theory Moment+
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imports Main+
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begin+
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+
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fun firstn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where+
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  "firstn 0 s = []" |+
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  "firstn (Suc n) [] = []" |+
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  "firstn (Suc n) (e#s) = e#(firstn n s)"+
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+
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fun restn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where "restn n s = rev (firstn (length s - n) (rev s))"+
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+
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definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where "moment n s = rev (firstn n (rev s))"+
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+
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definition restm :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where "restm n s = rev (restn n (rev s))"+
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+
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definition from_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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  where "from_to i j s = firstn (j - i) (restn i s)"+
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+
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definition down_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"+
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where "down_to j i s = rev (from_to i j (rev s))"+
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+
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(*+
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value "down_to 6 2 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"+
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value "from_to 2 6 [0, 1, 2, 3, 4, 5, 6, 7]"+
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*)+
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+
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lemma length_eq_elim_l: "\<lbrakk>length xs = length ys; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"+
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  by auto+
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+
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lemma length_eq_elim_r: "\<lbrakk>length us = length vs; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"+
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  by simp+
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+
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lemma firstn_nil [simp]: "firstn n [] = []"+
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  by (cases n, simp+)+
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+
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(*+
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value "from_to 0 2 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] @ +
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       from_to 2 5 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"+
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*)+
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+
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lemma firstn_le: "\<And> n s'. n \<le> length s \<Longrightarrow> firstn n (s@s') = firstn n s"+
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proof (induct s, simp)+
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  fix a s n s'+
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  assume ih: "\<And>n s'. n \<le> length s \<Longrightarrow> firstn n (s @ s') = firstn n s"+
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  and le_n: " n \<le> length (a # s)"+
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  show "firstn n ((a # s) @ s') = firstn n (a # s)"+
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  proof(cases n, simp)+
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    fix k+
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    assume eq_n: "n = Suc k"+
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    with le_n have "k \<le> length s" by auto+
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    from ih [OF this] and eq_n+
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    show "firstn n ((a # s) @ s') = firstn n (a # s)" by auto+
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  qed+
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qed+
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+
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lemma firstn_ge [simp]: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"+
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proof(induct s, simp)+
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  fix a s n+
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  assume ih: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"+
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    and le: "length (a # s) \<le> n"+
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  show "firstn n (a # s) = a # s"+
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  proof(cases n)+
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    assume eq_n: "n = 0" with le show ?thesis by simp+
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  next+
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    fix k+
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    assume eq_n: "n = Suc k"+
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    with le have le_k: "length s \<le> k" by simp+
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    from ih [OF this] have "firstn k s = s" .+
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    from eq_n and this+
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    show ?thesis by simp+
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  qed+
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qed+
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+
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lemma firstn_eq [simp]: "firstn (length s) s = s"+
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  by simp+
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+
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lemma firstn_restn_s: "(firstn n (s::'a list)) @ (restn n s) = s"+
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proof(induct n arbitrary:s, simp)+
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  fix n s+
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  assume ih: "\<And>t. firstn n (t::'a list) @ restn n t = t"+
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  show "firstn (Suc n) (s::'a list) @ restn (Suc n) s = s"+
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  proof(cases s, simp)+
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    fix x xs+
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    assume eq_s: "s = x#xs"+
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    show "firstn (Suc n) s @ restn (Suc n) s = s"+
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    proof -+
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      have "firstn (Suc n) s @ restn (Suc n) s =  x # (firstn n xs @ restn n xs)"+
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      proof -+
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        from eq_s have "firstn (Suc n) s =  x # firstn n xs" by simp+
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        moreover have "restn (Suc n) s = restn n xs"+
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        proof -+
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          from eq_s have "restn (Suc n) s = rev (firstn (length xs - n) (rev xs @ [x]))" by simp+
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          also have "\<dots> = restn n xs"+
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          proof -+
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            have "(firstn (length xs - n) (rev xs @ [x])) = (firstn (length xs - n) (rev xs))"+
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              by(rule firstn_le, simp)+
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            hence "rev (firstn (length xs - n) (rev xs @ [x])) = +
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              rev (firstn (length xs - n) (rev xs))" by simp+
