# Theory BExp

```subsection "Boolean Expressions"

theory BExp imports AExp begin

datatype bexp = Bc bool | Not bexp | And bexp bexp | Less aexp aexp

text_raw‹\snip{BExpbvaldef}{1}{2}{%›
fun bval :: "bexp ⇒ state ⇒ bool" where
"bval (Bc v) s = v" |
"bval (Not b) s = (¬ bval b s)" |
"bval (And b⇩1 b⇩2) s = (bval b⇩1 s ∧ bval b⇩2 s)" |
"bval (Less a⇩1 a⇩2) s = (aval a⇩1 s < aval a⇩2 s)"
text_raw‹}%endsnip›

value "bval (Less (V ''x'') (Plus (N 3) (V ''y'')))
<''x'' := 3, ''y'' := 1>"

subsection "Constant Folding"

text‹Optimizing constructors:›

text_raw‹\snip{BExplessdef}{0}{2}{%›
fun less :: "aexp ⇒ aexp ⇒ bexp" where
"less (N n⇩1) (N n⇩2) = Bc(n⇩1 < n⇩2)" |
"less a⇩1 a⇩2 = Less a⇩1 a⇩2"
text_raw‹}%endsnip›

lemma [simp]: "bval (less a1 a2) s = (aval a1 s < aval a2 s)"
apply(induction a1 a2 rule: less.induct)
apply simp_all
done

text_raw‹\snip{BExpanddef}{2}{2}{%›
fun "and" :: "bexp ⇒ bexp ⇒ 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 b⇩1 b⇩2 = And b⇩1 b⇩2"
text_raw‹}%endsnip›

lemma bval_and[simp]: "bval (and b1 b2) s = (bval b1 s ∧ bval b2 s)"
apply(induction b1 b2 rule: and.induct)
apply simp_all
done

text_raw‹\snip{BExpnotdef}{2}{2}{%›
fun not :: "bexp ⇒ bexp" where
"not (Bc True) = Bc False" |
"not (Bc False) = Bc True" |
"not b = Not b"
text_raw‹}%endsnip›

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:›

text_raw‹\snip{BExpbsimpdef}{0}{2}{%›
fun bsimp :: "bexp ⇒ bexp" where
"bsimp (Bc v) = Bc v" |
"bsimp (Not b) = not(bsimp b)" |
"bsimp (And b⇩1 b⇩2) = and (bsimp b⇩1) (bsimp b⇩2)" |
"bsimp (Less a⇩1 a⇩2) = less (asimp a⇩1) (asimp a⇩2)"
text_raw‹}%endsnip›

value "bsimp (And (Less (N 0) (N 1)) b)"

value "bsimp (And (Less (N 1) (N 0)) (Bc True))"

theorem "bval (bsimp b) s = bval b s"
apply(induction b)
apply simp_all
done

end
```