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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 8: Abstract Interpretation and Dataflow Analysis
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap Imp.
Export Imp.
Set Implicit Arguments.
(* Reduced version of code from AbstractInterpretation.v *)
Record absint := {
Domain :> Set;
Top : Domain;
Constant : nat -> Domain;
Add : Domain -> Domain -> Domain;
Subtract : Domain -> Domain -> Domain;
Multiply : Domain -> Domain -> Domain;
Join : Domain -> Domain -> Domain;
Represents : nat -> Domain -> Prop
}.
Record absint_sound (a : absint) : Prop := {
TopSound : forall n, a.(Represents) n a.(Top);
ConstSound : forall n, a.(Represents) n (a.(Constant) n);
AddSound : forall n na m ma, a.(Represents) n na
-> a.(Represents) m ma
-> a.(Represents) (n + m) (a.(Add) na ma);
SubtractSound: forall n na m ma, a.(Represents) n na
-> a.(Represents) m ma
-> a.(Represents) (n - m) (a.(Subtract) na ma);
MultiplySound : forall n na m ma, a.(Represents) n na
-> a.(Represents) m ma
-> a.(Represents) (n * m) (a.(Multiply) na ma);
AddMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
-> (forall n, a.(Represents) n (a.(Add) na ma)
-> a.(Represents) n (a.(Add) na' ma'));
SubtractMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
-> (forall n, a.(Represents) n (a.(Subtract) na ma)
-> a.(Represents) n (a.(Subtract) na' ma'));
MultiplyMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
-> (forall n, a.(Represents) n (a.(Multiply) na ma)
-> a.(Represents) n (a.(Multiply) na' ma'));
JoinSoundLeft : forall x y n, a.(Represents) n x
-> a.(Represents) n (a.(Join) x y);
JoinSoundRight : forall x y n, a.(Represents) n y
-> a.(Represents) n (a.(Join) x y)
}.
Global Hint Resolve TopSound ConstSound AddSound SubtractSound MultiplySound
AddMonotone SubtractMonotone MultiplyMonotone
JoinSoundLeft JoinSoundRight : core.
Definition astate (a : absint) := fmap var a.
Fixpoint absint_interp (e : arith) a (s : astate a) : a :=
match e with
| Const n => a.(Constant) n
| Var x => match s $? x with
| None => a.(Top)
| Some xa => xa
end
| Plus e1 e2 => a.(Add) (absint_interp e1 s) (absint_interp e2 s)
| Minus e1 e2 => a.(Subtract) (absint_interp e1 s) (absint_interp e2 s)
| Times e1 e2 => a.(Multiply) (absint_interp e1 s) (absint_interp e2 s)
end.
Definition merge_astate a : astate a -> astate a -> astate a :=
merge (fun x y =>
match x with
| None => None
| Some x' =>
match y with
| None => None
| Some y' => Some (a.(Join) x' y')
end
end).
Definition subsumed a (s1 s2 : astate a) :=
forall x, match s1 $? x with
| None => s2 $? x = None
| Some xa1 =>
forall xa2, s2 $? x = Some xa2
-> forall n, a.(Represents) n xa1
-> a.(Represents) n xa2
end.
Theorem subsumed_refl : forall a (s : astate a),
subsumed s s.
Proof.
unfold subsumed; simplify.
cases (s $? x); equality.
Qed.
Global Hint Resolve subsumed_refl : core.
Lemma subsumed_use : forall a (s s' : astate a) x n t0 t,
s $? x = Some t0
-> subsumed s s'
-> s' $? x = Some t
-> Represents a n t0
-> Represents a n t.
Proof.
unfold subsumed; simplify.
specialize (H0 x).
rewrite H in H0.
eauto.
Qed.
Lemma subsumed_use_empty : forall a (s s' : astate a) x n t0 t,
s $? x = None
-> subsumed s s'
-> s' $? x = Some t
-> Represents a n t0
-> Represents a n t.
Proof.
unfold subsumed; simplify.
specialize (H0 x).
rewrite H in H0.
equality.
Qed.
Global Hint Resolve subsumed_use subsumed_use_empty : core.
Lemma subsumed_trans : forall a (s1 s2 s3 : astate a),
subsumed s1 s2
-> subsumed s2 s3
-> subsumed s1 s3.
