--- a/prio/Moment.thy Fri Apr 13 13:12:43 2012 +0000
+++ b/prio/Moment.thy Sun Apr 15 21:53:12 2012 +0000
@@ -112,12 +112,37 @@
qed
lemma moment_restm_s: "(restm n s)@(moment n s) = s"
-by (metis firstn_restn_s moment_def restm_def rev_append rev_rev_ident)
+proof -
+ have " rev ((firstn n (rev s)) @ (restn n (rev s))) = s" (is "rev ?x = s")
+ proof -
+ have "?x = rev s" by (simp only:firstn_restn_s)
+ thus ?thesis by auto
+ qed
+ thus ?thesis
+ by (auto simp:restm_def moment_def)
+qed
declare restn.simps [simp del] firstn.simps[simp del]
lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"
-by (metis firstn_ge)
+proof(induct n arbitrary:s, simp add:firstn.simps)
+ case (Suc k)
+ assume ih: "\<And> s. length (s::'a list) \<le> k \<Longrightarrow> length (firstn k s) = length s"
+ and le: "length s \<le> Suc k"
+ show ?case
+ proof(cases s)
+ case Nil
+ from Nil show ?thesis by simp
+ next
+ case (Cons x xs)
+ from le and Cons have "length xs \<le> k" by simp
+ from ih [OF this] have "length (firstn k xs) = length xs" .
+ moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
+ by (simp add:firstn.simps)
+ moreover note Cons
+ ultimately show ?thesis by simp
+ qed
+qed
lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"
proof(induct n arbitrary:s, simp add:firstn.simps)
@@ -141,26 +166,78 @@
lemma app_firstn_restn:
fixes s1 s2
shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"
-by (metis append_eq_conv_conj firstn_ge firstn_le firstn_restn_s le_refl)
+proof(rule length_eq_elim_l)
+ have "length s1 \<le> length (s1 @ s2)" by simp
+ from length_firstn_le [OF this]
+ show "length s1 = length (firstn (length s1) (s1 @ s2))" by simp
+next
+ from firstn_restn_s
+ show "s1 @ s2 = firstn (length s1) (s1 @ s2) @ restn (length s1) (s1 @ s2)"
+ by metis
+qed
+
+
lemma length_moment_le:
fixes k s
assumes le_k: "k \<le> length s"
shows "length (moment k s) = k"
-by (metis assms length_firstn_le length_rev moment_def)
+proof -
+ have "length (rev (firstn k (rev s))) = k"
+ proof -
+ have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
+ also have "\<dots> = k"
+ proof(rule length_firstn_le)
+ from le_k show "k \<le> length (rev s)" by simp
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:moment_def)
+qed
lemma app_moment_restm:
fixes s1 s2
shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"
-by (metis app_firstn_restn length_rev moment_def restm_def rev_append rev_rev_ident)
+proof(rule length_eq_elim_r)
+ have "length s2 \<le> length (s1 @ s2)" by simp
+ from length_moment_le [OF this]
+ show "length s2 = length (moment (length s2) (s1 @ s2))" by simp
+next
+ from moment_restm_s
+ show "s1 @ s2 = restm (length s2) (s1 @ s2) @ moment (length s2) (s1 @ s2)"
+ by metis
+qed
lemma length_moment_ge:
fixes k s
assumes le_k: "length s \<le> k"
shows "length (moment k s) = (length s)"
-by (metis assms firstn_ge length_rev moment_def)
+proof -
+ have "length (rev (firstn k (rev s))) = length s"
+ proof -
+ have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
+ also have "\<dots> = length s"
+ proof -
+ have "\<dots> = length (rev s)"
+ proof(rule length_firstn_ge)
+ from le_k show "length (rev s) \<le> k" by simp
+ qed
+ also have "\<dots> = length s" by simp
+ finally show ?thesis .
