--- a/red_1.thy Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,359 +0,0 @@
-section {*
- This file contains lemmas used to guide the recalculation of current precedence
- after every system call (or system operation)
-*}
-theory CpsG
-imports PrioG Max RTree
-begin
-
-
-definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
-
-definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
-
-definition "tRAG s = wRAG s O hRAG s"
-
-definition "pairself f = (\<lambda>(a, b). (f a, f b))"
-
-definition "rel_map f r = (pairself f ` r)"
-
-fun the_thread :: "node \<Rightarrow> thread" where
- "the_thread (Th th) = th"
-
-definition "tG s = rel_map the_thread (tRAG s)"
-
-locale pip =
- fixes s
- assumes vt: "vt s"
-
-
-lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
- by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
- s_holding_abv cs_RAG_def, auto)
-
-lemma relpow_mult:
- "((r::'a rel) ^^ m) ^^ n = r ^^ (m*n)"
-proof(induct n arbitrary:m)
- case (Suc k m)
- thus ?case (is "?L = ?R")
- proof -
- have h: "(m * k + m) = (m + m * k)" by auto
- show ?thesis
- apply (simp add:Suc relpow_add[symmetric])
- by (unfold h, simp)
- qed
-qed simp
-
-lemma compose_relpow_2:
- assumes "r1 \<subseteq> r"
- and "r2 \<subseteq> r"
- shows "r1 O r2 \<subseteq> r ^^ (2::nat)"
-proof -
- { fix a b
- assume "(a, b) \<in> r1 O r2"
- then obtain e where "(a, e) \<in> r1" "(e, b) \<in> r2"
- by auto
- with assms have "(a, e) \<in> r" "(e, b) \<in> r" by auto
- hence "(a, b) \<in> r ^^ (Suc (Suc 0))" by auto
- } thus ?thesis by (auto simp:numeral_2_eq_2)
-qed
-
-
-lemma acyclic_compose:
- assumes "acyclic r"
- and "r1 \<subseteq> r"
- and "r2 \<subseteq> r"
- shows "acyclic (r1 O r2)"
-proof -
- { fix a
- assume "(a, a) \<in> (r1 O r2)^+"
- from trancl_mono[OF this compose_relpow_2[OF assms(2, 3)]]
- have "(a, a) \<in> (r ^^ 2) ^+" .
- from trancl_power[THEN iffD1, OF this]
- obtain n where h: "(a, a) \<in> (r ^^ 2) ^^ n" "n > 0" by blast
- from this(1)[unfolded relpow_mult] have h2: "(a, a) \<in> r ^^ (2 * n)" .
- have "(a, a) \<in> r^+"
- proof(cases rule:trancl_power[THEN iffD2])
- from h(2) h2 show "\<exists>n>0. (a, a) \<in> r ^^ n"
- by (rule_tac x = "2*n" in exI, auto)
- qed
- with assms have "False" by (auto simp:acyclic_def)
- } thus ?thesis by (auto simp:acyclic_def)
-qed
-
-lemma range_tRAG: "Range (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
- have "Range (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
- proof -
- have "?L \<subseteq> Range (hRAG s)" by auto
- also have "... \<subseteq> ?R"
- by (unfold hRAG_def, auto)
- finally show ?thesis by auto
- qed
- thus ?thesis by (simp add:tRAG_def)
-qed
-
-lemma domain_tRAG: "Domain (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
- have "Domain (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
- proof -
- have "?L \<subseteq> Domain (wRAG s)" by auto
- also have "... \<subseteq> ?R"
- by (unfold wRAG_def, auto)
- finally show ?thesis by auto
- qed
- thus ?thesis by (simp add:tRAG_def)
-qed
-
-lemma rel_mapE:
- assumes "(a, b) \<in> rel_map f r"
- obtains c d
- where "(c, d) \<in> r" "(a, b) = (f c, f d)"
- using assms
- by (unfold rel_map_def pairself_def, auto)
-
-lemma rel_mapI:
- assumes "(a, b) \<in> r"
- and "c = f a"
- and "d = f b"
- shows "(c, d) \<in> rel_map f r"
- using assms
- by (unfold rel_map_def pairself_def, auto)
-
-lemma map_appendE:
- assumes "map f zs = xs @ ys"
- obtains xs' ys'
- where "zs = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
-proof -
- have "\<exists> xs' ys'. zs = xs' @ ys' \<and> xs = map f xs' \<and> ys = map f ys'"
- using assms
- proof(induct xs arbitrary:zs ys)
- case (Nil zs ys)
