begin proving stuff

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+Require Import Coqlib Maps Errors Integers Floats Lattice Kildall.
+Require Import AST Linking.
+Require Import Memory Registers Op RTL Maps.
+
+Require Import Globalenvs Values.
+
+(* Static analysis *)
+
+Inductive sym_val : Type :=
+| SMove (src : reg)
+| SOp (op : operation) (args : list reg).
+
+Definition eq_args (x y : list reg) : { x = y } + { x <> y } :=
+  list_eq_dec peq x y.
+
+Definition eq_sym_val : forall x y : sym_val,
+    {x = y} + { x <> y }.
+Proof.
+  generalize eq_operation.
+  generalize eq_args.
+  generalize peq.
+  decide equality.
+Defined.
+
+Module RELATION.
+  
+Definition t := (PTree.t sym_val).
+Definition eq (r1 r2 : t) :=
+  forall x, (PTree.get x r1) = (PTree.get x r2).
+
+Definition top : t := PTree.empty sym_val.
+
+Lemma eq_refl: forall x, eq x x.
+Proof.
+  unfold eq.
+  intros; reflexivity.
+Qed.
+
+Lemma eq_sym: forall x y, eq x y -> eq y x.
+Proof.
+  unfold eq.
+  intros; eauto.
+Qed.
+
+Lemma eq_trans: forall x y z, eq x y -> eq y z -> eq x z.
+Proof.
+  unfold eq.
+  intros; congruence.
+Qed.
+
+Definition sym_val_beq (x y : sym_val) :=
+  if eq_sym_val x y then true else false.
+
+Definition beq (r1 r2 : t) := PTree.beq sym_val_beq r1 r2.
+
+Lemma beq_correct: forall r1 r2, beq r1 r2 = true -> eq r1 r2.
+Proof.
+  unfold beq, eq. intros r1 r2 EQ x.
+  pose proof (PTree.beq_correct sym_val_beq r1 r2) as CORRECT.
+  destruct CORRECT as [CORRECTF CORRECTB].
+  pose proof (CORRECTF EQ x) as EQx.
+  clear CORRECTF CORRECTB EQ.
+  unfold sym_val_beq in *.
+  destruct (r1 ! x) as [R1x | ] in *;
+    destruct (r2 ! x) as [R2x | ] in *;
+    trivial; try contradiction.
+  destruct (eq_sym_val R1x R2x) in *; congruence.
+Qed.
+
+Definition ge (r1 r2 : t) :=
+  forall x,
+    match PTree.get x r1 with
+    | None => True
+    | Some v => (PTree.get x r2) = Some v
+    end.
+
+Lemma ge_refl: forall r1 r2, eq r1 r2 -> ge r1 r2.
+Proof.
+  unfold eq, ge.
+  intros r1 r2 EQ x.
+  pose proof (EQ x) as EQx.
+  clear EQ.
+  destruct (r1 ! x).
+  - congruence.
+  - trivial.
+Qed.
+
+Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
+Proof.
+  unfold ge.
+  intros r1 r2 r3 GE12 GE23 x.
+  pose proof (GE12 x) as GE12x; clear GE12.
+  pose proof (GE23 x) as GE23x; clear GE23.
+  destruct (r1 ! x); trivial.
+  destruct (r2 ! x); congruence.
+Qed.
+
+Definition lub (r1 r2 : t) :=
+  PTree.combine
+    (fun ov1 ov2 =>
+       match ov1, ov2 with
+       | (Some v1), (Some v2) =>
+         if eq_sym_val v1 v2
+         then ov1
+         else None
+       | None, _
+       | _, None => None
+       end)
+    r1 r2.
+
+Lemma ge_lub_left: forall x y, ge (lub x y) x.
+Proof.
+  unfold ge, lub.
+  intros r1 r2 x.
+  rewrite PTree.gcombine by reflexivity.
+  destruct (_ ! _); trivial.
+  destruct (_ ! _); trivial.
+  destruct (eq_sym_val _ _); trivial.
+Qed.
+
+Lemma ge_lub_right: forall x y, ge (lub x y) y.
+Proof.
+  unfold ge, lub.
+  intros r1 r2 x.
+  rewrite PTree.gcombine by reflexivity.
+  destruct (_ ! _); trivial.
+  destruct (_ ! _); trivial.
+  destruct (eq_sym_val _ _); trivial.
+  congruence.
+Qed.
+
+End RELATION.
+
+Module Type SEMILATTICE_WITHOUT_BOTTOM.
+
+  Parameter t: Type.
+  Parameter eq: t -> t -> Prop.
+  Axiom eq_refl: forall x, eq x x.
+  Axiom eq_sym: forall x y, eq x y -> eq y x.
+  Axiom eq_trans: forall x y z, eq x y -> eq y z -> eq x z.
+  Parameter beq: t -> t -> bool.
+  Axiom beq_correct: forall x y, beq x y = true -> eq x y.
+  Parameter ge: t -> t -> Prop.
+  Axiom ge_refl: forall x y, eq x y -> ge x y.
+  Axiom ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
+  Parameter lub: t -> t -> t.