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            also have "\<dots> = rev (firstn (length (rev xs) - n) (rev xs))" by simp+
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            finally show ?thesis by simp+
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          qed+
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          finally show ?thesis by simp+
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        qed+
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        ultimately show ?thesis by simp+
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      qed with ih eq_s show ?thesis by simp+
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    qed+
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  qed+
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qed+
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+
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lemma moment_restm_s: "(restm n s)@(moment n s) = s"+
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by (metis firstn_restn_s moment_def restm_def rev_append rev_rev_ident)+
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+
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declare restn.simps [simp del] firstn.simps[simp del]+
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+
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lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"+
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by (metis firstn_ge)+
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+
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lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"+
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proof(induct n arbitrary:s, simp add:firstn.simps)+
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  case (Suc k)+
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  assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"+
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    and le: "Suc k \<le> length s"+
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  show ?case+
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  proof(cases s)+
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    case Nil+
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    from Nil and le show ?thesis by auto+
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  next+
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    case (Cons x xs)+
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    from le and Cons have "k \<le> length xs" by simp+
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    from ih [OF this] have "length (firstn k xs) = k" .+
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    moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))" +
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      by (simp add:firstn.simps)+
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    ultimately show ?thesis by simp+
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  qed+
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qed+
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+
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lemma app_firstn_restn: +
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  fixes s1 s2+
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  shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"+
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by (metis append_eq_conv_conj firstn_ge firstn_le firstn_restn_s le_refl)+
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lemma length_moment_le:+
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  fixes k s+
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  assumes le_k: "k \<le> length s"+
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  shows "length (moment k s) = k"+
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by (metis assms length_firstn_le length_rev moment_def)+
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+
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lemma app_moment_restm: +
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  fixes s1 s2+
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  shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"+
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by (metis app_firstn_restn length_rev moment_def restm_def rev_append rev_rev_ident)+
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+
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lemma length_moment_ge:+
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  fixes k s+
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  assumes le_k: "length s \<le> k"+
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  shows "length (moment k s) = (length s)"+
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by (metis assms firstn_ge length_rev moment_def)+
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+
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lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"+
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by (metis length_firstn_ge length_firstn_le nat_le_linear)+
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+
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lemma firstn_conc: +
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  fixes m n+
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  assumes le_mn: "m \<le> n"+
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  shows "firstn m s = firstn m (firstn n  s)"+
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proof(cases "m \<le> length s")+
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  case True+
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  have "s = (firstn n s) @ (restn n s)" by (simp add:firstn_restn_s)+
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  hence "firstn m s = firstn m \<dots>" by simp+
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  also have "\<dots> = firstn m (firstn n s)" +
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  proof -+
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    from length_firstn [of n s]+
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    have "m \<le> length (firstn n s)"+
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    proof+
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      assume "length (firstn n s) = length s" with True show ?thesis by simp+
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    next+
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      assume "length (firstn n s) = n " with le_mn show ?thesis by simp+
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    qed+
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    from firstn_le [OF this, of "restn n s"]+
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    show ?thesis .+
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  qed+
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  finally show ?thesis by simp+
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next+
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  case False+
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  from False and le_mn have "length s \<le> n"  by simp+
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  from firstn_ge [OF this] show ?thesis by simp+
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qed+
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+
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lemma restn_conc: +
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  fixes i j k s+
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  assumes eq_k: "j + i = k"+
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  shows "restn k s = restn j (restn i s)"+
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by (metis app_moment_restm append_take_drop_id assms drop_drop length_drop moment_def restn.simps)+
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+
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(*+
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value "down_to 2 0 [5, 4, 3, 2, 1, 0]"+
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value "moment 2 [5, 4, 3, 2, 1, 0]"+
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*)+
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+
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lemma from_to_firstn: "from_to 0 k s = firstn k s"+
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by (simp add:from_to_def restn.simps)+
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+
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lemma moment_app [simp]:+
