tuned proofs -- clarified flow of facts wrt. calculation;
(* Author: Tobias Nipkow *)
theory Def_Init_Exp
imports Vars
begin
subsection "Initialization-Sensitive Expressions Evaluation"
type_synonym state = "vname \<Rightarrow> val option"
fun aval :: "aexp \<Rightarrow> state \<Rightarrow> val option" where
"aval (N i) s = Some i" |
"aval (V x) s = s x" |
"aval (Plus a\<^sub>1 a\<^sub>2) s =
(case (aval a\<^sub>1 s, aval a\<^sub>2 s) of
(Some i\<^sub>1,Some i\<^sub>2) \<Rightarrow> Some(i\<^sub>1+i\<^sub>2) | _ \<Rightarrow> None)"
fun bval :: "bexp \<Rightarrow> state \<Rightarrow> bool option" where
"bval (Bc v) s = Some v" |
"bval (Not b) s = (case bval b s of None \<Rightarrow> None | Some bv \<Rightarrow> Some(\<not> bv))" |
"bval (And b\<^sub>1 b\<^sub>2) s = (case (bval b\<^sub>1 s, bval b\<^sub>2 s) of
(Some bv\<^sub>1, Some bv\<^sub>2) \<Rightarrow> Some(bv\<^sub>1 & bv\<^sub>2) | _ \<Rightarrow> None)" |
"bval (Less a\<^sub>1 a\<^sub>2) s = (case (aval a\<^sub>1 s, aval a\<^sub>2 s) of
(Some i\<^sub>1, Some i\<^sub>2) \<Rightarrow> Some(i\<^sub>1 < i\<^sub>2) | _ \<Rightarrow> None)"
lemma aval_Some: "vars a \<subseteq> dom s \<Longrightarrow> \<exists> i. aval a s = Some i"
by (induct a) auto
lemma bval_Some: "vars b \<subseteq> dom s \<Longrightarrow> \<exists> bv. bval b s = Some bv"
by (induct b) (auto dest!: aval_Some)
end