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:moment_def)
+qed
lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"
-by (metis length_firstn_ge length_firstn_le nat_le_linear)
+proof(cases "n \<le> length s")
+ case True
+ from length_firstn_le [OF True] show ?thesis by auto
+next
+ case False
+ from False have "length s \<le> n" by simp
+ from firstn_ge [OF this] show ?thesis by auto
+qed
lemma firstn_conc:
fixes m n
@@ -193,7 +270,45 @@
fixes i j k s
assumes eq_k: "j + i = k"
shows "restn k s = restn j (restn i s)"
-by (metis app_moment_restm append_take_drop_id assms drop_drop length_drop moment_def restn.simps)
+proof -
+ have "(firstn (length s - k) (rev s)) =
+ (firstn (length (rev (firstn (length s - i) (rev s))) - j)
+ (rev (rev (firstn (length s - i) (rev s)))))"
+ proof -
+ have "(firstn (length s - k) (rev s)) =
+ (firstn (length (rev (firstn (length s - i) (rev s))) - j)
+ (firstn (length s - i) (rev s)))"
+ proof -
+ have " (length (rev (firstn (length s - i) (rev s))) - j) = length s - k"
+ proof -
+ have "(length (rev (firstn (length s - i) (rev s))) - j) = (length s - i) - j"
+ proof -
+ have "(length (rev (firstn (length s - i) (rev s))) - j) =
+ length ((firstn (length s - i) (rev s))) - j"
+ by simp
+ also have "\<dots> = length ((firstn (length (rev s) - i) (rev s))) - j" by simp
+ also have "\<dots> = (length (rev s) - i) - j"
+ proof -
+ have "length ((firstn (length (rev s) - i) (rev s))) = (length (rev s) - i)"
+ by (rule length_firstn_le, simp)
+ thus ?thesis by simp
+ qed
+ also have "\<dots> = (length s - i) - j" by simp
+ finally show ?thesis .
+ qed
+ with eq_k show ?thesis by auto
+ qed
+ moreover have "(firstn (length s - k) (rev s)) =
+ (firstn (length s - k) (firstn (length s - i) (rev s)))"
+ proof(rule firstn_conc)
+ from eq_k show "length s - k \<le> length s - i" by simp
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by simp
+ qed
+ thus ?thesis by (simp only:restn.simps)
+qed
(*
value "down_to 2 0 [5, 4, 3, 2, 1, 0]"
@@ -204,12 +319,28 @@
by (simp add:from_to_def restn.simps)
lemma moment_app [simp]:
- assumes ile: "i \<le> length s"
+ assumes
+ ile: "i \<le> length s"
shows "moment i (s'@s) = moment i s"
-by (metis assms firstn_le length_rev moment_def rev_append)
+proof -
+ have "moment i (s'@s) = rev (firstn i (rev (s'@s)))" by (simp add:moment_def)
+ moreover have "firstn i (rev (s'@s)) = firstn i (rev s @ rev s')" by simp
+ moreover have "\<dots> = firstn i (rev s)"
+ proof(rule firstn_le)
+ have "length (rev s) = length s" by simp
+ with ile show "i \<le> length (rev s)" by simp
+ qed
+ ultimately show ?thesis by (simp add:moment_def)
+qed
lemma moment_eq [simp]: "moment (length s) (s'@s) = s"
-by (metis app_moment_restm)
+proof -
+ have "length s \<le> length s" by simp
+ from moment_app [OF this, of s']
+ have " moment (length s) (s' @ s) = moment (length s) s" .
+ moreover have "\<dots> = s" by (simp add:moment_def)
+ ultimately show ?thesis by simp
+qed
lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
by (unfold moment_def, simp)
@@ -403,16 +534,42 @@
assumes le_ij: "i \<le> j"
and le_jk: "j \<le> k"
shows "down_to k j s @ down_to j i s = down_to k i s"
-by (metis down_to_def from_to_conc le_ij le_jk rev_append)
+proof -
+ have "rev (from_to j k (rev s)) @ rev (from_to i j (rev s)) = rev (from_to i k (rev s))"
+ (is "?L = ?R")
+ proof -
+ have "?L = rev (from_to i j (rev s) @ from_to j k (rev s))" by simp
+ also have "\<dots> = ?R" (is "rev ?x = rev ?y")
+ proof -
+ have "?x = ?y" by (rule from_to_conc[OF le_ij le_jk])
+ thus ?thesis by simp
+ qed
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:down_to_def)
+qed
lemma restn_ge:
fixes s k
assumes le_k: "length s \<le> k"
shows "restn k s = []"
-by (metis assms diff_is_0_eq moment_def moment_zero restn.simps)
+proof -
+ from firstn_restn_s [of k s, symmetric] have "s = (firstn k s) @ (restn k s)" .
