Misc language clean up.

algebra/cofe.v

@@ -422,7 +422,6 @@ Proof.
   apply cofe_morC_map_ne; apply cFunctor_contractive=>i ?; split; by apply Hfg.
 Qed.
 
-
 (** Discrete cofe *)
 Section discrete_cofe.
   Context `{Equiv A, @Equivalence A (≡)}.

heap_lang/lang.v

@@ -228,14 +228,12 @@ Program Fixpoint wexpr {X Y} (H : X `included` Y) (e : expr X) : expr Y :=
   end.
 Solve Obligations with set_solver.
 
-Program Definition wexpr' {X} (e : expr []) : expr X :=
-  wexpr _ e.
-Solve Obligations with set_solver.
+Definition wexpr' {X} (e : expr []) : expr X := wexpr (included_nil _) e.
 
 Definition of_val' {X} (v : val) : expr X := wexpr (included_nil _) (of_val v).
 
 Lemma wsubst_rec_true_prf {X Y x} (H : X `included` x :: Y) {f y}
-    (Hfy :BNamed x ≠ f ∧ BNamed x ≠ y) :
+    (Hfy : BNamed x ≠ f ∧ BNamed x ≠ y) :
   f :b: y :b: X `included` x :: f :b: y :b: Y.
 Proof. set_solver. Qed.
 Lemma wsubst_rec_false_prf {X Y x} (H : X `included` x :: Y) {f y}
@@ -413,21 +411,6 @@ Proof.
   apply wsubst_closed, not_elem_of_nil.
 Qed.
 
-Lemma of_val'_closed (v : val) :
-  of_val' v = of_val v.
-Proof. by rewrite /of_val' wexpr_id. Qed.
-
-(** to_val propagation.
-    TODO: automatically appliy in wp_tactics? *)
-Lemma to_val_InjL e v : to_val e = Some v → to_val (InjL e) = Some (InjLV v).
-Proof. move=>H. simpl. by rewrite H. Qed.
-Lemma to_val_InjR e v : to_val e = Some v → to_val (InjR e) = Some (InjRV v).
-Proof. move=>H. simpl. by rewrite H. Qed.
-Lemma to_val_Pair e1 e2 v1 v2 :
-  to_val e1 = Some v1 → to_val e2 = Some v2 →
-  to_val (Pair e1 e2) = Some (PairV v1 v2).
-Proof. move=>H1 H2. simpl. by rewrite H1 H2. Qed.
-
 (** Basic properties about the language *)
 Lemma to_of_val v : to_val (of_val v) = Some v.
 Proof. by induction v; simplify_option_eq. Qed.
@@ -452,10 +435,6 @@ Proof. destruct Ki; intros ???; simplify_eq/=; auto with f_equal. Qed.
 Instance fill_inj K : Inj (=) (=) (fill K).
 Proof. red; induction K as [|Ki K IH]; naive_solver. Qed.
 
-Lemma fill_inj' K e1 e2 :
-  fill K e1 = fill K e2 → e1 = e2.
-Proof. eapply fill_inj. Qed.
-
 Lemma fill_val K e : is_Some (to_val (fill K e)) → is_Some (to_val e).
 Proof.
   intros [v' Hv']; revert v' Hv'.
@@ -574,16 +553,16 @@ Instance val_inhabited : Inhabited val := populate (LitV LitUnit).
 End heap_lang.
 
 (** Language *)
-Program Canonical Structure heap_ectx_lang :
-  ectx_language (heap_lang.expr []) heap_lang.val heap_lang.ectx heap_lang.state
-  := {|
-    of_val := heap_lang.of_val; to_val := heap_lang.to_val;
-    empty_ectx := []; comp_ectx := app; fill := heap_lang.fill; 
-    atomic := heap_lang.atomic; head_step := heap_lang.head_step;
-  |}.
+Program Instance heap_ectx_lang :
+  EctxLanguage
+    (heap_lang.expr []) heap_lang.val heap_lang.ectx heap_lang.state := {|
+  of_val := heap_lang.of_val; to_val := heap_lang.to_val;
+  empty_ectx := []; comp_ectx := (++); fill := heap_lang.fill; 
+  atomic := heap_lang.atomic; head_step := heap_lang.head_step;
+|}.
 Solve Obligations with eauto using heap_lang.to_of_val, heap_lang.of_to_val,
   heap_lang.val_stuck, heap_lang.atomic_not_val, heap_lang.atomic_step,
-  heap_lang.fill_inj', heap_lang.fill_not_val, heap_lang.atomic_fill,
+  heap_lang.fill_not_val, heap_lang.atomic_fill,
   heap_lang.step_by_val, fold_right_app, app_eq_nil.
 
