(* Author: Tobias Nipkow *)
subsection "Collecting Semantics of Commands"
theory Collecting
imports Complete_Lattice Big_Step ACom
begin
subsubsection "The generic Step function"
notation
sup (infixl "\<squnion>" 65) and
inf (infixl "\<sqinter>" 70) and
bot ("\<bottom>") and
top ("\<top>")
context
fixes f :: "vname \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a::sup"
fixes g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"
begin
fun Step :: "'a \<Rightarrow> 'a acom \<Rightarrow> 'a acom" where
"Step S (SKIP {Q}) = (SKIP {S})" |
"Step S (x ::= e {Q}) =
x ::= e {f x e S}" |
"Step S (C1;; C2) = Step S C1;; Step (post C1) C2" |
"Step S (IF b THEN {P1} C1 ELSE {P2} C2 {Q}) =
IF b THEN {g b S} Step P1 C1 ELSE {g (Not b) S} Step P2 C2
{post C1 \<squnion> post C2}" |
"Step S ({I} WHILE b DO {P} C {Q}) =
{S \<squnion> post C} WHILE b DO {g b I} Step P C {g (Not b) I}"
end
lemma strip_Step[simp]: "strip(Step f g S C) = strip C"
by(induct C arbitrary: S) auto
subsubsection "Annotated commands as a complete lattice"
instantiation acom :: (order) order
begin
definition less_eq_acom :: "('a::order)acom \<Rightarrow> 'a acom \<Rightarrow> bool" where
"C1 \<le> C2 \<longleftrightarrow> strip C1 = strip C2 \<and> (\<forall>p<size(annos C1). anno C1 p \<le> anno C2 p)"
definition less_acom :: "'a acom \<Rightarrow> 'a acom \<Rightarrow> bool" where
"less_acom x y = (x \<le> y \<and> \<not> y \<le> x)"
instance
proof (standard, goal_cases)
case 1 show ?case by(simp add: less_acom_def)
next
case 2 thus ?case by(auto simp: less_eq_acom_def)
next
case 3 thus ?case by(fastforce simp: less_eq_acom_def size_annos)
next
case 4 thus ?case
by(fastforce simp: le_antisym less_eq_acom_def size_annos
eq_acom_iff_strip_anno)
qed
end
lemma less_eq_acom_annos:
"C1 \<le> C2 \<longleftrightarrow> strip C1 = strip C2 \<and> list_all2 (\<le>) (annos C1) (annos C2)"
by(auto simp add: less_eq_acom_def anno_def list_all2_conv_all_nth size_annos_same2)
lemma SKIP_le[simp]: "SKIP {S} \<le> c \<longleftrightarrow> (\<exists>S'. c = SKIP {S'} \<and> S \<le> S')"
by (cases c) (auto simp:less_eq_acom_def anno_def)
lemma Assign_le[simp]: "x ::= e {S} \<le> c \<longleftrightarrow> (\<exists>S'. c = x ::= e {S'} \<and> S \<le> S')"
by (cases c) (auto simp:less_eq_acom_def anno_def)
lemma Seq_le[simp]: "C1;;C2 \<le> C \<longleftrightarrow> (\<exists>C1' C2'. C = C1';;C2' \<and> C1 \<le> C1' \<and> C2 \<le> C2')"
apply (cases C)
apply(auto simp: less_eq_acom_annos list_all2_append size_annos_same2)
done
lemma If_le[simp]: "IF b THEN {p1} C1 ELSE {p2} C2 {S} \<le> C \<longleftrightarrow>
(\<exists>p1' p2' C1' C2' S'. C = IF b THEN {p1'} C1' ELSE {p2'} C2' {S'} \<and>
p1 \<le> p1' \<and> p2 \<le> p2' \<and> C1 \<le> C1' \<and> C2 \<le> C2' \<and> S \<le> S')"
apply (cases C)
apply(auto simp: less_eq_acom_annos list_all2_append size_annos_same2)
done
lemma While_le[simp]: "{I} WHILE b DO {p} C {P} \<le> W \<longleftrightarrow>
(\<exists>I' p' C' P'. W = {I'} WHILE b DO {p'} C' {P'} \<and> C \<le> C' \<and> p \<le> p' \<and> I \<le> I' \<and> P \<le> P')"
apply (cases W)
apply(auto simp: less_eq_acom_annos list_all2_append size_annos_same2)
done
lemma mono_post: "C \<le> C' \<Longrightarrow> post C \<le> post C'"
using annos_ne[of C']
by(auto simp: post_def less_eq_acom_def last_conv_nth[OF annos_ne] anno_def
dest: size_annos_same)
definition Inf_acom :: "com \<Rightarrow> 'a::complete_lattice acom set \<Rightarrow> 'a acom" where
"Inf_acom c M = annotate (\<lambda>p. INF C\<in>M. anno C p) c"
global_interpretation
Complete_Lattice "{C. strip C = c}" "Inf_acom c" for c
proof (standard, goal_cases)
case 1 thus ?case
by(auto simp: Inf_acom_def less_eq_acom_def size_annos intro:INF_lower)
next
case 2 thus ?case
by(auto simp: Inf_acom_def less_eq_acom_def size_annos intro:INF_greatest)
next
case 3 thus ?case by(auto simp: Inf_acom_def)
qed
subsubsection "Collecting semantics"
definition "step = Step (\<lambda>x e S. {s(x := aval e s) |s. s \<in> S}) (\<lambda>b S. {s:S. bval b s})"
