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Commit adedfdce authored by Ralf Jung's avatar Ralf Jung
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Set Automatic Coercions Import.
Require Import ssreflect ssrfun ssrbool eqtype seq fintype.
Require Import ssreflect ssrfun ssrbool eqtype.
Require Import core_lang masks iris_wp.
Require Import ModuRes.PCM ModuRes.UPred ModuRes.BI ModuRes.PreoMet ModuRes.Finmap.
......@@ -15,14 +15,40 @@ Module Unsafety (RL : PCM_T) (C : CORE_LANG).
Local Open Scope lang_scope.
Local Open Scope iris_scope.
Implicit Types (P Q R : Props) (i n k : nat) (safe : bool) (m : mask) (e : expr) (φ : value -n> Props) (r : res) (w : Wld).
(* PDS: Why isn't this inferred automatically? If necessary, move this to ris_core.v *)
Global Program Instance res_preorder : PreOrder (pcm_ord res) := @preoPO res (PCM_preo res).
(* PDS: Move to iris_core.v *)
Lemma ownL_timeless {r : option RL.res} :
valid(timeless(ownL r)).
Proof. intros w n _ w' k r' HSW HLE; now destruct r. Qed.
(* PDS: Hoist, somewhere. *)
Program Definition restrictV (Q : expr -n> Props) : vPred :=
n[(fun v => Q (` v))].
Solve Obligations using resp_set.
Next Obligation.
move=> v v' Hv w k r; move: Hv.
case: n=>[_ Hk | n]; first by exfalso; omega.
by move=> /= ->.
Qed.
(* PDS: masks.v *)
Lemma mask_full_disjoint m (HD : m # mask_full) :
meq m mask_emp.
Proof.
move=> i; move: {HD} (HD i) => HD; split; last done.
move=> Hm; exfalso; exact: HD.
Qed.
Implicit Types (P : Props) (i n k : nat) (safe : bool) (m : mask) (e : expr) (Q : vPred) (r : res) (w : Wld) (σ : state).
(* PDS: Move to iris_wp.v *)
Lemma htUnsafe {m P e φ} : ht true m P e φ ⊑ ht false m P e φ.
Lemma htUnsafe {m P e Q} : ht true m P e Q ⊑ ht false m P e Q.
Proof.
move=> wz nz rz He w HSw n r HLe Hr HP.
move: {He P wz nz rz HSw HLe Hr HP} (He _ HSw _ _ HLe Hr HP).
move: n e φ w r; elim/wf_nat_ind; move=> n IH e φ w r He.
move: n e Q w r; elim/wf_nat_ind; move=> n IH e Q w r He.
rewrite unfold_wp; move=> w' k rf mf σ HSw HLt HD Hw.
move: {IH} (IH _ HLt) => IH.
move: He => /unfold_wp He; move: {He HSw HLt HD Hw} (He _ _ _ _ _ HSw HLt HD Hw) => [HV [HS [HF _] ] ].
......@@ -48,17 +74,21 @@ Module Unsafety (RL : PCM_T) (C : CORE_LANG).
Notation "a ⊑%pcm b" := (pcm_ord _ a b) (at level 70, no associativity) : pcm_scope.
Lemma wpO {safe m e φ w r} : wp safe m e φ w O r.
Lemma wpO {safe m e Q w r} : wp safe m e Q w O r.
Proof.
rewrite unfold_wp.
move=> w' k rf mf σ HSw HLt HD HW.
by absurd (k < 0); omega.
by exfalso; omega.
Qed.
(* Easier to apply (with SSR, at least) than wsat_not_empty. *)
Lemma wsat_nz {σ m w k} : ~ wsat σ m 0 w (S k) tt.
Proof. by move=> [rs [HD _] ]; exact: HD. Qed.
(* Leibniz equality arise from SSR's case tactic. *)
Lemma equivP {T : Type} `{eqT : Setoid T} {a b : T} : a = b -> a == b.
Proof. by move=>->; reflexivity. Qed.
(*
Simple monotonicity tactics for props and wsat.
......@@ -85,235 +115,164 @@ Module Unsafety (RL : PCM_T) (C : CORE_LANG).
Ltac wsatM H := solve [done | apply (wsatM H); solve [done | omega] ].
Notation "'ℊ' a" := (pcm_unit(pcm_res_ex state), a)
(at level 41, format "ℊ a") : iris_scope.
(*
* Robust safety:
*
* Assume E : Exp → Prop satisfies
*
* E(fork e) = E e
* E(K[e]) = E e * E(K[()])
*
* and there exist Γ₀, Θ₀ s.t.