 Canonical Structure heap_lang := ectx_lang heap_ectx_lang.

heap_lang/lib/spawn.v

@@ -74,12 +74,11 @@ Proof.
     solve_sep_entails.
   - wp_focus (f _). rewrite wp_frame_r wp_frame_l.
     rewrite (of_to_val e) //. apply wp_mono=>v.
-    eapply (inv_fsa (wp_fsa _)) with (N0:=N); simpl;
-      (* TODO: Collect these in some Hint DB? Or add to an existing one? *)
-      eauto using to_val_InjR,to_val_InjL,to_of_val with I ndisj.
+    eapply (inv_fsa (wp_fsa _)) with (N0:=N);
+      rewrite /= ?to_of_val; eauto with I ndisj.
     apply wand_intro_l. rewrite /spawn_inv {1}later_exist !sep_exist_r.
     apply exist_elim=>lv. rewrite later_sep.
-    eapply wp_store; eauto using to_val_InjR,to_val_InjL,to_of_val with I ndisj.
+    eapply wp_store; rewrite /= ?to_of_val; eauto with I ndisj.
     cancel [▷ (l ↦ lv)]%I. strip_later. apply wand_intro_l.
     rewrite right_id -later_intro -{2}[(∃ _, _ ↦ _ ★ _)%I](exist_intro (InjRV v)).
     ecancel [l ↦ _]%I. apply or_intro_r'. rewrite sep_elim_r sep_elim_r sep_elim_l.
@@ -115,5 +114,4 @@ Proof.
     wp_case. wp_let. ewp (eapply wp_value; wp_done).
     rewrite (forall_elim v). rewrite !assoc. eapply wand_apply_r'; eauto with I.
 Qed.
-
 End proof.

heap_lang/lifting.v

@@ -3,7 +3,7 @@ From iris.program_logic Require Import ownership. (* for ownP *)
 From iris.heap_lang Require Export lang.
 From iris.heap_lang Require Import tactics.
 Import uPred.
-Local Hint Extern 0 (head_reducible _ _) => do_step eauto 2.
+Local Hint Extern 0 (head_reducible _ _) => do_head_step eauto 2.
 
 Section lifting.
 Context {Σ : iFunctor}.
@@ -27,7 +27,7 @@ Proof.
   intros. set (φ (e' : expr []) σ' ef := ∃ l,
     ef = None ∧ e' = Loc l ∧ σ' = <[l:=v]>σ ∧ σ !! l = None).
   rewrite -(wp_lift_atomic_head_step (Alloc e) φ σ) // /φ;
-    last (by intros; inv_step; eauto 8); last (by simpl; eauto).
+    last (by intros; inv_head_step; eauto 8); last (by simpl; eauto).
   apply sep_mono, later_mono; first done.
   apply forall_intro=>v2; apply forall_intro=>σ2; apply forall_intro=>ef.
   apply wand_intro_l.
@@ -43,7 +43,7 @@ Lemma wp_load_pst E σ l v Φ :
   (▷ ownP σ ★ ▷ (ownP σ -★ Φ v)) ⊢ WP Load (Loc l) @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ v σ None) ?right_id //;
-    last (by intros; inv_step; eauto using to_of_val); simpl; by eauto.
+    last (by intros; inv_head_step; eauto using to_of_val); simpl; by eauto.
 Qed.
 
 Lemma wp_store_pst E σ l e v v' Φ :
@@ -52,7 +52,7 @@ Lemma wp_store_pst E σ l e v v' Φ :
   ⊢ WP Store (Loc l) e @ E {{ Φ }}.
 Proof.
   intros. rewrite-(wp_lift_atomic_det_head_step σ (LitV LitUnit) (<[l:=v]>σ) None)
-    ?right_id //; last (by intros; inv_step; eauto); simpl; by eauto.
+    ?right_id //; last (by intros; inv_head_step; eauto); simpl; by eauto.
 Qed.
 
