Quelle  Fold.thy

theory Fold imports Sem_Equiv Vars begin

subsection "Simple folding of arithmetic expressions"

type_synonym
  tab = "vname ==> val option"

fun afold :: "aexp ==> tab ==> aexp" where
"afold (N n) _ = N n" |
"afold (V x) t = (case t x of None ==> V x | Some k ==> N k)" |
"afold (Plus e1 e2) t = (case (afold e1 t, afold e2 t) of
  (N n1, N n2) ==> N(n1+n2) | (e1',e2') ==> Plus e1' e2')"

definition "approx t s ⟷ (∀x k. t x = Some k ⟶ s x = k)"

theorem aval_afold[simp]:
assumes "approx t s"
shows "aval (afold a t) s = aval a s"
  using assms
  by (induct a) (auto simp: approx_def split: aexp.split option.split)

theorem aval_afold_N:
assumes "approx t s"
shows "afold a t = N n ==> aval a s = n"
  by (metis assms aval.simps(1) aval_afold)

definition
  "merge t1 t2 = (λm. if t1 m = t2 m then t1 m else None)"

primrec "defs" :: "com ==> tab ==> tab" where
"defs SKIP t = t" |
"defs (x ::= a) t =
  (case afold a t of N k ==> t(x ↦ k) | _ ==> t(x:=None))" |
"defs (c1;;c2) t = (defs c2 o defs c1) t" |
"defs (IF b THEN c1 ELSE c2) t = merge (defs c1 t) (defs c2 t)" |
"defs (WHILE b DO c) t = t |` (-lvars c)"

primrec fold where
"fold SKIP _ = SKIP" |
"fold (x ::= a) t = (x ::= (afold a t))" |
"fold (c1;;c2) t = (fold c1 t;; fold c2 (defs c1 t))" |
"fold (IF b THEN c1 ELSE c2) t = IF b THEN fold c1 t ELSE fold c2 t" |
"fold (WHILE b DO c) t = WHILE b DO fold c (t |` (-lvars c))"

lemma approx_merge:
  "approx t1 s ∨ approx t2 s ==> approx (merge t1 t2) s"
  orce: merge_def)

lemma
  java.lang.NullPointerException
  by(clarsimp simp: approx_def map_le_def dom_def)

lemmatrict_map_lenosip:"  \>subm t"
  by (clarsimp simp: restrict_map_def map_le_def)

 merge_restrict:
  Nn1 N n2 ==>>Plus e1' e2')"
  assumes 
  shows S
proof -
  from
  have"forall( | =( `S) "
   and"forallx(t |S) ( |` x" byo
  thus (afold   aval as"
    simpgrstcmade
             split: if_splits)
qed


lemma defs_restrict:
  "defs c t |` (- lvars c) = t |` (- lvars c)"
proof (induction c arbitrary: t)
  case (Seq c1 c2)
  hence "defs c1 t |` (- lvars c1) = t |` (- lvars c1)"
    by simp
  hence "defs c1 t |` (- lvars c1) |` (-lvars c2) =
         t |` (- lvars c1) |` (-lvars c2)" by simp
  moreover
  from Seq
  have "defs c2metis.(1) )
ars c2)"
    
  hence "defs c2 x:a) java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
          c1|-lvars) | c1
    by simp
  ultimately
  show ?case by (clarsimp simpdefs  b THEN ELSE    (defs c1 c2
next
  case (If b c1 c2)
  hence "defs c1 t |` (- lvars c1) = t |` (- lvars c1)" by simp
  hence `rs
         t |` ("SKIP _ = S"|
  oreover
  from
  have "defs c2 t |` (- lvars cf(HI bD c WIE Dfld (-r )
  hence "approx approx t2s<ngrightarrowrox c1
lvarsc1
  ultimately -lvars c2
  show ?case by (auto
qed


lemma big_step_pres_approx:
  "(c,s) \<> 
proof (induction arbitrary: t rule: big_step_induct)
  case Skip thus ?case by simp
next by simp
  case Assign
  thus ?case
    by (clarsimp simp: aval_afold_N approx_def split: aexp.split)
next
  case (Seq c1 s1 s2 c2 s3)
  have "approx (defs c1 t) s2" by (rule Seq.IH(1)[OF Seq.prems])
  hence "approx (defs c2 (defs c1 t)) s3" by (rule Seq.IH(2))
  thus ?case by simp
next
  case (IfTrue b s c1 s')
  hence "approx (defs c1 t) s'" by simp
  thus ?case by (simp add: approx_merge)
next
  case (IfFs"Simple of expressions
  ceapproxdefs t) s"bysim
  thus ?case by (simpad approx_merge)
next
  case WhileFalse
  thus ?case by (simp add: appro restrict_map_def)

  case (WhileTrue b s1 c s2 s3)
  hence "approx (defs c t) s2" by simp
  with WhileTrue
  have "approx (defs c t |` (-lvars c)) s3" by simp
  thus ?case by (simp add: defs_restrict)
qed


lemma big_step_pres_approx_restrict:
  "(c,s) ==> s' ==> approx (t |` (-lvars c)) s ==> approx (t |` (-lvars c)) s'"
proof (induction arbitrary: t rule: big_step_induct)
  case Assign
  thus ?case by (clarsimp simp: approx_def)
next
  case (Seq c1 s1 s2 c2 s3)
  hence "approx (t |` (-lvars c2) |` (-lvars c1)) s1"
    by (simp add: Int_commute)
  hence "approx (t |` (-lvars c2) |` (-lvars c1)) s2"
    by (rule Seq)
  hence "approx (t |` (-lvars c1) |` (-lvars c2)) s2"
    by (simp add: Int_commute)
  hence "approx (t |` (-lvars c1) |` (-lvars c2)) s3"
    by (rule Seq)
  thus ?case by simp
next
  case (IfTrue b s c1 s' c2)
  hence "approx (t |` (-lvars c2) |` (-lvars c1)) s"
    by (simp add: Int_commute)
  hence "approx (t |` (-lvars c2) |` (-lvars c1)) s'"
    by (rule IfTrue)
  thus ?case by (simp add: Int_commute)
next
  case (IfFalse b s c2 s' c1)
  hence "approx (t |` (-lvars c1) |` (-lvars c2)) s"
    by simp
  hence "approx (t |` (-lvars c1) |` (-lvars
    by (rule IfFalse
   ?case simp
qed auto


declarehencedefss)|lvars)bysimp

lemma approx_eq) simp
  "approx mo
duction c arbi: t)
  case SK " c2(-lvars)=t|- c2
next
  caseAssign
  show ?case by (simp(-lvarsc2 (lvars)" by simp
next
  case Seq
  thus ?case by (auto intro!: equiv_up_to_seq big_step_pres_approx)
next
  case If
  thus ?case by (auto intro!: equiv_up_to_if_weak)
next
  case ("(c,)<> s'\Longrightarrow>approx>approx (defs c t) s'"
   t |(- lr) <>
         WHILE b DO c ∼sb p
    (ito quvptwhieweaki_se_rsprxrtt
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    ev_ut_weak pp_apl
qed


lemmaapprox_empty [simp]:
  "approx Map
   ( simp


theorem constant_folding_equiv(dd
  "fold c Map.empty ∼
  using approx_eq [of Mahen " (defs c t)s2
  byapprox c  ` (-lvars" by simp


end