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  assumes ile: "i \<le> length s"+
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  shows "moment i (s'@s) = moment i s"+
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by (metis assms firstn_le length_rev moment_def rev_append)+
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+
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lemma moment_eq [simp]: "moment (length s) (s'@s) = s"+
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by (metis app_moment_restm)+
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+
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lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"+
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  by (unfold moment_def, simp)+
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+
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lemma moment_zero [simp]: "moment 0 s = []"+
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  by (simp add:moment_def firstn.simps)+
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+
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lemma p_split_gen: +
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  "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>+
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  (\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"+
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proof (induct s, simp)+
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  fix a s+
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  assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>+
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           \<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"+
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    and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"+
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  have le_k: "k \<le> length s"+
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  proof -+
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    { assume "length s < k"+
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      hence "length (a#s) \<le> k" by simp+
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      from moment_ge [OF this] and nq and qa+
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      have "False" by auto+
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    } thus ?thesis by arith+
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  qed+
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  have nq_k: "\<not> Q (moment k s)"+
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  proof -+
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    have "moment k (a#s) = moment k s"+
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    proof -+
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      from moment_app [OF le_k, of "[a]"] show ?thesis by simp+
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    qed+
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    with nq show ?thesis by simp+
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  qed+
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  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)))"+
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  proof -+
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    { assume "Q s"+
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      from ih [OF this nq_k]+
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      obtain i where lti: "i < length s" +
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        and nq: "\<not> Q (moment i s)" +
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        and rst: "\<forall>i'>i. Q (moment i' s)" +
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        and lki: "k \<le> i" by auto+
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      have ?thesis +
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      proof -+
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        from lti have "i < length (a # s)" by auto+
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        moreover have " \<not> Q (moment i (a # s))"+
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        proof -+
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          from lti have "i \<le> (length s)" by simp+
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          from moment_app [OF this, of "[a]"]+
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          have "moment i (a # s) = moment i s" by simp+
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          with nq show ?thesis by auto+
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        qed+
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        moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"+
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        proof -+
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          {+
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            fix i'+
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            assume lti': "i < i'"+
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            have "Q (moment i' (a # s))"+
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            proof(cases "length (a#s) \<le> i'")+
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              case True+
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              from True have "moment i' (a#s) = a#s" by simp+
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              with qa show ?thesis by simp+
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            next+
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              case False+
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              from False have "i' \<le> length s" by simp+
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              from moment_app [OF this, of "[a]"]+
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              have "moment i' (a#s) = moment i' s" by simp+
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              with rst lti' show ?thesis by auto+
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            qed+
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          } thus ?thesis by auto+
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        qed+
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        moreover note lki+
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        ultimately show ?thesis by auto+
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      qed+
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    } moreover {+
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      assume ns: "\<not> Q s"+
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      have ?thesis+
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      proof -+
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        let ?i = "length s"+
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        have "\<not> Q (moment ?i (a#s))"+
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        proof -+
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          have "?i \<le> length s" by simp+
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          from moment_app [OF this, of "[a]"]+
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          have "moment ?i (a#s) = moment ?i s" by simp+
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          moreover have "\<dots> = s" by simp+
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          ultimately show ?thesis using ns by auto+
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        qed+
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        moreover have "\<forall> i' > ?i. Q (moment i' (a#s))" +
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        proof -+
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          { fix i'+
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            assume "i' > ?i"+
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            hence "length (a#s) \<le> i'" by simp+
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            from moment_ge [OF this] +
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            have " moment i' (a # s) = a # s" .+
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            with qa have "Q (moment i' (a#s))" by simp+
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          } thus ?thesis by auto+
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        qed+
