# HG changeset patch
# User nipkow
# Date 1371569871 -7200
# Node ID 3971dd9ca831ed1223e44b8e161aedf1263eee3f
# Parent  b5b510c686cb32c46433fa2446539ef2b47e8650
Added continuity and determinism proof

diff -r b5b510c686cb -r 3971dd9ca831 src/HOL/IMP/Denotation.thy
--- a/src/HOL/IMP/Denotation.thy	Tue Jun 18 10:04:06 2013 +0200
+++ b/src/HOL/IMP/Denotation.thy	Tue Jun 18 17:37:51 2013 +0200
@@ -9,51 +9,153 @@
 definition W :: "bexp \<Rightarrow> com_den \<Rightarrow> (com_den \<Rightarrow> com_den)" where
 "W b rc = (\<lambda>rw. {(s,t). if bval b s then (s,t) \<in> rc O rw else s=t})"
 
-fun C :: "com \<Rightarrow> com_den" where
-"C SKIP   = {(s,t). s=t}" |
-"C (x ::= a) = {(s,t). t = s(x := aval a s)}" |
-"C (c0;;c1)  = C(c0) O C(c1)" |
-"C (IF b THEN c1 ELSE c2)
- = {(s,t). if bval b s then (s,t) \<in> C c1 else (s,t) \<in> C c2}" |
-"C(WHILE b DO c) = lfp (W b (C c))"
+fun D :: "com \<Rightarrow> com_den" where
+"D SKIP   = Id" |
+"D (x ::= a) = {(s,t). t = s(x := aval a s)}" |
+"D (c0;;c1)  = D(c0) O D(c1)" |
+"D (IF b THEN c1 ELSE c2)
+ = {(s,t). if bval b s then (s,t) \<in> D c1 else (s,t) \<in> D c2}" |
+"D (WHILE b DO c) = lfp (W b (D c))"
 
 lemma W_mono: "mono (W b r)"
 by (unfold W_def mono_def) auto
 
-lemma C_While_If:
-  "C(WHILE b DO c) = C(IF b THEN c;;WHILE b DO c ELSE SKIP)"
-apply simp
-apply (subst lfp_unfold [OF W_mono])  --{*lhs only*}
-apply (simp add: W_def)
-done
+lemma D_While_If:
+  "D(WHILE b DO c) = D(IF b THEN c;;WHILE b DO c ELSE SKIP)"
+proof-
+  let ?w = "WHILE b DO c"
+  have "D ?w = lfp (W b (D c))" by simp
+  also have "\<dots> = W b (D c) (lfp (W b (D c)))" by(rule lfp_unfold [OF W_mono])
+  also have "\<dots> = D(IF b THEN c;;?w ELSE SKIP)" by (simp add: W_def)
+  finally show ?thesis .
+qed
 
 text{* Equivalence of denotational and big-step semantics: *}
 
-lemma C_if_big_step:  "(c,s) \<Rightarrow> t \<Longrightarrow> (s,t) \<in> C(c)"
+lemma D_if_big_step:  "(c,s) \<Rightarrow> t \<Longrightarrow> (s,t) \<in> D(c)"
 proof (induction rule: big_step_induct)
   case WhileFalse
-  with C_While_If show ?case by auto
+  with D_While_If show ?case by auto
 next
   case WhileTrue
-  show ?case unfolding C_While_If using WhileTrue by auto
+  show ?case unfolding D_While_If using WhileTrue by auto
 qed auto
 
 abbreviation Big_step :: "com \<Rightarrow> com_den" where
 "Big_step c \<equiv> {(s,t). (c,s) \<Rightarrow> t}"
 
-lemma Big_step_if_C:  "(s,t) \<in> C(c) \<Longrightarrow> (s,t) : Big_step c"
+lemma Big_step_if_D:  "(s,t) \<in> D(c) \<Longrightarrow> (s,t) : Big_step c"
 proof (induction c arbitrary: s t)
   case Seq thus ?case by fastforce
 next
   case (While b c)
   let ?B = "Big_step (WHILE b DO c)"
-  have "W b (C c) ?B <= ?B" using While.IH by (auto simp: W_def)
-  from lfp_lowerbound[where ?f = "W b (C c)", OF this] While.prems
+  have "W b (D c) ?B <= ?B" using While.IH by (auto simp: W_def)
+  from lfp_lowerbound[where ?f = "W b (D c)", OF this] While.prems
   show ?case by auto
 qed (auto split: if_splits)
 
