Theory Procs_Stat_Vars_Dyn

theory Procs_Stat_Vars_Dyn imports Procs
begin

subsubsection "Static Scoping of Procedures, Dynamic of Variables"

type_synonym penv = "(pname × com) list"

inductive
  big_step :: "penv => com × state => state => bool" ("_ \<turnstile> _ => _" [60,0,60] 55)
where
Skip:    "pe \<turnstile> (SKIP,s) => s" |
Assign:  "pe \<turnstile> (x ::= a,s) => s(x := aval a s)" |
Seq:     "[| pe \<turnstile> (c⇣1,s⇣1) => s⇣2;  pe \<turnstile> (c⇣2,s⇣2) => s⇣3 |] ==>
          pe \<turnstile> (c⇣1;c⇣2, s⇣1) => s⇣3" |

IfTrue:  "[| bval b s;  pe \<turnstile> (c⇣1,s) => t |] ==>
         pe \<turnstile> (IF b THEN c⇣1 ELSE c⇣2, s) => t" |
IfFalse: "[| ¬bval b s;  pe \<turnstile> (c⇣2,s) => t |] ==>
         pe \<turnstile> (IF b THEN c⇣1 ELSE c⇣2, s) => t" |

WhileFalse: "¬bval b s ==> pe \<turnstile> (WHILE b DO c,s) => s" |
WhileTrue:
  "[| bval b s⇣1;  pe \<turnstile> (c,s⇣1) => s⇣2;  pe \<turnstile> (WHILE b DO c, s⇣2) => s⇣3 |] ==>
   pe \<turnstile> (WHILE b DO c, s⇣1) => s⇣3" |

Var: "pe \<turnstile> (c,s) => t  ==>  pe \<turnstile> ({VAR x;; c}, s) => t(x := s x)" |

Call1: "(p,c)#pe \<turnstile> (c, s) => t  ==>  (p,c)#pe \<turnstile> (CALL p, s) => t" |
Call2: "[| p' ≠ p;  pe \<turnstile> (CALL p, s) => t |] ==>
       (p',c)#pe \<turnstile> (CALL p, s) => t" |

Proc: "(p,cp)#pe \<turnstile> (c,s) => t  ==>  pe \<turnstile> ({PROC p = cp;; c}, s) => t"

code_pred big_step .

values "{map t [''x'', ''y'', ''z''] |t. 
            [] \<turnstile> (CALL ''p'', <''x'' := 42, ''y'' := 43>) => t}"

values "{map t [''x'', ''y'', ''z''] |t. [] \<turnstile> (test_com, <>) => t}"

end