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From iris.proofmode Require Import tactics intro_patterns.
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Set Default Proof Using "Type".
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Section tests.
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Context {PROP : bi}.
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Implicit Types P Q R : PROP.
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Lemma test_eauto_emp_isplit_biwand P : emp  P - P.
Proof. eauto 6. Qed.

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Lemma test_eauto_isplit_biwand P :  P - P.
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Proof. eauto. Qed.
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Fixpoint test_fixpoint (n : nat) {struct n} : True  emp @{PROP}  (n + 0)%nat = n .
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Proof.
  case: n => [|n] /=; first (iIntros (_) "_ !%"; reflexivity).
  iIntros (_) "_".
  by iDestruct (test_fixpoint with "[//]") as %->.
Qed.

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Check "demo_0".
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Lemma demo_0 P Q :  (P  Q) - ( x, x = 0  x = 1)  (Q  P).
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Proof.
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  iIntros "H #H2". Show. iDestruct "H" as "###H".
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  (* should remove the disjunction "H" *)
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  iDestruct "H" as "[#?|#?]"; last by iLeft. Show.
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  (* should keep the disjunction "H" because it is instantiated *)
  iDestruct ("H2" $! 10) as "[%|%]". done. done.
Qed.

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Lemma demo_2 P1 P2 P3 P4 Q (P5 : nat  PROP) `{!Affine P4, !Absorbing P2} :
  P2  (P3  Q)  True  P1  P2  (P4  ( x:nat, P5 x  P3))  emp -
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    P1 - (True  True) -
  (((P2  False  P2  0 = 0)  P3)  Q  P1  True) 
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     (P2  False)  (False  P5 0).
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Proof.
  (* Intro-patterns do something :) *)
  iIntros "[H2 ([H3 HQ]&?&H1&H2'&foo&_)] ? [??]".
  (* To test destruct: can also be part of the intro-pattern *)
  iDestruct "foo" as "[_ meh]".
  repeat iSplit; [|by iLeft|iIntros "#[]"].
  iFrame "H2".
  (* split takes a list of hypotheses just for the LHS *)
  iSplitL "H3".
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  - iFrame "H3". iRight. auto.
  - iSplitL "HQ". iAssumption. by iSplitL "H1".
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Qed.

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Lemma demo_3 P1 P2 P3 :
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  P1  P2  P3 - P1   (P2   x, (P3  x = 0)  P3).
Proof. iIntros "($ & $ & $)". iNext. by iExists 0. Qed.
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Lemma test_pure_space_separated P1 :
  <affine> True  P1 - P1.
Proof.
  (* [% H] should be parsed as two separate patterns and not the pure name
  [H] *)
  iIntros "[% H] //".
Qed.

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Definition foo (P : PROP) := (P - P)%I.
Definition bar : PROP := ( P, foo P)%I.
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Lemma test_unfold_constants :  bar.
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Proof. iIntros (P) "HP //". Qed.
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Check "test_iStopProof".
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Lemma test_iStopProof Q : emp - Q - Q.
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Proof. iIntros "#H1 H2". Show. iStopProof. Show. by rewrite bi.sep_elim_r. Qed.
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Lemma test_iRewrite `{!BiInternalEq PROP} {A : ofeT} (x y : A) P :
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   ( z, P - <affine> (z  y)) - (P - P  (x,x)  (y,x)).
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Proof.
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  iIntros "#H1 H2".
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  iRewrite (internal_eq_sym x x with "[# //]").
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  iRewrite -("H1" $! _ with "[- //]").
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  auto.
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Qed.

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Check "test_iDestruct_and_emp".
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Lemma test_iDestruct_and_emp P Q `{!Persistent P, !Persistent Q} :
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  P  emp - emp  Q - <affine> (P  Q).
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Proof. iIntros "[#? _] [_ #?]". Show. auto. Qed.
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Lemma test_iIntros_persistent P Q `{!Persistent Q} :  (P  Q  P  Q).
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Proof. iIntros "H1 #H2". by iFrame "∗#". Qed.
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Lemma test_iDestruct_intuitionistic_1 P Q `{!Persistent P}:
  Q   (Q - P) - P  Q.
Proof. iIntros "[HQ #HQP]". iDestruct ("HQP" with "HQ") as "#HP". by iFrame. Qed.

Lemma test_iDestruct_intuitionistic_2 P Q `{!Persistent P, !Affine P}:
  Q  (Q - P) - P.
Proof. iIntros "[HQ HQP]". iDestruct ("HQP" with "HQ") as "#HP". done. Qed.

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Lemma test_iDestruct_intuitionistic_affine_bi `{!BiAffine PROP} P Q `{!Persistent P}:
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  Q  (Q - P) - P  Q.
Proof. iIntros "[HQ HQP]". iDestruct ("HQP" with "HQ") as "#HP". by iFrame. Qed.

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Check "test_iDestruct_spatial".
Lemma test_iDestruct_spatial Q :  Q - Q.
Proof. iIntros "#HQ". iDestruct "HQ" as "-#HQ". Show. done. Qed.

Check "test_iDestruct_spatial_affine".
Lemma test_iDestruct_spatial_affine Q `{!Affine Q} :  Q - Q.
Proof.
  iIntros "#-#HQ".
  (* Since [Q] is affine, it should not add an <affine> modality *)
  Show. done.
Qed.

