(* Authors: Heiko Loetzbeyer, Robert Sandner, Tobias Nipkow *)
header "Denotational Semantics of Commands"
theory Denotation imports Big_Step begin
type_synonym com_den = "(state \<times> state) set"
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 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 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 D_if_big_step: "(c,s) \<Rightarrow> t \<Longrightarrow> (s,t) \<in> D(c)"
proof (induction rule: big_step_induct)
case WhileFalse
with D_While_If show ?case by auto
next
case WhileTrue
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_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 (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> 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*}
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