
diff -r 12e9aa68d5db -r 4190df6f4488 prio/Lsp.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/Lsp.thy	Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,323 @@
+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




