# Module Fixpoints

From Coq Require Import Arith ZArith Psatz Bool String List FMaps Program Wellfounded.
Require Import Sequences IMP Compil Constprop.

## 9.1 From Knaster-Tarski to effective computation of fixpoints

Section FIXPOINT.

Consider a type A equipped with a decidable equality eq and a transitive ordering le.

Variable A: Type.

Variable eq: A -> A -> Prop.
Variable beq: A -> A -> bool.
Hypothesis beq_eq: forall x y, beq x y = true -> eq x y.
Hypothesis beq_neq: forall x y, beq x y = false -> ~eq x y.

Variable le: A -> A -> Prop.
Hypothesis le_trans: forall x y z, le x y -> le y z -> le x z.

This is the strict order induced by le. We assume it is well-founded: all strictly ascending chains are finite.

Definition gt (x y: A) := le y x /\ ~eq y x.
Hypothesis gt_wf: well_founded gt.

Let bot be a smallest element of A.
Variable bot: A.
Hypothesis bot_smallest: forall x, le bot x.

Let F be a monotonic function from A to A.

Variable F: A -> A.
Hypothesis F_mon: forall x y, le x y -> le (F x) (F y).

Let's show the existence of a fixpoint, as in the Knaster-Tarski theorem. The proof is by well-founded induction.

Lemma fixpoint_exists_1: exists x, eq x (F x).
Proof.
assert(REC: forall x, le x (F x) -> exists y, eq y (F y)).
{
induction x using (well_founded_ind gt_wf).
intros LE.
set (x' := F x).
destruct (beq x x') eqn:EQ.
- exists x. apply beq_eq. auto.
- apply H with x'.
+ split. apply LE. apply beq_neq. auto.
+ apply F_mon. apply LE.
}
apply REC with (x := bot). apply bot_smallest.
Qed.

Unfortunately, we cannot extract the value of the fixpoint.

Fail Definition fixpoint_1: A :=
match fixpoint_exists_1 with ex_intro _ x P => x end.

To compute the fixpoint effectively, we need to use Sigma-types { x : A | P x } instead of "there exists" quantifiers exists x : A, P x .

Lemma fixpoint_exists_2: { x : A | eq x (F x) }.
Proof.
assert(REC: forall x, le x (F x) -> { y : A | eq y (F y) } ).
{
induction x using (Fix gt_wf).
intros LE.
set (x' := F x).
destruct (beq x x') eqn:EQ.
- exists x. apply beq_eq. auto.
- apply X with x'.
+ split. apply LE. apply beq_neq. auto.
+ apply F_mon. apply LE.
}
apply REC with (x := bot). apply bot_smallest.
Defined.

Now we can extract the value of the fixpoint we found, and a proof that it is a fixpoint.

Definition fixpoint_2 : A := proj1_sig fixpoint_exists_2.

Lemma fixpoint_2_correct: eq fixpoint_2 (F fixpoint_2).
Proof.
unfold fixpoint_2. apply proj2_sig.
Qed.

However, since the program fixpoint_exists_2 was synthesized by Coq's proof tactics, we don't know exactly what algorithm is used to compute the fixpoint, and whether it is efficient. We can look at the term that Coq synthesized, but it's completely unreadable!

Alternatively, we can use the Program Fixpoint mechanism to write the iteration algorithm explicitly, leaving holes ``_'' where proofs are needed. Program Fixpoint then drops into interactive proof mode to let us construct those proofs.

The algorithm is simple: we iterate F starting from a pre-fixpoint x. The iterate function takes as argument not just x, but also two proofs:
• that x is a pre-fixpoint, i.e. le x (F x)
• that x is below any post-fixpoint z.
Likewise, it returns as result not just the fixpoint y, but also two proofs:
• that y is a fixpoint, i.e. eq y (F y)
• that y is below any post-fixpoint z.

