tactics.v 36.8 KB
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From iris.proofmode Require Import coq_tactics intro_patterns spec_patterns.
From iris.algebra Require Export upred.
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From iris.proofmode Require Export classes notation.
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From iris.prelude Require Import stringmap hlist.
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Declare Reduction env_cbv := cbv [
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  env_lookup env_fold env_lookup_delete env_delete env_app
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    env_replace env_split_go env_split
  decide (* operational classes *)
  sumbool_rec sumbool_rect (* sumbool *)
  bool_eq_dec bool_rec bool_rect bool_dec eqb andb (* bool *)
  assci_eq_dec ascii_to_digits Ascii.ascii_dec Ascii.ascii_rec Ascii.ascii_rect
  string_eq_dec string_rec string_rect (* strings *)
  env_persistent env_spatial envs_persistent
  envs_lookup envs_lookup_delete envs_delete envs_app
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    envs_simple_replace envs_replace envs_split envs_clear_spatial].
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Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.

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(** * Misc *)
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Ltac iFresh :=
  lazymatch goal with
  |- of_envs ?Δ  _ =>
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     (* [vm_compute fails] if any of the hypotheses in [Δ] contain evars, so
     first use [cbv] to compute the domain of [Δ] *)
     let Hs := eval cbv in (envs_dom Δ) in
     eval vm_compute in (fresh_string_of_set "~" (of_list Hs))
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  | _ => constr:"~"
  end.

Tactic Notation "iTypeOf" constr(H) tactic(tac):=
  let Δ := match goal with |- of_envs ?Δ  _ => Δ end in
  match eval env_cbv in (envs_lookup H Δ) with
  | Some (?p,?P) => tac p P
  end.

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Ltac iMatchGoal tac :=
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

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(** * Start a proof *)
Tactic Notation "iProof" :=
  lazymatch goal with
  | |- of_envs _  _ => fail "iProof: already in Iris proofmode"
  | |- True  _ => apply tac_adequate
  | |- _  _ => apply uPred.wand_entails, tac_adequate
  end.

(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
    [env_cbv; reflexivity || fail "iRename:" H1 "not found"
    |env_cbv; reflexivity || fail "iRename:" H2 "not fresh"|].

Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
    match Hs with
    | [] => idtac
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    | "★" :: ?Hs => eapply tac_clear_spatial; [env_cbv; reflexivity|go Hs]
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    | ?H :: ?Hs =>
       eapply tac_clear with _ H _ _; (* (i:=H) *)
         [env_cbv; reflexivity || fail "iClear:" H "not found"|go Hs]
    end in
  let Hs := words Hs in go Hs.

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
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  eapply tac_assumption with H _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iExact:" H "not found"
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    |let P := match goal with |- FromAssumption _ ?P _ => P end in
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     apply _ || fail "iExact:" H ":" P "does not match goal"].
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Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
    match Γ with
    | Esnoc ?Γ ?j ?Q => first [unify P Q; unify i j| find Γ i P]
    end in
  match goal with
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
     first [is_evar i; fail 1 | env_cbv; reflexivity]
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
     is_evar i; first [find Γp i P | find Γs i P]; env_cbv; reflexivity
  end.
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Tactic Notation "iAssumption" :=
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  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
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       [pose proof (_ : FromAssumption p P Q) as Hass;
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        apply (tac_assumption _ j p P); [env_cbv; reflexivity|apply Hass]
       |find p Γ Q]
    end in
  match goal with
  | |- of_envs (Envs ?Γp ?Γs)  ?Q =>
     first [find true Γp Q | find false Γs Q
           |fail "iAssumption:" Q "not found"]
  end.
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(** * False *)
Tactic Notation "iExFalso" := apply tac_ex_falso.

(** * Making hypotheses persistent or pure *)
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Local Tactic Notation "iPersistent" constr(H) :=
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  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iPersistent:" H "not found"
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    |let Q := match goal with |- IntoPersistentP ?Q _ => Q end in
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     apply _ || fail "iPersistent:" H ":" Q "not persistent"
    |env_cbv; reflexivity|].

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Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
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  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
    [env_cbv; reflexivity || fail "iPure:" H "not found"
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    |let P := match goal with |- IntoPure ?P _ => P end in
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     apply _ || fail "iPure:" H ":" P "not pure"
    |intros pat].