*
* (i) for every pure reduction ς; e → ς; e',
* Γ₀ | □Θ₀ ⊢ E e ==>>_⊤ E e'
*
* (ii) for every atomic reduction ς; e → ς'; e',
* Γ₀ | □Θ₀ ⊢ [E e] e [v. E v]_⊤
*
* Then, for every e, Γ₀ | □Θ₀ ⊢ [E e] e [v. E v]_⊤.
*
* The theorem has applications to security (hence the name).
*)
Section RobustSafety.
(* The adversarial resources in e. *)
Variable prim_of : expr -> RL.res.
Variable prim_dup : forall {e}, (* Irrelevant to robust_safety. *)
Some(prim_of e) == Some(prim_of e) · Some(prim_of e).
(* Compatibility. *)
Hypothesis prim_of_fork : forall {e},
prim_of (fork e) == prim_of e.
Hypothesis prim_of_fork_ret : (* Irrelevant to robust_safety. *)
prim_of fork_ret == pcm_unit RL.res.
Hypothesis prim_of_split : forall {K e},
Some(prim_of (K [[e]])) == Some(prim_of e) · Some(prim_of (K [[fork_ret]])).
(*
* Adversarial steps need only adversarial resources and preserve
* any frame and all invariants.
*)
Notation "'ɑ' e" := (ℊ (prim_of e))
(at level 41, format "ɑ e") : iris_scope. (* K[[e]] at level 40 *)
Hypothesis adv_step : forall {e σ e' σ' rf w k},
prim_step (e,σ) (e',σ') ->
wsat σ mask_full (Some (ɑ e) · rf) w @ S k ->
exists w', w ⊑ w' /\
wsat σ' mask_full (Some (ɑ e') · rf) w' @ k.
(*
* Lift compatibility to full resources. (Trivial.)
* Assume primitive reductions are either pure (do not change the
* state) or atomic (step to a value).
*
* PDS: I suspect we need these to prove the lifting lemmas. If
* so, they should be in core_lang.v.
*)
Lemma res_of_fork {e} :
ɑ fork e == ɑ e.
Proof. by rewrite prim_of_fork. Qed.
Lemma res_of_fork_ret : (* Irrelevant to robust_safety. *)
ɑ fork_ret == ℊ pcm_unit(RL.res).
Proof. by rewrite prim_of_fork_ret. Qed.
Axiom atomic_dec : forall e, atomic e + ~atomic e.
(* PDS: Is there a cleaner way? *)
Lemma prim_res_eq_hack {a b : option RL.res} : a == b -> a = b.
Proof.
rewrite/=/opt_eq.
by case Ha: a=>[a' |]; case Hb: b=>[b' |]; first by move=>->.
Qed.
Axiom pure_step : forall e σ e' σ',
~ atomic e ->
prim_step (e, σ) (e', σ') ->
σ = σ'.
Lemma res_of_split {K e} :
Some (ɑ K [[e]]) == Some(ɑ e) · Some(ɑ K [[fork_ret]]).
Proof.
rewrite /pcm_op/res_op/pcm_op_prod.
case Hp: (_ · _)=>[p |]; last done.
rewrite /pcm_op/= in Hp; case: Hp=><-.
(*
rewrite -prim_of_split.
Maybe I'm missing some instances (rewrite, erewrite also fail):
Error:
Tactic failure:Unable to satisfy the rewriting constraints.
Unable to satisfy the following constraints:
EVARS:
?8556==[K e p _pattern_value_ _rewrite_rule_
(do_subrelation:=do_subrelation)
|- Proper (eq ==> flip impl) (SetoidClass.equiv (Some (ɑ K [[e]])))]
(internal placeholder)
?8555==[K e p _pattern_value_ _rewrite_rule_
|- subrelation SetoidClass.equiv eq] (internal placeholder)
But I can hack around it...
*)
move: (@prim_of_split K e) => /prim_res_eq_hack Hsplit.
by rewrite -Hsplit.
Qed.
Parameter E : expr -n> Props.
(*
* The predicate adv e internalizes ɑ e ⊑ -.
*)
Definition advP e r := ɑ e ⊑ r.
Program Definition adv : expr -n> Props :=
n[(fun e => m[(fun w => mkUPred (fun n r => advP e r) _)])].
