Theory AExp

theory AExp
imports Main
section "Arithmetic and Boolean Expressions"

theory AExp imports Main begin

subsection "Arithmetic Expressions"

type_synonym vname = string
type_synonym val = int
type_synonym state = "vname ⇒ val"

datatype aexp = N int | V vname | Plus aexp aexp

fun aval :: "aexp ⇒ state ⇒ val" where
"aval (N n) s = n" |
"aval (V x) s = s x" |
"aval (Plus a1 a2) s = aval a1 s + aval a2 s"

value "aval (Plus (V ''x'') (N 5)) (λx. if x = ''x'' then 7 else 0)"

text ‹The same state more concisely:›
value "aval (Plus (V ''x'') (N 5)) ((λx. 0) (''x'':= 7))"

text ‹A little syntax magic to write larger states compactly:›

definition null_state ("<>") where
  "null_state ≡ λx. 0"
  "_State" :: "updbinds => 'a" ("<_>")
  "_State ms" == "_Update <> ms"
  "_State (_updbinds b bs)" <= "_Update (_State b) bs"

text ‹\noindent
  We can now write a series of updates to the function @{text "λx. 0"} compactly:
lemma "<a := 1, b := 2> = (<> (a := 1)) (b := (2::int))"
  by (rule refl)

value "aval (Plus (V ''x'') (N 5)) <''x'' := 7>"

text ‹In  the @{term[source] "<a := b>"} syntax, variables that are not mentioned are 0 by default:
value "aval (Plus (V ''x'') (N 5)) <''y'' := 7>"

text‹Note that this @{text"<…>"} syntax works for any function space
@{text"τ1 ⇒ τ2"} where @{text "τ2"} has a @{text 0}.›

subsection "Constant Folding"

text‹Evaluate constant subsexpressions:›

fun asimp_const :: "aexp ⇒ aexp" where
"asimp_const (N n) = N n" |
"asimp_const (V x) = V x" |
"asimp_const (Plus a1 a2) =
  (case (asimp_const a1, asimp_const a2) of
    (N n1, N n2) ⇒ N(n1+n2) |
    (b1,b2) ⇒ Plus b1 b2)"

theorem aval_asimp_const:
  "aval (asimp_const a) s = aval a s"
apply(induction a)
apply (auto split: aexp.split)

text‹Now we also eliminate all occurrences 0 in additions. The standard
method: optimized versions of the constructors:›

fun plus :: "aexp ⇒ aexp ⇒ aexp" where
"plus (N i1) (N i2) = N(i1+i2)" |
"plus (N i) a = (if i=0 then a else Plus (N i) a)" |
"plus a (N i) = (if i=0 then a else Plus a (N i))" |
"plus a1 a2 = Plus a1 a2"

lemma aval_plus[simp]:
  "aval (plus a1 a2) s = aval a1 s + aval a2 s"
apply(induction a1 a2 rule: plus.induct)
apply simp_all (* just for a change from auto *)

fun asimp :: "aexp ⇒ aexp" where
"asimp (N n) = N n" |
"asimp (V x) = V x" |
"asimp (Plus a1 a2) = plus (asimp a1) (asimp a2)"

text‹Note that in @{const asimp_const} the optimized constructor was
inlined. Making it a separate function @{const plus} improves modularity of
the code and the proofs.›

value "asimp (Plus (Plus (N 0) (N 0)) (Plus (V ''x'') (N 0)))"

theorem aval_asimp[simp]:
  "aval (asimp a) s = aval a s"
apply(induction a)
apply simp_all