tactics.v 63.7 KB
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From iris.proofmode Require Import coq_tactics.
From iris.proofmode Require Import intro_patterns spec_patterns sel_patterns.
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From iris.base_logic Require Export base_logic.
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From iris.proofmode Require Export classes notation.
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From iris.proofmode Require Import class_instances.
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From stdpp Require Import stringmap hlist.
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From iris.proofmode Require Import strings.
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Set Default Proof Using "Type".
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Declare Reduction env_cbv := cbv [
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  beq ascii_beq string_beq
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  env_lookup env_lookup_delete env_delete env_app env_replace env_dom
  env_persistent env_spatial env_spatial_is_nil envs_dom
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  envs_lookup envs_lookup_delete envs_delete envs_snoc envs_app
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    envs_simple_replace envs_replace envs_split
    envs_clear_spatial envs_clear_persistent
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    envs_split_go envs_split].
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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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(* Tactic Notation tactics cannot return terms *)
Ltac iFresh' H :=
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  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
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     eval vm_compute in (fresh_string_of_set H (of_list Hs))
  | _ => H
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  end.
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Ltac iFresh := iFresh' "~".
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Tactic Notation "iTypeOf" constr(H) tactic(tac):=
  let Δ := match goal with |- of_envs ?Δ  _ => Δ end in
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  lazymatch eval env_cbv in (envs_lookup H Δ) with
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  | Some (?p,?P) => tac p P
  end.

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

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

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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.

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

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Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
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(** * 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:" Q "not persistent"
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    |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:" P "not pure"
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    |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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(** 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_cbv; reflexivity || fail "iFrame:" H "not found"
    |let R := match goal with |- Frame ?R _ _ => R end in
     apply _ || 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 :=
    match Hs with
    | [] => idtac
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    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
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    end
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  in let Hs := sel_pat.parse Hs in 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.

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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 )" :=
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  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
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Notation "( H $! x1 .. xn 'with' pat )" :=
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  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
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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"
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      |let P := match goal with |- ForallSpecialize _ ?P _ => P end in
       apply _ || fail 1 "iSpecialize:" P "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:" P "not an implication/wand" in
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  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
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    | SForall :: ?pats =>
       idtac "the * specialization pattern is deprecated because it is applied implicitly";
       go H1 pats
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    | SName ?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" P "with" Q
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         |env_cbv; reflexivity|go H1 pats]
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    | SGoalPersistent :: ?pats =>
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       eapply tac_specialize_assert_persistent 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 |- PersistentP ?Q => Q end in
          apply _ || fail "iSpecialize:" Q "not persistent"
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         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
    | SGoalPure :: ?pats =>
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       eapply tac_specialize_assert_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 (SpecGoal ?m ?lr ?Hs_frame ?Hs) :: ?pats =>
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       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
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       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 m with
          | false => apply elim_modal_dummy
          | true => apply _ || fail "iSpecialize: goal not a modality"
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          end
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         |env_cbv; reflexivity || fail "iSpecialize:" Hs "not found"
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         |iFrame Hs_frame (*goal*)
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         |go H1 pats]
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    end in let pats := spec_pat.parse pat in go H pats.