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Tactic Notation "iPureIntro" :=
  eapply tac_pure_intro;
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    [let P := match goal with |- FromPure ?P _ => P end in
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     apply _ || fail "iPureIntro:" P "not pure"|].
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(** * Specialize *)
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Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 0).
Notation "( H $! x1 .. xn 'with' pat )" :=
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 0).
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

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Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
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  match xs with
  | hnil => idtac
  | _ =>
    eapply tac_forall_specialize with _ H _ _ _ xs; (* (i:=H) (a:=x) *)
      [env_cbv; reflexivity || fail 1 "iSpecialize:" H "not found"
      |apply _ || fail 1 "iSpecialize:" H "not a forall of the right arity or type"
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      |cbn [himpl hcurry]; reflexivity|]
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  end.
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Local Tactic Notation "iSpecializePat" constr(H) constr(pat) :=
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  let solve_to_wand H1 :=
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    let P := match goal with |- IntoWand ?P _ _ => P end in
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    apply _ || fail "iSpecialize:" H1 ":" P "not an implication/wand" in
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
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    | SForall :: ?pats => try (iSpecializeArgs H1 (hcons _ _)); go H1 pats
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    | SName false ?H2 :: ?pats =>
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       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
         [env_cbv; reflexivity || fail "iSpecialize:" H2 "not found"
         |env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
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         |let P := match goal with |- IntoWand ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand ?P ?Q _ => Q end in
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          apply _ || fail "iSpecialize: cannot instantiate" H1 ":" P "with" H2 ":" Q
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         |env_cbv; reflexivity|go H1 pats]
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    | SName true ?H2 :: ?pats =>
       eapply tac_specialize_persistent with _ _ H1 _ _ _ _;
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         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
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         |env_cbv; reflexivity
         |iExact H2 || fail "iSpecialize:" H2 "not found or wrong type"
         |let Q1 := match goal with |- PersistentP ?Q1  _ => Q1 end in
          let Q2 := match goal with |- _  PersistentP ?Q2 => Q2 end in
          first [left; apply _ | right; apply _]
            || fail "iSpecialize:" Q1 "nor" Q2 "persistent"
         |go H1 pats]
    | SGoalPersistent :: ?pats =>
       eapply tac_specialize_persistent with _ _ H1 _ _ _ _;
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |env_cbv; reflexivity
         |(*goal*)
         |let Q1 := match goal with |- PersistentP ?Q1  _ => Q1 end in
          let Q2 := match goal with |- _  PersistentP ?Q2 => Q2 end in
          first [left; apply _ | right; apply _]
            || fail "iSpecialize:" Q1 "nor" Q2 "persistent"
         |go H1 pats]
    | SGoalPure :: ?pats =>
       eapply tac_specialize_pure with _ H1 _ _ _ _ _;
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         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
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         |let Q := match goal with |- FromPure ?Q _ => Q end in
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          apply _ || fail "iSpecialize:" Q "not pure"
         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
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    | SGoal ?k ?lr ?Hs :: ?pats =>
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs _ _ _ _;
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         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
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         |match k with
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          | GoalStd => apply into_assert_default
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          | GoalPvs => apply _ || fail "iSpecialize: cannot generate pvs goal"
          end
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         |env_cbv; reflexivity || fail "iSpecialize:" Hs "not found"
         |(*goal*)
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         |go H1 pats]
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    end in let pats := spec_pat.parse pat in go H pats.

Tactic Notation "iSpecialize" open_constr(t) :=
  match t with
  | ITrm ?H ?xs ?pat => iSpecializeArgs H xs; iSpecializePat H pat
  end.
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(** * Pose proof *)
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(* The tactic [iIntoEntails] tactic solves a goal [True ⊢ Q]. The arguments [t]
is a Coq term whose type is of the following shape:

- [∀ (x_1 : A_1) .. (x_n : A_n), True ⊢ Q]
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -★ P2]
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊣⊢ P2], in which case [Q] becomes [P1 ↔ P2]