definition CS :: "com \<Rightarrow> state set acom" where
"CS c = lfp c (step UNIV)"
lemma mono2_Step: fixes C1 C2 :: "'a::semilattice_sup acom"
assumes "!!x e S1 S2. S1 \<le> S2 \<Longrightarrow> f x e S1 \<le> f x e S2"
"!!b S1 S2. S1 \<le> S2 \<Longrightarrow> g b S1 \<le> g b S2"
shows "C1 \<le> C2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> Step f g S1 C1 \<le> Step f g S2 C2"
proof(induction S1 C1 arbitrary: C2 S2 rule: Step.induct)
case 1 thus ?case by(auto)
next
case 2 thus ?case by (auto simp: assms(1))
next
case 3 thus ?case by(auto simp: mono_post)
next
case 4 thus ?case
by(auto simp: subset_iff assms(2))
(metis mono_post le_supI1 le_supI2)+
next
case 5 thus ?case
by(auto simp: subset_iff assms(2))
(metis mono_post le_supI1 le_supI2)+
qed
lemma mono2_step: "C1 \<le> C2 \<Longrightarrow> S1 \<subseteq> S2 \<Longrightarrow> step S1 C1 \<le> step S2 C2"
unfolding step_def by(rule mono2_Step) auto
lemma mono_step: "mono (step S)"
by(blast intro: monoI mono2_step)
lemma strip_step: "strip(step S C) = strip C"
by (induction C arbitrary: S) (auto simp: step_def)
lemma lfp_cs_unfold: "lfp c (step S) = step S (lfp c (step S))"
apply(rule lfp_unfold[OF _ mono_step])
apply(simp add: strip_step)
done
lemma CS_unfold: "CS c = step UNIV (CS c)"
by (metis CS_def lfp_cs_unfold)
lemma strip_CS[simp]: "strip(CS c) = c"
by(simp add: CS_def index_lfp[simplified])
subsubsection "Relation to big-step semantics"
lemma asize_nz: "asize(c::com) \<noteq> 0"
by (metis length_0_conv length_annos_annotate annos_ne)
lemma post_Inf_acom:
"\<forall>C\<in>M. strip C = c \<Longrightarrow> post (Inf_acom c M) = \<Inter>(post ` M)"
apply(subgoal_tac "\<forall>C\<in>M. size(annos C) = asize c")
apply(simp add: post_anno_asize Inf_acom_def asize_nz neq0_conv[symmetric])
apply(simp add: size_annos)
done
lemma post_lfp: "post(lfp c f) = (\<Inter>{post C|C. strip C = c \<and> f C \<le> C})"
by(auto simp add: lfp_def post_Inf_acom)
lemma big_step_post_step:
"\<lbrakk> (c, s) \<Rightarrow> t; strip C = c; s \<in> S; step S C \<le> C \<rbrakk> \<Longrightarrow> t \<in> post C"
proof(induction arbitrary: C S rule: big_step_induct)
case Skip thus ?case by(auto simp: strip_eq_SKIP step_def post_def)
next
case Assign thus ?case
by(fastforce simp: strip_eq_Assign step_def post_def)
next
case Seq thus ?case
by(fastforce simp: strip_eq_Seq step_def post_def last_append annos_ne)
next
case IfTrue thus ?case apply(auto simp: strip_eq_If step_def post_def)
by (metis (lifting,full_types) mem_Collect_eq subsetD)
next
case IfFalse thus ?case apply(auto simp: strip_eq_If step_def post_def)
by (metis (lifting,full_types) mem_Collect_eq subsetD)
next
case (WhileTrue b s1 c' s2 s3)
from WhileTrue.prems(1) obtain I P C' Q where "C = {I} WHILE b DO {P} C' {Q}" "strip C' = c'"
by(auto simp: strip_eq_While)
from WhileTrue.prems(3) \<open>C = _\<close>
have "step P C' \<le> C'" "{s \<in> I. bval b s} \<le> P" "S \<le> I" "step (post C') C \<le> C"
by (auto simp: step_def post_def)
have "step {s \<in> I. bval b s} C' \<le> C'"
by (rule order_trans[OF mono2_step[OF order_refl \<open>{s \<in> I. bval b s} \<le> P\<close>] \<open>step P C' \<le> C'\<close>])
have "s1 \<in> {s\<in>I. bval b s}" using \<open>s1 \<in> S\<close> \<open>S \<subseteq> I\<close> \<open>bval b s1\<close> by auto
note s2_in_post_C' = WhileTrue.IH(1)[OF \<open>strip C' = c'\<close> this \<open>step {s \<in> I. bval b s} C' \<le> C'\<close>]
from WhileTrue.IH(2)[OF WhileTrue.prems(1) s2_in_post_C' \<open>step (post C') C \<le> C\<close>]
show ?case .
next
case (WhileFalse b s1 c') thus ?case
by (force simp: strip_eq_While step_def post_def)
qed
lemma big_step_lfp: "\<lbrakk> (c,s) \<Rightarrow> t; s \<in> S \<rbrakk> \<Longrightarrow> t \<in> post(lfp c (step S))"
by(auto simp add: post_lfp intro: big_step_post_step)
lemma big_step_CS: "(c,s) \<Rightarrow> t \<Longrightarrow> t \<in> post(CS c)"
by(simp add: CS_def big_step_lfp)
end