Proof.
unfold subsumed; simplify.
specialize (H x); specialize (H0 x).
cases (s1 $? x); simplify.
cases (s2 $? x); eauto.
cases (s2 $? x); eauto.
equality.
Qed.
Lemma subsumed_merge_left : forall a, absint_sound a
-> forall s1 s2 : astate a,
subsumed s1 (merge_astate s1 s2).
Proof.
unfold subsumed, merge_astate; simplify.
cases (s1 $? x); trivial.
cases (s2 $? x); simplify; try equality.
invert H0; eauto.
Qed.
Global Hint Resolve subsumed_merge_left : core.
Lemma subsumed_add : forall a, absint_sound a
-> forall (s1 s2 : astate a) x v1 v2,
subsumed s1 s2
-> (forall n, a.(Represents) n v1 -> a.(Represents) n v2)
-> subsumed (s1 $+ (x, v1)) (s2 $+ (x, v2)).
Proof.
unfold subsumed; simplify.
cases (x ==v x0); subst; simplify; eauto.
invert H2; eauto.
specialize (H0 x0); eauto.
Qed.
Global Hint Resolve subsumed_add : core.
(** * Flow-sensitive analysis *)
Definition compatible a (s : astate a) (v : valuation) : Prop :=
forall x xa, s $? x = Some xa
-> exists n, v $? x = Some n
/\ a.(Represents) n xa.
Lemma compatible_add : forall a (s : astate a) v x na n,
compatible s v
-> a.(Represents) n na
-> compatible (s $+ (x, na)) (v $+ (x, n)).
Proof.
unfold compatible; simplify.
cases (x ==v x0); simplify; eauto.
invert H1; eauto.
Qed.
Global Hint Resolve compatible_add : core.
(* A similar result follows about soundness of expression interpretation. *)
Theorem absint_interp_ok : forall a, absint_sound a
-> forall (s : astate a) v e,
compatible s v
-> a.(Represents) (interp e v) (absint_interp e s).
Proof.
induct e; simplify; eauto.
cases (s $? x); auto.
unfold compatible in H0.
apply H0 in Heq.
invert Heq.
propositional.
rewrite H2.
assumption.
Qed.
Global Hint Resolve absint_interp_ok : core.
Definition astates (a : absint) := fmap cmd (astate a).
Fixpoint absint_step a (s : astate a) (c : cmd) (wrap : cmd -> cmd) : option (astates a) :=
match c with
| Skip => None
| Assign x e => Some ($0 $+ (wrap Skip, s $+ (x, absint_interp e s)))
| Sequence c1 c2 =>
match absint_step s c1 (fun c => wrap (Sequence c c2)) with
| None => Some ($0 $+ (wrap c2, s))
| v => v
end
| If _ then_ else_ => Some ($0 $+ (wrap then_, s) $+ (wrap else_, s))
| While e body => Some ($0 $+ (wrap Skip, s) $+ (wrap (Sequence body (While e body)), s))
end.
Lemma command_equal : forall c1 c2 : cmd, sumbool (c1 = c2) (c1 <> c2).
Proof.
repeat decide equality.
Qed.
Theorem absint_step_ok : forall a, absint_sound a
-> forall (s : astate a) v, compatible s v
-> forall c v' c', step (v, c) (v', c')
-> forall wrap, exists ss s', absint_step s c wrap = Some ss
/\ ss $? wrap c' = Some s'
/\ compatible s' v'.
Proof.
induct 2; simplify.
do 2 eexists; propositional.
simplify; equality.
eauto.
eapply IHstep in H0; auto.
invert H0.
invert H2.
propositional.
rewrite H2.
eauto.
do 2 eexists; propositional.
simplify; equality.
assumption.
do 2 eexists; propositional.
cases (command_equal (wrap c') (wrap else_)).
simplify; equality.
simplify; equality.
assumption.
do 2 eexists; propositional.
simplify; equality.
assumption.
do 2 eexists; propositional.
simplify; equality.
assumption.
do 2 eexists; propositional.
cases (command_equal (wrap Skip) (wrap (body;; while e loop body done))).
simplify; equality.
simplify; equality.
assumption.
Qed.