The tactic instantiates each dependent argument [x_i] with an evar and generates
a goal [P] for non-dependent arguments [x_i : P]. *)
Tactic Notation "iIntoEntails" open_constr(t) :=
  let rec go t :=
    lazymatch type of t with
    | True  _ => apply t
    | _  _ => apply (uPred.entails_wand _ _ t)
    | _  _ => apply (uPred.equiv_iff _ _ t)
    | ?P  ?Q => let H := fresh in assert P as H; [|go uconstr:(t H)]
    |  _ : ?T, _ =>
       (* Put [T] inside an [id] to avoid TC inference from being invoked. *)
       (* This is a workarround for Coq bug #4969. *)
       let e := fresh in evar (e:id T);
       let e' := eval unfold e in e in clear e; go (t e')
    end
  in go t.

Tactic Notation "iPoseProofCore" open_constr(lem) "as" tactic(tac) :=
  let pose_trm t tac :=
    let Htmp := iFresh in
    lazymatch type of t with
    | string =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
         [env_cbv; reflexivity || fail "iPoseProof:" t "not found"
         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |tac Htmp]
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
         [iIntoEntails t
         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |tac Htmp]
    end;
    try (apply _) (* solve TC constraints. It is essential that this happens
    after the continuation [tac] has been called. *)
  in lazymatch lem with
  | ITrm ?t ?xs ?pat =>
     pose_trm t ltac:(fun Htmp =>
       iSpecializeArgs Htmp xs; iSpecializePat Htmp pat; last (tac Htmp))
  | _ => pose_trm lem tac
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  end.

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Tactic Notation "iPoseProof" open_constr(lem) "as" constr(H) :=
  iPoseProofCore lem as (fun Htmp => iRename Htmp into H).
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Tactic Notation "iPoseProof" open_constr(lem) :=
  iPoseProofCore lem as (fun _ => idtac).
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(** * Apply *)
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Tactic Notation "iApply" open_constr(lem) :=
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  let finish H := first
    [iExact H
    |eapply tac_apply with _ H _ _ _;
       [env_cbv; reflexivity || fail 1 "iApply:" H "not found"
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       |let P := match goal with |- IntoWand ?P _ _ => P end in
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        apply _ || fail 1 "iApply: cannot apply" H ":" P
       |lazy beta (* reduce betas created by instantiation *)]] in
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  lazymatch lem with
  | ITrm ?t ?xs ?pat =>
     iPoseProofCore t as (fun Htmp =>
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       iSpecializeArgs Htmp xs;
       try (iSpecializeArgs Htmp (hcons _ _));
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       iSpecializePat Htmp pat; last finish Htmp)
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  | _ =>
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     iPoseProofCore lem as (fun Htmp =>
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       try (iSpecializeArgs Htmp (hcons _ _));
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       finish Htmp)
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  end.
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(** * Revert *)
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Local Tactic Notation "iForallRevert" ident(x) :=
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  let A := type of x in
  lazymatch type of A with
  | Prop => revert x; apply tac_pure_revert
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  | _ => revert x; apply tac_forall_revert
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  end || fail "iRevert: cannot revert" x.
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Tactic Notation "iRevert" constr(Hs) :=
  let rec go H2s :=
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    match H2s with
    | [] => idtac
    | "★" :: ?H2s => go H2s; eapply tac_revert_spatial; env_cbv
    | ?H2 :: ?H2s =>
       go H2s;
       eapply tac_revert with _ H2 _ _; (* (i:=H2) *)
         [env_cbv; reflexivity || fail "iRevert:" H2 "not found"
         |env_cbv]
    end in
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  let Hs := words Hs in go Hs.

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Tactic Notation "iRevert" "(" ident(x1) ")" :=
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  iForallRevert x1.
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Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" :=
  iForallRevert x2; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" :=
  iForallRevert x3; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")" :=
  iForallRevert x4; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" :=
  iForallRevert x5; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" :=
  iForallRevert x6; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" :=
  iForallRevert x7; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" :=
  iForallRevert x8; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).

