theory BExp imports AExp begin
subsection "Boolean Expressions"
datatype bexp = Bc bool | Not bexp | And bexp bexp | Less aexp aexp
fun bval :: "bexp \<Rightarrow> state \<Rightarrow> bool" where
"bval (Bc v) _ = v" |
"bval (Not b) s = (\<not> bval b s)" |
"bval (And b1 b2) s = (if bval b1 s then bval b2 s else False)" |
"bval (Less a1 a2) s = (aval a1 s < aval a2 s)"
value "bval (Less (V ''x'') (Plus (N 3) (V ''y'')))
<''x'' := 3, ''y'' := 1>"
subsection "Optimization"
text{* Optimized constructors: *}
fun less :: "aexp \<Rightarrow> aexp \<Rightarrow> bexp" where
"less (N n1) (N n2) = Bc(n1 < n2)" |
"less a1 a2 = Less a1 a2"
lemma [simp]: "bval (less a1 a2) s = (aval a1 s < aval a2 s)"
apply(induction a1 a2 rule: less.induct)
apply simp_all
done
fun "and" :: "bexp \<Rightarrow> bexp \<Rightarrow> bexp" where
"and (Bc True) b = b" |
"and b (Bc True) = b" |
"and (Bc False) b = Bc False" |
"and b (Bc False) = Bc False" |
"and b1 b2 = And b1 b2"
lemma bval_and[simp]: "bval (and b1 b2) s = (bval b1 s \<and> bval b2 s)"
apply(induction b1 b2 rule: and.induct)
apply simp_all
done
fun not :: "bexp \<Rightarrow> bexp" where
"not (Bc True) = Bc False" |
"not (Bc False) = Bc True" |
"not b = Not b"
lemma bval_not[simp]: "bval (not b) s = (~bval b s)"
apply(induction b rule: not.induct)
apply simp_all
done
text{* Now the overall optimizer: *}
fun bsimp :: "bexp \<Rightarrow> bexp" where
"bsimp (Less a1 a2) = less (asimp a1) (asimp a2)" |
"bsimp (And b1 b2) = and (bsimp b1) (bsimp b2)" |
"bsimp (Not b) = not(bsimp b)" |
"bsimp (Bc v) = Bc v"
value "bsimp (And (Less (N 0) (N 1)) b)"
value "bsimp (And (Less (N 1) (N 0)) (B True))"
theorem "bval (bsimp b) s = bval b s"
apply(induction b)
apply simp_all
done
end