Inductive abs_step a : astate a * cmd -> astate a * cmd -> Prop :=
| AbsStep : forall s c ss s' c',
absint_step s c (fun x => x) = Some ss
-> ss $? c' = Some s'
-> abs_step (s, c) (s', c').
Global Hint Constructors abs_step : core.
Definition absint_trsys a (c : cmd) := {|
Initial := {($0, c)};
Step := abs_step (a := a)
|}.
Inductive Rabsint a : valuation * cmd -> astate a * cmd -> Prop :=
| RAbsint : forall v s c,
compatible s v
-> Rabsint (v, c) (s, c).
Global Hint Constructors abs_step Rabsint : core.
Theorem absint_simulates : forall a v c,
absint_sound a
-> simulates (Rabsint (a := a)) (trsys_of v c) (absint_trsys a c).
Proof.
simplify.
constructor; simplify.
exists ($0, c); propositional.
subst.
constructor.
unfold compatible.
simplify.
equality.
invert H0.
cases st1'.
eapply absint_step_ok in H1; eauto.
invert H1.
invert H0.
propositional.
eauto.
Qed.
Definition merge_astates a : astates a -> astates a -> astates a :=
merge (fun x y =>
match x with
| None => y
| Some x' =>
match y with
| None => Some x'
| Some y' => Some (merge_astate x' y')
end
end).
Inductive oneStepClosure a : astates a -> astates a -> Prop :=
| OscNil :
oneStepClosure $0 $0
| OscCons : forall ss c s ss' ss'',
oneStepClosure ss ss'
-> match absint_step s c (fun x => x) with
| None => ss'
| Some ss'' => merge_astates ss'' ss'
end = ss''
-> oneStepClosure (ss $+ (c, s)) ss''.
Definition subsumeds a (ss1 ss2 : astates a) :=
forall c s1, ss1 $? c = Some s1
-> exists s2, ss2 $? c = Some s2
/\ subsumed s1 s2.
Theorem subsumeds_refl : forall a (ss : astates a),
subsumeds ss ss.
Proof.
unfold subsumeds; simplify; eauto.
Qed.
Global Hint Resolve subsumeds_refl : core.
Lemma subsumeds_add : forall a (ss1 ss2 : astates a) c s1 s2,
subsumeds ss1 ss2
-> subsumed s1 s2
-> subsumeds (ss1 $+ (c, s1)) (ss2 $+ (c, s2)).
Proof.
unfold subsumeds; simplify.
cases (command_equal c c0); subst; simplify; eauto.
invert H1; eauto.
Qed.
Global Hint Resolve subsumeds_add : core.
Lemma subsumeds_empty : forall a (ss : astates a),
subsumeds $0 ss.
Proof.
unfold subsumeds; simplify.
equality.
Qed.
Lemma subsumeds_add_left : forall a (ss1 ss2 : astates a) c s,
ss2 $? c = Some s
-> subsumeds ss1 ss2
-> subsumeds (ss1 $+ (c, s)) ss2.
Proof.
unfold subsumeds; simplify.
cases (command_equal c c0); subst; simplify; eauto.
invert H1; eauto.
Qed.
Inductive interpret a : astates a -> astates a -> astates a -> Prop :=
| InterpretDone : forall ss1 any ss2,
oneStepClosure ss1 ss2
-> subsumeds ss2 ss1
-> interpret ss1 any ss1
| InterpretStep : forall ss worklist ss' ss'',
oneStepClosure worklist ss'
-> interpret (merge_astates ss ss') ss' ss''
-> interpret ss worklist ss''.
Lemma oneStepClosure_sound : forall a, absint_sound a
-> forall ss ss' : astates a, oneStepClosure ss ss'
-> forall c s s' c', ss $? c = Some s
-> abs_step (s, c) (s', c')
-> exists s'', ss' $? c' = Some s''
/\ subsumed s' s''.
Proof.
induct 2; simplify.
equality.
cases (command_equal c c0); subst; simplify.
invert H2.
invert H3.
rewrite H5.
unfold merge_astates; simplify.
rewrite H7.
cases (ss' $? c').
eexists; propositional.
unfold subsumed; simplify.
unfold merge_astate; simplify.
cases (s' $? x); try equality.
cases (a0 $? x); simplify; try equality.
invert H1; eauto.
eauto.
apply IHoneStepClosure in H3; auto.
invert H3; propositional.
cases (absint_step s c (fun x => x)); eauto.
unfold merge_astates; simplify.
rewrite H3.
cases (a0 $? c'); eauto.
eexists; propositional.
unfold subsumed; simplify.
unfold merge_astate; simplify.
specialize (H4 x0).
cases (s' $? x0).
cases (a1 $? x0); try equality.
cases (x $? x0); try equality.
invert 1.
eauto.
rewrite H4.
cases (a1 $? x0); equality.
