Remove dependent types in heap_lang representation.

.gitlab-ci.yml

@@ -11,6 +11,7 @@ buildjob:
   only:
   - master
   - jh_simplified_resources
+  - rk/substitition
   artifacts:
     paths:
     - build-time.txt

heap_lang/derived.v

@@ -19,29 +19,34 @@ Implicit Types Φ : val → iProp heap_lang Σ.
 (** Proof rules for the sugar *)
 Lemma wp_lam E x ef e v Φ :
   to_val e = Some v →
+  Closed (x :b: []) ef →
   ▷ WP subst' x e ef @ E {{ Φ }} ⊢ WP App (Lam x ef) e @ E {{ Φ }}.
 Proof. intros. by rewrite -(wp_rec _ BAnon) //. Qed.
 
 Lemma wp_let E x e1 e2 v Φ :
   to_val e1 = Some v →
+  Closed (x :b: []) e2 →
   ▷ WP subst' x e1 e2 @ E {{ Φ }} ⊢ WP Let x e1 e2 @ E {{ Φ }}.
 Proof. apply wp_lam. Qed.
 
 Lemma wp_seq E e1 e2 v Φ :
   to_val e1 = Some v →
+  Closed [] e2 →
   ▷ WP e2 @ E {{ Φ }} ⊢ WP Seq e1 e2 @ E {{ Φ }}.
-Proof. intros ?. by rewrite -wp_let. Qed.
+Proof. intros ??. by rewrite -wp_let. Qed.
 
 Lemma wp_skip E Φ : ▷ Φ (LitV LitUnit) ⊢ WP Skip @ E {{ Φ }}.
 Proof. rewrite -wp_seq // -wp_value //. Qed.
 
 Lemma wp_match_inl E e0 v0 x1 e1 x2 e2 Φ :
   to_val e0 = Some v0 →
+  Closed (x1 :b: []) e1 →
   ▷ WP subst' x1 e0 e1 @ E {{ Φ }} ⊢ WP Match (InjL e0) x1 e1 x2 e2 @ E {{ Φ }}.
 Proof. intros. by rewrite -wp_case_inl // -[X in _ ⊢ X]later_intro -wp_let. Qed.
 
 Lemma wp_match_inr E e0 v0 x1 e1 x2 e2 Φ :
   to_val e0 = Some v0 →
+  Closed (x2 :b: []) e2 →
   ▷ WP subst' x2 e0 e2 @ E {{ Φ }} ⊢ WP Match (InjR e0) x1 e1 x2 e2 @ E {{ Φ }}.
 Proof. intros. by rewrite -wp_case_inr // -[X in _ ⊢ X]later_intro -wp_let. Qed.
 

heap_lang/lib/assert.v

 From iris.heap_lang Require Export derived.
 From iris.heap_lang Require Import wp_tactics substitution notation.
 
-Definition Assert {X} (e : expr X) : expr X :=
+Definition Assert (e : expr) : expr :=
   if: e then #() else #0 #0. (* #0 #0 is unsafe *)
 
-Instance do_wexpr_assert {X Y} (H : X `included` Y) e er :
-  WExpr H e er → WExpr H (Assert e) (Assert er) := _.
-Instance do_wsubst_assert {X Y} x es (H : X `included` x :: Y) e er :
-  WSubst x es H e er → WSubst x es H (Assert e) (Assert er).
-Proof. intros; red. by rewrite /Assert /wsubst -/wsubst; f_equal/=. Qed.
+Instance closed_assert X e : Closed X e → Closed X (Assert e) := _.
+Instance do_subst_assert x es e er :
+  Subst x es e er → Subst x es (Assert e) (Assert er).
+Proof. intros; red. by rewrite /Assert /subst -/subst; f_equal/=. Qed.
 Typeclasses Opaque Assert.
 
 Lemma wp_assert {Σ} (Φ : val → iProp heap_lang Σ) :

heap_lang/lib/barrier/barrier.v

 From iris.heap_lang Require Export notation.
 