Program Fixpoint iterate
(x: A) (PRE: le x (F x)) (SMALL: forall z, le (F z) z -> le x z)
{wf gt x}
: {y : A | eq y (F y) /\ forall z, le (F z) z -> le y z } :=
let x' := F x in
match beq x x' with
| true => x
| false => iterate x' _ _
end.
All proof obligations corresponding to ``holes'' were solved automatically, except one: that the argument x' to the recursive call is strictly greater than the parameter x. This we prove interactively.
Next Obligation.
split.
- auto.
- apply beq_neq. auto.
Qed.

The fixpoint is obtained by iterating from bot.

Program Definition fixpoint : A := iterate bot _ _.

It is therefore both a fixpoint and the smallest post-fixpoint.

Theorem fixpoint_correct:
eq fixpoint (F fixpoint) /\ forall z, le (F z) z -> le fixpoint z.
Proof.
unfold fixpoint. apply proj2_sig.
Qed.

End FIXPOINT.

## 9.2 Application to constant propagation analysis

First, we equip the type of abstract stores with the required equality and ordering predicates. le and beq are defined in Constprop, under the names le' and equal'.

Definition Eq (S1 S2: Store) : Prop :=
IdentMap.Equal S1 S2.

Lemma Eq_sym: forall S1 S2, Eq S1 S2 -> Eq S2 S1.
Proof.
intros. apply IMFact.Equal_sym. auto.
Qed.

Lemma Equal_Eq: forall S1 S2, Equal S1 S2 = true -> Eq S1 S2.
Proof.
intros. unfold Eq. apply <- IMFact.Equal_Equivb.
apply IdentMap.equal_2. eauto. apply Z.eqb_eq.
Qed.

Lemma Equal_nEq: forall S1 S2, Equal S1 S2 = false -> ~ Eq S1 S2.
Proof.
intros. unfold Eq. intros IE.
assert (Equal S1 S2 = true).
{ apply IdentMap.equal_1. apply IMFact.Equal_Equivb. apply Z.eqb_eq. auto. }
congruence.
Qed.

Lemma Eq_Le: forall S1 S2, Eq S1 S2 -> Le S1 S2.
Proof.
unfold Eq, Le; intros. rewrite H; auto.
Qed.

Lemma Le_trans: forall S1 S2 S3, Le S1 S2 -> Le S2 S3 -> Le S1 S3.
Proof.
unfold Le; auto.
Qed.

Definition Gt (S1 S2: Store) := Le S2 S1 /\ ~ Eq S2 S1.

We will use monotonically increasing functions a lot.

Definition Increasing (F: Store -> Store) : Prop :=
forall x y, Le x y -> Le (F x) (F y).

The really hard proof is to show that the strict order Gt is well-founded. For this we reason on the cardinal of the finite maps representing abstract stores, that is, the number of x = n facts contained in abstract stores.

Lemma Le_cardinal:
forall S T,
Le T S ->
IdentMap.cardinal S <= IdentMap.cardinal T
/\ (IdentMap.cardinal S = IdentMap.cardinal T -> IdentMap.Equal T S).
Proof.
induction S using IMProp.map_induction.
- intros. rewrite IMProp.cardinal_1 by auto. split.
+ lia.
+ intros.
assert (IdentMap.Empty T) by (apply IMProp.cardinal_inv_1; auto).
apply IMFact.Equal_mapsto_iff. intros.
specialize (H k e); specialize (H2 k e). tauto.
- intros T2 LE.
set (T1 := IdentMap.remove x T2).
assert (~ IdentMap.In x T1).
{ apply IdentMap.remove_1; auto. }
assert (IMProp.Add x e T1 T2).
{ intros y. unfold T1. rewrite IMFact.add_o, IMFact.remove_o.
destruct (IMProp.F.eq_dec x y); auto.
subst x. apply LE. rewrite H0. apply IMFact.add_eq_o. auto. }
assert (Le T1 S1).
{ red; intros. unfold T1; rewrite IMFact.remove_o.
destruct (IMProp.F.eq_dec x x0).
subst x0. exfalso; apply H. apply IMFact.in_find_iff. congruence.
apply LE. rewrite H0. rewrite IMFact.add_neq_o; auto.
}
rewrite (@IMProp.cardinal_2 _ S1 S2 x e) by auto.
rewrite (@IMProp.cardinal_2 _ T1 T2 x e) by auto.
destruct (IHS1 T1) as [A B]; auto.
split.
+ lia.
+ intros.
assert (IdentMap.Equal T1 S1) by (apply B; lia).
intros y. rewrite H0, H2. rewrite ! IMFact.add_o.
destruct (IMProp.F.eq_dec x y); auto.
Qed.