Tactic Notation "iRevert" "(" ident(x1) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")"
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    constr(Hs) :=
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  iRevert Hs; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 x8 ).
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(** * Disjunction *)
Tactic Notation "iLeft" :=
  eapply tac_or_l;
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    [let P := match goal with |- FromOr ?P _ _ => P end in
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     apply _ || fail "iLeft:" P "not a disjunction"|].
Tactic Notation "iRight" :=
  eapply tac_or_r;
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    [let P := match goal with |- FromOr ?P _ _ => P end in
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     apply _ || fail "iRight:" P "not a disjunction"|].

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Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
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  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_cbv; reflexivity || fail "iOrDestruct:" H "not found"
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    |let P := match goal with |- IntoOr ?P _ _ => P end in
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     apply _ || fail "iOrDestruct:" H ":" P "not a disjunction"
    |env_cbv; reflexivity || fail "iOrDestruct:" H1 "not fresh"
    |env_cbv; reflexivity || fail "iOrDestruct:" H2 "not fresh"| |].

(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
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  lazymatch goal with
  | |- _  _ =>
    eapply tac_and_split;
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      [let P := match goal with |- FromAnd ?P _ _ => P end in
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       apply _ || fail "iSplit:" P "not a conjunction"| |]
  | |- _  _ => apply (anti_symm ())
  end.
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Tactic Notation "iSplitL" constr(Hs) :=
  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
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    [let P := match goal with |- FromSep ?P _ _ => P end in
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     apply _ || fail "iSplitL:" P "not a separating conjunction"
    |env_cbv; reflexivity || fail "iSplitL: hypotheses" Hs "not found"| |].
Tactic Notation "iSplitR" constr(Hs) :=
  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
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    [let P := match goal with |- FromSep ?P _ _ => P end in
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     apply _ || fail "iSplitR:" P "not a separating conjunction"
    |env_cbv; reflexivity || fail "iSplitR: hypotheses" Hs "not found"| |].

Tactic Notation "iSplitL" := iSplitR "".
Tactic Notation "iSplitR" := iSplitL "".

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Local Tactic Notation "iSepDestruct" constr(H) "as" constr(H1) constr(H2) :=
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  eapply tac_sep_destruct with _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_cbv; reflexivity || fail "iSepDestruct:" H "not found"
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    |let P := match goal with |- IntoSep _ ?P _ _ => P end in
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     apply _ || fail "iSepDestruct:" H ":" P "not separating destructable"
    |env_cbv; reflexivity || fail "iSepDestruct:" H1 "or" H2 " not fresh"|].

Tactic Notation "iFrame" constr(Hs) :=
  let rec go Hs :=
    match Hs with
    | [] => idtac
    | ?H :: ?Hs =>
       eapply tac_frame with _ H _ _ _;
         [env_cbv; reflexivity || fail "iFrame:" H "not found"
         |let R := match goal with |- Frame ?R _ _ => R end in
          apply _ || fail "iFrame: cannot frame" R
         |lazy iota beta; go Hs]
    end
  in let Hs := words Hs in go Hs.

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Tactic Notation "iFrame" :=
  let rec go Hs :=
    match Hs with
    | [] => idtac
    | ?H :: ?Hs => try iFrame H; go Hs
    end in
  match goal with
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  | |- of_envs ?Δ  _ => let Hs := eval cbv in (env_dom (env_spatial Δ)) in go Hs
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  end.

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Tactic Notation "iCombine" constr(H1) constr(H2) "as" constr(H) :=
  eapply tac_combine with _ _ _ H1 _ _ H2 _ _ H _;
    [env_cbv; reflexivity || fail "iCombine:" H1 "not found"
    |env_cbv; reflexivity || fail "iCombine:" H2 "not found"
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    |let P1 := match goal with |- FromSep _ ?P1 _ => P1 end in
     let P2 := match goal with |- FromSep _ _ ?P2 => P2 end in
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     apply _ || fail "iCombine: cannot combine" H1 ":" P1 "and" H2 ":" P2
    |env_cbv; reflexivity || fail "iCombine:" H "not fresh"|].

(** * Existential *)
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Tactic Notation "iExists" uconstr(x1) :=
  eapply tac_exist;
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    [let P := match goal with |- FromExist ?P _ => P end in
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     apply _ || fail "iExists:" P "not an existential"
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    |cbv beta; eexists x1].
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
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  iExists x1; iExists x2.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
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  iExists x1; iExists x2, x3.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
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  iExists x1; iExists x2, x3, x4.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
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  iExists x1; iExists x2, x3, x4, x5.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
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  iExists x1; iExists x2, x3, x4, x5, x6.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
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  iExists x1; iExists x2, x3, x4, x5, x6, x7.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) ","
    uconstr(x8) :=
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  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.

