ltac_tactics.v 96.7 KB
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From iris.proofmode Require Import coq_tactics reduction.
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From iris.proofmode Require Import base intro_patterns spec_patterns sel_patterns.
From iris.bi Require Export bi.
From stdpp Require Import namespaces.
From iris.proofmode Require Export classes notation.
From stdpp Require Import hlist pretty.
Set Default Proof Using "Type".
Export ident.

(** For most of the tactics, we want to have tight control over the order and
way in which type class inference is performed. To that end, many tactics make
use of [notypeclasses refine] and the [iSolveTC] tactic to manually invoke type
class inference.

The tactic [iSolveTC] does not use [apply _], as that often leads to issues
because it will try to solve all evars whose type is a typeclass, in
dependency order (according to Matthieu). If one fails, it aborts. However, we
generally rely on progress on the main goal to be solved to make progress
elsewhere. With [typeclasses eauto], that seems to work better.

A drawback of [typeclasses eauto] is that it is multi-success, i.e. whenever
subsequent tactics fail, it will backtrack to [typeclasses eauto] to try the
next type class instance. This is almost always undesired and leads to poor
performance and horrible error messages, so we wrap it in a [once]. *)
Ltac iSolveTC :=
  solve [once (typeclasses eauto)].

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(** Tactic used for solving side-conditions arising from TC resolution in iMod
and iInv. *)
Ltac iSolveSideCondition :=
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  split_and?; try solve [ fast_done | solve_ndisj ].
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(** Used for printing [string]s and [ident]s. *)
Ltac pretty_ident H :=
  lazymatch H with
  | INamed ?H => H
  | ?H => H
  end.

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(** * Misc *)

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

Ltac iTypeOf H :=
  let Δ := match goal with |- envs_entails ?Δ _ => Δ end in
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  pm_eval (envs_lookup H Δ).
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Tactic Notation "iMatchHyp" tactic1(tac) :=
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

(** * Start a proof *)
Tactic Notation "iStartProof" :=
  lazymatch goal with
  | |- envs_entails _ _ => idtac
  | |- ?φ => notypeclasses refine (as_emp_valid_2 φ _ _);
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               [iSolveTC || fail "iStartProof: not a Bi entailment"
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               |apply tac_adequate]
  end.

(* Same as above, with 2 differences :
   - We can specify a BI in which we want the proof to be done
   - If the goal starts with a let or a ∀, they are automatically
     introduced. *)
Tactic Notation "iStartProof" uconstr(PROP) :=
  lazymatch goal with
  | |- @envs_entails ?PROP' _ _ =>
    (* This cannot be shared with the other [iStartProof], because
    type_term has a non-negligeable performance impact. *)
    let x := type_term (eq_refl : @eq Type PROP PROP') in idtac

  (* We eta-expand [as_emp_valid_2], in order to make sure that
     [iStartProof PROP] works even if [PROP] is the carrier type. In
     this case, typing this expression will end up unifying PROP with
     [bi_car _], and hence trigger the canonical structures mechanism
     to find the corresponding bi. *)
  | |- ?φ => notypeclasses refine ((λ P : PROP, @as_emp_valid_2 φ _ P) _ _ _);
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               [iSolveTC || fail "iStartProof: not a Bi entailment"
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               |apply tac_adequate]
  end.

(** * Generate a fresh identifier *)
(* Tactic Notation tactics cannot return terms *)
Ltac iFresh :=
  (* We need to increment the environment counter using [tac_fresh].
     But because [iFresh] returns a value, we have to let bind
     [tac_fresh] wrapped under a match to force evaluation of this
     side-effect. See https://stackoverflow.com/a/46178884 *)
  let do_incr :=
      lazymatch goal with
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      | _ => iStartProof; eapply tac_fresh; first by (pm_reflexivity)
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      end in
  lazymatch goal with
  |- envs_entails ?Δ _ =>
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    let n := pm_eval (env_counter Δ) in
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    constr:(IAnon n)
  end.

(** * Simplification *)
Tactic Notation "iEval" tactic(t) :=
  iStartProof;
  eapply tac_eval;
    [let x := fresh in intros x; t; unfold x; reflexivity
    |].

