--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IMP/Live_True.thy Sun Dec 11 18:22:06 2011 +0100
@@ -0,0 +1,219 @@
+(* Author: Tobias Nipkow *)
+
+theory Live_True
+imports "~~/src/HOL/Library/While_Combinator" Vars Big_Step
+begin
+
+subsection "True Liveness Analysis"
+
+fun L :: "com \<Rightarrow> vname set \<Rightarrow> vname set" where
+"L SKIP X = X" |
+"L (x ::= a) X = (if x:X then X-{x} \<union> vars a else X)" |
+"L (c\<^isub>1; c\<^isub>2) X = (L c\<^isub>1 \<circ> L c\<^isub>2) X" |
+"L (IF b THEN c\<^isub>1 ELSE c\<^isub>2) X = vars b \<union> L c\<^isub>1 X \<union> L c\<^isub>2 X" |
+"L (WHILE b DO c) X = lfp(%Y. vars b \<union> X \<union> L c Y)"
+
+lemma L_mono: "mono (L c)"
+proof-
+ { fix X Y have "X \<subseteq> Y \<Longrightarrow> L c X \<subseteq> L c Y"
+ proof(induction c arbitrary: X Y)
+ case (While b c)
+ show ?case
+ proof(simp, rule lfp_mono)
+ fix Z show "vars b \<union> X \<union> L c Z \<subseteq> vars b \<union> Y \<union> L c Z"
+ using While by auto
+ qed
+ next
+ case If thus ?case by(auto simp: subset_iff)
+ qed auto
+ } thus ?thesis by(rule monoI)
+qed
+
+lemma mono_union_L:
+ "mono (%Y. X \<union> L c Y)"
+by (metis (no_types) L_mono mono_def order_eq_iff set_eq_subset sup_mono)
+
+lemma L_While_unfold:
+ "L (WHILE b DO c) X = vars b \<union> X \<union> L c (L (WHILE b DO c) X)"
+by(metis lfp_unfold[OF mono_union_L] L.simps(5))
+
+
+subsection "Soundness"
+
+theorem L_sound:
+ "(c,s) \<Rightarrow> s' \<Longrightarrow> s = t on L c X \<Longrightarrow>
+ \<exists> t'. (c,t) \<Rightarrow> t' & s' = t' on X"
+proof (induction arbitrary: X t rule: big_step_induct)
+ case Skip then show ?case by auto
+next
+ case Assign then show ?case
+ by (auto simp: ball_Un)
+next
+ case (Semi c1 s1 s2 c2 s3 X t1)
+ from Semi.IH(1) Semi.prems obtain t2 where
+ t12: "(c1, t1) \<Rightarrow> t2" and s2t2: "s2 = t2 on L c2 X"
+ by simp blast
+ from Semi.IH(2)[OF s2t2] obtain t3 where
+ t23: "(c2, t2) \<Rightarrow> t3" and s3t3: "s3 = t3 on X"
+ by auto
+ show ?case using t12 t23 s3t3 by auto
+next
+ case (IfTrue b s c1 s' c2)
+ hence "s = t on vars b" "s = t on L c1 X" by auto
+ from bval_eq_if_eq_on_vars[OF this(1)] IfTrue(1) have "bval b t" by simp
+ from IfTrue(3)[OF `s = t on L c1 X`] obtain t' where
+ "(c1, t) \<Rightarrow> t'" "s' = t' on X" by auto
+ thus ?case using `bval b t` by auto
+next
+ case (IfFalse b s c2 s' c1)
+ hence "s = t on vars b" "s = t on L c2 X" by auto
+ from bval_eq_if_eq_on_vars[OF this(1)] IfFalse(1) have "~bval b t" by simp
+ from IfFalse(3)[OF `s = t on L c2 X`] obtain t' where
+ "(c2, t) \<Rightarrow> t'" "s' = t' on X" by auto
+ thus ?case using `~bval b t` by auto
+next
+ case (WhileFalse b s c)
+ hence "~ bval b t"
+ by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars)
+ thus ?case using WhileFalse.prems L_While_unfold[of b c X] by auto
+next
+ case (WhileTrue b s1 c s2 s3 X t1)
+ let ?w = "WHILE b DO c"
+ from `bval b s1` WhileTrue.prems have "bval b t1"
+ by (metis L_While_unfold UnI1 bval_eq_if_eq_on_vars)
+ have "s1 = t1 on L c (L ?w X)" using L_While_unfold WhileTrue.prems
+ by (blast)
+ from WhileTrue.IH(1)[OF this] obtain t2 where
+ "(c, t1) \<Rightarrow> t2" "s2 = t2 on L ?w X" by auto
+ from WhileTrue.IH(2)[OF this(2)] obtain t3 where "(?w,t2) \<Rightarrow> t3" "s3 = t3 on X"
+ by auto
+ with `bval b t1` `(c, t1) \<Rightarrow> t2` show ?case by auto
+qed
+
+
+instantiation com :: vars
+begin
+
+fun vars_com :: "com \<Rightarrow> vname set" where
+"vars SKIP = {}" |
+"vars (x::=e) = vars e" |
+"vars (c\<^isub>1; c\<^isub>2) = vars c\<^isub>1 \<union> vars c\<^isub>2" |
+"vars (IF b THEN c\<^isub>1 ELSE c\<^isub>2) = vars b \<union> vars c\<^isub>1 \<union> vars c\<^isub>2" |
+"vars (WHILE b DO c) = vars b \<union> vars c"
+
+instance ..