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Local Tactic Notation "iExistDestruct" constr(H)
    "as" simple_intropattern(x) constr(Hx) :=
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  eapply tac_exist_destruct with H _ Hx _ _; (* (i:=H) (j:=Hx) *)
    [env_cbv; reflexivity || fail "iExistDestruct:" H "not found"
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    |let P := match goal with |- IntoExist ?P _ => P end in
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     apply _ || fail "iExistDestruct:" H ":" P "not an existential"|];
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  let y := fresh in
  intros y; eexists; split;
    [env_cbv; reflexivity || fail "iExistDestruct:" Hx "not fresh"
    |revert y; intros x].
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(** * Basic destruct tactic *)
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Local Tactic Notation "iDestructHyp" constr(H) "as" constr(pat) :=
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  let rec go Hz pat :=
    lazymatch pat with
    | IAnom => idtac
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    | IAnomPure => iPure Hz as ?
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    | IDrop => iClear Hz
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    | IFrame => iFrame Hz
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    | IName ?y => iRename Hz into y
    | IPersistent ?pat => iPersistent Hz; go Hz pat
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    | IList [[]] => iExFalso; iExact Hz
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    | IList [[?pat1; ?pat2]] =>
       let Hy := iFresh in iSepDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
    | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]
    | _ => fail "iDestruct:" pat "invalid"
    end
  in let pat := intro_pat.parse_one pat in go H pat.

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Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1) ")"
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    constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as @ pat.
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Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) ")"
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    constr(pat) :=
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  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) simple_intropattern(x6) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7) ")"
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    constr(pat) :=
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  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 x7 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7)
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    simple_intropattern(x8) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 x7 x8 ) pat.
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(** * Always *)
Tactic Notation "iAlways":=
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  apply tac_always_intro;
    [reflexivity || fail "iAlways: spatial context non-empty"|].
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(** * Later *)
Tactic Notation "iNext":=
  eapply tac_next;
    [apply _
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    |let P := match goal with |- FromLater ?P _ => P end in
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     apply _ || fail "iNext:" P "does not contain laters"|].

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Tactic Notation "iTimeless" constr(H) :=
  eapply tac_timeless with _ H _ _;
    [let Q := match goal with |- TimelessElim ?Q => Q end in
     apply _ || fail "iTimeless: cannot eliminate later in goal" Q
    |env_cbv; reflexivity || fail "iTimeless:" H "not found"
    |let P := match goal with |- TimelessP ?P => P end in
     apply _ || fail "iTimeless: " P "not timeless"
    |env_cbv; reflexivity|].

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(** * Introduction tactic *)
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Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" := first
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  [ (* (∀ _, _) *) apply tac_forall_intro; intros x
  | (* (?P → _) *) eapply tac_impl_intro_pure;
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     [let P := match goal with |- IntoPure ?P _ => P end in
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      apply _ || fail "iIntro:" P "not pure"
     |intros x]
  | (* (?P -★ _) *) eapply tac_wand_intro_pure;
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     [let P := match goal with |- IntoPure ?P _ => P end in
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      apply _ || fail "iIntro:" P "not pure"
     |intros x]
  |intros x].

