diff -r 2c56b20032a7 -r 0679a84b11ad prio/Attic/Lsp.thy
--- a/prio/Attic/Lsp.thy	Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,323 +0,0 @@
-theory Lsp
-imports Main
-begin
-
-fun lsp :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list)"
-where 
-   "lsp f [] = ([], [], [])" |
-   "lsp f [x] = ([], [x], [])" |
-   "lsp f (x#xs) = (case (lsp f xs) of
-                     (l, [], r) \<Rightarrow> ([], [x], []) |
-                     (l, y#ys, r) \<Rightarrow> if f x \<ge> f y then ([], [x], xs) else (x#l, y#ys, r))"
-
-inductive lsp_p :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> ('a list \<times> 'a list \<times> 'a list) \<Rightarrow> bool"
-for f :: "('a \<Rightarrow> ('b::linorder))"
-where
-  lsp_nil [intro]: "lsp_p f [] ([], [], [])" |
-  lsp_single [intro]: "lsp_p f [x] ([], [x], [])" |
-  lsp_cons_1 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x \<ge> f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) ([], [x], xs)" |
-  lsp_cons_2 [intro]: "\<lbrakk>xs \<noteq> []; lsp_p f xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> lsp_p f (x#xs) (x#l, [m], r)"
-
-lemma lsp_p_lsp_1: "lsp_p f x y \<Longrightarrow> y = lsp f x"
-proof (induct rule:lsp_p.induct)
-  case (lsp_cons_1 xs  l m r x)
-  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
-    and le_mx: "f m \<le> f x"
-  show ?case (is "?L = ?R")
-  proof(cases xs, simp)
-    case (Cons v vs)
-    show ?thesis
-      apply (simp add:Cons)
-      apply (fold Cons)
-      by (simp add:lsp_xs le_mx)
-  qed
-next
-  case (lsp_cons_2 xs l m r x)
-  assume lsp_xs [symmetric]: "(l, [m], r) = lsp f xs"
-    and lt_xm: "f x < f m"
-  show ?case (is "?L = ?R")
-  proof(cases xs)
-    case (Cons v vs)
-    show ?thesis
-      apply (simp add:Cons)
-      apply (fold Cons)
-      apply (simp add:lsp_xs)
-      by (insert lt_xm, auto)
-  next
-    case Nil
-    from prems show ?thesis by simp
-  qed
-qed auto
-
-lemma lsp_mid_nil: "lsp f xs = (a, [], c) \<Longrightarrow> xs = []"
-  apply (induct xs arbitrary:a c, auto)
-  apply (case_tac xs, auto)
-  by (case_tac "(lsp f (ab # list))", auto split:if_splits list.splits)
-
-
-lemma lsp_mid_length: "lsp f x = (u, v, w) \<Longrightarrow> length v \<le> 1"
-proof(induct x arbitrary:u v w, simp)
-  case (Cons x xs)
-  assume ih: "\<And> u v w. lsp f xs = (u, v, w) \<Longrightarrow> length v \<le> 1"
-  and h: "lsp f (x # xs) = (u, v, w)"
-  show "length v \<le> 1" using h
-  proof(cases xs, simp add:h)
-    case (Cons z zs)
-    assume eq_xs: "xs = z # zs"
-    show ?thesis
-    proof(cases "lsp f xs")
-      fix l m r
-      assume eq_lsp: "lsp f xs = (l, m, r)"
-      show ?thesis
-      proof(cases m)
-        case Nil
-        from Nil and eq_lsp have "lsp f xs = (l, [], r)" by simp
-        from lsp_mid_nil [OF this] have "xs = []" .
-        with h show ?thesis by auto
-      next
-        case (Cons y ys)
-        assume eq_m: "m = y # ys"
-        from ih [OF eq_lsp] have eq_xs_1: "length m \<le> 1" .