+  Axiom ge_lub_left: forall x y, ge (lub x y) x.
+  Axiom ge_lub_right: forall x y, ge (lub x y) y.
+
+End SEMILATTICE_WITHOUT_BOTTOM.
+
+Module ADD_BOTTOM(L : SEMILATTICE_WITHOUT_BOTTOM).
+  Definition t := option L.t.
+  Definition eq (a b : t) :=
+    match a, b with
+    | None, None => True
+    | Some x, Some y => L.eq x y
+    | Some _, None | None, Some _ => False
+    end.
+  
+  Lemma eq_refl: forall x, eq x x.
+  Proof.
+    unfold eq; destruct x; trivial.
+    apply L.eq_refl.
+  Qed.
+
+  Lemma eq_sym: forall x y, eq x y -> eq y x.
+  Proof.
+    unfold eq; destruct x; destruct y; trivial.
+    apply L.eq_sym.
+  Qed.
+  
+  Lemma eq_trans: forall x y z, eq x y -> eq y z -> eq x z.
+  Proof.
+    unfold eq; destruct x; destruct y; destruct z; trivial.
+    - apply L.eq_trans.
+    - contradiction.
+  Qed.
+  
+  Definition beq (x y : t) :=
+    match x, y with
+    | None, None => true
+    | Some x, Some y => L.beq x y
+    | Some _, None | None, Some _ => false
+    end.
+  
+  Lemma beq_correct: forall x y, beq x y = true -> eq x y.
+  Proof.
+    unfold beq, eq.
+    destruct x; destruct y; trivial; try congruence.
+    apply L.beq_correct.
+  Qed.
+  
+  Definition ge (x y : t) :=
+    match x, y with
+    | None, Some _ => False
+    | _, None => True
+    | Some a, Some b => L.ge a b
+    end.
+  
+  Lemma ge_refl: forall x y, eq x y -> ge x y.
+  Proof.
+    unfold eq, ge.
+    destruct x; destruct y; trivial.
+    apply L.ge_refl.
+  Qed.
+  
+  Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
+  Proof.
+    unfold ge.
+    destruct x; destruct y; destruct z; trivial; try contradiction.
+    apply L.ge_trans.
+  Qed.
+  
+  Definition bot: t := None.
+  Lemma ge_bot: forall x, ge x bot.
+  Proof.
+    unfold ge, bot.
+    destruct x; trivial.
+  Qed.
+  
+  Definition lub (a b : t) :=
+    match a, b with
+    | None, _ => b
+    | _, None => a
+    | (Some x), (Some y) => Some (L.lub x y)
+    end.
+
+  Lemma ge_lub_left: forall x y, ge (lub x y) x.
+  Proof.
+    unfold ge, lub.
+    destruct x; destruct y; trivial.
+    - apply L.ge_lub_left.
+    - apply L.ge_refl.
+      apply L.eq_refl.
+  Qed.
+  
+  Lemma ge_lub_right: forall x y, ge (lub x y) y.
+  Proof.
+    unfold ge, lub.
+    destruct x; destruct y; trivial.
+    - apply L.ge_lub_right.
+    - apply L.ge_refl.
+      apply L.eq_refl.
+  Qed.
+End ADD_BOTTOM.
+
+Module RB := ADD_BOTTOM(RELATION).
+Module DS := Dataflow_Solver(RB)(NodeSetForward).
+
+Definition kill_sym_val (dst : reg) (sv : sym_val) :=
+  match sv with
+  | SMove src => if peq dst src then true else false
+  | SOp op args => List.existsb (peq dst) args
+  end.
+                                                 
+Definition kill (dst : reg) (rel : RELATION.t) :=
+  PTree.filter1 (fun x => negb (kill_sym_val dst x))
+                (PTree.remove dst rel).
+
+Lemma args_unaffected:
+  forall rs : regset,
+  forall dst : reg,
+  forall v,
+  forall args : list reg,
+    existsb (fun y : reg => peq dst y) args = false ->
+    (rs # dst <- v ## args) = (rs ## args).
+Proof.
+  induction args; simpl; trivial.
+  destruct (peq dst a) as [EQ | NEQ]; simpl.
+  { discriminate.
+  }
+  intro EXIST.
+  f_equal.
+  {
+    apply Regmap.gso.
+    congruence.
+  }
+  apply IHargs.
+  assumption.
+Qed.
+  
+Section SOUNDNESS.
+  Parameter F V : Type.
+  Parameter genv: Genv.t F V.
+  Parameter sp : val.
+  Parameter m : mem.
+                
+Definition sem_reg (rel : RELATION.t) (x : reg) (rs : regset) : option val :=
+  match PTree.get x rel with
+  | None => Some (rs # x)
+  | Some (SMove src) => Some (rs # src)
+  | Some (SOp op args) =>
+    eval_operation genv sp op (rs ## args) m
+  end.
+
+Definition sem_rel (rel : RELATION.t) (rs : regset) :=
+  forall x : reg, (sem_reg rel x rs) = Some (rs # x).