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Φ :
@@ -61,7 +61,7 @@ Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Φ :
   ⊢ WP CAS (Loc l) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool false) σ None)
-    ?right_id //; last (by intros; inv_step; eauto);
+    ?right_id //; last (by intros; inv_head_step; eauto);
     simpl; split_and?; by eauto.
 Qed.
 
@@ -71,7 +71,7 @@ Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Φ :
   ⊢ WP CAS (Loc l) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool true)
-    (<[l:=v2]>σ) None) ?right_id //; last (by intros; inv_step; eauto);
+    (<[l:=v2]>σ) None) ?right_id //; last (by intros; inv_head_step; eauto);
     simpl; split_and?; by eauto.
 Qed.
 
@@ -80,7 +80,7 @@ Lemma wp_fork E e Φ :
   (▷ Φ (LitV LitUnit) ★ ▷ WP e {{ λ _, True }}) ⊢ WP Fork e @ E {{ Φ }}.
 Proof.
   rewrite -(wp_lift_pure_det_head_step (Fork e) (Lit LitUnit) (Some e)) //=;
-    last by intros; inv_step; eauto.
+    last by intros; inv_head_step; eauto.
   rewrite later_sep -(wp_value _ _ (Lit _)) //.
 Qed.
 
@@ -91,7 +91,7 @@ Lemma wp_rec E f x e1 e2 v Φ :
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (App _ _)
     (subst' x e2 (subst' f (Rec f x e1) e1)) None) //= ?right_id;
-    intros; inv_step; eauto.
+    intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_rec' E f x erec e1 e2 v2 Φ :
@@ -106,7 +106,7 @@ Lemma wp_un_op E op l l' Φ :
   ▷ Φ (LitV l') ⊢ WP UnOp op (Lit l) @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (UnOp op _) (Lit l') None)
-    ?right_id -?wp_value //; intros; inv_step; eauto.
+    ?right_id -?wp_value //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_bin_op E op l1 l2 l' Φ :
@@ -114,21 +114,21 @@ Lemma wp_bin_op E op l1 l2 l' Φ :
   ▷ Φ (LitV l') ⊢ WP BinOp op (Lit l1) (Lit l2) @ E {{ Φ }}.
 Proof.
   intros Heval. rewrite -(wp_lift_pure_det_head_step (BinOp op _ _) (Lit l') None)
-    ?right_id -?wp_value //; intros; inv_step; eauto.
+    ?right_id -?wp_value //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_if_true E e1 e2 Φ :
   ▷ WP e1 @ E {{ Φ }} ⊢ WP If (Lit (LitBool true)) e1 e2 @ E {{ Φ }}.
 Proof.
   rewrite -(wp_lift_pure_det_head_step (If _ _ _) e1 None)
-    ?right_id //; intros; inv_step; eauto.
+    ?right_id //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_if_false E e1 e2 Φ :
   ▷ WP e2 @ E {{ Φ }} ⊢ WP If (Lit (LitBool false)) e1 e2 @ E {{ Φ }}.
 Proof.
   rewrite -(wp_lift_pure_det_head_step (If _ _ _) e2 None)
-    ?right_id //; intros; inv_step; eauto.
+    ?right_id //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_fst E e1 v1 e2 v2 Φ :
@@ -136,7 +136,7 @@ Lemma wp_fst E e1 v1 e2 v2 Φ :
   ▷ Φ v1 ⊢ WP Fst (Pair e1 e2) @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (Fst _) e1 None)
-    ?right_id -?wp_value //; intros; inv_step; eauto.
+    ?right_id -?wp_value //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_snd E e1 v1 e2 v2 Φ :
@@ -144,7 +144,7 @@ Lemma wp_snd E e1 v1 e2 v2 Φ :
   ▷ Φ v2 ⊢ WP Snd (Pair e1 e2) @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (Snd _) e2 None)
-    ?right_id -?wp_value //; intros; inv_step; eauto.
+    ?right_id -?wp_value //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_case_inl E e0 v0 e1 e2 Φ :
@@ -152,7 +152,7 @@ Lemma wp_case_inl E e0 v0 e1 e2 Φ :
   ▷ WP App e1 e0 @ E {{ Φ }} ⊢ WP Case (InjL e0) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
-    (App e1 e0) None) ?right_id //; intros; inv_step; eauto.
+    (App e1 e0) None) ?right_id //; intros; inv_head_step; eauto.
 Qed.
 