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(* The argument [p] denotes whether the conclusion of the specialized term is
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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
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`#` 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). *)
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Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
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  let p := intro_pat_persistent p in
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  let t :=
    match type of t with string => constr:(ITrm t hnil "") | _ => t end in
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  lazymatch t with
  | ITrm ?H ?xs ?pat =>
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    let pat := spec_pat.parse pat in
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    lazymatch type of H with
    | string =>
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      lazymatch eval compute in (p && negb (existsb spec_pat_modal pat)) with
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      | true =>
         eapply tac_specialize_persistent_helper with _ H _ _ _;
           [env_cbv; reflexivity || fail "iSpecialize:" H "not found"
           |iSpecializeArgs H xs; iSpecializePat H pat; last (iExact H)
           |let Q := match goal with |- PersistentP ?Q => Q end in
            apply _ || fail "iSpecialize:" Q "not persistent"
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           |env_cbv; reflexivity|(* goal *)]
      | false => iSpecializeArgs H xs; iSpecializePat H pat
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      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
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    end
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  | _ => fail "iSpecialize:" t "should be a proof mode term"
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  end.
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Tactic Notation "iSpecialize" open_constr(t) :=
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  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.
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(** * Pose proof *)
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(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
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- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
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- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
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- [∀ (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]. *)
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Tactic Notation "iIntoValid" open_constr(t) :=
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  let rec go t :=
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    let tT := type of t in
    lazymatch eval hnf in tT with
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    | True  _ => apply t
    | _  _ => apply (uPred.entails_wand _ _ t)
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    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply (uPred.equiv_iff _ _ t)
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    | ?P  ?Q => let H := fresh in assert P as H; [|go uconstr:(t H); clear H]
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    |  _ : ?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')
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    end in
  go t.
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(* 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 :=
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    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"
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         |goal_tac ()]
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    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
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         [iIntoValid t
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         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
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         |goal_tac ()]
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    end;
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    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)
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  end.

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Tactic Notation "iPoseProof" open_constr(lem) "as" constr(H) :=
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  iPoseProofCore lem as false false (fun Htmp => iRename Htmp into H).
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(** * Apply *)
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Tactic Notation "iApply" open_constr(lem) :=
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  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
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      [env_cbv; reflexivity
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      |apply _
      |lazy beta (* reduce betas created by instantiation *)]
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    |iSpecializePat H "[]"; last go H] in
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  iPoseProofCore lem as false true (fun H =>
    first [iExact H|go H|iTypeOf H (fun Q => fail 1 "iApply: cannot apply" Q)]).
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(** * Revert *)
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Local Tactic Notation "iForallRevert" ident(x) :=
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  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
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  let A := type of x in
  lazymatch type of A with
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  | Prop => revert x; first [apply tac_pure_revert|err x]
  | _ => revert x; first [apply tac_forall_revert|err x]
  end.
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Tactic Notation "iRevert" constr(Hs) :=
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  let rec go Hs :=
    lazymatch Hs with
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    | [] => idtac
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    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
         [env_cbv; reflexivity || fail "iRevert:" H "not found"
         |env_cbv; go Hs]
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    end in
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  iElaborateSelPat Hs go.
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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" :=
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  iStartProof;
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  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" :=
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  iStartProof;
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  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: cannot destruct" P
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    |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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  iStartProof;
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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"| |]
  end.
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Tactic Notation "iSplitL" constr(Hs) :=
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  iStartProof;
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  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"
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    |env_cbv; reflexivity || fail "iSplitL: hypotheses" Hs
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                                  "not found in the context"| |].
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Tactic Notation "iSplitR" constr(Hs) :=
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  iStartProof;
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  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"
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    |env_cbv; reflexivity || fail "iSplitR: hypotheses" Hs
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                                  "not found in the context"| |].
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Tactic Notation "iSplitL" := iSplitR "".
Tactic Notation "iSplitR" := iSplitL "".

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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_cbv; reflexivity || fail "iAndDestruct:" H "not found"
    |let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     apply _ || fail "iAndDestruct: cannot destruct" P
    |env_cbv; 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' _ _ _;
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    [env_cbv; reflexivity || fail "iAndDestructChoice:" H "not found"
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    |let P := match goal with |- IntoAnd _ ?P _ _ => P end in
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     apply _ || fail "iAndDestructChoice: cannot destruct" P
    |env_cbv; reflexivity || fail "iAndDestructChoice:" H' " not fresh"|].
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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" P1 "and" P2
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    |env_cbv; reflexivity || fail "iCombine:" H "not fresh"|].

(** * Existential *)
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Tactic Notation "iExists" uconstr(x1) :=
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  iStartProof;
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  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: cannot destruct" P|];
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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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(** * Always *)
Tactic Notation "iAlways":=
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  iStartProof;
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  apply tac_always_intro;
    [reflexivity || fail "iAlways: spatial context non-empty"|].

(** * Later *)
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Tactic Notation "iNext" open_constr(n) :=
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  iStartProof;
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  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|].

