theory Def_Init
imports Vars Com
begin
subsection "Definite Initialization Analysis"
inductive D :: "vname set \<Rightarrow> com \<Rightarrow> vname set \<Rightarrow> bool" where
Skip: "D A SKIP A" |
Assign: "vars a \<subseteq> A \<Longrightarrow> D A (x ::= a) (insert x A)" |
Seq: "\<lbrakk> D A\<^isub>1 c\<^isub>1 A\<^isub>2; D A\<^isub>2 c\<^isub>2 A\<^isub>3 \<rbrakk> \<Longrightarrow> D A\<^isub>1 (c\<^isub>1;; c\<^isub>2) A\<^isub>3" |
If: "\<lbrakk> vars b \<subseteq> A; D A c\<^isub>1 A\<^isub>1; D A c\<^isub>2 A\<^isub>2 \<rbrakk> \<Longrightarrow>
D A (IF b THEN c\<^isub>1 ELSE c\<^isub>2) (A\<^isub>1 Int A\<^isub>2)" |
While: "\<lbrakk> vars b \<subseteq> A; D A c A' \<rbrakk> \<Longrightarrow> D A (WHILE b DO c) A"
inductive_cases [elim!]:
"D A SKIP A'"
"D A (x ::= a) A'"
"D A (c1;;c2) A'"
"D A (IF b THEN c1 ELSE c2) A'"
"D A (WHILE b DO c) A'"
lemma D_incr:
"D A c A' \<Longrightarrow> A \<subseteq> A'"
by (induct rule: D.induct) auto
end