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From iris.proofmode Require Import tactics.
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From stdpp Require Import gmap.
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Set Default Proof Using "Type".
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Section tests.
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Context {PROP : sbi}.
Implicit Types P Q R : PROP.
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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". iDestruct "H" as "###H".
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  (* should remove the disjunction "H" *)
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  iDestruct "H" as "[#?|#?]"; last by iLeft.
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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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Definition foo (P : PROP) := (P - P)%I.
Definition bar : PROP := ( P, foo P)%I.
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Lemma test_unfold_constants : bar.
Proof. iIntros (P) "HP //". Qed.
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Lemma test_iRewrite {A : ofeT} (x y : A) P :
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   ( z, P - bi_affinely (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 (bi.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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Lemma test_iDestruct_and_emp P Q `{!Persistent P, !Persistent Q} :
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  P  emp - emp  Q - bi_affinely (P  Q).
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Proof. iIntros "[#? _] [_ #?]". auto. Qed.

Lemma test_iIntros_persistent P Q `{!Persistent Q} : (P  Q  P  Q)%I.
Proof. iIntros "H1 #H2". by iFrame. Qed.
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Lemma test_iIntros_pure (ψ φ : Prop) P : ψ  ( φ   P   φ  ψ   P)%I.
Proof. iIntros (??) "H". auto. Qed.

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Lemma test_iIntros_pure_not : ( ¬False  : PROP)%I.
Proof. by iIntros (?). Qed.

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Lemma test_fast_iIntros P Q :
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  ( x y z : nat,
    x = plus 0 x  y = 0  z = 0  P   Q  foo (x  x))%I.
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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)%I.
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Proof. by iIntros. 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_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_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_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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(* 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 `{Collection A C} P : ( x1 x2 : gset positive, P - P)%I.
Proof. iExists {[ 1%positive ]}, . auto. Qed.

Lemma test_iSpecialize_tc P : ( x y z : gset positive, P) - P.
Proof. iIntros "H". iSpecialize ("H" $!  {[ 1%positive ]} ). done. Qed.

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Lemma test_iFrame_pure {A : ofeT} (φ : Prop) (y z : A) :
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  φ  bi_affinely y  z - ( φ    φ   y  z : PROP).
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Proof. iIntros (Hv) "#Hxy". iFrame (Hv) "Hxy". Qed.

Lemma test_iAssert_modality P :  False -  P.
Proof.
  iIntros "HF".
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  iAssert (bi_affinely False)%I with "[> -]" as %[].
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  by iMod "HF".
Qed.
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Lemma test_iMod_affinely_timeless P `{!Timeless P} :
  bi_affinely ( P) -  bi_affinely 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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(* 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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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_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
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    (R1 := (P  Q)%I) (R2 := ( P   Q)%I) :
  ( P   Q)  R1  R2 -  (P  Q)   R1  R2.
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 test_iFrame_persistent (P Q : PROP) :
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   P - Q - bi_persistently (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 - bi_persistently (P  P).
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Proof. iIntros "#?". by iSplit. Qed.
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Lemma test_iSpecialize_persistent P Q :  P - (bi_persistently 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 P : ( n, ^n P)%I.
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.
  rewrite (inj_iff S). by iApply ("IH" with "[%]"); first omega.
Qed.

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Lemma test_iIntros_start_proof :
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  (True : PROP)%I.
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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_foo P Q :
  bi_affinely ( (Q  P)) - bi_affinely ( Q) - bi_affinely ( 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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  (bi_persistently (   x : nat,  x = 0    x = 0  - False - P - P))%I.
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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 P Q :
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  bi_affinely ( (Q  P)) - bi_affinely ( Q) - bi_affinely ( 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 - bi_persistently Q  R - bi_persistently (bi_affinely (bi_affinely P))   Q.
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Proof. iIntros "#HP #HQ HR". iSplitL. iAlways. done. iAlways. 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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End tests.