regexp: comparison prio/Moment.thy

equal deleted inserted replaced

-:e7504bfdbd50
+:b3add51e2e0f
 qed
 qed
 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)
 case (Suc k)
 assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"
 qed
 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
 assumes le_mn: "m \<le> n"
 shows "firstn m s = firstn m (firstn n  s)"
 lemma restn_conc:
 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]"
 value "moment 2 [5, 4, 3, 2, 1, 0]"
 *)
 lemma from_to_firstn: "from_to 0 k s = firstn k s"
 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)
 lemma moment_zero [simp]: "moment 0 s = []"
 lemma down_to_conc:
 fixes i j k s
 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]"
 value "restn 2  [0, 1, 2, 3, 4]"
 *)
 lemma from_to_restn:
 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
 shows "down_to (m + i) i s = moment m (restm i s)"
 by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)
 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

changeset 339	b3add51e2e0f
parent 336	f9e0d3274c14
child 347	73127f5db18f