Tactic Notation "iEval" tactic(t) "in" constr(H) :=
  iStartProof;
  eapply tac_eval_in with _ H _ _ _;
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    [pm_reflexivity || fail "iEval:" H "not found"
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    |let x := fresh in intros x; t; unfold x; reflexivity
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    |pm_reflexivity
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    |].

Tactic Notation "iSimpl" := iEval simpl.
Tactic Notation "iSimpl" "in" constr(H) := iEval simpl in H.

(* It would be nice to also have an `iSsrRewrite`, however, for this we need to
pass arguments to Ssreflect's `rewrite` like `/= foo /bar` in Ltac, see:

  https://sympa.inria.fr/sympa/arc/coq-club/2018-01/msg00000.html

PMP told me (= Robbert) in person that this is not possible today, but may be
possible in Ltac2. *)

(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
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    [pm_reflexivity || fail "iRename:" H1 "not found"
    |pm_reflexivity || fail "iRename:" H2 "not fresh"|].
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Local Inductive esel_pat :=
  | ESelPure
  | ESelIdent : bool  ident  esel_pat.

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Local Ltac iElaborateSelPat_go pat Δ Hs :=
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  lazymatch pat with
  | [] => eval cbv in Hs
  | SelPure :: ?pat =>  iElaborateSelPat_go pat Δ (ESelPure :: Hs)
  | SelPersistent :: ?pat =>
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    let Hs' := pm_eval (env_dom (env_intuitionistic Δ)) in
    let Δ' := pm_eval (envs_clear_persistent Δ) in
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    iElaborateSelPat_go pat Δ' ((ESelIdent true <$> Hs') ++ Hs)
  | SelSpatial :: ?pat =>
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    let Hs' := pm_eval (env_dom (env_spatial Δ)) in
    let Δ' := pm_eval (envs_clear_spatial Δ) in
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    iElaborateSelPat_go pat Δ' ((ESelIdent false <$> Hs') ++ Hs)
  | SelIdent ?H :: ?pat =>
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    lazymatch pm_eval (envs_lookup_delete false H Δ) with
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    | Some (?p,_,?Δ') =>  iElaborateSelPat_go pat Δ' (ESelIdent p H :: Hs)
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    | None =>
      let H := pretty_ident H in
      fail "iElaborateSelPat:" H "not found"
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    end
  end.
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Ltac iElaborateSelPat pat :=
  lazymatch goal with
  | |- envs_entails ?Δ _ =>
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    let pat := sel_pat.parse pat in iElaborateSelPat_go pat Δ (@nil esel_pat)
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  end.

Local Ltac iClearHyp H :=
  eapply tac_clear with _ H _ _; (* (i:=H) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iClear:" H "not found"
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    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iClear:" H ":" P "not affine and the goal not absorbing"
    |].

Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
    lazymatch Hs with
    | [] => idtac
    | ESelPure :: ?Hs => clear; go Hs
    | ESelIdent _ ?H :: ?Hs => iClearHyp H; go Hs
    end in
  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.

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) *)
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    [pm_reflexivity ||
     fail "iExact:" H "not found"
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    |iSolveTC ||
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     let P := match goal with |- FromAssumption _ ?P _ => P end in
     fail "iExact:" H ":" P "does not match goal"
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    |pm_reduce; iSolveTC ||
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     fail "iExact:" H "not absorbing and the remaining hypotheses not affine"].

Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
    lazymatch 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) =>
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     first [is_evar i; fail 1 | pm_reflexivity]
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  | |- envs_lookup ?i (Envs ?Γp ?Γs _) = Some (_, ?P) =>
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     is_evar i; first [find Γp i P | find Γs i P]; pm_reflexivity
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  | |- envs_lookup_delete _ ?i (Envs ?Γp ?Γs _) = Some (_, ?P, _) =>
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     first [is_evar i; fail 1 | pm_reflexivity]
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  | |- envs_lookup_delete _ ?i (Envs ?Γp ?Γs _) = Some (_, ?P, _) =>
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     is_evar i; first [find Γp i P | find Γs i P]; pm_reflexivity
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  end.