Local Tactic Notation "iIntro" constr(H) := first
  [ (* (?Q → _) *)
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    eapply tac_impl_intro with _ H; (* (i:=H) *)
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      [reflexivity || fail 1 "iIntro: introducing" H
                             "into non-empty spatial context"
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      |env_cbv; reflexivity || fail "iIntro:" H "not fresh"|]
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  | (* (_ -★ _) *)
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    eapply tac_wand_intro with _ H; (* (i:=H) *)
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      [env_cbv; reflexivity || fail 1 "iIntro:" H "not fresh"|]
  | fail 1 "iIntro: nothing to introduce" ].
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Local Tactic Notation "iIntro" "#" constr(H) := first
  [ (* (?P → _) *)
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    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
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      [let P := match goal with |- IntoPersistentP ?P _ => P end in
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       apply _ || fail 1 "iIntro: " P " not persistent"
      |env_cbv; reflexivity || fail 1 "iIntro:" H "not fresh"|]
  | (* (?P -★ _) *)
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    eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
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      [let P := match goal with |- IntoPersistentP ?P _ => P end in
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       apply _ || fail 1 "iIntro: " P " not persistent"
      |env_cbv; reflexivity || fail 1 "iIntro:" H "not fresh"|]
  | fail 1 "iIntro: nothing to introduce" ].
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Local Tactic Notation "iIntroForall" :=
  lazymatch goal with
  | |-  _, ?P => fail
  | |-  _, _ => intro
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  | |- _  ( x : _, _) => iIntro (x)
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  end.
Local Tactic Notation "iIntro" :=
  lazymatch goal with
  | |- _  ?P => intro
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  | |- _  (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  | |- _  (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
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  end.

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Tactic Notation "iIntros" constr(pat) :=
  let rec go pats :=
    lazymatch pats with
    | [] => idtac
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    | IForall :: ?pats => repeat iIntroForall; go pats
    | IAll :: ?pats => repeat (iIntroForall || iIntro); go pats
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    | ISimpl :: ?pats => simpl; go pats
    | IAlways :: ?pats => iAlways; go pats
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    | INext :: ?pats => iNext; go pats
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    | IClear ?cpats :: ?pats =>
       let rec clr cpats :=
         match cpats with
         | [] => go pats
         | (false,?H) :: ?cpats => iClear H; clr cpats
         | (true,?H) :: ?cpats => iFrame H; clr cpats
         end in clr cpats
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    | IPersistent (IName ?H) :: ?pats => iIntro #H; go pats
    | IName ?H :: ?pats => iIntro H; go pats
    | IPersistent IAnom :: ?pats => let H := iFresh in iIntro #H; go pats
    | IAnom :: ?pats => let H := iFresh in iIntro H; go pats
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    | IAnomPure :: ?pats => iIntro (?); go pats
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    | IPersistent ?pat :: ?pats =>
       let H := iFresh in iIntro #H; iDestructHyp H as pat; go pats
    | ?pat :: ?pats =>
       let H := iFresh in iIntro H; iDestructHyp H as pat; go pats
    | _ => fail "iIntro: failed with" pats
    end
  in let pats := intro_pat.parse pat in try iProof; go pats.
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Tactic Notation "iIntros" := iIntros "**".
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Tactic Notation "iIntros" "(" simple_intropattern(x1) ")" :=
  try iProof; iIntro ( x1 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1)
    simple_intropattern(x2) ")" :=
  iIntros ( x1 ); iIntro ( x2 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    simple_intropattern(x3) ")" :=
  iIntros ( x1 x2 ); iIntro ( x3 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    simple_intropattern(x3) simple_intropattern(x4) ")" :=
  iIntros ( x1 x2 x3 ); iIntro ( x4 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5) ")" :=
  iIntros ( x1 x2 x3 x4 ); iIntro ( x5 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    simple_intropattern(x6) ")" :=
  iIntros ( x1 x2 x3 x4 x5 ); iIntro ( x6 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    simple_intropattern(x6) simple_intropattern(x7) ")" :=
  iIntros ( x1 x2 x3 x4 x5 x6 ); iIntro ( x7 ).
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    simple_intropattern(x6) simple_intropattern(x7) simple_intropattern(x8) ")" :=
  iIntros ( x1 x2 x3 x4 x5 x6 x7 ); iIntro ( x8 ).

