tactics.v

From iris.proofmode Require Import coq_tactics.
From iris.proofmode Require Import intro_patterns spec_patterns sel_patterns.
From iris.base_logic Require Export base_logic.
From iris.proofmode Require Export classes notation.
From iris.proofmode Require Import class_instances.
From stdpp Require Import stringmap hlist.
From iris.proofmode Require Import strings.
Set Default Proof Using "Type".

Declare Reduction env_cbv := cbv [
  beq ascii_beq string_beq
  env_lookup env_lookup_delete env_delete env_app env_replace env_dom
  env_persistent env_spatial env_spatial_is_nil envs_dom
  envs_lookup envs_lookup_delete envs_delete envs_snoc envs_app
    envs_simple_replace envs_replace envs_split
    envs_clear_spatial envs_clear_persistent
    envs_split_go envs_split].
Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.
Ltac env_reflexivity := env_cbv; reflexivity.

(** * Misc *)
(* Tactic Notation tactics cannot return terms *)
Ltac iFresh' H :=
  lazymatch goal with
  |- of_envs ?Δ ⊢ _ =>
     (* [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 H (of_list Hs))
  | _ => H
  end.
Ltac iFresh := iFresh' "~".

Ltac iMissingHyps Hs :=
  let Δ :=
    lazymatch goal with
    | |- of_envs ?Δ ⊢ _ => Δ
    | |- context[ envs_split _ _ ?Δ ] => Δ
    end in
  let Hhyps := eval env_cbv in (envs_dom Δ) in
  eval vm_compute in (list_difference Hs Hhyps).

Ltac iTypeOf H :=
  let Δ := match goal with |- of_envs ?Δ ⊢ _ => Δ end in
  eval env_cbv in (envs_lookup H Δ).

Tactic Notation "iMatchHyp" tactic1(tac) :=
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

(** * Start a proof *)
Ltac iStartProof :=
  lazymatch goal with
  | |- of_envs _ ⊢ _ => idtac
  | |- ?P =>
    lazymatch eval hnf in P with
    (* need to use the unfolded version of [uPred_valid] due to the hnf *)
    | True ⊢ _ => apply tac_adequate
    | _ ⊢ _ => apply uPred.wand_entails, tac_adequate
    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply uPred.iff_equiv, tac_adequate
    | _ => fail "iStartProof: not a uPred"
    end
  end.

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

Local Inductive esel_pat :=
  | ESelPure
  | ESelName : bool → string → esel_pat.

Ltac iElaborateSelPat pat tac :=
  let rec go pat Δ Hs :=
    lazymatch pat with
    | [] => let Hs' := eval cbv in Hs in tac Hs'
    | SelPure :: ?pat => go pat Δ (ESelPure :: Hs)
    | SelPersistent :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_persistent Δ)) in
       let Δ' := eval env_cbv in (envs_clear_persistent Δ) in
       go pat Δ' ((ESelName true <$> Hs') ++ Hs)
    | SelSpatial :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_spatial Δ)) in
       let Δ' := eval env_cbv in (envs_clear_spatial Δ) in
       go pat Δ' ((ESelName false <$> Hs') ++ Hs)
    | SelName ?H :: ?pat =>
       lazymatch eval env_cbv in (envs_lookup_delete H Δ) with
       | Some (?p,_,?Δ') => go pat Δ' (ESelName p H :: Hs)
       | None => fail "iElaborateSelPat:" H "not found"
       end
    end in
  lazymatch goal with
  | |- of_envs ?Δ ⊢ _ =>
    let pat := sel_pat.parse pat in go pat Δ (@nil esel_pat)
  end.

Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
    lazymatch Hs with
    | [] => idtac
    | ESelPure :: ?Hs => clear; go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_clear with _ H _ _; (* (i:=H) *)
         [env_reflexivity || fail "iClear:" H "not found"|go Hs]
    end in
  iElaborateSelPat Hs go.

Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
  eapply tac_assumption with H _ _; (* (i:=H) *)
    [env_reflexivity || fail "iExact:" H "not found"
    |apply _ ||
     let P := match goal with |- FromAssumption _ ?P _ => P end in
     fail "iExact:" H ":" P "does not match goal"].

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_reflexivity]
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
     first [is_evar i; fail 1 | env_reflexivity]
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
  end.