Lemma Gt_cardinal:
forall S S', Gt S S' -> IdentMap.cardinal S < IdentMap.cardinal S'.
Proof.
intros S S' [LE NEQ].
edestruct Le_cardinal as [A B]. eauto.
assert (IdentMap.cardinal S <> IdentMap.cardinal S').
{ intros EQ; apply NEQ; apply B; auto. }
lia.
Qed.

Lemma Gt_wf: well_founded Gt.
Proof.
apply wf_incl with (ltof Store (@IdentMap.cardinal Z)).
- intros S S' GT. apply Gt_cardinal; auto.
- apply well_founded_ltof.
Qed.

Another difficulty is that our type of abstract stores does not have a smallest element bot. But for the kind of functions we take fixpoints of, we know a pre-fixpoint we can start iterating with.

Section FIXPOINT_JOIN.

Variable Init: Store.
Variable F: Store -> Store.
Hypothesis F_incr: Increasing F.

Program Definition fixpoint_join : Store :=
iterate Store Eq Equal Equal_Eq Equal_nEq Le Le_trans Gt_wf
(fun x => Join Init (F x)) _ Init _ _.
Next Obligation.
unfold Le; intros.
apply Join_characterization in H0. destruct H0 as [FIND1 FIND2].
apply Join_characterization. split; auto. apply (F_incr _ _ H). auto.
Qed.
Next Obligation.
apply Le_Join_l.
Qed.
Next Obligation.
eapply Le_trans. apply Le_Join_l. eauto.
Qed.

Lemma fixpoint_join_eq:
Eq (Join Init (F fixpoint_join)) fixpoint_join.
Proof.
apply Eq_sym. unfold fixpoint_join.
destruct iterate as (X & EQ & SMALL). auto.
Qed.

Lemma fixpoint_join_sound:
Le Init fixpoint_join /\ Le (F fixpoint_join) fixpoint_join.
Proof.
assert (LE: Le (Join Init (F fixpoint_join)) fixpoint_join).
{ apply Eq_Le. apply fixpoint_join_eq. }
split.
- eapply Le_trans. apply Le_Join_l. eauto.
- eapply Le_trans. apply Le_Join_r. eauto.
Qed.

Lemma fixpoint_join_smallest:
forall S, Le (Join Init (F S)) S -> Le fixpoint_join S.
Proof.
unfold fixpoint_join. destruct iterate as (X & EQ & SMALL).
auto.
Qed.

End FIXPOINT_JOIN.

Now we can try to use the fixpoint_join function above in the Cexec static analyzer. But we are in for a surprise!

Fail Fixpoint Cexec (S: Store) (c: com) : Store :=
match c with
| SKIP => S
| ASSIGN x a => Update x (Aeval S a) S
| SEQ c1 c2 => Cexec (Cexec S c1) c2
| IFTHENELSE b c1 c2 =>
match Beval S b with
| Some true => Cexec S c1
| Some false => Cexec S c2
| None => Join (Cexec S c1) (Cexec S c2)
end
| WHILE b c1 =>
fixpoint_join S (fun x => Cexec x c1) _
end.

It says: "Cannot infer this placeholder of type Increasing (fun x : Store => Cexec x c1)". Of course! We can only take fixpoints of increasing functions! So, we need to find a way to define the Cexec function and SIMULTANEOUSLY show that it is increasing. This can be done, but we'll need a lot of lemmas about increasing functions first.