+
+end
+
+lemma L_subset_vars: "L c X \<subseteq> vars c \<union> X"
+proof(induction c arbitrary: X)
+ case (While b c)
+ have "lfp(%Y. vars b \<union> X \<union> L c Y) \<subseteq> vars b \<union> vars c \<union> X"
+ using While.IH[of "vars b \<union> vars c \<union> X"]
+ by (auto intro!: lfp_lowerbound)
+ thus ?case by simp
+qed auto
+
+lemma afinite[simp]: "finite(vars(a::aexp))"
+by (induction a) auto
+
+lemma bfinite[simp]: "finite(vars(b::bexp))"
+by (induction b) auto
+
+lemma cfinite[simp]: "finite(vars(c::com))"
+by (induction c) auto
+
+(* move to Inductive; call Kleene? *)
+lemma lfp_finite_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
+shows "lfp f = (f^^k) bot"
+proof(rule antisym)
+ show "lfp f \<le> (f^^k) bot"
+ proof(rule lfp_lowerbound)
+ show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
+ qed
+next
+ show "(f^^k) bot \<le> lfp f"
+ proof(induction k)
+ case 0 show ?case by simp
+ next
+ case Suc
+ from monoD[OF assms(1) Suc] lfp_unfold[OF assms(1)]
+ show ?case by simp
+ qed
+qed
+
+(* move to While_Combinator *)
+lemma while_option_stop2:
+ "while_option b c s = Some t \<Longrightarrow> EX k. t = (c^^k) s \<and> \<not> b t"
+apply(simp add: while_option_def split: if_splits)
+by (metis (lam_lifting) LeastI_ex)
+(* move to While_Combinator *)
+lemma while_option_finite_subset_Some: fixes C :: "'a set"
+ assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
+ shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
+proof(rule measure_while_option_Some[where
+ f= "%A::'a set. card C - card A" and P= "%A. A \<subseteq> C \<and> A \<subseteq> f A" and s= "{}"])
+ fix A assume A: "A \<subseteq> C \<and> A \<subseteq> f A" "f A \<noteq> A"
+ show "(f A \<subseteq> C \<and> f A \<subseteq> f (f A)) \<and> card C - card (f A) < card C - card A"
+ (is "?L \<and> ?R")
+ proof
+ show ?L by(metis A(1) assms(2) monoD[OF `mono f`])
+ show ?R by (metis A assms(2,3) card_seteq diff_less_mono2 equalityI linorder_le_less_linear rev_finite_subset)
+ qed
+qed simp
+(* move to While_Combinator *)
+lemma lfp_eq_while_option:
+ assumes "mono f" and "!!X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C" and "finite C"
+ shows "lfp f = the(while_option (\<lambda>A. f A \<noteq> A) f {})"
+proof-
+ obtain P where "while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
+ using while_option_finite_subset_Some[OF assms] by blast
+ with while_option_stop2[OF this] lfp_finite_iter[OF assms(1)]
+ show ?thesis by auto
+qed
+
+text{* For code generation: *}
+lemma L_While: fixes b c X
+assumes "finite X" defines "f == \<lambda>A. vars b \<union> X \<union> L c A"
+shows "L (WHILE b DO c) X = the(while_option (\<lambda>A. f A \<noteq> A) f {})" (is "_ = ?r")
+proof -
+ let ?V = "vars b \<union> vars c \<union> X"
+ have "lfp f = ?r"
+ proof(rule lfp_eq_while_option[where C = "?V"])
+ show "mono f" by(simp add: f_def mono_union_L)
+ next
+ fix Y show "Y \<subseteq> ?V \<Longrightarrow> f Y \<subseteq> ?V"
+ unfolding f_def using L_subset_vars[of c] by blast
+ next
+ show "finite ?V" using `finite X` by simp
+ qed
+ thus ?thesis by (simp add: f_def)
+qed
+
+text{* An approximate computation of the WHILE-case: *}
+
+fun iter :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"
+where
+"iter f 0 p d = d" |
+"iter f (Suc n) p d = (if f p = p then p else iter f n (f p) d)"
+
+lemma iter_pfp:
+ "f d \<le> d \<Longrightarrow> mono f \<Longrightarrow> x \<le> f x \<Longrightarrow> f(iter f i x d) \<le> iter f i x d"
+apply(induction i arbitrary: x)
+ apply simp
+apply (simp add: mono_def)
+done
+
+lemma iter_While_pfp:
+fixes b c X W k f
+defines "f == \<lambda>A. vars b \<union> X \<union> L c A" and "W == vars b \<union> vars c \<union> X"
+and "P == iter f k {} W"
+shows "f P \<subseteq> P"
+proof-
+ have "f W \<subseteq> W" unfolding f_def W_def using L_subset_vars[of c] by blast
+ have "mono f" by(simp add: f_def mono_union_L)
+ from iter_pfp[of f, OF `f W \<subseteq> W` `mono f` empty_subsetI]
+ show ?thesis by(simp add: P_def)
+qed
+
+end