Next Obligation.
move=> n k r r' HLe [α Hr'] [β Hr]; rewrite/advP.
move: Hr'; move: Hr<-;
rewrite (* α · (β · e) *)
[Some β · _]pcm_op_comm (* α · (e · β) *)
pcm_op_assoc (* (α · e) · β *)
[Some α · _]pcm_op_comm (* (e · α) · β *)
-pcm_op_assoc (* e · (α · β) *)
pcm_op_comm (* (α · β) · e *)
=> Hr'.
move: Hr'; case Hαβ: (Some α · _) => [αβ |] Hr'; last done.
by exists αβ.
Qed.
Next Obligation. done. Qed.
Next Obligation. by move=> w w' HSw. Qed.
Next Obligation. (* use the def of discrete e = n = e' *)
move=> e e' HEq w k r HLt; rewrite/=/advP.
move: HEq HLt; case: n=>[| n'] /= HEq HLt.
- by absurd(k < 0); omega.
by rewrite HEq.
Qed.
Lemma robust_safety {e} : valid (ht false mask_full (adv e) e (umconst ⊤)).
Proof.
move=> wz nz rz w HSw n r HLe _ He.
move: {HSw HLe} He; move: n e w r; elim/wf_nat_ind; move=> {wz nz rz} n IH e w r He.
rewrite unfold_wp; move=> w' k rf mf σ _ HLt _ HW.
split; [| split; [| split; [| done] ] ].
(* e value *)
- by move=> {IH HLt} HV; exists w' r; split; [by reflexivity | done].
(* e steps *)
- move=> σ' ei ei' K HDec HStep.
move: He; move: HDec->; move=> [r' He].
move: He; (* r' · K[ei] *)
rewrite
pcm_op_comm (* K[ei] · r' *)
res_of_split (* (ei · K) · r' *)
-pcm_op_assoc (* ei · (K · r') *)
=> He.
move: HW; rewrite {1}mask_full_union => HW.
(* Bug?: Error: tampering with discharged assumptions of "in" tactical
rewrite mask_full_union in HW.
*)
move: HW; rewrite -He -pcm_op_assoc; move=> {He} HW.
move: {HStep HW} (adv_step _ _ _ _ _ _ _ HStep HW) => [w'' [HSW' HW'] ].
move: HW'; (* ei' · ((K · r') · rf) *)
rewrite
pcm_op_assoc (* ei' · (K · (r' · rf)) *)
pcm_op_assoc (* ((ei' · K) · r') · rf *)
-res_of_split (* (K[ei]' · r') · rf *)
=> HW'.
move: HW' HLt; case HEq: k=>[| k'] HW' HLt.
+ by exists w' r'; split; [by reflexivity | split; [exact: wpO| done] ].
move: HW'; case Hr': (Some _ · Some _) => [r'' |] HW'; last first.
+ by rewrite pcm_op_zero in HW'; exfalso; exact: (wsat_nz HW').
exists w'' r''; split; [done | split; [| by rewrite mask_full_union] ].
apply: (IH _ HLt _ _); rewrite/=/advP/=/pcm_ord; exists r'.
by rewrite pcm_op_comm -Hr'; reflexivity.
(* e forks *)
move=> e' K HDec.
move: He; move: HDec->; move=> [r' He].
move: He; (* r' · K[fork e'] *)
rewrite
res_of_split (* r' · (fork e' · K) *)
res_of_fork (* r' · (e' · K) *)
pcm_op_comm (* (e' · K) · r' *)
-pcm_op_assoc. (* e' · (K · r') *)
case Hrret: (_ · Some r') => [rret |] He; last done.
exists w' (ɑ e') rret; split; first by reflexivity.
have {IH} IH: forall e r, ɑ e ⊑ r -> (wp false mask_full e (umconst top)) w' k r.
+ by move=> e0 r0 He0; apply: (IH _ HLt); rewrite/=/advP.
split; [| split ].
+ by apply IH; rewrite/=; exists r'; rewrite pcm_op_comm Hrret.
+ by apply IH; reflexivity.
by rewrite He; wsatM HW.
Qed.
(*
* Facts about adv.
*)
Lemma adv_timeless {e} :
valid(timeless(adv e)).
Proof. by move=> w n _ w' k r HSW HLe; rewrite/=/advP. Qed.
(* Compatibility for those expressions wp cares about. *)
Axiom forkE : forall e, E (fork e) == E e.
Axiom fillE : forall K e, E (K [[e]]) == E e * E (K [[fork_ret]]).
(* One can prove forkE, fillE as valid internal equalities. *)
Remark valid_intEq {P P' : Props} (H : valid(P === P')) : P == P'.
Proof. move=> w n r; exact: H. Qed.