 Definition newbarrier : val := λ: <>, ref #0.
-Definition signal : val := λ: "x", '"x" <- #1.
+Definition signal : val := λ: "x", "x" <- #1.
 Definition wait : val :=
-  rec: "wait" "x" := if: !'"x" = #1 then #() else '"wait" '"x".
+  rec: "wait" "x" := if: !"x" = #1 then #() else "wait" "x".
 Global Opaque newbarrier signal wait.

heap_lang/lib/counter.v

@@ -8,9 +8,9 @@ Import uPred.
 Definition newcounter : val := λ: <>, ref #0.
 Definition inc : val :=
   rec: "inc" "l" :=
-    let: "n" := !'"l" in
-    if: CAS '"l" '"n" (#1 + '"n") then #() else '"inc" '"l".
-Definition read : val := λ: "l", !'"l".
+    let: "n" := !"l" in
+    if: CAS "l" "n" (#1 + "n") then #() else "inc" "l".
+Definition read : val := λ: "l", !"l".
 Global Opaque newcounter inc get.
 
 (** The CMRA we need. *)

heap_lang/lib/lock.v

@@ -6,8 +6,8 @@ Import uPred.
 Definition newlock : val := λ: <>, ref #false.
 Definition acquire : val :=
   rec: "acquire" "l" :=
-    if: CAS '"l" #false #true then #() else '"acquire" '"l".
-Definition release : val := λ: "l", '"l" <- #false.
+    if: CAS "l" #false #true then #() else "acquire" "l".
+Definition release : val := λ: "l", "l" <- #false.
 Global Opaque newlock acquire release.
 
 (** The CMRA we need. *)

heap_lang/lib/par.v

@@ -2,18 +2,14 @@ From iris.heap_lang Require Export spawn.
 From iris.heap_lang Require Import proofmode notation.
 Import uPred.
 
-Definition par {X} : expr X :=
+Definition par : val :=
   λ: "fs",
-    let: "handle" := ^spawn (Fst '"fs") in
-    let: "v2" := Snd '"fs" #() in
-    let: "v1" := ^join '"handle" in
-    Pair '"v1" '"v2".
+    let: "handle" := spawn (Fst "fs") in
+    let: "v2" := Snd "fs" #() in
+    let: "v1" := join "handle" in
+    Pair "v1" "v2".
 Notation Par e1 e2 := (par (Pair (λ: <>, e1) (λ: <>, e2)))%E.
 Infix "||" := Par : expr_scope.
-
-Instance do_wexpr_par {X Y} (H : X `included` Y) : WExpr H par par := _.
-Instance do_wsubst_par {X Y} x es (H : X `included` x :: Y) :
-  WSubst x es H par par := do_wsubst_closed _ x es H _.
 Global Opaque par.
 
 Section proof.
@@ -36,13 +32,14 @@ Proof.
   iSpecialize ("HΦ" with "* [-]"); first by iSplitL "H1". by wp_let.
 Qed.
 
-Lemma wp_par (Ψ1 Ψ2 : val → iProp) (e1 e2 : expr []) (Φ : val → iProp) :
+Lemma wp_par (Ψ1 Ψ2 : val → iProp) (e1 e2 : expr) `{!Closed [] e1, Closed [] e2}
+    (Φ : val → iProp) :
   heapN ⊥ N →
   (heap_ctx heapN ★ WP e1 {{ Ψ1 }} ★ WP e2 {{ Ψ2 }} ★
    ∀ v1 v2, Ψ1 v1 ★ Ψ2 v2 -★ ▷ Φ (v1,v2)%V)
   ⊢ WP e1 || e2 {{ Φ }}.
 Proof.
-  iIntros (?) "(#Hh&H1&H2&H)". iApply (par_spec Ψ1 Ψ2); auto.
+  iIntros (?) "(#Hh&H1&H2&H)". iApply (par_spec Ψ1 Ψ2); auto. apply is_value.
   iFrame "Hh H". iSplitL "H1"; by wp_let.
 Qed.
 End proof.