 Lemma wp_case_inr E e0 v0 e1 e2 Φ :
@@ -160,7 +160,6 @@ Lemma wp_case_inr E e0 v0 e1 e2 Φ :
   ▷ WP App e2 e0 @ E {{ Φ }} ⊢ WP Case (InjR e0) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
-    (App e2 e0) None) ?right_id //; intros; inv_step; eauto.
+    (App e2 e0) None) ?right_id //; intros; inv_head_step; eauto.
 Qed.
-
 End lifting.

heap_lang/tactics.v

@@ -2,18 +2,18 @@ From iris.heap_lang Require Export substitution.
 From iris.prelude Require Import fin_maps.
 Import heap_lang.
 
-(** The tactic [inv_step] performs inversion on hypotheses of the
+(** The tactic [inv_head_step] performs inversion on hypotheses of the
 shape [head_step]. The tactic will discharge head-reductions starting
 from values, and simplifies hypothesis related to conversions from and
 to values, and finite map operations. This tactic is slightly ad-hoc
 and tuned for proving our lifting lemmas. *)
-Ltac inv_step :=
+Ltac inv_head_step :=
   repeat match goal with
   | _ => progress simplify_map_eq/= (* simplify memory stuff *)
   | H : to_val _ = Some _ |- _ => apply of_to_val in H
   | H : context [to_val (of_val _)] |- _ => rewrite to_of_val in H
   | H : head_step ?e _ _ _ _ |- _ =>
-     try (is_var e; fail 1); (* inversion yields many goals if e is a variable
+     try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
      and can thus better be avoided. *)
      inversion H; subst; clear H
   end.
@@ -63,18 +63,15 @@ Ltac reshape_expr e tac :=
   | CAS ?e0 ?e1 ?e2 => go (CasLCtx e1 e2 :: K) e0
   end in go (@nil ectx_item) e.
 
-(** The tactic [do_step tac] solves goals of the shape
-[head_reducible] and [head_step] by performing a reduction step and
-uses [tac] to solve any side-conditions generated by individual
-steps. *)
-Tactic Notation "do_step" tactic3(tac) :=
+(** The tactic [do_head_step tac] solves goals of the shape [head_reducible] and
+[head_step] by performing a reduction step and uses [tac] to solve any
+side-conditions generated by individual steps. *)
+Tactic Notation "do_head_step" tactic3(tac) :=
   try match goal with |- head_reducible _ _ => eexists _, _, _ end;
   simpl;
   match goal with
   | |- head_step ?e1 ?σ1 ?e2 ?σ2 ?ef =>
      first [apply alloc_fresh|econstructor];
-     (* If there is at least one goal left now, then do the last
-        goal last -- it may rely on evars being instantiaed elsewhere. *)
-     first [ fail
-           | rewrite ?to_of_val; [tac..|]; tac; fast_done ]
+       (* solve [to_val] side-conditions *)
+       first [rewrite ?to_of_val; reflexivity|simpl_subst; tac; fast_done]
   end.

program_logic/ectx_language.v

@@ -3,7 +3,7 @@ From iris.program_logic Require Export language.
 
 (* We need to make thos arguments indices that we want canonical structure
    inference to use a keys. *)
-Class ectx_language (expr val ectx state : Type) := EctxLanguage {
+Class EctxLanguage (expr val ectx state : Type) := {
   of_val : val → expr;
   to_val : expr → option val;
   empty_ectx : ectx;
@@ -18,14 +18,14 @@ Class ectx_language (expr val ectx state : Type) := EctxLanguage {
 
   fill_empty e : fill empty_ectx e = e;
   fill_comp K1 K2 e : fill K1 (fill K2 e) = fill (comp_ectx K1 K2) e;
-  fill_inj K e1 e2 : fill K e1 = fill K e2 → e1 = e2;
+  fill_inj K :> Inj (=) (=) (fill K);
   fill_not_val K e : to_val e = None → to_val (fill K e) = None;
 