-        show ?thesis
-        proof(cases "f x \<ge> f y")
-          case True
-          from eq_xs eq_xs_1 True h eq_lsp show ?thesis 
-            by (auto split:list.splits if_splits)
-        next
-          case False
-          from eq_xs eq_xs_1 False h eq_lsp show ?thesis 
-             by (auto split:list.splits if_splits)
-        qed
-      qed
-    qed
-  next
-    assume "[] = u \<and> [x] = v \<and> [] = w"
-    hence "v = [x]" by simp
-    thus "length v \<le> Suc 0" by simp
-  qed
-qed
-
-lemma lsp_p_lsp_2: "lsp_p f x (lsp f x)"
-proof(induct x, auto)
-  case (Cons x xs)
-  assume ih: "lsp_p f xs (lsp f xs)"
-  show ?case
-  proof(cases xs)
-    case Nil
-    thus ?thesis by auto
-  next
-    case (Cons v vs)
-    show ?thesis
-    proof(cases "xs")
-      case Nil
-      thus ?thesis by auto
-    next
-      case (Cons v vs)
-      assume eq_xs: "xs = v # vs"
-      show ?thesis
-      proof(cases "lsp f xs")
-        fix l m r
-        assume eq_lsp_xs: "lsp f xs = (l, m, r)"
-        show ?thesis
-        proof(cases m)
-          case Nil
-          from eq_lsp_xs and Nil have "lsp f xs = (l, [], r)" by simp
-          from lsp_mid_nil [OF this] have eq_xs: "xs = []" .
-          hence "lsp f (x#xs) = ([], [x], [])" by simp
-          with eq_xs show ?thesis by auto
-        next
-          case (Cons y ys)
-          assume eq_m: "m = y # ys"
-          show ?thesis
-          proof(cases "f x \<ge> f y")
-            case True
-            from eq_xs eq_lsp_xs Cons True
-            have eq_lsp: "lsp f (x#xs) = ([], [x], v # vs)" by simp
-            show ?thesis
-            proof (simp add:eq_lsp)
-              show "lsp_p f (x # xs) ([], [x], v # vs)"
-              proof(fold eq_xs, rule lsp_cons_1 [OF _])
-                from eq_xs show "xs \<noteq> []" by simp
-              next
-                from lsp_mid_length [OF eq_lsp_xs] and Cons
-                have "m = [y]" by simp
-                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
-                with ih show "lsp_p f xs (l, [y], r)" by simp
-              next
-                from True show "f y \<le> f x" by simp
-              qed
-            qed
-          next
-            case False
-            from eq_xs eq_lsp_xs Cons False
-            have eq_lsp: "lsp f (x#xs) = (x # l, y # ys, r) " by simp
-            show ?thesis
-            proof (simp add:eq_lsp)
-              from lsp_mid_length [OF eq_lsp_xs] and eq_m
-              have "ys = []" by simp
-              moreover have "lsp_p f (x # xs) (x # l, [y], r)"
-              proof(rule lsp_cons_2)
-                from eq_xs show "xs \<noteq> []" by simp
-              next
-                from lsp_mid_length [OF eq_lsp_xs] and Cons
-                have "m = [y]" by simp
-                with eq_lsp_xs have "lsp f xs = (l, [y], r)" by simp
-                with ih show "lsp_p f xs (l, [y], r)" by simp
-              next
-                from False show "f x < f y" by simp
-              qed
-              ultimately show "lsp_p f (x # xs) (x # l, y # ys, r)" by simp
-            qed
-          qed
-        qed
-      qed
-    qed
-  qed
-qed
-
-lemma lsp_induct:
-  fixes f x1 x2 P
-  assumes h: "lsp f x1 = x2"
-  and p1: "P [] ([], [], [])"
-  and p2: "\<And>x. P [x] ([], [x], [])"
-  and p3: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f m \<le> f x\<rbrakk> \<Longrightarrow> P (x # xs) ([], [x], xs)"
-  and p4: "\<And>xs l m r x. \<lbrakk>xs \<noteq> []; lsp f xs = (l, [m], r); P xs (l, [m], r); f x < f m\<rbrakk> \<Longrightarrow> P (x # xs) (x # l, [m], r)"
-  shows "P x1 x2"
-proof(rule lsp_p.induct)
-  from lsp_p_lsp_2 and h
-  show "lsp_p f x1 x2" by metis
-next
-  from p1 show "P [] ([], [], [])" by metis
-next
-  from p2 show "\<And>x. P [x] ([], [x], [])" by metis
-next
-  fix xs l m r x 
-  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f m \<le> f x"
-  show "P (x # xs) ([], [x], xs)" 
-  proof(rule p3 [OF h1 _ h3 h4])
-    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
-  qed
-next
-  fix xs l m r x 
-  assume h1: "xs \<noteq> []" and h2: "lsp_p f xs (l, [m], r)" and h3: "P xs (l, [m], r)" and h4: "f x < f m"
-  show "P (x # xs) (x # l, [m], r)"
-  proof(rule p4 [OF h1 _ h3 h4])
-    from h2 and lsp_p_lsp_1 show "lsp f xs = (l, [m], r)" by metis
-  qed
-qed
-
-lemma lsp_set_eq: 
-  fixes f x u v w
-  assumes h: "lsp f x = (u, v, w)"
-  shows "x = u@v@w"
-proof -
-  have "\<And> f x r. lsp f x = r \<Longrightarrow> \<forall> u v w. (r = (u, v, w) \<longrightarrow> x = u@v@w)" 
-    by (erule lsp_induct, simp+)
-  from this [rule_format, OF h] show ?thesis by simp
-qed
-
-lemma lsp_set: 
-  assumes h: "(u, v, w) = lsp f x"
-  shows "set (u@v@w) = set x"
-proof -
-  from lsp_set_eq [OF h[symmetric]] 
-  show ?thesis by simp
-qed
-
-lemma max_insert_gt:
-  fixes S fx
-  assumes h: "fx < Max S"
-  and np: "S \<noteq> {}"
-  and fn: "finite S" 
-  shows "Max S = Max (insert fx S)"
-proof -
-  from Max_insert [OF fn np]
-  have "Max (insert fx S) = max fx (Max S)" .