heap_lang/lib/spawn.v

@@ -6,12 +6,12 @@ Import uPred.
 Definition spawn : val :=
   λ: "f",
     let: "c" := ref (InjL #0) in
-    Fork ('"c" <- InjR ('"f" #())) ;; '"c".
+    Fork ("c" <- InjR ("f" #())) ;; "c".
 Definition join : val :=
   rec: "join" "c" :=
-    match: !'"c" with
-      InjR "x" => '"x"
-    | InjL <>  => '"join" '"c"
+    match: !"c" with
+      InjR "x" => "x"
+    | InjL <>  => "join" "c"
     end.
 Global Opaque spawn join.
 

heap_lang/lifting.v

@@ -10,7 +10,7 @@ Section lifting.
 Context {Σ : iFunctor}.
 Implicit Types P Q : iProp heap_lang Σ.
 Implicit Types Φ : val → iProp heap_lang Σ.
-Implicit Types ef : option (expr []).
+Implicit Types ef : option expr.
 
 (** Bind. This bundles some arguments that wp_ectx_bind leaves as indices. *)
 Lemma wp_bind {E e} K Φ :
@@ -84,9 +84,10 @@ Qed.
 Lemma wp_rec E f x erec e1 e2 v2 Φ :
   e1 = Rec f x erec →
   to_val e2 = Some v2 →
+  Closed (f :b: x :b: []) erec →
   ▷ WP subst' x e2 (subst' f e1 erec) @ E {{ Φ }} ⊢ WP App e1 e2 @ E {{ Φ }}.
 Proof.
-  intros -> ?. rewrite -(wp_lift_pure_det_head_step (App _ _)
+  intros -> ??. rewrite -(wp_lift_pure_det_head_step (App _ _)
     (subst' x e2 (subst' f (Rec f x erec) erec)) None) //= ?right_id;
     intros; inv_head_step; eauto.
 Qed.

heap_lang/notation.v

@@ -24,6 +24,8 @@ Coercion LitLoc : loc >-> base_lit.
 Coercion App : expr >-> Funclass.
 Coercion of_val : val >-> expr.
 
+Coercion Var : string >-> expr.
+
 Coercion BNamed : string >-> binder.
 Notation "<>" := BAnon : binder_scope.
 
@@ -32,9 +34,6 @@ properly. *)
 Notation "# l" := (LitV l%Z%V) (at level 8, format "# l").
 Notation "# l" := (Lit l%Z%V) (at level 8, format "# l") : expr_scope.
 
-Notation "' x" := (Var x) (at level 8, format "' x") : expr_scope.
-Notation "^ e" := (wexpr' e) (at level 8, format "^ e") : expr_scope.
-
 (** Syntax inspired by Coq/Ocaml. Constructions with higher precedence come
     first. *)
 Notation "( e1 , e2 , .. , en )" := (Pair .. (Pair e1 e2) .. en) : expr_scope.

heap_lang/substitution.v

@@ -2,196 +2,99 @@ From iris.heap_lang Require Export lang.
 Import heap_lang.
 
 (** The tactic [simpl_subst] performs substitutions in the goal. Its behavior
-can be tuned by declaring [WExpr] and [WSubst] instances. *)
-
-(** * Weakening *)
-Class WExpr {X Y} (H : X `included` Y) (e : expr X) (er : expr Y) :=
-  do_wexpr : wexpr H e = er.
-Hint Mode WExpr + + + + - : typeclass_instances.
+can be tuned by declaring [Subst] instances. *)
+(** * Substitution *)
+Class Subst (x : string) (es : expr) (e : expr) (er : expr) :=
+  do_subst : subst x es e = er.
+Hint Mode Subst + + + - : typeclass_instances.
 