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

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

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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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    | IDrop => iClear Hz
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    | IFrame => iFrame Hz
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    | IName ?y => iRename Hz into y
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    | IList [[]] => iExFalso; iExact Hz
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    | IList [[?pat1; IDrop]] => iAndDestructChoice Hz as true Hz; go Hz pat1
    | IList [[IDrop; ?pat2]] => iAndDestructChoice Hz as false Hz; go Hz pat2
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    | IList [[?pat1; ?pat2]] =>
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       let Hy := iFresh in iAndDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
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    | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]
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    | IPureElim => iPure Hz as ?
    | IAlwaysElim ?pat => iPersistent Hz; go Hz pat
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    | IModalElim ?pat => iModCore Hz; go Hz pat
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    | _ => fail "iDestruct:" pat "invalid"
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    end in
  let rec find_pat found pats :=
    lazymatch pats with
    | [] =>
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      lazymatch found with
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      | true => idtac
      | false => fail "iDestruct:" pat "should contain exactly one proper introduction pattern"
      end
    | ISimpl :: ?pats => simpl; find_pat found pats
    | IClear ?H :: ?pats => iClear H; find_pat found pats
    | IClearFrame ?H :: ?pats => iFrame H; find_pat found pats
    | ?pat :: ?pats =>
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       lazymatch found with
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       | false => go H pat; find_pat true pats
       | true => fail "iDestruct:" pat "should contain exactly one proper introduction pattern"
       end
    | _ => fail "hallo" pats
    end in
  let pats := intro_pat.parse pat in
  find_pat false pats.
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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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(** * Introduction tactic *)
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Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
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  try iStartProof;
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  try first
    [(* (∀ _, _) *) apply tac_forall_intro
    |(* (?P → _) *) eapply tac_impl_intro_pure;
      [let P := match goal with |- IntoPure ?P _ => P end in
       apply _ || fail "iIntro:" P "not pure"|]
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    |(* (?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"|]
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    |(* ⌜∀ _, _⌝ *) apply tac_pure_forall_intro
    |(* ⌜_ → _⌝ *) apply tac_pure_impl_intro];
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  intros x.
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Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
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  [ (* (?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) :=
  iStartProof;
  first
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  [ (* (?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"|]
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  | (* (?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 "iIntro" "_" :=
  try iStartProof;
  first
  [ (* (?Q → _) *) apply tac_impl_intro_drop
  | (* (_ -∗ _) *) apply tac_wand_intro_drop
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

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Local Tactic Notation "iIntroForall" :=
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  try iStartProof;
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  lazymatch goal with
  | |-  _, ?P => fail
  | |-  _, _ => intro
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  | |- _  ( x : _, _) => let x' := fresh x in iIntro (x')
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  end.
Local Tactic Notation "iIntro" :=
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  try iStartProof;
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  lazymatch goal with
  | |- _  ?P => intro
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  | |- _  (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
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  | |- _  (_  _) => 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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    (* Optimizations to avoid generating fresh names *)
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    | IPureElim :: ?pats => iIntro (?); go pats
    | IAlwaysElim (IName ?H) :: ?pats => iIntro #H; go pats
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    | IDrop :: ?pats => iIntro _; go pats
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    | IName ?H :: ?pats => iIntro H; go pats
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    (* Introduction patterns that can only occur at the top-level *)
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    | IPureIntro :: ?pats => iPureIntro; go pats
    | IAlwaysIntro :: ?pats => iAlways; go pats
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    | IModalIntro :: ?pats => iModIntro; go pats
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    | IForall :: ?pats => repeat iIntroForall; go pats
    | IAll :: ?pats => repeat (iIntroForall || iIntro); go pats
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    (* Clearing and simplifying introduction patterns *)
    | ISimpl :: ?pats => simpl; go pats
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    | IClear ?H :: ?pats => iClear H; go pats
    | IClearFrame ?H :: ?pats => iFrame H; go pats
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    (* Introduction + destruct *)
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    | IAlwaysElim ?pat :: ?pats =>
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       let H := iFresh in iIntro