Theory Def_Init_Exp

(* Author: Tobias Nipkow *)

theory Def_Init_Exp
imports Vars
begin

subsection "Initialization-Sensitive Expressions Evaluation"

type_synonym state = "vname => val option"


fun aval :: "aexp => state => val option" where
"aval (N i) s = Some i" |
"aval (V x) s = s x" |
"aval (Plus a⇣1 a⇣2) s =
  (case (aval a⇣1 s, aval a⇣2 s) of
     (Some i⇣1,Some i⇣2) => Some(i⇣1+i⇣2) | _ => None)"


fun bval :: "bexp => state => bool option" where
"bval (Bc v) s = Some v" |
"bval (Not b) s = (case bval b s of None => None | Some bv => Some(¬ bv))" |
"bval (And b⇣1 b⇣2) s = (case (bval b⇣1 s, bval b⇣2 s) of
  (Some bv⇣1, Some bv⇣2) => Some(bv⇣1 & bv⇣2) | _ => None)" |
"bval (Less a⇣1 a⇣2) s = (case (aval a⇣1 s, aval a⇣2 s) of
 (Some i⇣1, Some i⇣2) => Some(i⇣1 < i⇣2) | _ => None)"


lemma aval_Some: "vars a ⊆ dom s ==> ∃ i. aval a s = Some i"
by (induct a) auto

lemma bval_Some: "vars b ⊆ dom s ==> ∃ bv. bval b s = Some bv"
by (induct b) (auto dest!: aval_Some)

end