-  (* There are a whole lot of sensible axioms we could demand for comp_ectx
-     and empty_ectx. However, this one is enough. *)
+  (* There are a whole lot of sensible axioms (like associativity, and left and
+  right identity, we could demand for [comp_ectx] and [empty_ectx]. However,
+  positivity suffices. *)
   ectx_positive K1 K2 :
-    empty_ectx = comp_ectx K1 K2 →
-    K1 = empty_ectx ∧ K2 = empty_ectx;
+    comp_ectx K1 K2 = empty_ectx → K1 = empty_ectx ∧ K2 = empty_ectx;
 
   step_by_val K K' e1 e1' σ1 e2 σ2 ef :
     fill K e1 = fill K' e1' →
@@ -43,6 +43,7 @@ Class ectx_language (expr val ectx state : Type) := EctxLanguage {
     to_val e = None →
     K = empty_ectx;
 }.
+
 Arguments of_val {_ _ _ _ _} _.
 Arguments to_val {_ _ _ _ _} _.
 Arguments empty_ectx {_ _ _ _ _}.
@@ -56,7 +57,6 @@ Arguments of_to_val {_ _ _ _ _} _ _ _.
 Arguments val_stuck {_ _ _ _ _} _ _ _ _ _ _.
 Arguments fill_empty {_ _ _ _ _} _.
 Arguments fill_comp {_ _ _ _ _} _ _ _.
-Arguments fill_inj {_ _ _ _ _} _ _ _ _.
 Arguments fill_not_val {_ _ _ _ _} _ _ _.
 Arguments ectx_positive {_ _ _ _ _} _ _ _.
 Arguments step_by_val {_ _ _ _ _} _ _ _ _ _ _ _ _ _ _ _.
@@ -65,57 +65,57 @@ Arguments atomic_step {_ _ _ _ _} _ _ _ _ _ _ _.
 Arguments atomic_fill {_ _ _ _ _} _ _ _ _.
 
 (* From an ectx_language, we can construct a language. *)
-Section Language.
-  Context {expr val ectx state : Type} {Λ : ectx_language expr val ectx state}.
+Section ectx_language.
+  Context {expr val ectx state} {Λ : EctxLanguage expr val ectx state}.
   Implicit Types (e : expr) (K : ectx).
 
   Definition head_reducible (e : expr) (σ : state) :=
     ∃ e' σ' ef, head_step e σ e' σ' ef.
 
   Inductive prim_step (e1 : expr) (σ1 : state)
-    (e2 : expr) (σ2: state) (ef: option expr) : Prop :=
-  Ectx_step K e1' e2' :
-    e1 = fill K e1' → e2 = fill K e2' →
-    head_step e1' σ1 e2' σ2 ef → prim_step e1 σ1 e2 σ2 ef.
+      (e2 : expr) (σ2 : state) (ef : option expr) : Prop :=
+    Ectx_step K e1' e2' :
+      e1 = fill K e1' → e2 = fill K e2' →
+      head_step e1' σ1 e2' σ2 ef → prim_step e1 σ1 e2 σ2 ef.
 
   Lemma val_prim_stuck e1 σ1 e2 σ2 ef :
     prim_step e1 σ1 e2 σ2 ef → to_val e1 = None.
   Proof. intros [??? -> -> ?]; eauto using fill_not_val, val_stuck. Qed.
 
   Lemma atomic_prim_step e1 σ1 e2 σ2 ef :
-  atomic e1 → prim_step e1 σ1 e2 σ2 ef → is_Some (to_val e2).
+    atomic e1 → prim_step e1 σ1 e2 σ2 ef → is_Some (to_val e2).
   Proof.
     intros Hatomic [K e1' e2' -> -> Hstep].
     assert (K = empty_ectx) as -> by eauto 10 using atomic_fill, val_stuck.
-    eapply atomic_step; first done. by rewrite !fill_empty.
+    revert Hatomic; rewrite !fill_empty. eauto using atomic_step.
   Qed.
 
   Canonical Structure ectx_lang : language := {|
     language.expr := expr; language.val := val; language.state := state;
     language.of_val := of_val; language.to_val := to_val;
-    language.atomic := atomic; language.prim_step := prim_step;
+    language.atomic := atomic;
+    language.prim_step := prim_step;
     language.to_of_val := to_of_val; language.of_to_val := of_to_val;
-    language.val_stuck := val_prim_stuck; language.atomic_not_val := atomic_not_val;
+    language.val_stuck := val_prim_stuck;
+    language.atomic_not_val := atomic_not_val;
     language.atomic_step := atomic_prim_step
   |}.
 