-  moreover have "\<dots> = Max S"
-  proof(cases "fx \<le> Max S")
-    case False
-    with h
-    show ?thesis by (simp add:max_def)
-  next
-    case True
-    thus ?thesis by (simp add:max_def)
-  qed
-  ultimately show ?thesis by simp
-qed
-
-lemma max_insert_le: 
-  fixes S fx
-  assumes h: "Max S \<le> fx"
-  and fn: "finite S"
-  shows "fx = Max (insert fx S)"
-proof(cases "S = {}")
-  case True
-  thus ?thesis by simp
-next
-  case False
-  from Max_insert [OF fn False]
-  have "Max (insert fx S) = max fx (Max S)" .
-  moreover have "\<dots> = fx"
-  proof(cases "fx \<le> Max S")
-    case False
-    thus ?thesis by (simp add:max_def)
-  next
-    case True
-    have hh: "\<And> x y. \<lbrakk> x \<le> (y::('a::linorder)); y \<le> x\<rbrakk> \<Longrightarrow> x = y" by auto
-    from hh [OF True h]
-    have "fx = Max S" .
-    thus ?thesis by simp
-  qed
-  ultimately show ?thesis by simp
-qed
-  
-lemma lsp_max: 
-  fixes f x u m w
-  assumes h: "lsp f x = (u, [m], w)"
-  shows "f m = Max (f ` (set x))"
-proof -
-  { fix y
-    have "lsp f x = y \<Longrightarrow> \<forall> u m w. y = (u, [m], w) \<longrightarrow> f m = Max (f ` (set x))"
-    proof(erule lsp_induct, simp)
-      { fix x u m w
-        assume "(([]::'a list), ([x]::'a list), ([]::'a list)) = (u, [m], w)"
-        hence "f m = Max (f ` set [x])"  by simp
-      } thus "\<And>x. \<forall>u m w. ([], [x], []) = (u, [m], w) \<longrightarrow> f m = Max (f ` set [x])" by simp
-    next
-      fix xs l m r x
-      assume h1: "xs \<noteq> []"
-        and h2: " lsp f xs = (l, [m], r)"
-        and h3: "\<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
-        and h4: "f m \<le> f x"
-      show " \<forall>u m w. ([], [x], xs) = (u, [m], w) \<longrightarrow> f m = Max (f ` set (x # xs))"
-      proof -
-        have "f x = Max (f ` set (x # xs))"
-        proof -
-          from h2 h3 have "f m = Max (f ` set xs)" by simp
-          with h4 show ?thesis
-            apply auto
-            by (rule_tac max_insert_le, auto)
-        qed
-        thus ?thesis by simp
-      qed
-    next
-      fix xs l m r x
-      assume h1: "xs \<noteq> []"
-        and h2: " lsp f xs = (l, [m], r)"
-        and h3: " \<forall>u ma w. (l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set xs)"
-        and h4: "f x < f m"
-      show "\<forall>u ma w. (x # l, [m], r) = (u, [ma], w) \<longrightarrow> f ma = Max (f ` set (x # xs))"
-      proof -
-        from h2 h3 have "f m = Max (f ` set xs)" by simp
-        with h4
-        have "f m =  Max (f ` set (x # xs))"
-          apply auto
-          apply (rule_tac max_insert_gt, simp+)
-          by (insert h1, simp+)
-        thus ?thesis by auto
-      qed
-    qed
-  } with h show ?thesis by metis
-qed
-
-end