Lemma test_iDestruct_spatial_noop Q : Q - Q.
Proof. iIntros "-#HQ". done. Qed.

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Lemma test_iDestruct_exists (Φ: nat  PROP) :
  ( y, Φ y) -  n, Φ n.
Proof. iIntros "H". iDestruct "H" as (y) "H". by iExists y. Qed.

Lemma test_iDestruct_exists_automatic (Φ: nat  PROP) :
  ( y, Φ y) -  n, Φ n.
Proof.
  iIntros "H".
  iDestruct "H" as (?) "H".
  (* the automatic name should by [y] *)
  by iExists y.
Qed.

Lemma test_iDestruct_exists_automatic_multiple (Φ: nat  PROP) :
  ( y n baz, Φ (y+n+baz)) -  n, Φ n.
Proof. iDestruct 1 as (???) "H". by iExists (y+n+baz). Qed.

Lemma test_iDestruct_exists_freshen (y:nat) (Φ: nat  PROP) :
  ( y, Φ y) -  n, Φ n.
Proof.
  iIntros "H".
  iDestruct "H" as (?) "H".
  (* the automatic name is the freshened form of [y] *)
  by iExists y0.
Qed.

Check "test_iDestruct_exists_not_exists".
Lemma test_iDestruct_exists_not_exists P :
  P - P.
Proof. Fail iDestruct 1 as (?) "H". Abort.

Lemma test_iDestruct_exists_explicit_name (Φ: nat  PROP) :
  ( y, Φ y) -  n, Φ n.
Proof.
  (* give an explicit name that isn't the binder name *)
  iDestruct 1 as (foo) "?".
  by iExists foo.
Qed.

Lemma test_iDestruct_exists_pure (Φ: nat  Prop) :
   y, Φ y @{PROP}  n, ⌜Φ n.
Proof.
  iDestruct 1 as (?) "H".
  by iExists y.
Qed.

Lemma test_iDestruct_exists_and_pure (H: True) P :
  False  P - False.
Proof.
  (* this automatic name uses [fresh H] as a sensible default (it's a hypothesis
  in [Prop] and the user cannot supply a name in their code) *)
  iDestruct 1 as (?) "H".
  contradict H0.
Qed.

Check "test_iDestruct_exists_intuitionistic".
Lemma test_iDestruct_exists_intuitionistic P (Φ: nat  PROP) :
   ( y, Φ y  P) - P.
Proof.
  iDestruct 1 as (?) "#H". Show.
  iDestruct "H" as "[_ $]".
Qed.

Lemma test_iDestruct_exists_freshen_local_name (Φ: nat  PROP) :
  let y := 0 in
   ( y, Φ y) -  n, Φ (y+n).
Proof.
  iIntros (y) "#H".
  iDestruct "H" as (?) "H".
  iExists y0; auto.
Qed.

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(* regression test for #337 *)
Check "test_iDestruct_exists_anonymous".
Lemma test_iDestruct_exists_anonymous P Φ :
  ( _:nat, P)  ( x:nat, Φ x) - P   x, Φ x.
Proof.
  iIntros "[HP HΦ]".
  (* this should not use [x] as the default name for the unnamed binder *)
  iDestruct "HP" as (?) "$". Show.
  iDestruct "HΦ" as (x) "HΦ".
  by iExists x.
Qed.

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Definition an_exists P : PROP := ( (an_exists_name:nat), ^an_exists_name P)%I.

(* should use the name from within [an_exists] *)
Lemma test_iDestruct_exists_automatic_def P :
  an_exists P -  k, ^k P.
Proof. iDestruct 1 as (?) "H". by iExists an_exists_name. Qed.

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Lemma test_iIntros_pure (ψ φ : Prop) P : ψ    φ   P   φ  ψ   P.
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Proof. iIntros (??) "H". auto. Qed.

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Check "test_iIntros_forall_pure".
Lemma test_iIntros_forall_pure (Ψ: nat  PROP) :
    x : nat, Ψ x  Ψ x.
Proof.
  iIntros "%".
  (* should be a trivial implication now *)
  Show. auto.
Qed.

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Lemma test_iIntros_pure_not `{!BiPureForall PROP} : @{PROP}  ¬False .
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Proof. by iIntros (?). Qed.

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Lemma test_fast_iIntros `{!BiInternalEq PROP} P Q :
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    x y z : nat,
    x = plus 0 x  y = 0  z = 0  P   Q  foo (x  x).
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Proof.
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  iIntros (a) "*".
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  iIntros "#Hfoo **".
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  iIntros "_ //".
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Qed.
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Lemma test_very_fast_iIntros P :
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   x y : nat,   x = y   P - P.
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Proof. by iIntros. Qed.

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Lemma test_iIntros_automatic_name (Φ: nat  PROP) :
   y, Φ y -  x, Φ x.
Proof. iIntros (?) "H". by iExists y. Qed.

Lemma test_iIntros_automatic_name_proofmode_intro (Φ: nat  PROP) :
   y, Φ y -  x, Φ x.
Proof. iIntros "% H". by iExists y. Qed.