 (* Variables *)
-Hint Extern 0 (WExpr _ (Var ?y) _) =>
-  apply var_proof_irrel : typeclass_instances.
-
-(* Rec *)
-Instance do_wexpr_rec_true {X Y f y e} {H : X `included` Y} er :
-  WExpr (wexpr_rec_prf H) e er → WExpr H (Rec f y e) (Rec f y er) | 10.
-Proof. intros; red; f_equal/=. by etrans; [apply wexpr_proof_irrel|]. Qed.
-
-(* Values *)
-Instance do_wexpr_wexpr X Y Z (H1 : X `included` Y) (H2 : Y `included` Z) e er :
-  WExpr (transitivity H1 H2) e er → WExpr H2 (wexpr H1 e) er | 0.
-Proof. by rewrite /WExpr wexpr_wexpr'. Qed.
-Instance do_wexpr_closed_closed (H : [] `included` []) e : WExpr H e e | 1.
-Proof. apply wexpr_id. Qed.
-Instance do_wexpr_closed_wexpr Y (H : [] `included` Y) e :
-  WExpr H e (wexpr' e) | 2.
-Proof. apply wexpr_proof_irrel. Qed.
-
-(* Boring connectives *)
-Section do_wexpr.
-Context {X Y : list string} (H : X `included` Y).
-Notation W := (WExpr H).
-
-(* Ground terms *)
-Global Instance do_wexpr_lit l : W (Lit l) (Lit l).
-Proof. done. Qed.
-Global Instance do_wexpr_app e1 e2 e1r e2r :
-  W e1 e1r → W e2 e2r → W (App e1 e2) (App e1r e2r).
-Proof. intros; red; f_equal/=; apply: do_wexpr. Qed.
-Global Instance do_wexpr_unop op e er : W e er → W (UnOp op e) (UnOp op er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_binop op e1 e2 e1r e2r :
-  W e1 e1r → W e2 e2r → W (BinOp op e1 e2) (BinOp op e1r e2r).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_if e0 e1 e2 e0r e1r e2r :
-  W e0 e0r → W e1 e1r → W e2 e2r → W (If e0 e1 e2) (If e0r e1r e2r).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_pair e1 e2 e1r e2r :
-  W e1 e1r → W e2 e2r → W (Pair e1 e2) (Pair e1r e2r).
-Proof. by intros ??; red; f_equal/=. Qed.
-Global Instance do_wexpr_fst e er : W e er → W (Fst e) (Fst er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_snd e er : W e er → W (Snd e) (Snd er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_injL e er : W e er → W (InjL e) (InjL er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_injR e er : W e er → W (InjR e) (InjR er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_case e0 e1 e2 e0r e1r e2r :
-  W e0 e0r → W e1 e1r → W e2 e2r → W (Case e0 e1 e2) (Case e0r e1r e2r).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_fork e er : W e er → W (Fork e) (Fork er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_alloc e er : W e er → W (Alloc e) (Alloc er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_load e er : W e er → W (Load e) (Load er).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_store e1 e2 e1r e2r :
-  W e1 e1r → W e2 e2r → W (Store e1 e2) (Store e1r e2r).
-Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wexpr_cas e0 e1 e2 e0r e1r e2r :
-  W e0 e0r → W e1 e1r → W e2 e2r → W (CAS e0 e1 e2) (CAS e0r e1r e2r).
-Proof. by intros; red; f_equal/=. Qed.
-End do_wexpr.
+Lemma do_subst_var_eq x er : Subst x er (Var x) er.
+Proof. intros; red; simpl. by case_decide. Qed.
+Lemma do_subst_var_neq x y er : bool_decide (x ≠ y) → Subst x er (Var y) (Var y).
+Proof. rewrite bool_decide_spec. intros; red; simpl. by case_decide. Qed.
 