- thus ?case by auto
- next
- case (Cons x xs zs ys)
- note h = this
- show ?case
- proof(cases zs)
- case (Cons e es)
- with h have eq_x: "map f es = xs @ ys" "x = f e" by auto
- from h(1)[OF this(1)]
- obtain xs' ys' where "es = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
- by blast
- with Cons eq_x
- have "zs = (e#xs') @ ys' \<and> x # xs = map f (e#xs') \<and> ys = map f ys'" by auto
- thus ?thesis by metis
- qed (insert h, auto)
- qed
- thus ?thesis by (auto intro!:that)
-qed
-
-lemma rel_map_mono:
- assumes "r1 \<subseteq> r2"
- shows "rel_map f r1 \<subseteq> rel_map f r2"
- using assms
- by (auto simp:rel_map_def pairself_def)
-
-lemma rel_map_compose [simp]:
- shows "rel_map f1 (rel_map f2 r) = rel_map (f1 o f2) r"
- by (auto simp:rel_map_def pairself_def)
-
-lemma edges_on_map: "edges_on (map f xs) = rel_map f (edges_on xs)"
-proof -
- { fix a b
- assume "(a, b) \<in> edges_on (map f xs)"
- then obtain l1 l2 where eq_map: "map f xs = l1 @ [a, b] @ l2"
- by (unfold edges_on_def, auto)
- hence "(a, b) \<in> rel_map f (edges_on xs)"
- by (auto elim!:map_appendE intro!:rel_mapI simp:edges_on_def)
- } moreover {
- fix a b
- assume "(a, b) \<in> rel_map f (edges_on xs)"
- then obtain c d where
- h: "(c, d) \<in> edges_on xs" "(a, b) = (f c, f d)"
- by (elim rel_mapE, auto)
- then obtain l1 l2 where
- eq_xs: "xs = l1 @ [c, d] @ l2"
- by (auto simp:edges_on_def)
- hence eq_map: "map f xs = map f l1 @ [f c, f d] @ map f l2" by auto
- have "(a, b) \<in> edges_on (map f xs)"
- proof -
- from h(2) have "[f c, f d] = [a, b]" by simp
- from eq_map[unfolded this] show ?thesis by (auto simp:edges_on_def)
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma plus_rpath:
- assumes "(a, b) \<in> r^+"
- obtains xs where "rpath r a xs b" "xs \<noteq> []"
-proof -
- from assms obtain m where h: "(a, m) \<in> r" "(m, b) \<in> r^*"
- by (auto dest!:tranclD)
- from star_rpath[OF this(2)] obtain xs where "rpath r m xs b" by auto
- from rstepI[OF h(1) this] have "rpath r a (m # xs) b" .
- from that[OF this] show ?thesis by auto
-qed
-
-lemma edges_on_unfold:
- "edges_on (a # b # xs) = {(a, b)} \<union> edges_on (b # xs)" (is "?L = ?R")
-proof -
- { fix c d
- assume "(c, d) \<in> ?L"
- then obtain l1 l2 where h: "(a # b # xs) = l1 @ [c, d] @ l2"
- by (auto simp:edges_on_def)
- have "(c, d) \<in> ?R"
- proof(cases "l1")
- case Nil
- with h have "(c, d) = (a, b)" by auto
- thus ?thesis by auto
- next
- case (Cons e es)
- from h[unfolded this] have "b#xs = es@[c, d]@l2" by auto
- thus ?thesis by (auto simp:edges_on_def)
- qed
- } moreover
- { fix c d
- assume "(c, d) \<in> ?R"
- moreover have "(a, b) \<in> ?L"
- proof -
- have "(a # b # xs) = []@[a,b]@xs" by simp
- hence "\<exists> l1 l2. (a # b # xs) = l1@[a,b]@l2" by auto
- thus ?thesis by (unfold edges_on_def, simp)
- qed
- moreover {
- assume "(c, d) \<in> edges_on (b#xs)"
- then obtain l1 l2 where "b#xs = l1@[c, d]@l2" by (unfold edges_on_def, auto)
- hence "a#b#xs = (a#l1)@[c,d]@l2" by simp
- hence "\<exists> l1 l2. (a # b # xs) = l1@[c,d]@l2" by metis
- hence "(c,d) \<in> ?L" by (unfold edges_on_def, simp)
- }
- ultimately have "(c, d) \<in> ?L" by auto
- } ultimately show ?thesis by auto
-qed
-
-lemma edges_on_rpathI:
- assumes "edges_on (a#xs@[b]) \<subseteq> r"
- shows "rpath r a (xs@[b]) b"
- using assms
-proof(induct xs arbitrary: a b)
- case Nil
- moreover have "(a, b) \<in> edges_on (a # [] @ [b])"
- by (unfold edges_on_def, auto)
- ultimately have "(a, b) \<in> r" by auto
- thus ?case by auto
-next
- case (Cons x xs a b)
- from this(2) have "edges_on (x # xs @ [b]) \<subseteq> r" by (simp add:edges_on_unfold)
- from Cons(1)[OF this] have " rpath r x (xs @ [b]) b" .
