theory Procs_Stat_Vars_Stat imports Util Procs
begin
subsubsection "Static Scoping of Procedures and Variables"
type_synonym addr = nat
type_synonym venv = "name \<Rightarrow> addr"
type_synonym store = "addr \<Rightarrow> val"
type_synonym penv = "(name \<times> com \<times> venv) list"
fun venv :: "penv \<times> venv \<times> nat \<Rightarrow> venv" where
"venv(_,ve,_) = ve"
inductive
big_step :: "penv \<times> venv \<times> nat \<Rightarrow> com \<times> store \<Rightarrow> store \<Rightarrow> bool"
("_ \<turnstile> _ \<Rightarrow> _" [60,0,60] 55)
where
Skip: "e \<turnstile> (SKIP,s) \<Rightarrow> s" |
Assign: "(pe,ve,f) \<turnstile> (x ::= a,s) \<Rightarrow> s(ve x := aval a (s o ve))" |
Semi: "\<lbrakk> e \<turnstile> (c\<^isub>1,s\<^isub>1) \<Rightarrow> s\<^isub>2; e \<turnstile> (c\<^isub>2,s\<^isub>2) \<Rightarrow> s\<^isub>3 \<rbrakk> \<Longrightarrow>
e \<turnstile> (c\<^isub>1;c\<^isub>2, s\<^isub>1) \<Rightarrow> s\<^isub>3" |
IfTrue: "\<lbrakk> bval b (s \<circ> venv e); e \<turnstile> (c\<^isub>1,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow>
e \<turnstile> (IF b THEN c\<^isub>1 ELSE c\<^isub>2, s) \<Rightarrow> t" |
IfFalse: "\<lbrakk> \<not>bval b (s \<circ> venv e); e \<turnstile> (c\<^isub>2,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow>
e \<turnstile> (IF b THEN c\<^isub>1 ELSE c\<^isub>2, s) \<Rightarrow> t" |
WhileFalse: "\<not>bval b (s \<circ> venv e) \<Longrightarrow> e \<turnstile> (WHILE b DO c,s) \<Rightarrow> s" |
WhileTrue:
"\<lbrakk> bval b (s\<^isub>1 \<circ> venv e); e \<turnstile> (c,s\<^isub>1) \<Rightarrow> s\<^isub>2;
e \<turnstile> (WHILE b DO c, s\<^isub>2) \<Rightarrow> s\<^isub>3 \<rbrakk> \<Longrightarrow>
e \<turnstile> (WHILE b DO c, s\<^isub>1) \<Rightarrow> s\<^isub>3" |
Var: "(pe,ve(x:=f),f+1) \<turnstile> (c,s) \<Rightarrow> t \<Longrightarrow>
(pe,ve,f) \<turnstile> ({VAR x;; c}, s) \<Rightarrow> t(f := s f)" |
Call1: "((p,c,ve)#pe,ve,f) \<turnstile> (c, s) \<Rightarrow> t \<Longrightarrow>
((p,c,ve)#pe,ve',f) \<turnstile> (CALL p, s) \<Rightarrow> t" |
Call2: "\<lbrakk> p' \<noteq> p; (pe,ve,f) \<turnstile> (CALL p, s) \<Rightarrow> t \<rbrakk> \<Longrightarrow>
((p',c,ve')#pe,ve,f) \<turnstile> (CALL p, s) \<Rightarrow> t" |
Proc: "((p,cp,ve)#pe,ve,f) \<turnstile> (c,s) \<Rightarrow> t
\<Longrightarrow> (pe,ve,f) \<turnstile> ({PROC p = cp;; c}, s) \<Rightarrow> t"
code_pred big_step .
values "{map t [0,1] |t. ([], \<lambda>n. 0, 0) \<turnstile> (CALL ''p'', nth [42, 43]) \<Rightarrow> t}"
(* FIXME: syntax *)
values "{map t [0, 1, 2] |t. ([], (\<lambda>_. 3)(''x'' := 0, ''y'' := 1,''z'' := 2), 0) \<turnstile> (test_com, (%_. 0)) \<Rightarrow> t}"
end