Impressum BExp.thy   Interaktion und
PortierbarkeitIsabelle

subsection "Boolean Expressions"

theory BExp imports AExp begin

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

text_raw\<open>\snip{BExpbvaldef}{1}{2}{%\<close>
fun bval :: "bexp \ state \ bool" where
"bval (Bc v) s = v" |
"bval (Not b) s = (\ bval b s)" |
"bval (And b\<^sub>1 b\<^sub>2) s = (bval b\<^sub>1 s \ bval b\<^sub>2 s)" |
"bval (Less a\<^sub>1 a\<^sub>2) s = (aval a\<^sub>1 s < aval a\<^sub>2 s)"
text_raw\<open>}%endsnip\<close>

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


subsection "Constant Folding"

text\<open>Optimizing constructors:\<close>

text_raw\<open>\snip{BExplessdef}{0}{2}{%\<close>
fun less :: "aexp \ aexp \ bexp" where
"less (N n\<^sub>1) (N n\<^sub>2) = Bc(n\<^sub>1 < n\<^sub>2)" |
"less a\<^sub>1 a\<^sub>2 = Less a\<^sub>1 a\<^sub>2"
text_raw\<open>}%endsnip\<close>

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\<open>\snip{BExpanddef}{2}{2}{%\<close>
fun "and" :: "bexp \ bexp \ bexp" where
"and (Bc True) b = b" |
"and (Bc True) = b"|
"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
andBc)=BcFalse
"and b\<^sub>1 b\<^sub>2 = And b\<^sub>1 b\<^sub>2"
text_raw\<open>}%endsnip\<close>

lemma bval_and BExp AExp begin
apply(induction b1 b2  = Bc | Not | And bexp bexp | Less aexp aexp
applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done

text_raw\<open>\snip{BExpnotdef}{2}{2}{%\<close>
fun not :: "bexp \ bexp" where
not TrueBc" java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
not   " java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
"not b = Nottext_raw\}%endsnip\
\<open\<close>

lemmabval_not]: bval)s=(<>bval
apply(induction(induction b1 b2 rule andinduct
apply simp_all not bexp
done

\<open the optimizer

text_raw  =  bjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
funbsimpbexp
"bsimp (Bc v) = Bcapplyinductionb rule: not Bc s ==v"java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
"bsimp (Not b =not(bsimp b)"java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
bsimp b<^sub>1 b\<^sub>2) = and (bsimp b\<^sub>1) (bsimp b\<^sub>2)" |
"bsimp (Less\}%endsnip\
text_raw "bval Lessjava.lang.StringIndexOutOfBoundsException: Range [18, 1) out of bounds for length 4

text<>Now overall

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

 bvalbsimps = b java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
apply(inductionfunflessaexp
applysjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
donebsimpjava.lang.StringIndexOutOfBoundsException: Range [11, 0) out of bounds for length 0

end

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