(* even an object-level forall should get the right name *)
Lemma test_iIntros_object_forall P :
  P -  (y:unit), P.
Proof. iIntros "H". iIntros (?). destruct y. iAssumption. Qed.

Lemma test_iIntros_object_proofmode_intro (Φ: nat  PROP) :
    y, Φ y -  x, Φ x.
Proof. iIntros "% H". by iExists y. Qed.

Check "test_iIntros_pure_names".
Lemma test_iIntros_pure_names (H:True) P :
   x y : nat,   x = y   P - P.
Proof.
  iIntros (???).
  (* the pure hypothesis should get a sensible [H0] as its name *)
  Show. auto.
Qed.

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Definition tc_opaque_test : PROP := tc_opaque ( x : nat,  x = x )%I.
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Lemma test_iIntros_tc_opaque :  tc_opaque_test.
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Proof. by iIntros (x). Qed.
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(** Prior to 0b84351c this used to loop, now [iAssumption] instantiates [R] with
[False] and performs false elimination. *)
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Lemma test_iAssumption_evar_ex_false :  R, R   P, P.
Proof. eexists. iIntros "?" (P). iAssumption. Qed.

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Lemma test_iApply_evar P Q R : ( Q, Q - P) - R - P.
Proof. iIntros "H1 H2". iApply "H1". iExact "H2". Qed.

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Lemma test_iAssumption_affine P Q R `{!Affine P, !Affine R} : P - Q - R - Q.
Proof. iIntros "H1 H2 H3". iAssumption. Qed.

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Lemma test_done_goal_evar Q :  P, Q  P.
Proof. eexists. iIntros "H". Fail done. iAssumption. Qed.

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Lemma test_iDestruct_spatial_and P Q1 Q2 : P  (Q1  Q2) - P  Q1.
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Proof. iIntros "[H [? _]]". by iFrame. Qed.
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Lemma test_iAssert_persistent P Q : P - Q - True.
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Proof.
  iIntros "HP HQ".
  iAssert True%I as "#_". { by iClear "HP HQ". }
  iAssert True%I with "[HP]" as "#_". { Fail iClear "HQ". by iClear "HP". }
  iAssert True%I as %_. { by iClear "HP HQ". }
  iAssert True%I with "[HP]" as %_. { Fail iClear "HQ". by iClear "HP". }
  done.
Qed.
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Lemma test_iAssert_persistently P :  P - True.
Proof.
  iIntros "HP". iAssert ( P)%I with "[# //]" as "#H". done.
Qed.

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Lemma test_iSpecialize_auto_frame P Q R :
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  (P - True - True - Q - R) - P - Q - R.
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Proof. iIntros "H ? HQ". by iApply ("H" with "[$]"). Qed.
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Lemma test_iSpecialize_pure (φ : Prop) Q R :
  φ  (⌜φ⌝ - Q)   Q.
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Proof. iIntros (HP HPQ). iDestruct (HPQ $! HP) as "?". done. Qed.

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Lemma test_iSpecialize_pure_done (φ: Prop) Q :
  φ  (⌜φ⌝ - Q)  Q.
Proof. iIntros (HP) "HQ". iApply ("HQ" with "[% //]"). Qed.

Check "test_iSpecialize_pure_error".
Lemma test_iSpecialize_not_pure_error P Q :
  (P - Q)  Q.
Proof. iIntros "HQ". Fail iSpecialize ("HQ" with "[%]"). Abort.

Check "test_iSpecialize_pure_error".
Lemma test_iSpecialize_pure_done_error (φ: Prop) Q :
  (⌜φ⌝ - Q)  Q.
Proof. iIntros "HQ". Fail iSpecialize ("HQ" with "[% //]"). Abort.

Check "test_iSpecialize_done_error".
Lemma test_iSpecialize_done_error P Q :
  (P - Q)  Q.
Proof. iIntros "HQ". Fail iSpecialize ("HQ" with "[//]"). Abort.

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Lemma test_iSpecialize_Coq_entailment P Q R :
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  ( P)  (P - Q)  ( Q).
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Proof. iIntros (HP HPQ). iDestruct (HPQ $! HP) as "?". done. Qed.

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Lemma test_iSpecialize_intuitionistic P Q R :
   P -  (P - P - P - P -  P - P - Q) - R - R   (P  Q).
Proof.
  iIntros "#HP #H HR".
  (* Test that [H] remains in the intuitionistic context *)
  iSpecialize ("H" with "HP").
  iSpecialize ("H" with "[HP]"); first done.
  iSpecialize ("H" with "[]"); first done.
  iSpecialize ("H" with "[-HR]"); first done.
  iSpecialize ("H" with "[#]"); first done.
  iFrame "HR".
  iSpecialize ("H" with "[-]"); first done.
  by iFrame "#".
Qed.

Lemma test_iSpecialize_intuitionistic_2 P Q R :
   P -  (P - P - P - P -  P - P - Q) - R - R   (P  Q).
Proof.
  iIntros "#HP #H HR".
  (* Test that [H] remains in the intuitionistic context *)
  iSpecialize ("H" with "HP") as #.
  iSpecialize ("H" with "[HP]") as #; first done.
  iSpecialize ("H" with "[]") as #; first done.
  iSpecialize ("H" with "[-HR]") as #; first done.
  iSpecialize ("H" with "[#]") as #; first done.
  iFrame "HR".
  iSpecialize ("H" with "[-]") as #; first done.
  by iFrame "#".
Qed.