+ hence "length s = length \<dots>" by simp
+ also have "\<dots> = length (firstn k s) + length (restn k s)" by simp
+ finally have "length s = ..." by simp
+ moreover from length_firstn_ge and le_k
+ have "length (firstn k s) = length s" by simp
+ ultimately have "length (restn k s) = 0" by auto
+ thus ?thesis by auto
+qed
lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"
-by (metis firstn_nil from_to_def restn_ge)
+proof(simp only:from_to_def)
+ assume "length s \<le> k"
+ from restn_ge [OF this]
+ show "firstn (j - k) (restn k s) = []" by simp
+qed
(*
value "from_to 2 5 [0, 1, 2, 3, 4]"
@@ -423,31 +580,166 @@
fixes k j s
assumes le_j: "length s \<le> j"
shows "from_to k j s = restn k s"
-by (metis app_moment_restm append_Nil2 append_take_drop_id assms diff_is_0_eq' drop_take firstn_restn_s from_to_def length_drop moment_def moment_zero restn.simps)
+proof -
+ have "from_to 0 k s @ from_to k j s = from_to 0 j s"
+ proof(cases "k \<le> j")
+ case True
+ from from_to_conc True show ?thesis by auto
+ next
+ case False
+ from False le_j have lek: "length s \<le> k" by auto
+ from from_to_ge [OF this] have "from_to k j s = []" .
+ hence "from_to 0 k s @ from_to k j s = from_to 0 k s" by simp
+ also have "\<dots> = s"
+ proof -
+ from from_to_firstn [of k s]
+ have "\<dots> = firstn k s" .
+ also have "\<dots> = s" by (rule firstn_ge [OF lek])
+ finally show ?thesis .
+ qed
+ finally have "from_to 0 k s @ from_to k j s = s" .
+ moreover have "from_to 0 j s = s"
+ proof -
+ have "from_to 0 j s = firstn j s" by (simp add:from_to_firstn)
+ also have "\<dots> = s"
+ proof(rule firstn_ge)
+ from le_j show "length s \<le> j " by simp
+ qed
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis by auto
+ qed
+ also have "\<dots> = s"
+ proof -
+ from from_to_firstn have "\<dots> = firstn j s" .
+ also have "\<dots> = s"
+ proof(rule firstn_ge)
+ from le_j show "length s \<le> j" by simp
+ qed
+ finally show ?thesis .
+ qed
+ finally have "from_to 0 k s @ from_to k j s = s" .
+ moreover have "from_to 0 k s @ restn k s = s"
+ proof -
+ from from_to_firstn [of k s]
+ have "from_to 0 k s = firstn k s" .
+ thus ?thesis by (simp add:firstn_restn_s)
+ qed
+ ultimately have "from_to 0 k s @ from_to k j s =
+ from_to 0 k s @ restn k s" by simp
+ thus ?thesis by auto
+qed
lemma down_to_moment: "down_to k 0 s = moment k s"
-by (metis down_to_def from_to_firstn moment_def)
+proof -
+ have "rev (from_to 0 k (rev s)) = rev (firstn k (rev s))"
+ using from_to_firstn by metis
+ thus ?thesis by (simp add:down_to_def moment_def)
+qed
lemma down_to_restm:
assumes le_s: "length s \<le> j"
shows "down_to j k s = restm k s"
-by (metis assms down_to_def from_to_restn length_rev restm_def)
+proof -
+ have "rev (from_to k j (rev s)) = rev (restn k (rev s))" (is "?L = ?R")
+ proof -
+ from le_s have "length (rev s) \<le> j" by simp
+ from from_to_restn [OF this, of k] show ?thesis by simp
+ qed
+ thus ?thesis by (simp add:down_to_def restm_def)
+qed
lemma moment_split: "moment (m+i) s = down_to (m+i) i s @down_to i 0 s"
-by (metis down_to_conc down_to_moment le0 le_add1 nat_add_commute)
+proof -
+ have "moment (m + i) s = down_to (m+i) 0 s" using down_to_moment by metis
+ also have "\<dots> = (down_to (m+i) i s) @ (down_to i 0 s)"
+ by(rule down_to_conc[symmetric], auto)
+ finally show ?thesis .
+qed
lemma length_restn: "length (restn i s) = length s - i"
-by (metis diff_le_self length_firstn_le length_rev restn.simps)
+proof(cases "i \<le> length s")
+ case True
+ from length_firstn_le [OF this] have "length (firstn i s) = i" .