- moreover from Cons(2) have "(a, x) \<in> r" by (auto simp:edges_on_unfold)
- ultimately show ?case by (auto intro!:rstepI)
-qed
-
-lemma image_id:
- assumes "\<And> x. x \<in> A \<Longrightarrow> f x = x"
- shows "f ` A = A"
- using assms by (auto simp:image_def)
-
-lemma rel_map_inv_id:
- assumes "inj_on f ((Domain r) \<union> (Range r))"
- shows "(rel_map (inv_into ((Domain r) \<union> (Range r)) f \<circ> f) r) = r"
-proof -
- let ?f = "(inv_into (Domain r \<union> Range r) f \<circ> f)"
- {
- fix a b
- assume h0: "(a, b) \<in> r"
- have "pairself ?f (a, b) = (a, b)"
- proof -
- from assms h0 have "?f a = a" by (auto intro:inv_into_f_f)
- moreover have "?f b = b"
- by (insert h0, simp, intro inv_into_f_f[OF assms], auto intro!:RangeI)
- ultimately show ?thesis by (auto simp:pairself_def)
- qed
- } thus ?thesis by (unfold rel_map_def, intro image_id, case_tac x, auto)
-qed
-
-lemma rel_map_acyclic:
- assumes "acyclic r"
- and "inj_on f ((Domain r) \<union> (Range r))"
- shows "acyclic (rel_map f r)"
-proof -
- let ?D = "Domain r \<union> Range r"
- { fix a
- assume "(a, a) \<in> (rel_map f r)^+"
- from plus_rpath[OF this]
- obtain xs where rp: "rpath (rel_map f r) a xs a" "xs \<noteq> []" by auto
- from rpath_nnl_lastE[OF this] obtain xs' where eq_xs: "xs = xs'@[a]" by auto
- from rpath_edges_on[OF rp(1)]
- have h: "edges_on (a # xs) \<subseteq> rel_map f r" .
- from edges_on_map[of "inv_into ?D f" "a#xs"]
- have "edges_on (map (inv_into ?D f) (a # xs)) = rel_map (inv_into ?D f) (edges_on (a # xs))" .
- with rel_map_mono[OF h, of "inv_into ?D f"]
- have "edges_on (map (inv_into ?D f) (a # xs)) \<subseteq> rel_map ((inv_into ?D f) o f) r" by simp
- from this[unfolded eq_xs]
- have subr: "edges_on (map (inv_into ?D f) (a # xs' @ [a])) \<subseteq> rel_map (inv_into ?D f \<circ> f) r" .
- have "(map (inv_into ?D f) (a # xs' @ [a])) = (inv_into ?D f a) # map (inv_into ?D f) xs' @ [inv_into ?D f a]"
- by simp
- from edges_on_rpathI[OF subr[unfolded this]]
- have "rpath (rel_map (inv_into ?D f \<circ> f) r)
- (inv_into ?D f a) (map (inv_into ?D f) xs' @ [inv_into ?D f a]) (inv_into ?D f a)" .
- hence "(inv_into ?D f a, inv_into ?D f a) \<in> (rel_map (inv_into ?D f \<circ> f) r)^+"
- by (rule rpath_plus, simp)
- moreover have "(rel_map (inv_into ?D f \<circ> f) r) = r" by (rule rel_map_inv_id[OF assms(2)])
- moreover note assms(1)
- ultimately have False by (unfold acyclic_def, auto)
- } thus ?thesis by (auto simp:acyclic_def)
-qed
-
-context pip
-begin
-
-interpretation rtree_RAG: rtree "RAG s"
-proof
- show "single_valued (RAG s)"
- by (unfold single_valued_def, auto intro: unique_RAG[OF vt])
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG[OF vt])
-qed
-
-lemma sgv_wRAG:
- shows "single_valued (wRAG s)"
- using waiting_unique[OF vt]
- by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG:
- shows "single_valued (hRAG s)"
- using held_unique
- by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: shows "single_valued (tRAG s)"
- by (unfold tRAG_def, rule Relation.single_valued_relcomp,
- insert sgv_hRAG sgv_wRAG, auto)
-
-lemma acyclic_hRAG:
- shows "acyclic (hRAG s)"
- by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_wRAG:
- shows "acyclic (wRAG s)"
- by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_tRAG:
- shows "acyclic (tRAG s)"
- by (unfold tRAG_def, rule acyclic_compose[OF acyclic_RAG[OF vt]],
- unfold RAG_split, auto)
-
-lemma acyclic_tG:
- shows "acyclic (tG s)"
-proof(unfold tG_def, rule rel_map_acyclic[OF acyclic_tRAG])
- show "inj_on the_thread (Domain (tRAG s) \<union> Range (tRAG s))"
- proof(rule subset_inj_on)
- show " inj_on the_thread {Th th |th. True}" by (unfold inj_on_def, auto)
- next
- from domain_tRAG range_tRAG
- show " Domain (tRAG s) \<union> Range (tRAG s) \<subseteq> {Th th |th. True}" by auto
- qed
-qed
-
-end