-(** * WSubstitution *)
-Class WSubst {X Y} (x : string) (es : expr []) H (e : expr X) (er : expr Y) :=
-  do_wsubst : wsubst x es H e = er.
-Hint Mode WSubst + + + + + + - : typeclass_instances.
-
-Lemma do_wsubst_closed (e: ∀ {X}, expr X) {X Y} x es (H : X `included` x :: Y) :
-  (∀ X, WExpr (included_nil X) e e) → WSubst x es H e e.
-Proof.
-  rewrite /WSubst /WExpr=> He. rewrite -(He X) wsubst_wexpr'.
-  by rewrite (wsubst_closed _ _ _ _ _ (included_nil _)); last set_solver.
-Qed.
-
-(* Variables *)
-Lemma do_wsubst_var_eq {X Y x es} {H : X `included` x :: Y} `{VarBound x X} er :
-  WExpr (included_nil _) es er → WSubst x es H (Var x) er.
-Proof.
-  intros; red; simpl. case_decide; last done.
-  by etrans; [apply wexpr_proof_irrel|].
-Qed.
-Hint Extern 0 (WSubst ?x ?v _ (Var ?y) _) => first
-  [ apply var_proof_irrel
-  | apply do_wsubst_var_eq ] : typeclass_instances.
+Hint Extern 0 (Subst ?x ?v (Var ?y) _) =>
+  first [apply do_subst_var_eq
+        |apply do_subst_var_neq, I] : typeclass_instances.
 
 (** Rec *)
-Lemma do_wsubst_rec_true {X Y x es f y e} {H : X `included` x :: Y}
-    (Hfy : BNamed x ≠ f ∧ BNamed x ≠ y) er :
-  WSubst x es (wsubst_rec_true_prf H Hfy) e er →
-  WSubst x es H (Rec f y e) (Rec f y er).
-Proof.
-  intros ?; red; f_equal/=; case_decide; last done.
-  by etrans; [apply wsubst_proof_irrel|].
-Qed.
-Lemma do_wsubst_rec_false {X Y x es f y e} {H : X `included` x :: Y}
-    (Hfy : ¬(BNamed x ≠ f ∧ BNamed x ≠ y)) er :
-  WExpr (wsubst_rec_false_prf H Hfy) e er →
-  WSubst x es H (Rec f y e) (Rec f y er).
-Proof.
-  intros; red; f_equal/=; case_decide; first done.
-  by etrans; [apply wexpr_proof_irrel|].
-Qed.
-
-Ltac bool_decide_no_check := apply (bool_decide_unpack _); vm_cast_no_check I.
-Hint Extern 0 (WSubst ?x ?v _ (Rec ?f ?y ?e) _) =>
+Lemma do_subst_rec_true {x es f y e er} :
+  bool_decide (BNamed x ≠ f ∧ BNamed x ≠ y) →
+  Subst x es e er → Subst x es (Rec f y e) (Rec f y er).
+Proof. rewrite bool_decide_spec. intros; red; f_equal/=; by case_decide. Qed.
+Lemma do_subst_rec_false {x es f y e} :
+  bool_decide (¬(BNamed x ≠ f ∧ BNamed x ≠ y)) →
+  Subst x es (Rec f y e) (Rec f y e).
+Proof. rewrite bool_decide_spec. intros; red; f_equal/=; by case_decide. Qed.
+
+Local Ltac bool_decide_no_check := vm_cast_no_check I.
+Hint Extern 0 (Subst ?x ?v (Rec ?f ?y ?e) _) =>
   match eval vm_compute in (bool_decide (BNamed x ≠ f ∧ BNamed x ≠ y)) with
-  | true => eapply (do_wsubst_rec_true ltac:(bool_decide_no_check))
-  | false => eapply (do_wsubst_rec_false ltac:(bool_decide_no_check))
+  | true => eapply (do_subst_rec_true ltac:(bool_decide_no_check))
+  | false => eapply (do_subst_rec_false ltac:(bool_decide_no_check))
   end : typeclass_instances.
 