Tactic Notation "iAssumption" :=
  let Hass := fresh in
  let rec find p Γ Q :=
    lazymatch Γ with
    | Esnoc ?Γ ?j ?P => first
       [pose proof (_ : FromAssumption p P Q) as Hass;
        eapply (tac_assumption _ _ j p P);
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          [pm_reflexivity
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          |apply Hass
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          |pm_reduce; iSolveTC ||
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           fail 1 "iAssumption:" j "not absorbing and the remaining hypotheses not affine"]
       |assert (P = False%I) as Hass by reflexivity;
        apply (tac_false_destruct _ j p P);
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          [pm_reflexivity
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          |exact Hass]
       |find p Γ Q]
    end in
  lazymatch goal with
  | |- envs_entails (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) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iPersistent:" H "not found"
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    |iSolveTC ||
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     let P := match goal with |- IntoPersistent _ ?P _ => P end in
     fail "iPersistent:" P "not persistent"
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    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iPersistent:" P "not affine and the goal not absorbing"
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    |pm_reflexivity|].
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Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iPure:" H "not found"
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    |iSolveTC ||
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     let P := match goal with |- IntoPure ?P _ => P end in
     fail "iPure:" P "not pure"
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    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iPure:" P "not affine and the goal not absorbing"
    |intros pat].

Tactic Notation "iEmpIntro" :=
  iStartProof;
  eapply tac_emp_intro;
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    [pm_reduce; iSolveTC ||
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     fail "iEmpIntro: spatial context contains non-affine hypotheses"].

Tactic Notation "iPureIntro" :=
  iStartProof;
  eapply tac_pure_intro;
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    [pm_reflexivity
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    |iSolveTC ||
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     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
  | |- envs_entails _ True => by iPureIntro
  | |- envs_entails _ emp => iEmpIntro
  end.

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

Local Ltac iFrameHyp H :=
  iStartProof;
  eapply tac_frame with _ H _ _ _;
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iFrame:" H "not found"
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    |iSolveTC ||
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     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 :=
  iStartProof;
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => repeat iFrameHyp H; go Hs end in
  match goal with
  | |- envs_entails ?Δ _ =>
     let Hs := eval cbv in (env_dom (env_intuitionistic Δ)) in go Hs
  end.

Local Ltac iFrameAnySpatial :=
  iStartProof;
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
  | |- envs_entails ?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 ).

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Local Ltac iFrame_go Hs :=
  lazymatch Hs with
  | [] => idtac
  | SelPure :: ?Hs => iFrameAnyPure; iFrame_go Hs
  | SelPersistent :: ?Hs => iFrameAnyPersistent; iFrame_go Hs
  | SelSpatial :: ?Hs => iFrameAnySpatial; iFrame_go Hs
  | SelIdent ?H :: ?Hs => iFrameHyp H; iFrame_go Hs
  end.

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Tactic Notation "iFrame" constr(Hs) :=
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  let Hs := sel_pat.parse Hs in iFrame_go Hs.
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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) ")" :=
  (* In the case the goal starts with an [let x := _ in _], we do not
     want to unfold x and start the proof mode. Instead, we want to
     use intros. So [iStartProof] has to be called only if [intros]
     fails *)
  intros x ||
    (iStartProof;
     lazymatch goal with
     | |- envs_entails _ _ =>
       eapply tac_forall_intro;
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       [iSolveTC ||
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              let P := match goal with |- FromForall ?P _ => P end in
              fail "iIntro: cannot turn" P "into a universal quantifier"
       |lazy beta; intros x]
     end).

Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
  [ (* (?Q → _) *)
    eapply tac_impl_intro with _ H _ _ _; (* (i:=H) *)
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      [iSolveTC
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      |pm_reduce; iSolveTC ||
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       let P := lazymatch goal with |- Persistent ?P => P end in
       fail 1 "iIntro: introducing non-persistent" H ":" P
              "into non-empty spatial context"
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      |pm_reflexivity ||
       let H := pretty_ident H in
       fail 1 "iIntro:" H "not fresh"
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      |iSolveTC
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      |]
  | (* (_ -∗ _) *)
    eapply tac_wand_intro with _ H _ _; (* (i:=H) *)
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      [iSolveTC
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      | pm_reflexivity ||
        let H := pretty_ident H in
        fail 1 "iIntro:" H "not fresh"
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      |]
  | fail "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
  [ (* (?P → _) *)
    eapply tac_impl_intro_persistent with _ H _ _ _; (* (i:=H) *)
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      [iSolveTC
      |iSolveTC ||
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       let P := match goal with |- IntoPersistent _ ?P _ => P end in
       fail 1 "iIntro:" P "not persistent"
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      |pm_reflexivity ||
       let H := pretty_ident H in
       fail 1 "iIntro:" H "not fresh"
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      |]
  | (* (?P -∗ _) *)
    eapply tac_wand_intro_persistent with _ H _ _ _; (* (i:=H) *)
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      [ iSolveTC
      | iSolveTC ||
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       let P := match goal with |- IntoPersistent _ ?P _ => P end in
       fail 1 "iIntro:" P "not persistent"
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      |iSolveTC ||
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       let P := match goal with |- TCOr (Affine ?P) _ => P end in
       fail 1 "iIntro:" P "not affine and the goal not absorbing"
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      |pm_reflexivity ||
       let H := pretty_ident H in
       fail 1 "iIntro:" H "not fresh"
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      |]
  | fail "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "_" :=
  first
  [ (* (?Q → _) *)
    iStartProof; eapply tac_impl_intro_drop;
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    [ iSolveTC | ]
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  | (* (_ -∗ _) *)
    iStartProof; eapply tac_wand_intro_drop;
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      [ iSolveTC
      | iSolveTC ||
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       let P := match goal with |- TCOr (Affine ?P) _ => P end in
       fail 1 "iIntro:" P "not affine and the goal not absorbing"
      |]
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntroForall" :=
  lazymatch goal with
  | |-  _, ?P => fail (* actually an →, this is handled by iIntro below *)
  | |-  _, _ => intro
  | |- let _ := _ in _ => intro
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ ( x : _, _) => let x' := fresh x in iIntro (x')
    end
  end.
Local Tactic Notation "iIntro" :=
  lazymatch goal with
  | |- _  ?P => intro
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    | |- envs_entails _ (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    end
  end.

(** * Specialize *)
Record iTrm {X As S} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : S }.
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).

(** There is some hacky stuff going on here: because of Coq bug #6583, unresolved
type classes in the arguments `xs` are resolved at arbitrary moments. Tactics
like `apply`, `split` and `eexists` wrongly trigger type class search to resolve
these holes. To avoid TC being triggered too eagerly, this tactic uses `refine`
at most places instead of `apply`. *)
Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
  let rec go xs :=
    lazymatch xs with
    | hnil => idtac
    | hcons ?x ?xs =>
       notypeclasses refine (tac_forall_specialize _ _ H _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H := pretty_ident H in
          fail "iSpecialize:" H "not found"
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         |iSolveTC ||
          let P := match goal with |- IntoForall ?P _ => P end in
          fail "iSpecialize: cannot instantiate" P "with" x
         |lazymatch goal with (* Force [A] in [ex_intro] to deal with coercions. *)
          | |-  _ : ?A, _ =>
            notypeclasses refine (@ex_intro A _ x (conj _ _))
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          end; [shelve..|pm_reflexivity|go xs]]
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    end in
  go xs.

