SSL Vars.thy

(* Author: Tobias Nipkow, Max Haslbeck *)

subsection "The Variables in an Expression"

theory Vars imports Com
begin


text‹We need to collect the variables in both arithmetic and boolean
 . For a change we do not introduce two functions, e.g.\ ‹avars› and ‹bvars›, but we overload the name ‹vars›
  a \emph{type class}, a device that originated with Haskell:›
 
class vars =
fixes vars :: "'a ==> vname set"

text‹This defines a type class ``vars'' with a single
  of (coincidentally) the same name. Then we define two separated
  of the class, one for @{typ aexp} and one for @{typ bexp}:›

instantiation aexp :: vars
begin

fun vars_aexp :: "aexp ==> vname set" where
"vars (N n) = {}" |
"vars (V x) = {x}" |
"vars (Plus a₁ a₂) = vars a₁ ∪ vars a₂"  |
"vars (Times a₁ a₂) = vars a₁ ∪ vars a₂"  |
"vars (Div a₁ a₂) = vars a₁ ∪ vars a₂" 

instance ..

end

value "vars (Plus (V ''x'') (V ''y''))"

instantiation bexp :: vars
begin

fun vars_bexp :: "bexp ==> vname set" where
"vars (Bc v) = {}" |
"vars (Not b) = vars b" |
"vars (And b₁ b₂) = vars b₁ ∪ vars b₂" |
"vars (Less a₁ a₂) = vars a₁ ∪ vars a₂"

instance ..

end

value "vars (Less (Plus (V ''z'') (V ''y'')) (V ''x''))"

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]:
  "s₁ = s₂ on vars a ==> aval a s₁ = aval a s₂"
apply(induction a)
apply simp_all
done          

lemma bval_eq_if_eq_on_vars:
  "s₁ = s₂ on vars b ==> bval b s₁ = bval b s₂"
proof(induction b)
  case (Less a1 a2)
  hence "aval a1 s₁ = aval a1 s₂" and java.lang.NullPointerException
   ?case by simpjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed simp_all

 

fun lvars :: "com< vnameset where
"lvars SKIP = {}" |
"lvars (x::=e) = {x}" |
 (c1;  lvars aa\mph{typeass  evicehatginatedd thllclose
java.lang.NullPointerException
"lvars (WHILE b DO c) = lvars c"

fun rvars :: "com ==> vname set" where
"rvars SKIP = {}" |
"rvars (x::=e) = vars e" |
"rvars (c1;;c2 = rvars rvars c2" |
"rvars (IF b THEN c1 ELSE c2) = vars eg\ <open and ‹name \ <>vars
  (WHILE b Oc) vr \union rvars c"

  com :: vars
 

  "vars_com c = lvars c ∪ rvars c"

  ..

 

  vars_com_simps[simp]:
 "vars SKIP = {}"
 "vars (x::=e) = {x} \<> 
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 "vars (IF b THEN c1 ELSE c2) = vars b ∪ vars c1 ∪ vars c2"
 "vars (WHILE b DO c) = vars b ∪ vars c"
 (auto simp: vars_com_def)