Lemma adv_dup {e} :
valid(adv e → adv e * adv e).
Proof.
move=> _ _ _ w' _ k r' _ _ [β Hβ]; rewrite/=/advP.
have Hdup: Some(ɑ e) == Some(ɑ e)· Some(ɑ e).
- rewrite/pcm_op/res_op/pcm_op_prod pcm_op_unit.
by move/prim_res_eq_hack: (prim_dup e) => <-.
move: Hβ; rewrite Hdup pcm_op_assoc.
case Hβe: (Some _ · _) => [βe |]; last done.
case Hβee: (_ · _) => [βee |] Hβ; last done.
exists βe (ɑ e); split; [| split].
- by move: Hβee Hβ; rewrite /= => -> ->.
- by rewrite/=; exists β; move: Hβe; rewrite /= => ->.
by reflexivity.
Qed.
(* View shifts or atomic triples for every primitive reduction. *)
Parameter w₀ : Wld.
Definition valid₀ P := forall {w n r} (HSw₀ : w₀ ⊑ w), P w n r.
Axiom pureE : forall {e σ e'},
prim_step (e,σ) (e',σ) ->
valid₀ (vs mask_full mask_full (E e) (E e')).
Lemma adv_fork {e} :
valid(adv (fork e) ↔ adv e).
Proof. by move=> w n r; rewrite/=/advP prim_of_fork; tauto. Qed.
Axiom atomicE : forall {e},
atomic e ->
valid₀ (ht false mask_full (E e) e (restrictV E)).
Lemma adv_fork_ret :
valid(adv fork_ret).
Lemma robust_safety {e} : valid₀(ht false mask_full (E e) e (restrictV E)).
Proof.
move=> w n r; rewrite/=/advP prim_of_fork_ret /=.
by exists r; rewrite pcm_op_comm pcm_op_unit.
move=> wz nz rz HSw₀ w HSw n r HLe _ He.
have {HSw₀ HSw} HSw₀ : w₀ ⊑ w by transitivity wz.
(* For e = K[fork e'] we'll have to prove wp(e', ⊤), so the IH takes a post. *)
pose post Q := forall (v : value) w n r, (E (`v)) w n r -> (Q v) w n r.
set Q := restrictV E; have HQ: post Q by done.
move: {HLe} HSw₀ He HQ; move: n e w r Q; elim/wf_nat_ind;
move=> {wz nz rz} n IH e w r Q HSw₀ He HQ.
apply unfold_wp; move=> w' k rf mf σ HSw HLt HD HW.
split; [| split; [| split; [| done] ] ]; first 2 last.
(* e forks: fillE, IH (twice), forkE *)
- move=> e' K HDec.
have {He} He: (E e) w' k r by propsM He.
move: He; rewrite HDec fillE; move=> [re' [rK [Hr [He' HK] ] ] ].
exists w' re' rK; split; first by reflexivity.
have {IH} IH: forall Q, post Q ->
forall e r, (E e) w' k r -> wp false mask_full e Q w' k r.
+ by move=> Q0 HQ0 e0 r0 He0; apply: (IH _ HLt); first by transitivity w.
split; [exact: IH | split]; last first.
+ by move: HW; rewrite -Hr => HW; wsatM HW.
have Htop: post (umconst ⊤) by done.
by apply: (IH _ Htop e' re'); move: He'; rewrite forkE.
(* e value: done *)
- move=> {IH} HV; exists w' r; split; [by reflexivity | split; [| done] ].
by apply: HQ; propsM He.
(* e steps: fillE, atomic_dec *)
move=> σ' ei ei' K HDec HStep.
have {HSw₀} HSw₀ : w₀ ⊑ w' by transitivity w.
move: He; rewrite HDec fillE; move=> [rei [rK [Hr [Hei HK] ] ] ].
move: HW; rewrite -Hr => HW.
(* bookkeeping common to both cases. *)
have {Hei} Hei: (E ei) w' (S k) rei by propsM Hei.
have HSw': w' ⊑ w' by reflexivity.
have HLe: S k <= S k by omega.
have H1ei: pcm_unit _ ⊑ rei by exact: unit_min.
have HLt': k < S k by omega.
move: HW; rewrite
{1}mask_full_union -{1}(mask_full_union mask_emp)
-pcm_op_assoc
=> HW.
case: (atomic_dec ei) => HA; last first.
