(* Author: Tobias Nipkow *) header "Definite Assignment Analysis" theory Vars imports Util BExp begin subsection "The Variables in an Expression" text{* We need to collect the variables in both arithmetic and boolean expressions. For a change we do not introduce two functions, e.g.\ @{text avars} and @{text bvars}, but we overload the name @{text vars} via a \emph{type class}, a device that originated with Haskell: *} class vars = fixes vars :: "'a \ name set" text{* This defines a type class ``vars'' with a single function of (coincidentally) the same name. Then we define two separated instances of the class, one for @{typ aexp} and one for @{typ bexp}: *} instantiation aexp :: vars begin fun vars_aexp :: "aexp \ name set" where "vars_aexp (N n) = {}" | "vars_aexp (V x) = {x}" | "vars_aexp (Plus a\<^isub>1 a\<^isub>2) = vars_aexp a\<^isub>1 \ vars_aexp a\<^isub>2" instance .. end value "vars(Plus (V 3) (V 2))" text{* We need to convert functions to lists before we can view them: *} value "list (vars(Plus (V 3) (V 2))) 4" instantiation bexp :: vars begin fun vars_bexp :: "bexp \ name set" where "vars_bexp (B bv) = {}" | "vars_bexp (Not b) = vars_bexp b" | "vars_bexp (And b\<^isub>1 b\<^isub>2) = vars_bexp b\<^isub>1 \ vars_bexp b\<^isub>2" | "vars_bexp (Less a\<^isub>1 a\<^isub>2) = vars a\<^isub>1 \ vars a\<^isub>2" instance .. end value "list (vars(Less (Plus (V 3) (V 2)) (V 1))) 5" abbreviation eq_on :: "('a \ 'b) \ ('a \ 'b) \ 'a set \ bool" ("(_ =/ _/ on _)" [50,0,50] 50) where "f = g on X == \ x \ X. f x = g x" lemma aval_eq_if_eq_on_vars[simp]: "s\<^isub>1 = s\<^isub>2 on vars a \ aval a s\<^isub>1 = aval a s\<^isub>2" apply(induct a) apply simp_all done lemma bval_eq_if_eq_on_vars: "s\<^isub>1 = s\<^isub>2 on vars b \ bval b s\<^isub>1 = bval b s\<^isub>2" proof(induct b) case (Less a1 a2) hence "aval a1 s\<^isub>1 = aval a1 s\<^isub>2" and "aval a2 s\<^isub>1 = aval a2 s\<^isub>2" by simp_all thus ?case by simp qed simp_all end