--- a/prio/Lsp.thy Sun Feb 05 14:29:08 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