Tactic Notation "iAssumption" :=
  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
       [pose proof (_ : FromAssumption p P Q) as Hass;
        apply (tac_assumption _ j p P); [env_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.

(** * False *)
Tactic Notation "iExFalso" := apply tac_ex_falso.

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

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

Tactic Notation "iPureIntro" :=
  iStartProof;
  eapply tac_pure_intro;
    [apply _ ||
     let P := match goal with |- FromPure ?P _ => P end in
     fail "iPureIntro:" P "not pure"
    |].

(** Framing *)
Local Ltac iFrameFinish :=
  lazy iota beta;
  try match goal with
  | |- _ ⊢ True => exact (uPred.pure_intro _ _ I)
  end.

Local Ltac iFramePure t :=
  let φ := type of t in
  eapply (tac_frame_pure _ _ _ _ t);
    [apply _ || fail "iFrame: cannot frame" φ
    |iFrameFinish].

Local Ltac iFrameHyp H :=
  eapply tac_frame with _ H _ _ _;
    [env_reflexivity || fail "iFrame:" H "not found"
    |apply _ ||
     let R := match goal with |- Frame _ ?R _ _ => R end in
     fail "iFrame: cannot frame" R
    |iFrameFinish].

Local Ltac iFrameAnyPure :=
  repeat match goal with H : _ |- _ => iFramePure H end.

Local Ltac iFrameAnyPersistent :=
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => repeat iFrameHyp H; go Hs end in
  match goal with
  | |- of_envs ?Δ ⊢ _ =>
     let Hs := eval cbv in (env_dom (env_persistent Δ)) in go Hs
  end.

Local Ltac iFrameAnySpatial :=
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
  | |- of_envs ?Δ ⊢ _ =>
     let Hs := eval cbv in (env_dom (env_spatial Δ)) in go Hs
  end.

Tactic Notation "iFrame" := iFrameAnySpatial.

Tactic Notation "iFrame" "(" constr(t1) ")" :=
  iFramePure t1.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" :=
  iFramePure t1; iFrame ( t2 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" :=
  iFramePure t1; iFrame ( t2 t3 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ).

Tactic Notation "iFrame" constr(Hs) :=
  let rec go Hs :=
    lazymatch Hs with
    | [] => idtac
    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
    end
  in let Hs := sel_pat.parse Hs in go Hs.
Tactic Notation "iFrame" "(" constr(t1) ")" constr(Hs) :=
  iFramePure t1; iFrame Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")"
    constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ) Hs.

(** * Basic introduction tactics *)
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
  try iStartProof;
  try first
    [(* (∀ _, _) *) apply tac_forall_intro
    |(* (?P → _) *) eapply tac_impl_intro_pure;
      [apply _ ||
       let P := match goal with |- IntoPure ?P _ => P end in
       fail "iIntro:" P "not pure"
      |]
    |(* (?P -∗ _) *) eapply tac_wand_intro_pure;
      [apply _ ||
       let P := match goal with |- IntoPure ?P _ => P end in
       fail "iIntro:" P "not pure"
      |]
    |(* ⌜∀ _, _⌝ *) apply tac_pure_forall_intro
    |(* ⌜_ → _⌝ *) apply tac_pure_impl_intro];
  intros x.

Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
  [ (* (?Q → _) *)
    eapply tac_impl_intro with _ H; (* (i:=H) *)
      [reflexivity || fail 1 "iIntro: introducing" H
                             "into non-empty spatial context"
      |env_reflexivity || fail "iIntro:" H "not fresh"
      |]
  | (* (_ -∗ _) *)
    eapply tac_wand_intro with _ H; (* (i:=H) *)
      [env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
  [ (* (?P → _) *)
    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
      [apply _ || 
       let P := match goal with |- IntoPersistentP ?P _ => P end in
       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
  | (* (?P -∗ _) *)
    eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
      [apply _ || 
       let P := match goal with |- IntoPersistentP ?P _ => P end in
       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "_" :=
  try iStartProof;
  first
  [ (* (?Q → _) *) apply tac_impl_intro_drop
  | (* (_ -∗ _) *) apply tac_wand_intro_drop
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntroForall" :=
  try iStartProof;
  lazymatch goal with
  | |- ∀ _, ?P => fail
  | |- ∀ _, _ => intro
  | |- _ ⊢ (∀ x : _, _) => let x' := fresh x in iIntro (x')
  end.
Local Tactic Notation "iIntro" :=
  try iStartProof;
  lazymatch goal with
  | |- _ → ?P => intro
  | |- _ ⊢ (_ -∗ _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  | |- _ ⊢ (_ → _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  end.