Lemma test_iSpecialize_intuitionistic_3 P Q R :
  P -  (P - Q) -  (P - <pers> Q) -  (Q - R) - P   (Q  R).
Proof.
  iIntros "HP #H1 #H2 #H3".
  (* Should fail, [Q] is not persistent *)
  Fail iSpecialize ("H1" with "HP") as #.
  (* Should succeed, [<pers> Q] is persistent *)
  iSpecialize ("H2" with "HP") as #.
  (* Should succeed, despite [R] not being persistent, no spatial premises are
  needed to prove [Q] *)
  iSpecialize ("H3" with "H2") as #.
  by iFrame "#".
Qed.

Check "test_iAssert_intuitionistic".
Lemma test_iAssert_intuitionistic `{!BiBUpd PROP} P :
   P -  |==> P.
Proof.
  iIntros "#HP".
  (* Test that [HPupd1] ends up in the intuitionistic context *)
  iAssert (|==> P)%I with "[]" as "#HPupd1"; first done.
  (* This should not work, [|==> P] is not persistent. *)
  Fail iAssert (|==> P)%I with "[#]" as "#HPupd2"; first done.
  done.
Qed.

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Lemma test_iSpecialize_evar P : ( R, R - R) - P - P.
Proof. iIntros "H HP". iApply ("H" with "[HP]"). done. Qed.

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Lemma test_iPure_intro_emp R `{!Affine R} :
  R - emp.
Proof. iIntros "HR". by iPureIntro. Qed.

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Lemma test_iEmp_intro P Q R `{!Affine P, !Persistent Q, !Affine R} :
  P - Q  R - emp.
Proof. iIntros "HP #HQ HR". iEmpIntro. Qed.

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Lemma test_iPure_intro (φ : nat  Prop) P Q R `{!Affine P, !Persistent Q, !Affine R} :
  φ 0  P - Q  R -  x : nat, <affine>  φ x    φ x .
Proof. iIntros (?) "HP #HQ HR". iPureIntro; eauto. Qed.
Lemma test_iPure_intro_2 (φ : nat  Prop) P Q R `{!Persistent Q} :
  φ 0  P - Q  R -  x : nat, <affine>  φ x    φ x .
Proof. iIntros (?) "HP #HQ HR". iPureIntro; eauto. Qed.

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Lemma test_fresh P Q:
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  (P  Q) - (P  Q).
Proof.
  iIntros "H".
  let H1 := iFresh in
  let H2 := iFresh in
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  let pat :=constr:(IList [cons (IIdent H1) (cons (IIdent H2) nil)]) in
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  iDestruct "H" as pat.
  iFrame.
Qed.

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(* Test for issue #288 *)
(* FIXME: Restore once we drop support for Coq 8.8 and Coq 8.9.
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Lemma test_iExists_unused : (∃ P : PROP, ∃ x : nat, P)%I.
Proof.
  iExists _.
  iExists 10.
  iAssert emp%I as "H"; first done.
  iExact "H".
Qed.
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*)
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(* Check coercions *)
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Lemma test_iExist_coercion (P : Z  PROP) : ( x, P x) -  x, P x.
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Proof. iIntros "HP". iExists (0:nat). iApply ("HP" $! (0:nat)). Qed.
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Lemma test_iExist_tc `{Set_ A C} P :   x1 x2 : gset positive, P - P.
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Proof. iExists {[ 1%positive ]}, . auto. Qed.

Lemma test_iSpecialize_tc P : ( x y z : gset positive, P) - P.
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Proof.
  iIntros "H".
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  (* FIXME: this [unshelve] and [apply _] should not be needed. *)
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  unshelve iSpecialize ("H" $!  {[ 1%positive ]} ); try apply _. done.
Qed.
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Lemma test_iFrame_pure `{!BiInternalEq PROP} {A : ofeT} (φ : Prop) (y z : A) :
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  φ  <affine> y  z - ( φ    φ   y  z : PROP).
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Proof. iIntros (Hv) "#Hxy". iFrame (Hv) "Hxy". Qed.

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Lemma test_iFrame_disjunction_1 P1 P2 Q1 Q2 :
  BiAffine PROP 
   P1 - Q2 - P2 - (P1  P2  False  P2)  (Q1  Q2).
Proof. intros ?. iIntros "#HP1 HQ2 HP2". iFrame "HP1 HQ2 HP2". Qed.
Lemma test_iFrame_disjunction_2 P : P - (True  True)  P.
Proof. iIntros "HP". iFrame "HP". auto. Qed.

Lemma test_iFrame_conjunction_1 P Q :
  P - Q - (P  Q)  (P  Q).
Proof. iIntros "HP HQ". iFrame "HP HQ". Qed.
Lemma test_iFrame_conjunction_2 P Q :
  P - Q - (P  P)  (Q  Q).
Proof. iIntros "HP HQ". iFrame "HP HQ". Qed.

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Lemma test_iFrame_later `{!BiAffine PROP} P Q : P - Q -  P  Q.
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Proof. iIntros "H1 H2". by iFrame "H1". Qed.