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Ltac iSpecializePat_go H1 pats :=
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  let solve_to_wand H1 :=
    iSolveTC ||
    let P := match goal with |- IntoWand _ _ ?P _ _ => P end in
    fail "iSpecialize:" P "not an implication/wand" in
  let solve_done d :=
    lazymatch d with
    | true =>
       done ||
       let Q := match goal with |- envs_entails _ ?Q => Q end in
       fail "iSpecialize: cannot solve" Q "using done"
    | false => idtac
    end in
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  lazymatch pats with
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    | [] => idtac
    | SForall :: ?pats =>
       idtac "[IPM] The * specialization pattern is deprecated because it is applied implicitly.";
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       iSpecializePat_go H1 pats
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    | SIdent ?H2 :: ?pats =>
       notypeclasses refine (tac_specialize _ _ _ H2 _ H1 _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H2 := pretty_ident H2 in
          fail "iSpecialize:" H2 "not found"
         |pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |iSolveTC ||
          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
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         |pm_reflexivity|iSpecializePat_go H1 pats]
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    | SPureGoal ?d :: ?pats =>
       notypeclasses refine (tac_specialize_assert_pure _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- FromPure _ ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
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         |pm_reflexivity
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         |solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | SGoal (SpecGoal GPersistent false ?Hs_frame [] ?d) :: ?pats =>
       notypeclasses refine (tac_specialize_assert_persistent _ _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- Persistent ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
         |iSolveTC
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         |pm_reflexivity
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         |iFrame Hs_frame; solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | 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
       notypeclasses refine (tac_specialize_assert _ _ _ _ H1 _ lr Hs' _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => notypeclasses refine (add_modal_id _ _)
          | GModal => iSolveTC || fail "iSpecialize: goal not a modality"
          end
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         |pm_reflexivity ||
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          let Hs' := iMissingHyps Hs' in
          fail "iSpecialize: hypotheses" Hs' "not found"
         |iFrame Hs_frame; solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | SAutoFrame GPersistent :: ?pats =>
       notypeclasses refine (tac_specialize_assert_persistent _ _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- Persistent ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |pm_reflexivity
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         |solve [iFrame "∗ #"]
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         |iSpecializePat_go H1 pats]
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    | SAutoFrame ?m :: ?pats =>
       notypeclasses refine (tac_specialize_frame _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => notypeclasses refine (add_modal_id _ _)
          | GModal => iSolveTC || fail "iSpecialize: goal not a modality"
          end
         |first
            [notypeclasses refine (tac_unlock_emp _ _ _)
            |notypeclasses refine (tac_unlock_True _ _ _)
            |iFrame "∗ #"; notypeclasses refine (tac_unlock _ _ _)
            |fail "iSpecialize: premise cannot be solved by framing"]
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         |exact eq_refl]; iIntro H1; iSpecializePat_go H1 pats
    end.

Local Tactic Notation "iSpecializePat" open_constr(H) constr(pat) :=
  let pats := spec_pat.parse pat in iSpecializePat_go H pats.
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(* 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(H)
    "with" open_constr(xs) open_constr(pat) "as" constr(p) :=
  let p := intro_pat_persistent p in
  let pat := spec_pat.parse pat in
  let H :=
    lazymatch type of H with
    | string => constr:(INamed H)
    | _ => H
    end in
  iSpecializeArgs H xs; [..|
  lazymatch type of H with
  | ident =>
    (* The lemma [tac_specialize_persistent_helper] allows one to use all
    spatial hypotheses for both proving the premises of the lemma we
    specialize as well as those of the remaining goal. We can only use it when
    the result of the specialization is persistent, and no modality is
    eliminated. As an optimization, we do not use this when only universal
    quantifiers are instantiated. *)
    let pat := spec_pat.parse pat in
    lazymatch eval compute in
      (p && bool_decide (pat  []) && negb (existsb spec_pat_modal pat)) with
    | true =>
       (* FIXME: do something reasonable when the BI is not affine *)
       notypeclasses refine (tac_specialize_persistent_helper _ _ H _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H := pretty_ident H in
          fail "iSpecialize:" H "not found"
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         |iSpecializePat H pat;
           [..
           |refine (tac_specialize_persistent_helper_done _ H _ _ _);
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            pm_reflexivity]
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         |iSolveTC ||
          let Q := match goal with |- IntoPersistent _ ?Q _ => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |pm_reduce; iSolveTC ||
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          let Q := match goal with |- TCAnd _ (Affine ?Q) => Q end in
          fail "iSpecialize:" Q "not affine"
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         |pm_reflexivity
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         |(* goal *)]
    | false => iSpecializePat H pat
    end
  | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
  end].

Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
  lazymatch type of t with
  | string => iSpecializeCore t with hnil "" as p
  | ident => iSpecializeCore t with hnil "" as p
  | _ =>
    lazymatch t with
    | ITrm ?H ?xs ?pat => iSpecializeCore H with xs pat as p
    | _ => fail "iSpecialize:" t "should be a proof mode term"
    end
  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 [iIntoEmpValid] tactic solves a goal [bi_emp_valid Q]. The
argument [t] must be a Coq term whose type is of the following shape:

[∀ (x_1 : A_1) .. (x_n : A_n), φ]

and so that we have an instance `AsValid φ Q`.