(** * Specialize *)
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 9).
Notation "( H $! x1 .. xn 'with' pat )" :=
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
  let rec go xs :=
    lazymatch xs with
    | hnil => idtac
    | hcons ?x ?xs =>
       eapply tac_forall_specialize with _ H _ _ _; (* (i:=H) (a:=x) *)
         [env_reflexivity || fail 1 "iSpecialize:" H "not found"
         |apply _ ||
          let P := match goal with |- IntoForall ?P _ => P end in
          fail 1 "iSpecialize: cannot instantiate" P "with" x
         |exists x; split; [env_reflexivity|go xs]]
    end in
  go xs.

Local Tactic Notation "iSpecializePat" constr(H) constr(pat) :=
  let solve_to_wand H1 :=
    apply _ ||
    let P := match goal with |- IntoWand ?P _ _ => P end in
    fail "iSpecialize:" P "not an implication/wand" in
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
    | SForall :: ?pats =>
       idtac "the * specialization pattern is deprecated because it is applied implicitly";
       go H1 pats
    | SName ?H2 :: ?pats =>
       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
         [env_reflexivity || fail "iSpecialize:" H2 "not found"
         |env_reflexivity || fail "iSpecialize:" H1 "not found"
         |apply _ ||
          let P := match goal with |- IntoWand ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand ?P ?Q _ => Q end in
          fail "iSpecialize: cannot instantiate" P "with" Q
         |env_reflexivity|go H1 pats]
    | SPureGoal ?d :: ?pats =>
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |apply _ ||
          let Q := match goal with |- FromPure ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
         |env_reflexivity
         |done_if d (*goal*)
         |go H1 pats]
    | SGoal (SpecGoal GPersistent false ?Hs_frame [] ?d) :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |apply _ ||
          let Q := match goal with |- PersistentP ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
         |env_reflexivity
         |iFrame Hs_frame; done_if d (*goal*)
         |go H1 pats]
    | SGoal (SpecGoal GPersistent _ _ _ _) :: ?pats =>
       fail "iSpecialize: cannot select hypotheses for persistent premise"
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs ?d) :: ?pats =>
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
          end
         |env_reflexivity ||
          let Hs' := iMissingHyps Hs' in
          fail "iSpecialize: hypotheses" Hs' "not found"
         |iFrame Hs_frame; done_if d (*goal*)
         |go H1 pats]
    | SAutoFrame GPersistent :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |apply _ ||
          let Q := match goal with |- PersistentP ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
         |env_reflexivity
         |solve [iFrame "∗ #"]
         |go H1 pats]
    | SAutoFrame ?m :: ?pats =>
       eapply tac_specialize_frame with _ H1 _ _ _ _ _ _;
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
          end
         |iFrame "∗ #"; apply tac_unlock ||
          fail "iSpecialize: premise cannot be solved by framing"
         |reflexivity]; iIntro H1; go H1 pats
    end in let pats := spec_pat.parse pat in go H pats.

(* The argument [p] denotes whether the conclusion of the specialized term is
persistent. If so, one can use all spatial hypotheses for both proving the
premises and the remaning goal. The argument [p] can either be a Boolean or an
introduction pattern, which will be coerced into [true] when it solely contains
`#` or `%` patterns at the top-level.

In case the specialization pattern in [t] states that the modality of the goal
should be kept for one of the premises (i.e. [>[H1 .. Hn]] is used) then [p]
defaults to [false] (i.e. spatial hypotheses are not preserved). *)
Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
  let p := intro_pat_persistent p in
  let t :=
    match type of t with string => constr:(ITrm t hnil "") | _ => t end in
  lazymatch t with
  | ITrm ?H ?xs ?pat =>
    let pat := spec_pat.parse pat in
    lazymatch type of H with
    | string =>
      lazymatch eval compute in (p && negb (existsb spec_pat_modal pat)) with
      | true =>
         eapply tac_specialize_persistent_helper with _ H _ _ _;
           [env_reflexivity || fail "iSpecialize:" H "not found"
           |iSpecializeArgs H xs; iSpecializePat H pat; last (iExact H)
           |apply _ ||
            let Q := match goal with |- PersistentP ?Q => Q end in
            fail "iSpecialize:" Q "not persistent"
           |env_reflexivity|(* goal *)]
      | false => iSpecializeArgs H xs; iSpecializePat H pat
      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
    end
  | _ => fail "iSpecialize:" t "should be a proof mode term"
  end.