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Lemma test_iFrame_affinely_1 P Q `{!Affine P} :
  P - <affine> Q - <affine> (P  Q).
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Proof. iIntros "HP HQ". iFrame "HQ". by iModIntro. Qed.
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Lemma test_iFrame_affinely_2 P Q `{!Affine P, !Affine Q} :
  P - Q - <affine> (P  Q).
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Proof. iIntros "HP HQ". iFrame "HQ". by iModIntro. Qed.
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Lemma test_iAssert_modality P :  False -  P.
Proof.
  iIntros "HF".
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  iAssert (<affine> False)%I with "[> -]" as %[].
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  by iMod "HF".
Qed.
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Lemma test_iMod_affinely_timeless P `{!Timeless P} :
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  <affine>  P -  <affine> P.
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Proof. iIntros "H". iMod "H". done. Qed.

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Lemma test_iAssumption_False P : False - P.
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Proof. iIntros "H". done. Qed.
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Lemma test_iAssumption_coq_1 P Q : ( Q)  <affine> P - Q.
Proof. iIntros (HQ) "_". iAssumption. Qed.

Lemma test_iAssumption_coq_2 P Q : (  Q)  <affine> P -  Q.
Proof. iIntros (HQ) "_". iAssumption. Qed.

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(* Check instantiation and dependent types *)
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Lemma test_iSpecialize_dependent_type (P :  n, vec nat n  PROP) :
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  ( n v, P n v) -  n v, P n v.
Proof.
  iIntros "H". iExists _, [#10].
  iSpecialize ("H" $! _ [#10]). done.
Qed.
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(* Check that typeclasses are not resolved too early *)
Lemma test_TC_resolution `{!BiAffine PROP} (Φ : nat  PROP) l x :
  x  l  ([ list] y  l, Φ y) - Φ x.
Proof.
  iIntros (Hp) "HT".
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  iDestruct (big_sepL_elem_of _ _ _ Hp with "HT") as "Hp".
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  done.
Qed.

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Lemma test_eauto_iFrame P Q R `{!Persistent R} :
  P - Q - R  R  Q  P  R  False.
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Proof. eauto 10 with iFrame. Qed.
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Lemma test_iCombine_persistent P Q R `{!Persistent R} :
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  P - Q - R  R  Q  P  R  False.
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Proof. iIntros "HP HQ #HR". iCombine "HR HQ HP HR" as "H". auto. Qed.
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Lemma test_iCombine_frame P Q R `{!Persistent R} :
  P - Q - R  R  Q  P  R.
Proof. iIntros "HP HQ #HR". iCombine "HQ HP HR" as "$". by iFrame. Qed.

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Lemma test_iNext_evar P : P - True.
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Proof.
  iIntros "HP". iAssert ( _ -  P)%I as "?"; last done.
  iIntros "?". iNext. iAssumption.
Qed.
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Lemma test_iNext_sep1 P Q (R1 := (P  Q)%I) :
  ( P   Q)  R1 -  ((P  Q)  R1).
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Proof.
  iIntros "H". iNext.
  rewrite {1 2}(lock R1). (* check whether R1 has not been unfolded *) done.
Qed.
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Lemma test_iNext_sep2 P Q :  P   Q -  (P  Q).
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Proof.
  iIntros "H". iNext. iExact "H". (* Check that the laters are all gone. *)
Qed.
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Lemma test_iNext_quantifier {A} (Φ : A  A  PROP) :
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  ( y,  x,  Φ x y) -  ( y,  x, Φ x y).
Proof. iIntros "H". iNext. done. Qed.

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Lemma text_iNext_Next `{!BiInternalEq PROP} {A B : ofeT} (f : A -n> A) x y :
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  Next x  Next y - (Next (f x)  Next (f y) : PROP).
Proof. iIntros "H". iNext. by iRewrite "H". Qed.

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Lemma test_iFrame_persistent (P Q : PROP) :
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   P - Q - <pers> (P  P)  (P  Q  Q).
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Proof. iIntros "#HP". iFrame "HP". iIntros "$". Qed.
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Lemma test_iSplit_persistently P Q :  P - <pers> (P  P).
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Proof. iIntros "#?". by iSplit. Qed.
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Lemma test_iSpecialize_persistent P Q :  P - (<pers> P  Q) - Q.
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Proof. iIntros "#HP HPQ". by iSpecialize ("HPQ" with "HP"). Qed.
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Lemma test_iDestruct_persistent P (Φ : nat  PROP) `{! x, Persistent (Φ x)}:
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   (P -  x, Φ x) -
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  P -  x, Φ x  P.
Proof.
  iIntros "#H HP". iDestruct ("H" with "HP") as (x) "#H2". eauto with iFrame.
Qed.

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Lemma test_iLöb `{!BiLöb PROP} P :   n, ^n P.
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Proof.
  iLöb as "IH". iDestruct "IH" as (n) "IH".
  by iExists (S n).
Qed.
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Lemma test_iInduction_wf (x : nat) P Q :
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   P - Q -  (x + 0 = x)%nat .
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Proof.
  iIntros "#HP HQ".
  iInduction (lt_wf x) as [[|x] _] "IH"; simpl; first done.
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  rewrite (inj_iff S). by iApply ("IH" with "[%]"); first lia.
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Qed.