+
+Lemma kill_sound :
+  forall rel : RELATION.t,
+  forall dst : reg,
+  forall rs,
+  forall v,
+    sem_rel rel rs ->
+    sem_rel (kill dst rel) (rs # dst <- v).
+Proof.
+  unfold sem_rel, kill, sem_reg.
+  intros until v.
+  intros REL x.
+  rewrite PTree.gfilter1.
+  destruct (Pos.eq_dec dst x).
+  {
+    subst x.
+    rewrite PTree.grs.
+    rewrite Regmap.gss.
+    reflexivity.
+  }
+  rewrite PTree.gro by congruence.
+  rewrite Regmap.gso by congruence.
+  destruct (rel ! x) as [relx | ] eqn:RELx.
+  2: reflexivity.
+  unfold kill_sym_val.
+  pose proof (REL x) as RELinstx.
+  rewrite RELx in RELinstx.
+  destruct relx eqn:SYMVAL.
+  {
+    destruct (peq dst src); simpl.
+    { reflexivity. }
+    rewrite Regmap.gso by congruence.
+    assumption.
+  }
+  { destruct existsb eqn:EXISTS; simpl.
+    { reflexivity. }
+    rewrite args_unaffected by exact EXISTS.
+    assumption.
+  }
+Qed.
+  
+Definition move (src dst : reg) (rel : RELATION.t) :=
+  PTree.set dst (match PTree.get src rel with
+                 | Some (SMove src') => SMove src'
+                 | _ => SMove src
+                 end) (kill dst rel).
+
+Fixpoint kill_builtin_res (res : builtin_res reg) (rel : RELATION.t) :=
+  match res with
+  | BR z => kill z rel
+  | BR_none => rel
+  | BR_splitlong hi lo => kill_builtin_res hi (kill_builtin_res lo rel)
+  end.
+
+Definition apply_instr instr x :=
+  match instr with
+  | Inop _
+  | Icond _ _ _ _
+  | Ijumptable _ _
+  | Istore _ _ _ _ _ => Some x
+  | Iop Omove (src :: nil) dst _ => Some (move src dst x)
+  | Iop _ _ dst _
+  | Iload _ _ _ _ dst _
+  | Icall _ _ _ dst _ => Some (kill dst x)
+  | Ibuiltin _ _ res _ => Some (RELATION.top) (* TODO (kill_builtin_res res x) *)
+  | Itailcall _ _ _ | Ireturn _ => RB.bot
+  end.
+
+Definition apply_instr' code (pc : node) (ro : RB.t) : RB.t :=
+  match ro with
+  | None => None
+  | Some x =>
+    match code ! pc with
+    | None => RB.bot
+    | Some instr => apply_instr instr x
+    end
+  end.
+
+Definition forward_map (f : RTL.function) := DS.fixpoint
+  (RTL.fn_code f) RTL.successors_instr
+  (apply_instr' (RTL.fn_code f)) (RTL.fn_entrypoint f) (Some RELATION.top).
+
+Definition get_r (rel : RELATION.t) (x : reg) :=
+  match PTree.get x rel with
+  | None => x
+  | Some src => src
+  end.
+
+Definition get_rb (rb : RB.t) (x : reg) :=
+  match rb with
+  | None => x
+  | Some rel => get_r rel x
+  end.
+
+Definition subst_arg (fmap : option (PMap.t RB.t)) (pc : node) (x : reg) : reg :=
+  match fmap with
+  | None => x
+  | Some inv => get_rb (PMap.get pc inv) x
+  end.
+
+Definition subst_args fmap pc := List.map (subst_arg fmap pc).
+
+(* Transform *)
+Definition transf_instr (fmap : option (PMap.t RB.t))
+           (pc: node) (instr: instruction) :=
+  match instr with
+  | Iop op args dst s =>
+    Iop op (subst_args fmap pc args) dst s
+  | Iload trap chunk addr args dst s =>
+    Iload trap chunk addr (subst_args fmap pc args) dst s
+  | Istore chunk addr args src s =>
+    Istore chunk addr (subst_args fmap pc args) src s
+  | Icall sig ros args dst s =>
+    Icall sig ros (subst_args fmap pc args) dst s
+  | Itailcall sig ros args =>
+    Itailcall sig ros (subst_args fmap pc args)
+  | Icond cond args s1 s2 =>
+    Icond cond (subst_args fmap pc args) s1 s2
+  | Ijumptable arg tbl =>
+    Ijumptable (subst_arg fmap pc arg) tbl
+  | Ireturn (Some arg) =>
+    Ireturn (Some (subst_arg fmap pc arg))
+  | _ => instr
+  end.
+
+Definition transf_function (f: function) : function :=
+  {| fn_sig := f.(fn_sig);
+     fn_params := f.(fn_params);
+     fn_stacksize := f.(fn_stacksize);
+     fn_code := PTree.map (transf_instr (forward_map f)) f.(fn_code);
+     fn_entrypoint := f.(fn_entrypoint) |}.
+
+
+Definition transf_fundef (fd: fundef) : fundef :=
+  AST.transf_fundef transf_function fd.
+
+Definition transf_program (p: program) : program :=
+  transform_program transf_fundef p.