   (* Some lemmas about this language *)
-  Lemma head_prim_reducible e σ :
-    head_reducible e σ → reducible e σ.
-  Proof.
-    intros (e'&?&?&?). do 3 eexists.
-    eapply Ectx_step with (K:=empty_ectx); rewrite ?fill_empty; done.
-  Qed.
+  Lemma head_prim_step e1 σ1 e2 σ2 ef :
+    head_step e1 σ1 e2 σ2 ef → prim_step e1 σ1 e2 σ2 ef.
+  Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.
+
+  Lemma head_prim_reducible e σ : head_reducible e σ → reducible e σ.
+  Proof. intros (e'&σ'&ef&?). eexists e', σ', ef. by apply head_prim_step. Qed.
 
   Lemma head_reducible_prim_step e1 σ1 e2 σ2 ef :
     head_reducible e1 σ1 → prim_step e1 σ1 e2 σ2 ef →
     head_step e1 σ1 e2 σ2 ef.
   Proof.
-    intros Hred Hstep. destruct Hstep as [? ? ? ? ? Hstep]; subst.
-    rename e1' into e1. rename e2' into e2.
-    destruct Hred as (e2'&σ2'&ef'&HstepK).
-    destruct (step_by_val K empty_ectx e1 (fill K e1) σ1 e2' σ2' ef')
-      as [K' [-> _]%ectx_positive];
+    intros (e2''&σ2''&ef''&?) [K e1' e2' -> -> Hstep].
+    destruct (step_by_val K empty_ectx e1' (fill K e1') σ1 e2'' σ2'' ef'')
+      as [K' [-> _]%symmetry%ectx_positive];
       eauto using fill_empty, fill_not_val, val_stuck.
     by rewrite !fill_empty.
   Qed.
@@ -129,16 +129,10 @@ Section Language.
       by exists (comp_ectx K K') e1' e2'; rewrite ?Heq1 ?Heq2 ?fill_comp.
     - intros e1 σ1 e2 σ2 ? Hnval [K'' e1'' e2'' Heq1 -> Hstep].
       destruct (step_by_val K K'' e1 e1'' σ1 e2'' σ2 ef) as [K' ->]; eauto.
-      rewrite -fill_comp in Heq1; apply fill_inj in Heq1.
+      rewrite -fill_comp in Heq1; apply (inj (fill _)) in Heq1.
       exists (fill K' e2''); rewrite -fill_comp; split; auto.
       econstructor; eauto.
   Qed.
-End Language.
-  
-Arguments ectx_lang {_ _ _ _} _ : clear implicits.
-
-
-
-
+End ectx_language.
 
-  
\ No newline at end of file
+Arguments ectx_lang {_ _ _ _} _.

program_logic/ectx_weakestpre.v

-(** Some derived lemmas for ectx-based languages *)
 From iris.program_logic Require Export ectx_language weakestpre lifting.
 From iris.program_logic Require Import ownership.
 
 Section wp.
-Context {expr val ectx state: Type} {Λ : ectx_language expr val ectx state}
-        {Σ : iFunctor}.
+Context {expr val ectx state} {Λ : EctxLanguage expr val ectx state}.
+Context {Σ : iFunctor}.
 Implicit Types P : iProp (ectx_lang Λ) Σ.
 Implicit Types Φ : val → iProp (ectx_lang Λ) Σ.
 Implicit Types v : val.
 Implicit Types e : expr.
+Hint Resolve head_prim_reducible head_reducible_prim_step.
 
 Notation wp_fork ef := (default True ef (flip (wp ⊤) (λ _, True)))%I.
 
@@ -24,20 +24,14 @@ Lemma wp_lift_head_step E1 E2
   (|={E1,E2}=> ▷ ownP σ1 ★ ▷ ∀ e2 σ2 ef,
     (■ φ e2 σ2 ef ∧ ownP σ2) -★ |={E2,E1}=> WP e2 @ E1 {{ Φ }} ★ wp_fork ef)
   ⊢ WP e1 @ E1 {{ Φ }}.
-Proof.
-  intros. apply wp_lift_step;
-            eauto using head_prim_reducible, head_reducible_prim_step.
-Qed.
+Proof. eauto using wp_lift_step. Qed.
 