Examples of such [φ]s are

- [bi_emp_valid P], in which case [Q] should be [P]
- [P1 ⊢ P2], in which case [Q] should be [P1 -∗ P2]
- [P1 ⊣⊢ P2], in which case [Q] should be [P1 ↔ P2]

The tactic instantiates each dependent argument [x_i] with an evar and generates
a goal [R] for each non-dependent argument [x_i : R].  For example, if the
original goal was [Q] and [t] has type [∀ x, P x → Q], then it generates an evar
[?x] for [x] and a subgoal [P ?x]. *)
Tactic Notation "iIntoEmpValid" open_constr(t) :=
  let rec go t :=
    (* We try two reduction tactics for the type of t before trying to
       specialize it. We first try the head normal form in order to
       unfold all the definition that could hide an entailment.  Then,
       we try the much weaker [eval cbv zeta], because entailment is
       not necessarilly opaque, and could be unfolded by [hnf].

       However, for calling type class search, we only use [cbv zeta]
       in order to make sure we do not unfold [bi_emp_valid]. *)
    let tT := type of t in
    first
      [ let tT' := eval hnf in tT in go_specialize t tT'
      | let tT' := eval cbv zeta in tT in go_specialize t tT'
      | let tT' := eval cbv zeta in tT in
        notypeclasses refine (as_emp_valid_1 tT _ _);
          [iSolveTC || fail "iPoseProof: not a BI assertion"
          |exact t]]
  with go_specialize t tT :=
    lazymatch tT with                (* We do not use hnf of tT, because, if
                                        entailment is not opaque, then it would
                                        unfold it. *)
    | ?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 #6583. *)
      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.

The tactic [iApply] uses laxy type class inference, so that evars can first be
instantiated by matching with the goal, whereas [iDestruct] does not, because
eliminations may not be performed when type classes have not been resolved.
*)
Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
  iStartProof;
  let Htmp := iFresh in
  let t := lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let t := lazymatch type of t with string => constr:(INamed t) | _ => t 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
    | ident =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
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         [pm_reflexivity ||
          let t := pretty_ident t in
          fail "iPoseProof:" t "not found"
         |pm_reflexivity ||
          let Htmp := pretty_ident Htmp in
          fail "iPoseProof:" Htmp "not fresh"
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         |goal_tac ()]
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
         [iIntoEmpValid t
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         |pm_reflexivity ||
          let Htmp := pretty_ident Htmp in
          fail "iPoseProof:" Htmp "not fresh"
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         |goal_tac ()]
    end;
    try iSolveTC 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 "iApplyHyp" constr(H) :=
  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
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      [pm_reflexivity
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      |iSolveTC
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      |pm_reduce (* reduce redexes created by instantiation *)]
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    |iSpecializePat H "[]"; last go H] in
  iExact H ||
  go H ||
  lazymatch iTypeOf H with
  | Some (_,?Q) => fail "iApply: cannot apply" Q
  end.

Tactic Notation "iApply" open_constr(lem) :=
  iPoseProofCore lem as false true (fun H => iApplyHyp H).

(** * 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
  iStartProof;
  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
    | ESelIdent _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
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         [pm_reflexivity ||
          let H := pretty_ident H in
          fail "iRevert:" H "not found"
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         |pm_reduce; go Hs]
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    end in
  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.

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;
    [iSolveTC ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
    |].
Tactic Notation "iRight" :=
  iStartProof;
  eapply tac_or_r;
    [iSolveTC ||
     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) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iOrDestruct:" H "not found"
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    |iSolveTC ||
     let P := match goal with |- IntoOr ?P _ _ => P end in
     fail "iOrDestruct: cannot destruct" P
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    |pm_reflexivity ||
     let H1 := pretty_ident H1 in
     fail "iOrDestruct:" H1 "not fresh"
    |pm_reflexivity ||
     let H2 := pretty_ident H2 in
     fail "iOrDestruct:" H2 "not fresh"
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    | |].