Tactic Notation "iSpecialize" open_constr(t) :=
  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.

(** * Pose proof *)
(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:

- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid 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 "iIntoValid" open_constr(t) :=
  let rec go t :=
    let tT := type of t in
    lazymatch eval hnf in tT with
    | True ⊢ _ => apply t
    | _ ⊢ _ => apply (uPred.entails_wand _ _ t)
    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply (uPred.equiv_iff _ _ t)
    | ?P → ?Q => let H := fresh in assert P as H; [|go uconstr:(t H); clear 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.

(* The tactic [tac] is called with a temporary fresh name [H]. The argument
[lazy_tc] denotes whether type class inference on the premises of [lem] should
be performed before (if false) or after (if true) [tac H] is called. *)
Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
  try iStartProof;
  let Htmp := iFresh in
  let t :=
    lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let spec_tac _ :=
    lazymatch lem with
    | ITrm ?t ?xs ?pat => iSpecializeCore (ITrm Htmp xs pat) as p
    | _ => idtac
    end in
  let go goal_tac :=
    lazymatch type of t with
    | string =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
         [env_reflexivity || fail "iPoseProof:" t "not found"
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |goal_tac ()]
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
         [iIntoValid t
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |goal_tac ()]
    end;
    try (apply _) in
  lazymatch eval compute in lazy_tc with
  | true => go ltac:(fun _ => spec_tac (); last (tac Htmp))
  | false => go spec_tac; last (tac Htmp)
  end.

(** * Apply *)
Tactic Notation "iApply" open_constr(lem) :=
  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
      [env_reflexivity
      |apply _
      |lazy beta (* reduce betas created by instantiation *)]
    |iSpecializePat H "[]"; last go H] in
  iPoseProofCore lem as false true (fun H => first
    [iExact H
    |go H
    |lazymatch iTypeOf H with
     | Some (_,?Q) => fail 1 "iApply: cannot apply" Q
     end]).

(** * Revert *)
Local Tactic Notation "iForallRevert" ident(x) :=
  let err x :=
    intros x;
    iMatchHyp (fun H P =>
      lazymatch P with
      | context [x] => fail 2 "iRevert:" x "is used in hypothesis" H
      end) in
  let A := type of x in
  lazymatch type of A with
  | Prop => revert x; first [apply tac_pure_revert|err x]
  | _ => revert x; first [apply tac_forall_revert|err x]
  end.

Tactic Notation "iRevert" constr(Hs) :=
  let rec go Hs :=
    lazymatch Hs with
    | [] => idtac
    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
         [env_reflexivity || fail "iRevert:" H "not found"
         |env_cbv; go Hs]
    end in
  iElaborateSelPat Hs go.

Tactic Notation "iRevert" "(" ident(x1) ")" :=
  iForallRevert x1.
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) ")"
    constr(Hs) :=
  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 ).

(** * Disjunction *)
Tactic Notation "iLeft" :=
  iStartProof;
  eapply tac_or_l;
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
    |].
Tactic Notation "iRight" :=
  iStartProof;
  eapply tac_or_r;
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iRight:" P "not a disjunction"
    |].

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

(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
  iStartProof;
  lazymatch goal with
  | |- _ ⊢ _ =>
    eapply tac_and_split;
      [apply _ ||
       let P := match goal with |- FromAnd _ ?P _ _ => P end in
       fail "iSplit:" P "not a conjunction"| |]
  end.

Tactic Notation "iSplitL" constr(Hs) :=
  iStartProof;
  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
    [apply _ ||
     let P := match goal with |- FromAnd _ ?P _ _ => P end in
     fail "iSplitL:" P "not a separating conjunction"
    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
     fail "iSplitL: hypotheses" Hs "not found"
    | |].

Tactic Notation "iSplitR" constr(Hs) :=
  iStartProof;
  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
    [apply _ ||
     let P := match goal with |- FromAnd _ ?P _ _ => P end in
     fail "iSplitR:" P "not a separating conjunction"
    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
     fail "iSplitR: hypotheses" Hs "not found"
    | |].