 Lemma wp_lift_pure_head_step E (φ : expr → option expr → Prop) Φ e1 :
   to_val e1 = None →
   (∀ σ1, head_reducible e1 σ1) →
   (∀ σ1 e2 σ2 ef, head_step e1 σ1 e2 σ2 ef → σ1 = σ2 ∧ φ e2 ef) →
   (▷ ∀ e2 ef, ■ φ e2 ef → WP e2 @ E {{ Φ }} ★ wp_fork ef) ⊢ WP e1 @ E {{ Φ }}.
-Proof.
-  intros. apply wp_lift_pure_step;
-            eauto using head_prim_reducible, head_reducible_prim_step.
-Qed.
+Proof. eauto using wp_lift_pure_step. Qed.
 
 Lemma wp_lift_atomic_head_step {E Φ} e1
     (φ : expr → state → option expr → Prop) σ1 :
@@ -47,10 +41,7 @@ Lemma wp_lift_atomic_head_step {E Φ} e1
     head_step e1 σ1 e2 σ2 ef → φ e2 σ2 ef) →
   (▷ ownP σ1 ★ ▷ ∀ v2 σ2 ef, ■ φ (of_val v2) σ2 ef ∧ ownP σ2 -★ Φ v2 ★ wp_fork ef)
   ⊢ WP e1 @ E {{ Φ }}.
-Proof.
-  intros. apply wp_lift_atomic_step;
-            eauto using head_prim_reducible, head_reducible_prim_step.
-Qed.
+Proof. eauto using wp_lift_atomic_step. Qed.
 
 Lemma wp_lift_atomic_det_head_step {E Φ e1} σ1 v2 σ2 ef :
   atomic e1 →
@@ -58,19 +49,12 @@ Lemma wp_lift_atomic_det_head_step {E Φ e1} σ1 v2 σ2 ef :
   (∀ e2' σ2' ef', head_step e1 σ1 e2' σ2' ef' →
     σ2 = σ2' ∧ to_val e2' = Some v2 ∧ ef = ef') →
   (▷ ownP σ1 ★ ▷ (ownP σ2 -★ Φ v2 ★ wp_fork ef)) ⊢ WP e1 @ E {{ Φ }}.
-Proof.
-  intros. apply wp_lift_atomic_det_step;
-            eauto using head_prim_reducible, head_reducible_prim_step.
-Qed.
+Proof. eauto using wp_lift_atomic_det_step. Qed.
 
 Lemma wp_lift_pure_det_head_step {E Φ} e1 e2 ef :
   to_val e1 = None →
   (∀ σ1, head_reducible e1 σ1) →
   (∀ σ1 e2' σ2 ef', head_step e1 σ1 e2' σ2 ef' → σ1 = σ2 ∧ e2 = e2' ∧ ef = ef')→
   ▷ (WP e2 @ E {{ Φ }} ★ wp_fork ef) ⊢ WP e1 @ E {{ Φ }}.
-Proof.
-  intros. apply wp_lift_pure_det_step;
-            eauto using head_prim_reducible, head_reducible_prim_step.
-Qed.
-
+Proof. eauto using wp_lift_pure_det_step. Qed.
 End wp.

tests/heap_lang.v

@@ -6,13 +6,13 @@ Import uPred.
 Section LangTests.
   Definition add : expr [] := (#21 + #21)%E.
   Goal ∀ σ, head_step add σ (#42) σ None.
-  Proof. intros; do_step done. Qed.
+  Proof. intros; do_head_step done. Qed.
   Definition rec_app : expr [] := ((rec: "f" "x" := '"f" '"x") #0)%E.
   Goal ∀ σ, head_step rec_app σ rec_app σ None.
-  Proof. intros. rewrite /rec_app. do_step simpl_subst. Qed.
+  Proof. intros. rewrite /rec_app. do_head_step done. Qed.
   Definition lam : expr [] := (λ: "x", '"x" + #21)%E.
   Goal ∀ σ, head_step (lam #21)%E σ add σ None.
-  Proof. intros. rewrite /lam. do_step done. Qed.
+  Proof. intros. rewrite /lam. do_head_step done. Qed.
 End LangTests.
 
 Section LiftingTests.