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

Tactic Notation "iSplitL" constr(Hs) :=
  iStartProof;
  let Hs := words Hs in
  let Hs := eval vm_compute in (INamed <$> Hs) in
  eapply tac_sep_split with _ _ Left Hs _ _; (* (js:=Hs) *)
    [iSolveTC ||
     let P := match goal with |- FromSep _ ?P _ _ => P end in
     fail "iSplitL:" P "not a separating conjunction"
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    |pm_reflexivity ||
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     let Hs := iMissingHyps Hs in
     fail "iSplitL: hypotheses" Hs "not found"
    | |].

Tactic Notation "iSplitR" constr(Hs) :=
  iStartProof;
  let Hs := words Hs in
  let Hs := eval vm_compute in (INamed <$> Hs) in
  eapply tac_sep_split with _ _ Right Hs _ _; (* (js:=Hs) *)
    [iSolveTC ||
     let P := match goal with |- FromSep _ ?P _ _ => P end in
     fail "iSplitR:" P "not a separating conjunction"
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    |pm_reflexivity ||
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     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) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iAndDestruct:" H "not found"
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    |pm_reduce; iSolveTC ||
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     let P :=
       lazymatch goal with
       | |- IntoSep ?P _ _ => P
       | |- IntoAnd _ ?P _ _ => P
       end in
     fail "iAndDestruct: cannot destruct" P
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    |pm_reflexivity ||
     let H1 := pretty_ident H1 in
     let H2 := pretty_ident H2 in
     fail "iAndDestruct:" H1 "or" H2 " not fresh"|].
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Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(d) constr(H') :=
  eapply tac_and_destruct_choice with _ H _ d H' _ _ _;
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    [pm_reflexivity || fail "iAndDestructChoice:" H "not found"
    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (IntoAnd _ ?P _ _) _ => P end in
     fail "iAndDestructChoice: cannot destruct" P
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    |pm_reflexivity ||
     let H' := pretty_ident H' in
     fail "iAndDestructChoice:" H' " not fresh"|].
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(** * Existential *)
Tactic Notation "iExists" uconstr(x1) :=
  iStartProof;
  eapply tac_exist;
    [iSolveTC ||
     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) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iExistDestruct:" H "not found"
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    |iSolveTC ||
     let P := match goal with |- IntoExist ?P _ => P end in
     fail "iExistDestruct: cannot destruct" P|];
  let y := fresh in
  intros y; eexists; split;
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    [pm_reflexivity ||
     let Hx := pretty_ident Hx in
     fail "iExistDestruct:" Hx "not fresh"
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    |revert y; intros x].

(** * Modality introduction *)
Tactic Notation "iModIntro" uconstr(sel) :=
  iStartProof;
  notypeclasses refine (tac_modal_intro _ sel _ _ _ _ _ _ _ _ _ _ _ _ _);
    [iSolveTC ||
     fail "iModIntro: the goal is not a modality"
    |iSolveTC ||
     let s := lazymatch goal with |- IntoModalPersistentEnv _ _ _ ?s => s end in
     lazymatch eval hnf in s with
     | MIEnvForall ?C => fail "iModIntro: persistent context does not satisfy" C
     | MIEnvIsEmpty => fail "iModIntro: persistent context is non-empty"
     end
    |iSolveTC ||
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     let s := lazymatch goal with |- IntoModalSpatialEnv _ _ _ ?s _ => s end in
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     lazymatch eval hnf in s with
     | MIEnvForall ?C => fail "iModIntro: spatial context does not satisfy" C
     | MIEnvIsEmpty => fail "iModIntro: spatial context is non-empty"
     end
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    |pm_reduce; iSolveTC ||
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     fail "iModIntro: cannot filter spatial context when goal is not absorbing"
    |].
Tactic Notation "iModIntro" := iModIntro _.
Tactic Notation "iAlways" := iModIntro.

(** * Later *)
Tactic Notation "iNext" open_constr(n) := iModIntro (^n _)%I.
Tactic Notation "iNext" := iModIntro (^_ _)%I.

(** * Update modality *)
Tactic Notation "iModCore" constr(H) :=
  eapply tac_modal_elim with _ H _ _ _ _ _ _;
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    [pm_reflexivity || fail "iMod:" H "not found"
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    |iSolveTC ||
     let P := match goal with |- ElimModal _ _ _ ?P _ _ _ => P end in
     let Q :=