(* ei pure: pureE, IH, fillE *)
- move: (pure_step _ _ _ _ HA HStep) => {HA} Hσ.
rewrite Hσ in HStep HW => {Hσ}.
move: (pureE HStep) => {HStep} He.
move: {He} (He w' (S k) r HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
move: {HD} (mask_emp_disjoint (mask_full ∪ mask_full)) => HD.
move: {He HSw' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [w'' [r' [HSw' [Hei' HW] ] ] ].
move: HW; rewrite pcm_op_assoc.
case Hα: (Some r' · Some rK)=>[α |] => HW; last by exfalso; exact: (wsat_nz HW).
have {HSw₀} HSw₀: w₀ ⊑ w'' by transitivity w'.
exists w'' α; split; [done| split]; last first.
+ by move: HW; rewrite 2! mask_full_union => HW; wsatM HW.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists r' rK; split; [exact: equivP | split; [by propsM Hei' |] ].
have {HSw} HSw: w ⊑ w'' by transitivity w'.
by propsM HK.
(* ei atomic: atomicE, IH, fillE *)
move: (atomic_step _ _ _ _ HA HStep) => HV.
move: (atomicE HA) => He {HA}.
move: {He} (He w' (S k) rei HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
(* unroll wp(ei,E)—step case—to get wp(ei',E) *)
move: He; rewrite {1}unfold_wp => He.
move: {HD} (mask_emp_disjoint (mask_full)) => HD.
move: {He HSw' HLt' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [_ [HS _] ].
have Hεei: ei = ε[[ei]] by move: (fill_empty ei)->.
move: {HS Hεei HStep} (HS _ _ _ _ Hεei HStep) => [w'' [r' [HSw' [He' HW] ] ] ].
(* unroll wp(ei',E)—value case—to get E ei' *)
move: He'; rewrite {1}unfold_wp => He'.
move: HW; case Hk': k => [| k'] HW.
- by exists w'' r'; split; [done | split; [exact: wpO | done] ].
have HSw'': w'' ⊑ w'' by reflexivity.
have HLt': k' < k by omega.
move: {He' HSw'' HLt' HD HW} (He' _ _ _ _ _ HSw'' HLt' HD HW) => [Hv _].
move: HV; rewrite -(fill_empty ei') => HV.
move: {Hv} (Hv HV) => [w''' [rei' [HSw'' [Hei' HW] ] ] ].
(* now IH *)
move: HW; rewrite pcm_op_assoc.
case Hα: (Some rei' · _) => [α |] HW; last first.
- rewrite pcm_op_zero in HW; exfalso; exact: (wsat_nz HW).
exists w''' α. split; first by transitivity w''.
split; last by rewrite mask_full_union -(mask_full_union mask_emp).
rewrite/= in Hei'; rewrite fill_empty -Hk' in Hei' * => {Hk'}.
have {HSw₀} HSw₀ : w₀ ⊑ w''' by transitivity w''; first by transitivity w'.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists rei' rK; split; [exact: equivP | split; [done |] ].
have {HSw HSw' HSw''} HSw: w ⊑ w''' by transitivity w''; first by transitivity w'.
by propsM HK.
Qed.
Lemma adv_split {K e} :
valid(adv (K [[e]]) ↔ adv e * adv (K [[fork_ret]])).
Proof.
move=> w n r; rewrite/=/advP; split; move=> {w n r} _ _ _ r _ _.
- move=> [β].
rewrite (* β · K[e] *)
res_of_split (* β · (e · K) *)
pcm_op_assoc. (* (β · e) · K) *)
case Hβe: (Some β · _)=>[βe |] Hβ; last done.
exists βe (ɑ K[[fork_ret]]); split; [done | split; [| by reflexivity] ].
+ by exists β; rewrite Hβe.
move=> [α [β [Hαβ [ [α' Hα'e] [β' Hβ'K] ] ] ] ].
move: Hαβ; move: Hβ'K <-; move: Hα'e <-.
rewrite (* (α' · e) · (β' · K) *)
[Some β' · _]pcm_op_comm (* (α' · e) · (K · β') *)
pcm_op_assoc (* ((α' · e) · K) · β') *)
-[_ · Some(ɑ K [[fork_ret]]) ]pcm_op_assoc (* (α' · (e · K)) · β' *)
-res_of_split (* (α' · K[e]) · β' *)
[Some α' · _]pcm_op_comm (* (K[e] · α) · β' *)
-pcm_op_assoc (* K[e] · (α' · β') *)
pcm_op_comm. (* (α' · β') · K[e] *)
case Hαβ: (Some α' · _) => [αβ |]; last done.
by exists αβ.
Qed.
End RobustSafety.
End Unsafety.
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