Lemma Join_increasing:
forall S1 S2 S3 S4,
Le S1 S2 -> Le S3 S4 -> Le (Join S1 S3) (Join S2 S4).
Proof.
unfold Le; intros. rewrite Join_characterization in *. destruct H1; split; auto.
Qed.

Lemma Aeval_increasing:
forall S1 S2, Le S1 S2 ->
forall a n, Aeval S2 a = Some n -> Aeval S1 a = Some n.
Proof.
intros S1 S2 LE; induction a; cbn; intros.
- auto.
- apply LE; auto.
- destruct (Aeval S2 a1), (Aeval S2 a2); cbn in H; inv H.
erewrite IHa1, IHa2 by eauto. auto.
- destruct (Aeval S2 a1), (Aeval S2 a2); cbn in H; inv H.
erewrite IHa1, IHa2 by eauto. auto.
Qed.

Lemma Beval_increasing:
forall S1 S2, Le S1 S2 ->
forall b n, Beval S2 b = Some n -> Beval S1 b = Some n.
Proof.
intros S1 S2 LE; induction b; cbn; intros.
- auto.
- auto.
- destruct (Aeval S2 a1) eqn:A1, (Aeval S2 a2) eqn:A2; cbn in H; inv H.
erewrite ! Aeval_increasing by eauto. auto.
- destruct (Aeval S2 a1) eqn:A1, (Aeval S2 a2) eqn:A2; cbn in H; inv H.
erewrite ! Aeval_increasing by eauto. auto.
- destruct (Beval S2 b); cbn in H; inv H.
erewrite IHb by eauto. auto.
- destruct (Beval S2 b1), (Beval S2 b2); cbn in H; inv H.
erewrite IHb1, IHb2 by eauto. auto.
Qed.

Lemma Update_increasing:
forall S1 S2 x a,
Le S1 S2 ->
Le (Update x (Aeval S1 a) S1) (Update x (Aeval S2 a) S2).
Proof.
intros; unfold Le; intros.
rewrite Update_characterization in *.
destruct (string_dec x x0); auto.
subst x0. apply Aeval_increasing with S2; auto.
Qed.

Lemma fixpoint_Join_increasing:
forall F (F_incr: Increasing F) S1 S2,
Le S1 S2 -> Le (fixpoint_join S1 F F_incr) (fixpoint_join S2 F F_incr).
Proof.
intros. apply fixpoint_join_smallest.
set (fix2 := fixpoint_join S2 F F_incr).
assert (Le (Join S2 (F fix2)) fix2) by (apply Eq_Le; apply fixpoint_join_eq).
eapply Le_trans; eauto.
apply Join_increasing; auto. unfold Le; auto.
Qed.

At long last, we can define Cexec while at the same time showing that it is increasing.

Program Fixpoint Cexec (c: com) : { F: Store -> Store | Increasing F } :=
match c with
| SKIP => fun S => S
| ASSIGN x a => fun S => Update x (Aeval S a) S
| SEQ c1 c2 => compose (Cexec c2) (Cexec c1)
| IFTHENELSE b c1 c2 =>
fun S =>
match Beval S b return _ with
| Some true => Cexec c1 S
| Some false => Cexec c2 S
| None => Join (Cexec c1 S) (Cexec c2 S)
end
| WHILE b c1 =>
fun S => fixpoint_join S (fun S => Cexec c1 S) _
end.
Next Obligation.
unfold Increasing. auto.
Defined.
Next Obligation.
intros y z LE. apply Update_increasing; auto.
Defined.
Next Obligation.
intros y z LE. unfold compose. auto.
Defined.
Next Obligation.
intros y z LE. destruct (Beval z b) as [bz|] eqn:BEV.
erewrite Beval_increasing by eauto. destruct bz; auto.
destruct (Beval y b) as [[]|].
apply Le_trans with (x z). auto. apply Le_Join_l.
apply Le_trans with (x0 z). auto. apply Le_Join_r.
apply Join_increasing; auto.
Defined.
Next Obligation.
intros y z LE. apply fixpoint_Join_increasing. auto.
Defined.