 theorem denotational_is_big_step:
-  "(s,t) \<in> C(c)  =  ((c,s) \<Rightarrow> t)"
-by (metis C_if_big_step Big_step_if_C[simplified])
+  "(s,t) \<in> D(c)  =  ((c,s) \<Rightarrow> t)"
+by (metis D_if_big_step Big_step_if_D[simplified])
+
+
+subsection "Continuity"
+
+definition chain :: "(nat \<Rightarrow> 'a set) \<Rightarrow> bool" where
+"chain S = (\<forall>i. S i \<subseteq> S(Suc i))"
+
+lemma chain_total: "chain S \<Longrightarrow> S i \<le> S j \<or> S j \<le> S i"
+by (metis chain_def le_cases lift_Suc_mono_le)
+
+definition cont :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool" where
+"cont f = (\<forall>S. chain S \<longrightarrow> f(UN n. S n) = (UN n. f(S n)))"
+
+lemma mono_if_cont: fixes f :: "'a set \<Rightarrow> 'a set"
+  assumes "cont f" shows "mono f"
+proof
+  fix a b :: "'a set" assume "a \<subseteq> b"
+  let ?S = "\<lambda>n::nat. if n=0 then a else b"
+  have "chain ?S" using `a \<subseteq> b` by(auto simp: chain_def)
+  hence "f(UN n. ?S n) = (UN n. f(?S n))" using assms by(simp add: cont_def)
+  moreover have "(UN n. ?S n) = b" using `a \<subseteq> b` by (auto split: if_splits)
+  moreover have "(UN n. f(?S n)) = f a \<union> f b" by (auto split: if_splits)
+  ultimately show "f a \<subseteq> f b" by (metis Un_upper1)
+qed
+
+lemma chain_iterates: fixes f :: "'a set \<Rightarrow> 'a set"
+  assumes "mono f" shows "chain(\<lambda>n. (f^^n) {})"
+proof-
+  { fix n have "(f ^^ n) {} \<subseteq> (f ^^ Suc n) {}" using assms
+    by(induction n) (auto simp: mono_def) }
+  thus ?thesis by(auto simp: chain_def)
+qed
+
+theorem lfp_if_cont:
+  assumes "cont f" shows "lfp f = (UN n. (f^^n) {})" (is "_ = ?U")
+proof
+  show "lfp f \<subseteq> ?U"
+  proof (rule lfp_lowerbound)
+    have "f ?U = (UN n. (f^^Suc n){})"
+      using chain_iterates[OF mono_if_cont[OF assms]] assms
+      by(simp add: cont_def)
+    also have "\<dots> = (f^^0){} \<union> \<dots>" by simp
+    also have "\<dots> = ?U"
+      by(auto simp del: funpow.simps) (metis not0_implies_Suc)
+    finally show "f ?U \<subseteq> ?U" by simp
+  qed
+next
+  { fix n p assume "f p \<subseteq> p"
+    have "(f^^n){} \<subseteq> p"
+    proof(induction n)
+      case 0 show ?case by simp
+    next
+      case Suc
+      from monoD[OF mono_if_cont[OF assms] Suc] `f p \<subseteq> p`
+      show ?case by simp
+    qed
+  }
+  thus "?U \<subseteq> lfp f" by(auto simp: lfp_def)
+qed
+
+lemma cont_W: "cont(W b r)"
+by(auto simp: cont_def W_def)
+
+subsection{*The denotational semantics is deterministic*}
+
+(* FIXME mv *)
+lemma simgle_valued_empty[simp]: "single_valued {}"
+by(simp add: single_valued_def)
+
+lemma single_valued_UN_chain:
+  assumes "chain S" "(\<And>n. single_valued (S n))"
+  shows "single_valued(UN n. S n)"
+proof(auto simp: single_valued_def)
+  fix m n x y z assume "(x, y) \<in> S m" "(x, z) \<in> S n"
+  with chain_total[OF assms(1), of m n] assms(2)
+  show "y = z" by (auto simp: single_valued_def)
+qed
+
+lemma single_valued_lfp: fixes f :: "com_den \<Rightarrow> com_den"
+assumes "cont f" "\<And>r. single_valued r \<Longrightarrow> single_valued (f r)"
+shows "single_valued(lfp f)"
+unfolding lfp_if_cont[OF assms(1)]
+proof(rule single_valued_UN_chain[OF chain_iterates[OF mono_if_cont[OF assms(1)]]])
+  fix n show "single_valued ((f ^^ n) {})"
+  by(induction n)(auto simp: assms(2))
+qed
+
+lemma single_valued_D: "single_valued (D c)"
+proof(induction c)
+  case Seq thus ?case by(simp add: single_valued_relcomp)
+next
+  case (While b c)
+  have "single_valued (lfp (W b (D c)))"
+  proof(rule single_valued_lfp[OF cont_W])
+    show "!!r. single_valued r \<Longrightarrow> single_valued (W b (D c) r)"
+      using While.IH by(force simp: single_valued_def W_def)
+  qed
+  thus ?case by simp
+qed (auto simp add: single_valued_def)
 
 end