Tactic Notation "iIntros" "(" simple_intropattern(x1) ")" constr(p) :=
  iIntros ( x1 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    ")" constr(p) :=
  iIntros ( x1 x2 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    simple_intropattern(x3) ")" constr(p) :=
  iIntros ( x1 x2 x3 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
    simple_intropattern(x3) simple_intropattern(x4) ")" constr(p) :=
  iIntros ( x1 x2 x3 x4 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    ")" constr(p) :=
  iIntros ( x1 x2 x3 x4 x5 ); iIntros p.
Tactic Notation "iIntros" "("simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    simple_intropattern(x6) ")" constr(p) :=
  iIntros ( x1 x2 x3 x4 x5 x6 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
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    simple_intropattern(x6) simple_intropattern(x7) ")" constr(p) :=
  iIntros ( x1 x2 x3 x4 x5 x6 x7 ); iIntros p.
Tactic Notation "iIntros" "(" simple_intropattern(x1) simple_intropattern(x2)
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    simple_intropattern(x3) simple_intropattern(x4) simple_intropattern(x5)
    simple_intropattern(x6) simple_intropattern(x7) simple_intropattern(x8)
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    ")" constr(p) :=
  iIntros ( x1 x2 x3 x4 x5 x6 x7 x8 ); iIntros p.
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(** * Destruct tactic *)
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Tactic Notation "iDestructCore" open_constr(lem) "as" tactic(tac) :=
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  let intro_destruct n :=
    let rec go n' :=
      lazymatch n' with
      | 0 => fail "iDestruct: cannot introduce" n "hypotheses"
      | 1 => repeat iIntroForall; let H := iFresh in iIntro H; tac H
      | S ?n' => repeat iIntroForall; let H := iFresh in iIntro H; go n'
      end in intros; try iProof; go n in
  lazymatch type of lem with
  | nat => intro_destruct lem
  | Z => (* to make it work in Z_scope. We should just be able to bind
     tactic notation arguments to notation scopes. *)
     let n := eval compute in (Z.to_nat lem) in intro_destruct n
  | string => tac lem
  | iTrm =>
     lazymatch lem with
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     | @iTrm string ?H _ hnil ?pat => iSpecializePat H pat; last (tac H)
     | _ => iPoseProofCore lem as tac
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     end
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  | _ => iPoseProofCore lem as tac
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  end.

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Tactic Notation "iDestruct" open_constr(lem) "as" constr(pat) :=
  iDestructCore lem as (fun H => iDestructHyp H as pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1) ")"
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    constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) ")" constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) ")" constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) ")"
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    constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 x4 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) ")" constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 x4 x5 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) simple_intropattern(x6) ")" constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 x4 x5 x6 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
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    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7) ")"
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    constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 x4 x5 x6 x7 ) pat).
Tactic Notation "iDestruct" open_constr(lem) "as" "(" simple_intropattern(x1)
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    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7)
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    simple_intropattern(x8) ")" constr(pat) :=
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  iDestructCore lem as (fun H => iDestructHyp H as ( x1 x2 x3 x4 x5 x6 x7 x8 ) pat).
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Tactic Notation "iDestruct" open_constr(lem) "as" "%" simple_intropattern(pat) :=
  iDestructCore lem as (fun H => iPure H as pat).
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(* This is pretty ugly, but without Ltac support for manipulating lists of
idents I do not know how to do this better. *)
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Local Ltac iLöbHelp IH tac_before tac_after :=
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  match goal with
  | |- of_envs ?Δ  _ =>
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     let Hs := constr:(reverse (env_dom (env_spatial Δ))) in
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     iRevert ["★"]; tac_before;
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     eapply tac_löb with _ IH;
       [reflexivity
       |env_cbv; reflexivity || fail "iLöb:" IH "not fresh"|];
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    tac_after; iIntros Hs
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  end.

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Tactic Notation "iLöb" "as" constr (IH) := iLöbHelp IH idtac idtac.
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Tactic Notation "iLöb" "(" ident(x1) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 )) ltac:(iIntros ( x1 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 )) ltac:(iIntros ( x1 x2 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 )) ltac:(iIntros ( x1 x2 x3 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")" "as"
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    constr (IH):=
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  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 x4 )) ltac:(iIntros ( x1 x2 x3 x4 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 x4 x5 ))
              ltac:(iIntros ( x1 x2 x3 x4 x5 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 x4 x5 x6 ))
              ltac:(iIntros ( x1 x2 x3 x4 x5 x6 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 x4 x5 x6 x7 ))
              ltac:(iIntros ( x1 x2 x3 x4 x5 x6 x7 )).
Tactic Notation "iLöb" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" "as" constr (IH) :=
  iLöbHelp IH ltac:(iRevert ( x1 x2 x3 x4 x5 x6 x7 x8 ))
              ltac