+ moreover have "length s = length (firstn i s) + length (restn i s)"
+ proof -
+ have "s = firstn i s @ restn i s" using firstn_restn_s by metis
+ hence "length s = length \<dots>" by simp
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis by simp
+next
+ case False
+ hence "length s \<le> i" by simp
+ from restn_ge [OF this] have "restn i s = []" .
+ with False show ?thesis by simp
+qed
lemma length_from_to_in:
fixes i j s
assumes le_ij: "i \<le> j"
and le_j: "j \<le> length s"
shows "length (from_to i j s) = j - i"
-by (metis diff_le_mono from_to_def le_j length_firstn_le length_restn)
+proof -
+ have "from_to 0 j s = from_to 0 i s @ from_to i j s"
+ by (rule from_to_conc[symmetric, OF _ le_ij], simp)
+ moreover have "length (from_to 0 j s) = j"
+ proof -
+ have "from_to 0 j s = firstn j s" using from_to_firstn by metis
+ moreover have "length \<dots> = j" by (rule length_firstn_le [OF le_j])
+ ultimately show ?thesis by simp
+ qed
+ moreover have "length (from_to 0 i s) = i"
+ proof -
+ have "from_to 0 i s = firstn i s" using from_to_firstn by metis
+ moreover have "length \<dots> = i"
+ proof (rule length_firstn_le)
+ from le_ij le_j show "i \<le> length s" by simp
+ qed
+ ultimately show ?thesis by simp
+ qed
+ ultimately show ?thesis by auto
+qed
lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"
-by (metis diff_add_inverse2 from_to_def)
+proof(cases "m+i \<le> length s")
+ case True
+ have "restn i s = from_to i (m+i) s @ from_to (m+i) (length s) s"
+ proof -
+ have "restn i s = from_to i (length s) s"
+ by(rule from_to_restn[symmetric], simp)
+ also have "\<dots> = from_to i (m+i) s @ from_to (m+i) (length s) s"
+ by(rule from_to_conc[symmetric, OF _ True], simp)
+ finally show ?thesis .
+ qed
+ hence "firstn m (restn i s) = firstn m \<dots>" by simp
+ moreover have "\<dots> = firstn (length (from_to i (m+i) s))
+ (from_to i (m+i) s @ from_to (m+i) (length s) s)"
+ proof -
+ have "length (from_to i (m+i) s) = m"
+ proof -
+ have "length (from_to i (m+i) s) = (m+i) - i"
+ by(rule length_from_to_in [OF _ True], simp)
+ thus ?thesis by simp
+ qed
+ thus ?thesis by simp
+ qed
+ ultimately show ?thesis using app_firstn_restn by metis
+next
+ case False
+ hence "length s \<le> m + i" by simp
+ from from_to_restn [OF this]
+ have "from_to i (m + i) s = restn i s" .
+ moreover have "firstn m (restn i s) = restn i s"
+ proof(rule firstn_ge)
+ show "length (restn i s) \<le> m"
+ proof -
+ have "length (restn i s) = length s - i" using length_restn by metis
+ with False show ?thesis by simp
+ qed
+ qed
+ ultimately show ?thesis by simp
+qed
lemma down_to_moment_restm:
fixes m i s
@@ -457,9 +749,25 @@
lemma moment_plus_split:
fixes m i s
shows "moment (m + i) s = moment m (restm i s) @ moment i s"
-by (metis down_to_moment down_to_moment_restm moment_split)
+proof -
+ from moment_split [of m i s]
+ have "moment (m + i) s = down_to (m + i) i s @ down_to i 0 s" .
+ also have "\<dots> = down_to (m+i) i s @ moment i s" using down_to_moment by simp
+ also from down_to_moment_restm have "\<dots> = moment m (restm i s) @ moment i s"
+ by simp
+ finally show ?thesis .
+qed
lemma length_restm: "length (restm i s) = length s - i"
-by (metis length_restn length_rev restm_def)
+proof -
+ have "length (rev (restn i (rev s))) = length s - i" (is "?L = ?R")
+ proof -
+ have "?L = length (restn i (rev s))" by simp
+ also have "\<dots> = length (rev s) - i" using length_restn by metis
+ also have "\<dots> = ?R" by simp
+ finally show ?thesis .
+ qed
+ thus ?thesis by (simp add:restm_def)
+qed
end
\ No newline at end of file