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Commit ea4d05c6 authored by Dan Frumin's avatar Dan Frumin Committed by Robbert Krebbers
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Formulate `is_closed_expr` in terms of `gset`s. 

And prove some additional lemmas.
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iris_heap_lang/metatheory.v
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From stdpp Require Import gmap. From stdpp Require Import gmap stringmap.
From iris.heap_lang Require Export lang. From iris.heap_lang Require Export lang.
From iris.prelude Require Import options. From iris.prelude Require Import options.
(* This file contains some metatheory about the heap_lang language, (* This file contains some metatheory about the heap_lang language,
which is not needed for verifying programs. *) which is not needed for verifying programs. *)
(* Closed expressions and values. *) (* Lifting `Insert` on strings to binders. *)
Fixpoint is_closed_expr (X : list string) (e : expr) : bool := Local Definition set_binder_insert (x : binder) (X : stringset) : stringset :=
match x with
| BAnon => X
| BNamed f => {[f]} ∪ X
end.
(* Check if expression [e] is closed w.r.t. the set [X] of variable names,
and that all the values in [e] are closed *)
Fixpoint is_closed_expr (X : stringset) (e : expr) : bool :=
match e with match e with
| Val v => is_closed_val v | Val v => is_closed_val v
| Var x => bool_decide (x ∈ X) | Var x => bool_decide (x ∈ X)
| Rec f x e => is_closed_expr (f :b: x :b: X) e | Rec f x e => is_closed_expr (set_binder_insert f (set_binder_insert x X)) e
| UnOp _ e | Fst e | Snd e | InjL e | InjR e | Fork e | Free e | Load e => | UnOp _ e | Fst e | Snd e | InjL e | InjR e | Fork e | Free e | Load e =>
is_closed_expr X e is_closed_expr X e
| App e1 e2 | BinOp _ e1 e2 | Pair e1 e2 | AllocN e1 e2 | Store e1 e2 | Xchg e1 e2 | FAA e1 e2 => | App e1 e2 | BinOp _ e1 e2 | Pair e1 e2 | AllocN e1 e2 | Store e1 e2 | Xchg e1 e2 | FAA e1 e2 =>
...@@ -22,19 +30,58 @@ Fixpoint is_closed_expr (X : list string) (e : expr) : bool := ...@@ -22,19 +30,58 @@ Fixpoint is_closed_expr (X : list string) (e : expr) : bool :=
with is_closed_val (v : val) : bool := with is_closed_val (v : val) : bool :=
match v with match v with
| LitV _ => true | LitV _ => true
| RecV f x e => is_closed_expr (f :b: x :b: []) e | RecV f x e => is_closed_expr (set_binder_insert f (set_binder_insert x ∅)) e
| PairV v1 v2 => is_closed_val v1 && is_closed_val v2 | PairV v1 v2 => is_closed_val v1 && is_closed_val v2
| InjLV v | InjRV v => is_closed_val v | InjLV v | InjRV v => is_closed_val v
end. end.
(* Parallel substitution *)
Fixpoint subst_map (vs : gmap string val) (e : expr) : expr :=
match e with
| Val _ => e
| Var y => if vs !! y is Some v then Val v else Var y
| Rec f y e => Rec f y (subst_map (binder_delete y (binder_delete f vs)) e)
| App e1 e2 => App (subst_map vs e1) (subst_map vs e2)
| UnOp op e => UnOp op (subst_map vs e)
| BinOp op e1 e2 => BinOp op (subst_map vs e1) (subst_map vs e2)
| If e0 e1 e2 => If (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| Pair e1 e2 => Pair (subst_map vs e1) (subst_map vs e2)
| Fst e => Fst (subst_map vs e)
| Snd e => Snd (subst_map vs e)
| InjL e => InjL (subst_map vs e)
| InjR e => InjR (subst_map vs e)
| Case e0 e1 e2 => Case (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| Fork e => Fork (subst_map vs e)
| AllocN e1 e2 => AllocN (subst_map vs e1) (subst_map vs e2)
| Free e => Free (subst_map vs e)
| Load e => Load (subst_map vs e)
| Store e1 e2 => Store (subst_map vs e1) (subst_map vs e2)
| Xchg e1 e2 => Xchg (subst_map vs e1) (subst_map vs e2)
| CmpXchg e0 e1 e2 => CmpXchg (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| FAA e1 e2 => FAA (subst_map vs e1) (subst_map vs e2)
| NewProph => NewProph
| Resolve e0 e1 e2 => Resolve (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
end.
(* Properties *)
Local Instance SetUnfoldElemOf_insert_binder x y X Q :
SetUnfoldElemOf y X Q →
SetUnfoldElemOf y (set_binder_insert x X) (Q ∨ BNamed y = x).
Proof.
intros H1. constructor.
destruct x as [|x]; set_solver.
Qed.