+Lemma do_subst_closed x es e : Closed [] e → Subst x es e e.
+Proof. apply closed_nil_subst. Qed.
+Hint Extern 10 (Subst ?x ?v ?e _) =>
+  is_var e; class_apply do_subst_closed : typeclass_instances.
+
 (* Values *)
-Instance do_wsubst_wexpr X Y Z x es
-    (H1 : X `included` Y) (H2 : Y `included` x :: Z) e er :
-  WSubst x es (transitivity H1 H2) e er → WSubst x es H2 (wexpr H1 e) er | 0.
-Proof. by rewrite /WSubst wsubst_wexpr'. Qed.
-Instance do_wsubst_closed_closed x es (H : [] `included` [x]) e :
-  WSubst x es H e e | 1.
-Proof. apply wsubst_closed_nil. Qed.
-Instance do_wsubst_closed_wexpr Y x es (H : [] `included` x :: Y) e :
-  WSubst x es H e (wexpr' e) | 2.
-Proof. apply wsubst_closed, not_elem_of_nil. Qed.
+Instance do_subst_of_val x es v : Subst x es (of_val v) (of_val v) | 0.
+Proof. eapply closed_nil_subst, of_val_closed. Qed.
 
 (* Boring connectives *)
-Section wsubst.
-Context {X Y} (x : string) (es : expr []) (H : X `included` x :: Y).
-Notation Sub := (WSubst x es H).
+Section subst.
+Context (x : string) (es : expr).
+Notation Sub := (Subst x es).
 
 (* Ground terms *)
-Global Instance do_wsubst_lit l : Sub (Lit l) (Lit l).
+Global Instance do_subst_lit l : Sub (Lit l) (Lit l).
 Proof. done. Qed.
-Global Instance do_wsubst_app e1 e2 e1r e2r :
+Global Instance do_subst_app e1 e2 e1r e2r :
   Sub e1 e1r → Sub e2 e2r → Sub (App e1 e2) (App e1r e2r).
-Proof. intros; red; f_equal/=; apply: do_wsubst. Qed.
-Global Instance do_wsubst_unop op e er : Sub e er → Sub (UnOp op e) (UnOp op er).
+Proof. intros; red; f_equal/=; apply: do_subst. Qed.
+Global Instance do_subst_unop op e er : Sub e er → Sub (UnOp op e) (UnOp op er).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_binop op e1 e2 e1r e2r :
+Global Instance do_subst_binop op e1 e2 e1r e2r :
   Sub e1 e1r → Sub e2 e2r → Sub (BinOp op e1 e2) (BinOp op e1r e2r).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_if e0 e1 e2 e0r e1r e2r :
+Global Instance do_subst_if e0 e1 e2 e0r e1r e2r :
   Sub e0 e0r → Sub e1 e1r → Sub e2 e2r → Sub (If e0 e1 e2) (If e0r e1r e2r).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_pair e1 e2 e1r e2r :
+Global Instance do_subst_pair e1 e2 e1r e2r :
   Sub e1 e1r → Sub e2 e2r → Sub (Pair e1 e2) (Pair e1r e2r).
 Proof. by intros ??; red; f_equal/=. Qed.
-Global Instance do_wsubst_fst e er : Sub e er → Sub (Fst e) (Fst er).
+Global Instance do_subst_fst e er : Sub e er → Sub (Fst e) (Fst er).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_snd e er : Sub e er → Sub (Snd e) (Snd er).
+Global Instance do_subst_snd e er : Sub e er → Sub (Snd e) (Snd er).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_injL e er : Sub e er → Sub (InjL e) (InjL er).
+Global Instance do_subst_injL e er : Sub e er → Sub (InjL e) (InjL er).
 Proof. by intros; red; f_equal/=. Qed.
-Global Instance do_wsubst_injR e er : Sub e er → Sub (InjR e) (InjR er).
+Global Instance do_subst_injR e er : Sub e er → Sub (InjR e) (InjR er).
 Proof. by intros; red; f_equal/=. Qed.