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Lemma test_iInduction_using (m : gmap nat nat) (Φ : nat  nat  PROP) y :
  ([ map] x  i  m, Φ y x) - ([ map] x  i  m, emp  Φ y x).
Proof.
  iIntros "Hm". iInduction m as [|i x m] "IH" using map_ind forall(y).
  - by rewrite !big_sepM_empty.
  - rewrite !big_sepM_insert //. iDestruct "Hm" as "[$ ?]".
    by iApply "IH".
Qed.

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Lemma test_iIntros_start_proof :
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  @{PROP} True.
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Proof.
  (* Make sure iIntros actually makes progress and enters the proofmode. *)
  progress iIntros. done.
Qed.

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Lemma test_True_intros : (True : PROP) - True.
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Proof.
  iIntros "?". done.
Qed.
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Lemma test_iPoseProof_let P Q :
  (let R := True%I in R  P  Q) 
  P  Q.
Proof.
  iIntros (help) "HP".
  iPoseProof (help with "[$HP]") as "?". done.
Qed.

Lemma test_iIntros_let P :
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   Q, let R := emp%I in P - R - Q - P  Q.
Proof. iIntros (Q R) "$ _ $". Qed.
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Lemma test_iNext_iRewrite `{!BiInternalEq PROP} P Q :
  <affine>  (Q  P) - <affine>  Q - <affine>  P.
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Proof.
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  iIntros "#HPQ HQ !>". iNext. by iRewrite "HPQ" in "HQ".
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Qed.

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Lemma test_iIntros_modalities `(!Absorbing P) :
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   <pers> (   x : nat,  x = 0    x = 0  - False - P - P).
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Proof.
  iIntros (x ??).
  iIntros "* **". (* Test that fast intros do not work under modalities *)
  iIntros ([]).
Qed.
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Lemma test_iIntros_rewrite P (x1 x2 x3 x4 : nat) :
  x1 = x2  ( x2 = x3    x3  x4   P) -  x1 = x4   P.
Proof. iIntros (?) "(-> & -> & $)"; auto. Qed.
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Lemma test_iNext_affine `{!BiInternalEq PROP} P Q :
  <affine>  (Q  P) - <affine>  Q - <affine>  P.
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Proof. iIntros "#HPQ HQ !>". iNext. by iRewrite "HPQ" in "HQ". Qed.
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Lemma test_iAlways P Q R :
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   P - <pers> Q  R - <pers> <affine> <affine> P   Q.
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Proof. iIntros "#HP #HQ HR". iSplitL. iModIntro. done. iModIntro. done. Qed.
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(* A bunch of test cases from #127 to establish that tactics behave the same on
`⌜ φ ⌝ → P` and `∀ _ : φ, P` *)
Lemma test_forall_nondep_1 (φ : Prop) :
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  φ  ( _ : φ, False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". by iApply "Hφ". Qed.
Lemma test_forall_nondep_2 (φ : Prop) :
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  φ  ( _ : φ, False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". iSpecialize ("Hφ" with "[% //]"). done. Qed.
Lemma test_forall_nondep_3 (φ : Prop) :
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  φ  ( _ : φ, False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". unshelve iSpecialize ("Hφ" $! _). done. done. Qed.
Lemma test_forall_nondep_4 (φ : Prop) :
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  φ  ( _ : φ, False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". iSpecialize ("Hφ" $! Hφ); done. Qed.

Lemma test_pure_impl_1 (φ : Prop) :
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  φ  (⌜φ⌝  False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". by iApply "Hφ". Qed.
Lemma test_pure_impl_2 (φ : Prop) :
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  φ  (⌜φ⌝  False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". iSpecialize ("Hφ" with "[% //]"). done. Qed.
Lemma test_pure_impl_3 (φ : Prop) :
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  φ  (⌜φ⌝  False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". unshelve iSpecialize ("Hφ" $! _). done. done. Qed.
Lemma test_pure_impl_4 (φ : Prop) :
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  φ  (⌜φ⌝  False : PROP) - False.
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Proof. iIntros (Hφ) "Hφ". iSpecialize ("Hφ" $! Hφ). done. Qed.

Lemma test_forall_nondep_impl2 (φ : Prop) P :
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  φ  P - ( _ : φ, P - False : PROP) - False.
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Proof.
  iIntros (Hφ) "HP Hφ".
  Fail iSpecialize ("Hφ" with "HP").
  iSpecialize ("Hφ" with "[% //] HP"). done.
Qed.

Lemma test_pure_impl2 (φ : Prop) P :
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  φ  P - (⌜φ⌝  P - False : PROP) - False.
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Proof.
  iIntros (Hφ) "HP Hφ".
  Fail iSpecialize ("Hφ" with "HP").
  iSpecialize ("Hφ" with "[% //] HP"). done.
Qed.

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Lemma demo_laterN_forall {A} (Φ Ψ: A  PROP) n: ( x, ^n Φ x) - ^n ( x, Φ x).
Proof.
  iIntros "H" (w). iApply ("H" $! w).
Qed.