Lemma is_closed_weaken X Y e : is_closed_expr X e → X ⊆ Y → is_closed_expr Y e. Lemma is_closed_weaken X Y e : is_closed_expr X e → X ⊆ Y → is_closed_expr Y e.
Proof. revert X Y; induction e; naive_solver (eauto; set_solver). Qed. Proof.
revert X Y; induction e; naive_solver (eauto; set_solver).
Qed.
Lemma is_closed_weaken_nil X e : is_closed_expr [] e → is_closed_expr X e. Lemma is_closed_weaken_empty X e : is_closed_expr ∅ e → is_closed_expr X e.
Proof. intros. by apply is_closed_weaken with [], list_subseteq_nil. Qed. Proof. intros. by apply is_closed_weaken with ∅, empty_subseteq. Qed.
Lemma is_closed_subst X e x v : Lemma is_closed_subst X e y v :
is_closed_val v → is_closed_expr (x :: X) e → is_closed_expr X (subst x v e). is_closed_val v → is_closed_expr ({[y]} ∪ X) e → is_closed_expr X (subst y v e).
Proof. Proof.
intros Hv. revert X. intros Hv. revert X.
induction e=> X /= ?; destruct_and?; split_and?; simplify_option_eq; induction e=> X /= ?; destruct_and?; split_and?; simplify_option_eq;
...@@ -43,24 +90,24 @@ Proof. ...@@ -43,24 +90,24 @@ Proof.
end; eauto using is_closed_weaken with set_solver. end; eauto using is_closed_weaken with set_solver.
Qed. Qed.
Lemma is_closed_subst' X e x v : Lemma is_closed_subst' X e x v :
is_closed_val v → is_closed_expr (x :b: X) e → is_closed_expr X (subst' x v e). is_closed_val v → is_closed_expr (set_binder_insert x X) e → is_closed_expr X (subst' x v e).
Proof. destruct x; eauto using is_closed_subst. Qed. Proof. destruct x; eauto using is_closed_subst. Qed.
(* Substitution *)
Lemma subst_is_closed X e x es : is_closed_expr X e → x ∉ X → subst x es e = e. Lemma subst_is_closed X e x es : is_closed_expr X e → x ∉ X → subst x es e = e.
Proof. Proof.
revert X. induction e=> X /=; rewrite ?bool_decide_spec ?andb_True=> ??; revert X. induction e=> X /=;
rewrite ?bool_decide_spec ?andb_True=> ??;
repeat case_decide; simplify_eq/=; f_equal; intuition eauto with set_solver. repeat case_decide; simplify_eq/=; f_equal; intuition eauto with set_solver.
Qed. Qed.
Lemma subst_is_closed_nil e x v : is_closed_expr [] e → subst x v e = e. Lemma subst_is_closed_empty e x v : is_closed_expr ∅ e → subst x v e = e.
Proof. intros. apply subst_is_closed with []; set_solver. Qed. Proof. intros. apply subst_is_closed with (∅:stringset); set_solver. Qed.
Lemma subst_subst e x v v' : Lemma subst_subst e x v v' :
subst x v (subst x v' e) = subst x v' e. subst x v (subst x v' e) = subst x v' e.
Proof. Proof.
intros. induction e; simpl; try (f_equal; by auto); intros. induction e; simpl; try (f_equal; by auto);
simplify_option_eq; auto using subst_is_closed_nil with f_equal. simplify_option_eq; auto using subst_is_closed_empty with f_equal.
Qed. Qed.
Lemma subst_subst' e x v v' : Lemma subst_subst' e x v v' :
subst' x v (subst' x v' e) = subst' x v' e. subst' x v (subst' x v' e) = subst' x v' e.
...@@ -70,7 +117,7 @@ Lemma subst_subst_ne e x y v v' : ...@@ -70,7 +117,7 @@ Lemma subst_subst_ne e x y v v' :
x ≠ y → subst x v (subst y v' e) = subst y v' (subst x v e). x ≠ y → subst x v (subst y v' e) = subst y v' (subst x v e).
Proof. Proof.
intros. induction e; simpl; try (f_equal; by auto); intros. induction e; simpl; try (f_equal; by auto);
simplify_option_eq; auto using eq_sym, subst_is_closed_nil with f_equal. simplify_option_eq; auto using eq_sym, subst_is_closed_empty with f_equal.
Qed. Qed.
Lemma subst_subst_ne' e x y v v' : Lemma subst_subst_ne' e x y v v' :
x ≠ y → subst' x v (subst' y v' e) = subst' y v' (subst' x v e). x ≠ y → subst' x v (subst' y v' e) = subst' y v' (subst' x v e).
...@@ -122,10 +169,10 @@ Qed. ...@@ -122,10 +169,10 @@ Qed.