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

Local Tactic Notation "iAndDestruct" constr(H) "as" constr(H1) constr(H2) :=
  eapply tac_and_destruct with _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_reflexivity || fail "iAndDestruct:" H "not found"
    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestruct: cannot destruct" P
    |env_reflexivity || fail "iAndDestruct:" H1 "or" H2 " not fresh"|].

Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(lr) constr(H') :=
  eapply tac_and_destruct_choice with _ H _ lr H' _ _ _;
    [env_reflexivity || fail "iAndDestructChoice:" H "not found"
    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestructChoice: cannot destruct" P
    |env_reflexivity || fail "iAndDestructChoice:" H' " not fresh"|].

(** * Combinining hypotheses *)
Tactic Notation "iCombine" constr(Hs) "as" constr(H) :=
  let Hs := words Hs in
  eapply tac_combine with _ _ Hs _ _ H _;
    [env_reflexivity ||
     let Hs := iMissingHyps Hs in
     fail "iCombine: hypotheses" Hs "not found"
    |apply _
    |env_reflexivity || fail "iCombine:" H "not fresh"|].

Tactic Notation "iCombine" constr(H1) constr(H2) "as" constr(H) :=
  iCombine [H1;H2] as H.

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

Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
  iExists x1; iExists x2.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
  iExists x1; iExists x2, x3.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
  iExists x1; iExists x2, x3, x4.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
  iExists x1; iExists x2, x3, x4, x5.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
  iExists x1; iExists x2, x3, x4, x5, x6.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
  iExists x1; iExists x2, x3, x4, x5, x6, x7.
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) ","
    uconstr(x8) :=
  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.

Local Tactic Notation "iExistDestruct" constr(H)
    "as" simple_intropattern(x) constr(Hx) :=
  eapply tac_exist_destruct with H _ Hx _ _; (* (i:=H) (j:=Hx) *)
    [env_reflexivity || fail "iExistDestruct:" H "not found"
    |apply _ ||
     let P := match goal with |- IntoExist ?P _ => P end in
     fail "iExistDestruct: cannot destruct" P|];
  let y := fresh in
  intros y; eexists; split;
    [env_reflexivity || fail "iExistDestruct:" Hx "not fresh"
    |revert y; intros x].

(** * Always *)
Tactic Notation "iAlways":=
  iStartProof;
  apply tac_always_intro; env_cbv.

(** * Later *)
Tactic Notation "iNext" open_constr(n) :=
  iStartProof;
  let P := match goal with |- _ ⊢ ?P => P end in
  try lazymatch n with 0 => fail 1 "iNext: cannot strip 0 laters" end;
  eapply (tac_next _ _ n);
    [apply _ || fail "iNext:" P "does not contain" n "laters"
    |lazymatch goal with
     | |- IntoLaterNEnvs 0 _ _ => fail "iNext:" P "does not contain laters"
     | _ => apply _
     end
    |lazy beta (* remove beta redexes caused by removing laters under binders*)].

Tactic Notation "iNext":= iNext _.

(** * Update modality *)
Tactic Notation "iModIntro" :=
  iStartProof;
  eapply tac_modal_intro;
    [apply _ ||
     let P := match goal with |- FromModal ?P _ => P end in
     fail "iModIntro:" P "not a modality"|].

Tactic Notation "iModCore" constr(H) :=
  eapply tac_modal_elim with _ H _ _ _ _;
    [env_reflexivity || fail "iMod:" H "not found"
    |apply _ ||
     let P := match goal with |- ElimModal ?P _ _ _ => P end in
     let Q := match goal with |- ElimModal _ _ ?Q _ => Q end in
     fail "iMod: cannot eliminate modality " P "in" Q
    |env_reflexivity|].

(** * Basic destruct tactic *)
Local Tactic Notation "iDestructHyp" constr(H) "as" constr(pat) :=
  let rec go Hz pat :=
    lazymatch pat with
    | IAnom => idtac
    | IDrop => iClear Hz
    | IFrame => iFrame Hz
    | IName ?y => iRename Hz into y
    | IList [[]] => iExFalso; iExact Hz
    | IList [[?pat1; IDrop]] => iAndDestructChoice Hz as true Hz; go Hz pat1
    | IList [[IDrop; ?pat2]] => iAndDestructChoice Hz as false Hz; go Hz pat2
    | IList [[?pat1; ?pat2]] =>
       let Hy := iFresh in iAndDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
    | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]