "moment ?i (a#s) = moment ?i s" by simp
+ moreover have "\<dots> = s" by simp
+ ultimately show ?thesis using ns by auto
+ qed
+ moreover have "\<forall> i' > ?i. Q (moment i' (a#s))"
+ proof -
+ { fix i'
+ assume "i' > ?i"
+ hence "length (a#s) \<le> i'" by simp
+ from moment_ge [OF this]
+ have " moment i' (a # s) = a # s" .
+ with qa have "Q (moment i' (a#s))" by simp
+ } thus ?thesis by auto
+ qed
+ moreover have "?i < length (a#s)" by simp
+ moreover note le_k
+ ultimately show ?thesis by auto
+ qed
+ } ultimately show ?thesis by auto
+ qed
+qed
+
+lemma p_split:
+ "\<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow>
+ (\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+proof -
+ fix s Q
+ assume qs: "Q s" and nq: "\<not> Q []"
+ from nq have "\<not> Q (moment 0 s)" by simp
+ from p_split_gen [of Q s 0, OF qs this]
+ show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
+ by auto
+qed
+
+lemma moment_Suc_tl:
+ assumes "Suc i \<le> length s"
+ shows "tl (moment (Suc i) s) = moment i s"
+ using assms unfolding moment_def rev_take
+by (simp, metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop)
+
+lemma moment_plus:
+ assumes "Suc i \<le> length s"
+ shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)"
+proof -
+ have "(moment (Suc i) s) \<noteq> []"
+ using assms by (auto simp add: moment_def)
+ hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) # tl (moment (Suc i) s)"
+ by auto
+<<<<<<< local
+ have "moment (i + length s) (t @ s) = moment i t @ moment (length s) (t @ s)"
+ by (simp add: moment_def)
+ with moment_app show ?thesis by auto
+qed
+
+lemma moment_Suc_tl:
+ assumes "Suc i \<le> length s"
+ shows "tl (moment (Suc i) s) = moment i s"
+ using assms unfolding moment_def rev_take
+ by (simp, metis Suc_diff_le diff_Suc_Suc drop_Suc tl_drop)
+
+lemma moment_plus':
+ assumes "Suc i \<le> length s"
+ shows "(moment (Suc i) s) = (hd (moment (Suc i) s)) # (moment i s)"
+proof -
+ have "(moment (Suc i) s) \<noteq> []"
+ using assms length_moment_le by fastforce
+ hence "(moment (Suc i) s) = (hd (moment (Suc i) s)) # tl (moment (Suc i) s)"
+ by auto
+ with moment_Suc_tl[OF assms]
+ show ?thesis by metis
+qed
+
+lemma moment_plus:
+ "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"
+proof(induct s, simp+)
+ fix a s
+ assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"
+ and le_i: "i \<le> length s"
+ show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"
+ proof(cases "i= length s")
+ case True
+ hence "Suc i = length (a#s)" by simp
+ with moment_eq have "moment (Suc i) (a#s) = a#s" by auto
+ moreover have "moment i (a#s) = s"
+ proof -
+ from moment_app [OF le_i, of "[a]"]
+ and True show ?thesis by simp
+ qed
+ ultimately show ?thesis by auto
+ next
+ case False
+ from False and le_i have lti: "i < length s" by arith
+ hence les_i: "Suc i \<le> length s" by arith
+ show ?thesis
+ proof -
+ from moment_app [OF les_i, of "[a]"]
+ have "moment (Suc i) (a # s) = moment (Suc i) s" by simp
+ moreover have "moment i (a#s) = moment i s"
+ proof -
+ from lti have "i \<le> length s" by simp
+ from moment_app [OF this, of "[a]"] show ?thesis by simp
+ qed
+ moreover note ih [OF les_i]
+ ultimately show ?thesis by auto
+ qed
+ qed
+=======
+ with moment_Suc_tl[OF assms]
+ show ?thesis by metis
+>>>>>>> other
+qed
+
+end
+