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        moreover have "?i < length (a#s)" by simp+
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        moreover note le_k+
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        ultimately show ?thesis by auto+
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      qed+
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    } ultimately show ?thesis by auto+
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  qed+
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qed+
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+
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lemma p_split: +
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  "\<And> s Q. \<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow> +
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       (\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"+
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proof -+
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  fix s Q+
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  assume qs: "Q s" and nq: "\<not> Q []"+
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  from nq have "\<not> Q (moment 0 s)" by simp+
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  from p_split_gen [of Q s 0, OF qs this]+
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  show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"+
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    by auto+
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qed+
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+
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lemma moment_plus: +
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  "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"+
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proof(induct s, simp+)+
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  fix a s+
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  assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"+
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    and le_i: "i \<le> length s"+
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  show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"+
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  proof(cases "i= length s")+
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    case True+
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    hence "Suc i = length (a#s)" by simp+
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    with moment_eq have "moment (Suc i) (a#s) = a#s" by auto+
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    moreover have "moment i (a#s) = s"+
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    proof -+
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      from moment_app [OF le_i, of "[a]"]+
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      and True show ?thesis by simp+
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    qed+
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    ultimately show ?thesis by auto+
−
  next+
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    case False+
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    from False and le_i have lti: "i < length s" by arith+
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    hence les_i: "Suc i \<le> length s" by arith+
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    show ?thesis +
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    proof -+
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      from moment_app [OF les_i, of "[a]"]+
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      have "moment (Suc i) (a # s) = moment (Suc i) s" by simp+
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      moreover have "moment i (a#s) = moment i s" +
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      proof -+
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        from lti have "i \<le> length s" by simp+
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        from moment_app [OF this, of "[a]"] show ?thesis by simp+
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      qed+
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      moreover note ih [OF les_i]+
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      ultimately show ?thesis by auto+
−
    qed+
−
  qed+
−
qed+
−
+
−
lemma from_to_conc:+
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  fixes i j k s+
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  assumes le_ij: "i \<le> j"+
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  and le_jk: "j \<le> k"+
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  shows "from_to i j s @ from_to j k s = from_to i k s"+
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proof -+
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  let ?ris = "restn i s"+
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  have "firstn (j - i) (restn i s) @ firstn (k - j) (restn j s) =+
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         firstn (k - i) (restn i s)" (is "?x @ ?y = ?z")+
−
  proof -+
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    let "firstn (k-j) ?u" = "?y"+
−
    let ?rst = " restn (k - j) (restn (j - i) ?ris)"+
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    let ?rst' = "restn (k - i) ?ris"+
−
    have "?u = restn (j-i) ?ris"+
−
    proof(rule restn_conc)+
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      from le_ij show "j - i + i = j" by simp+
−
    qed+
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    hence "?x @ ?y = ?x @ firstn (k-j) (restn (j-i) ?ris)" by simp+
−
    moreover have "firstn (k - j) (restn (j - i) (restn i s)) @ ?rst = +
−
                      restn (j-i) ?ris" by (simp add:firstn_restn_s)+
−
    ultimately have "?x @ ?y @ ?rst = ?x @ (restn (j-i) ?ris)" by simp+
−
    also have "\<dots> = ?ris" by (simp add:firstn_restn_s)+
−
    finally have "?x @ ?y @ ?rst = ?ris" .+
−
    moreover have "?z @ ?rst = ?ris"+
−
    proof -+
−
      have "?z @ ?rst' = ?ris" by (simp add:firstn_restn_s)+
−
      moreover have "?rst' = ?rst"+
−
      proof(rule restn_conc)+
−
        from le_ij le_jk show "k - j + (j - i) = k - i" by auto+
−
      qed+
−
      ultimately show ?thesis by simp+
−
    qed+
−
    ultimately have "?x @ ?y @ ?rst = ?z @ ?rst" by simp+
−
    thus ?thesis by auto    +
−
  qed+
−
  thus ?thesis by (simp only:from_to_def)+
−
qed+
−
+
−
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)+
−
+
−
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)+
−
+
−
lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"+
−
by (metis firstn_nil from_to_def restn_ge)+
−
+
−
(*+
−
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)+
−
+
−
lemma down_to_moment: "down_to k 0 s = moment k s"+
−
by (metis down_to_def from_to_firstn moment_def)+
−
+
−
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)+
−
+
−
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)+
−
+
−
lemma length_restn: "length (restn i s) = length s - i"+
−
by (metis diff_le_self length_firstn_le length_rev restn.simps)+
−
+
−
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)+
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+
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lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"+
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by (metis diff_add_inverse2 from_to_def)+
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+
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lemma down_to_moment_restm:+
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  fixes m i s+
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  shows "down_to (m + i) i s = moment m (restm i s)"+
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  by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)+
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+
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lemma moment_plus_split:+
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  fixes m i s+
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  shows "moment (m + i) s = moment m (restm i s) @ moment i s"+
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by (metis down_to_moment down_to_moment_restm moment_split)+
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+
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lemma length_restm: "length (restm i s) = length s - i"+
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by (metis length_restn length_rev restm_def)+
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+
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end+
−