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Lemma test_iNext_laterN_later P n :  ^n P - ^n  P.
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Proof. iIntros "H". iNext. by iNext. Qed.
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Lemma test_iNext_later_laterN P n : ^n  P -  ^n P.
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Proof. iIntros "H". iNext. by iNext. Qed.
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Lemma test_iNext_plus_1 P n1 n2 :  ^n1 ^n2 P - ^n1 ^n2  P.
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Proof. iIntros "H". iNext. iNext. by iNext. Qed.
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Lemma test_iNext_plus_2 P n m : ^n ^m P - ^(n+m) P.
Proof. iIntros "H". iNext. done. Qed.
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Check "test_iNext_plus_3".
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Lemma test_iNext_plus_3 P Q n m k :
  ^m ^(2 + S n + k) P - ^m  ^(2 + S n) Q - ^k  ^(S (S n + S m)) (P  Q).
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Proof. iIntros "H1 H2". iNext. iNext. iNext. iFrame. Show. iModIntro. done. Qed.
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Lemma test_iNext_unfold P Q n m (R := (^n P)%I) :
  R  ^m True.
Proof.
  iIntros "HR". iNext.
  match goal with |-  context [ R ] => idtac | |- _ => fail end.
  done.
Qed.

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Lemma test_iNext_fail P Q a b c d e f g h i j:
  ^(a + b) ^(c + d + e) P - ^(f + g + h + i + j) True.
Proof. iIntros "H". iNext. done. Qed.
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Lemma test_specialize_affine_pure (φ : Prop) P :
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  φ  (<affine> ⌜φ⌝ - P)  P.
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Proof.
  iIntros (Hφ) "H". by iSpecialize ("H" with "[% //]").
Qed.

Lemma test_assert_affine_pure (φ : Prop) P :
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  φ  P  P  <affine> ⌜φ⌝.
Proof. iIntros (Hφ). iAssert (<affine> ⌜φ⌝)%I with "[%]" as "$"; auto. Qed.
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Lemma test_assert_pure (φ : Prop) P :
  φ  P  P  ⌜φ⌝.
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Proof. iIntros (Hφ). iAssert ⌜φ⌝%I with "[%]" as "$"; auto with iFrame. Qed.
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Lemma test_specialize_very_nested (φ : Prop) P P2 Q R1 R2 :
  φ 
  P - P2 -
  (<affine>  φ  - P2 - Q) -
  (P - Q - R1) -
  (R1 - True - R2) -
  R2.
Proof.
  iIntros (?) "HP HP2 HQ H1 H2".
  by iApply ("H2" with "(H1 HP (HQ [% //] [-])) [//]").
Qed.

Lemma test_specialize_very_very_nested P1 P2 P3 P4 P5 :
   P1 -
   (P1 - P2) -
  (P2 - P2 - P3) -
  (P3 - P4) -
  (P4 - P5) -
  P5.
Proof.
  iIntros "#H #H1 H2 H3 H4".
  by iSpecialize ("H4" with "(H3 (H2 (H1 H) (H1 H)))").
Qed.

Check "test_specialize_nested_intuitionistic".
Lemma test_specialize_nested_intuitionistic (φ : Prop) P P2 Q R1 R2 :
  φ 
   P -  (P - Q) - (Q - Q - R2) - R2.
Proof.
  iIntros (?) "#HP #HQ HR".
  iSpecialize ("HR" with "(HQ HP) (HQ HP)").
  Show.
  done.
Qed.

Lemma test_specialize_intuitionistic P Q :
   P -  (P - Q) -  Q.
Proof. iIntros "#HP #HQ". iSpecialize ("HQ" with "HP"). done. Qed.

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Lemma test_iEval x y :  (y + x)%nat = 1  -  S (x + y) = 2%nat  : PROP.
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Proof.
  iIntros (H).
  iEval (rewrite (Nat.add_comm x y) // H).
  done.
Qed.

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Lemma test_iEval_precedence : True  True : PROP.
Proof.
  iIntros.
  (* Ensure that in [iEval (a); b], b is not parsed as part of the argument of [iEval]. *)
  iEval (rewrite /=); iPureIntro; exact I.
Qed.

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Check "test_iSimpl_in".
Lemma test_iSimpl_in x y :  (3 + x)%nat = y  -  S (S (S x)) = y  : PROP.
Proof. iIntros "H". iSimpl in "H". Show. done. Qed.

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Lemma test_iSimpl_in_2 x y z :
   (3 + x)%nat = y  -  (1 + y)%nat = z  -
   S (S (S x)) = y  : PROP.
Proof. iIntros "H1 H2". iSimpl in "H1 H2". Show. done. Qed.

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Lemma test_iSimpl_in3 x y z :
   (3 + x)%nat = y  -  (1 + y)%nat = z  -
   S (S (S x)) = y  : PROP.
Proof. iIntros "#H1 H2". iSimpl in "#". Show. done. Qed.

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Check "test_iSimpl_in4".
Lemma test_iSimpl_in4 x y :  (3 + x)%nat = y  -  S (S (S x)) = y  : PROP.
Proof. iIntros "H". Fail iSimpl in "%". by iSimpl in "H". Qed.

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Lemma test_iPureIntro_absorbing (φ : Prop) :
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  φ  @{PROP} <absorb> ⌜φ⌝.
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Proof. intros ?. iPureIntro. done. Qed.