Qed.
Lemma absint_step_monotone_None : forall a (s : astate a) c wrap,
absint_step s c wrap = None
-> forall s' : astate a, absint_step s' c wrap = None.
Proof.
induct c; simplify; try equality.
cases (absint_step s c1 (fun c => wrap (c;; c2))); equality.
Qed.
Lemma absint_interp_monotone : forall a, absint_sound a
-> forall (s : astate a) e s' n,
a.(Represents) n (absint_interp e s)
-> subsumed s s'
-> a.(Represents) n (absint_interp e s').
Proof.
induct e; simplify; eauto.
cases (s' $? x); eauto.
cases (s $? x); eauto.
Qed.
Global Hint Resolve absint_interp_monotone : core.
Lemma absint_step_monotone : forall a, absint_sound a
-> forall (s : astate a) c wrap ss,
absint_step s c wrap = Some ss
-> forall s', subsumed s s'
-> exists ss', absint_step s' c wrap = Some ss'
/\ subsumeds ss ss'.
Proof.
induct c; simplify.
equality.
invert H0.
eexists; propositional.
eauto.
apply subsumeds_add; eauto.
cases (absint_step s c1 (fun c => wrap (c;; c2))).
invert H0.
eapply IHc1 in Heq; eauto.
invert Heq; propositional.
rewrite H2; eauto.
invert H0.
eapply absint_step_monotone_None in Heq; eauto.
rewrite Heq; eauto.
invert H0; eauto.
invert H0; eauto.
Qed.
Lemma abs_step_monotone : forall a, absint_sound a
-> forall (s : astate a) c s' c',
abs_step (s, c) (s', c')
-> forall s1, subsumed s s1
-> exists s1', abs_step (s1, c) (s1', c')
/\ subsumed s' s1'.
Proof.
invert 2; simplify.
eapply absint_step_monotone in H4; eauto.
invert H4; propositional.
apply H3 in H6.
invert H6; propositional; eauto.
Qed.
Lemma interpret_sound' : forall c a, absint_sound a
-> forall ss worklist ss' : astates a, interpret ss worklist ss'
-> ss $? c = Some $0
-> invariantFor (absint_trsys a c) (fun p => exists s, ss' $? snd p = Some s
/\ subsumed (fst p) s).
Proof.
induct 2; simplify; subst.
apply invariant_induction; simplify; propositional; subst; simplify; eauto.
invert H3; propositional.
cases s.
cases s'.
simplify.
eapply abs_step_monotone in H4; eauto.
invert H4; propositional.
eapply oneStepClosure_sound in H4; eauto.
invert H4; propositional.
eapply H1 in H4.
invert H4; propositional.
eauto using subsumed_trans.
apply IHinterpret.
unfold merge_astates; simplify.
rewrite H2.
cases (ss' $? c); trivial.
unfold merge_astate; simplify; equality.
Qed.
Theorem interpret_sound : forall c a (ss : astates a),
absint_sound a
-> interpret ($0 $+ (c, $0)) ($0 $+ (c, $0)) ss
-> invariantFor (absint_trsys a c) (fun p => exists s, ss $? snd p = Some s
/\ subsumed (fst p) s).
Proof.
simplify.
eapply interpret_sound'; eauto.
simplify; equality.
Qed.
Ltac interpret_simpl := unfold merge_astates, merge_astate;
simplify; repeat simplify_map.
Ltac oneStepClosure := apply OscNil
|| (eapply OscCons; [ oneStepClosure
| interpret_simpl; reflexivity ]).
Ltac interpret1 := eapply InterpretStep; [ oneStepClosure | interpret_simpl ].
Ltac interpret_done := eapply InterpretDone; [ oneStepClosure
| repeat (apply subsumeds_add_left || apply subsumeds_empty); (simplify; equality) ].
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