(* The stepping relation preserves closedness *) (* The stepping relation preserves closedness *)
Lemma head_step_is_closed e1 σ1 obs e2 σ2 es : Lemma head_step_is_closed e1 σ1 obs e2 σ2 es :
is_closed_expr [] e1 → is_closed_expr ∅ e1 →
map_Forall (λ _ v, from_option is_closed_val true v) σ1.(heap) → map_Forall (λ _ v, from_option is_closed_val true v) σ1.(heap) →
head_step e1 σ1 obs e2 σ2 es → head_step e1 σ1 obs e2 σ2 es →
is_closed_expr [] e2 ∧ Forall (is_closed_expr []) es ∧ is_closed_expr ∅ e2 ∧ Forall (is_closed_expr ∅) es ∧
map_Forall (λ _ v, from_option is_closed_val true v) σ2.(heap). map_Forall (λ _ v, from_option is_closed_val true v) σ2.(heap).
Proof. Proof.
intros Cl1 Clσ1 STEP. intros Cl1 Clσ1 STEP.
...@@ -141,32 +188,6 @@ Proof. ...@@ -141,32 +188,6 @@ Proof.
- case_match; try apply map_Forall_insert_2; by naive_solver. - case_match; try apply map_Forall_insert_2; by naive_solver.
Qed. Qed.
Fixpoint subst_map (vs : gmap string val) (e : expr) : expr :=
match e with
| Val _ => e
| Var y => if vs !! y is Some v then Val v else Var y
| Rec f y e => Rec f y (subst_map (binder_delete y (binder_delete f vs)) e)
| App e1 e2 => App (subst_map vs e1) (subst_map vs e2)
| UnOp op e => UnOp op (subst_map vs e)
| BinOp op e1 e2 => BinOp op (subst_map vs e1) (subst_map vs e2)
| If e0 e1 e2 => If (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| Pair e1 e2 => Pair (subst_map vs e1) (subst_map vs e2)
| Fst e => Fst (subst_map vs e)
| Snd e => Snd (subst_map vs e)
| InjL e => InjL (subst_map vs e)
| InjR e => InjR (subst_map vs e)
| Case e0 e1 e2 => Case (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| Fork e => Fork (subst_map vs e)
| AllocN e1 e2 => AllocN (subst_map vs e1) (subst_map vs e2)
| Free e => Free (subst_map vs e)
| Load e => Load (subst_map vs e)
| Store e1 e2 => Store (subst_map vs e1) (subst_map vs e2)
| Xchg e1 e2 => Xchg (subst_map vs e1) (subst_map vs e2)
| CmpXchg e0 e1 e2 => CmpXchg (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
| FAA e1 e2 => FAA (subst_map vs e1) (subst_map vs e2)
| NewProph => NewProph
| Resolve e0 e1 e2 => Resolve (subst_map vs e0) (subst_map vs e1) (subst_map vs e2)
end.
Lemma subst_map_empty e : subst_map ∅ e = e. Lemma subst_map_empty e : subst_map ∅ e = e.
Proof. Proof.
...@@ -216,7 +237,6 @@ Proof. ...@@ -216,7 +237,6 @@ Proof.
by rewrite subst_map_binder_insert_2 !binder_delete_empty subst_map_empty. by rewrite subst_map_binder_insert_2 !binder_delete_empty subst_map_empty.
Qed. Qed.
(* subst_map on closed expressions *)
Lemma subst_map_is_closed X e vs : Lemma subst_map_is_closed X e vs :
is_closed_expr X e → is_closed_expr X e →
(∀ x, x ∈ X → vs !! x = None) → (∀ x, x ∈ X → vs !! x = None) →
...@@ -227,8 +247,16 @@ Proof. ...@@ -227,8 +247,16 @@ Proof.
x ∈ x2 :b: x1 :b: X → x ∈ x2 :b: x1 :b: X →
binder_delete x1 (binder_delete x2 vs) !! x = None). binder_delete x1 (binder_delete x2 vs) !! x = None).
{ intros x x1 x2 X vs ??. rewrite !lookup_binder_delete_None. set_solver. } { intros x x1 x2 X vs ??. rewrite !lookup_binder_delete_None. set_solver. }
induction e=> X vs /= ? HX; repeat case_match; naive_solver eauto with f_equal. induction e as [| |f x e IHe | | | | | | | | | | | | | | | | | | | |]=> X vs /= ? HX.
3:{ f_equal. eapply IHe; eauto.
intros x0 Hx0.
rewrite !lookup_binder_delete_None.
set_solver. }
all: repeat case_match; naive_solver eauto with f_equal.
Qed. Qed.
Lemma subst_map_is_closed_nil e vs : is_closed_expr [] e → subst_map vs e = e. Lemma subst_map_is_closed_empty e vs : is_closed_expr ∅ e → subst_map vs e = e.
Proof. intros. apply subst_map_is_closed with []; set_solver. Qed. Proof.
intros.
apply subst_map_is_closed with ∅; try setoid_rewrite elem_of_empty; set_solver.
Qed.
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