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Check "test_iFrame_later_1".
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Lemma test_iFrame_later_1 P Q : P   Q -  (P   Q).
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Proof. iIntros "H". iFrame "H". Show. auto. Qed.
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Check "test_iFrame_later_2".
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Lemma test_iFrame_later_2 P Q :  P   Q -  ( P   Q).
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Proof. iIntros "H". iFrame "H". Show. auto. Qed.
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Lemma test_with_ident P Q R : P - Q - (P - Q - R) - R.
Proof.
  iIntros "? HQ H".
  iMatchHyp (fun H _ =>
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    iApply ("H" with [spec_patterns.SIdent H []; spec_patterns.SIdent "HQ" []])).
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Qed.
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Lemma iFrame_with_evar_r P Q :
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   R, (P - Q - P  R)  R = Q.
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Proof.
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  eexists. split. iIntros "HP HQ". iFrame. iApply "HQ". done.
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Qed.
Lemma iFrame_with_evar_l P Q :
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   R, (P - Q - R  P)  R = Q.
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Proof.
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  eexists. split. iIntros "HP HQ". Fail iFrame "HQ".
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  iSplitR "HP"; iAssumption. done.
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Qed.
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Lemma iFrame_with_evar_persistent P Q :
   R, (P -  Q - P  R  Q)  R = emp%I.
Proof.
  eexists. split. iIntros "HP #HQ". iFrame "HQ HP". iEmpIntro. done.
Qed.

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Lemma test_iAccu P Q R S :
   PP, (P - Q - R - S - PP)  PP = (Q  R  S)%I.
Proof.
  eexists. split. iIntros "#? ? ? ?". iAccu. done.
Qed.

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Lemma test_iAssumption_evar P :  R, (R  P)  R = P.
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Proof.
  eexists. split.
  - iIntros "H". iAssumption.
  (* Now verify that the evar was chosen as desired (i.e., it should not pick False). *)
  - reflexivity.
Qed.

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Lemma test_iAssumption_False_no_loop :  R, R   P, P.
Proof. eexists. iIntros "?" (P). done. Qed.

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Lemma test_apply_affine_impl `{!BiPlainly PROP} (P : PROP) :
  P - ( Q : PROP,  (Q - <pers> Q)   (P - Q)  Q).
Proof. iIntros "HP" (Q) "_ #HPQ". by iApply "HPQ". Qed.

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Lemma test_apply_affine_wand `{!BiPlainly PROP} (P : PROP) :
  P - ( Q : PROP, <affine>  (Q - <pers> Q) - <affine>  (P - Q) - Q).
Proof. iIntros "HP" (Q) "_ #HPQ". by iApply "HPQ". Qed.

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Lemma test_and_sep (P Q R : PROP) : P  (Q   R)  (P  Q)   R.
Proof.
  iIntros "H". repeat iSplit.
  - iDestruct "H" as "[$ _]".
  - iDestruct "H" as "[_ [$ _]]".
  - iDestruct "H" as "[_ [_ #$]]".
Qed.

Lemma test_and_sep_2 (P Q R : PROP) `{!Persistent R, !Affine R} :
  P  (Q  R)  (P  Q)  R.
Proof.
  iIntros "H". repeat iSplit.
  - iDestruct "H" as "[$ _]".
  - iDestruct "H" as "[_ [$ _]]".
  - iDestruct "H" as "[_ [_ #$]]".
Qed.
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Check "test_and_sep_affine_bi".
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Lemma test_and_sep_affine_bi `{!BiAffine PROP} P Q :  P  Q   P  Q.
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Proof.
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  iIntros "[??]". iSplit; last done. Show. done.
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Qed.
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Check "test_big_sepL_simpl".
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Lemma test_big_sepL_simpl x (l : list nat) P :
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   P -
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  ([ list] ky  l, <affine>  y = y ) -
  ([ list] y  x :: l, <affine>  y = y ) -
  P.
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Proof. iIntros "HP ??". Show. simpl. Show. done. Qed.
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Check "test_big_sepL2_simpl".
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Lemma test_big_sepL2_simpl x1 x2 (l1 l2 : list nat) P :
  P -
  ([ list] ky1;y2  []; l2, <affine>  y1 = y2 ) -
  ([ list] y1;y2  x1 :: l1; (x2 :: l2) ++ l2, <affine>  y1 = y2 ) -
  P  ([ list] y1;y2  x1 :: l1; x2 :: l2, True).
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Proof. iIntros "HP ??". Show. simpl. Show. by iLeft. Qed.
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Check "test_big_sepL2_iDestruct".
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Lemma test_big_sepL2_iDestruct (Φ : nat  nat  PROP) x1 x2 (l1 l2 : list nat) :
  ([ list] y1;y2  x1 :: l1; x2 :: l2, Φ y1 y2) -
  <absorb> Φ x1 x2.
Proof. iIntros "[??]". Show. iFrame. Qed.

Lemma test_big_sepL2_iFrame (Φ : nat  nat  PROP) (l1 l2 : list nat) P :
  Φ 0 10 - ([ list] y1;y2  l1;l2, Φ y1 y2) -
  ([ list] y1;y2  (0 :: l1);(10 :: l2), Φ y1 y2).
Proof. iIntros "$ ?". iFrame. Qed.
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