(* tests/proofmode_ascii.v *)
From iris.proofmode Require Import proofmode monpred
From iris.base_logic Require Import base_logic
From iris.base_logic.lib Require Import invariants cancelable_invariants na_invariants
From iris.prelude Require Import options
From iris.bi Require Import ascii

Lemma test_random_stuff(P1 P2 P3 : nat -> uPred M) :
    |- forall (x y : nat) a b,
      x ≡ y ->
      <#> (uPred_ownM (a ⋅ b) -*
      (exists y1 y2 c, P1 ((x + y1) + y2) /\ True /\ <#> uPred_ownM c) -*
      <#> |> (forall z, P2 z ∨ True -> P2 z) -*
      |> (forall n m : nat, P1 n -> <#> (True /\ P2 n -> <#> (⌜n = n⌝ <-> P3 n))) -*
      |> ⌜x = 0⌝ \/ exists x z, |> P3 (x + z) ** uPred_ownM b ** uPred_ownM (core b)).
  Proof.
    iIntros (i [|j] a b ?) "!> [Ha Hb] H1 #H2 H3"; setoid_subst.
    { iLeft. by iNext. }
    iRight.
    iDestruct "H1" as (z1 z2 c) "(H1&_&#Hc)".
    iPoseProof "Hc" as "foo".
    iRevert (a b) "Ha Hb". iIntros (b a) "Hb {foo} Ha".
    iAssert (uPred_ownM (a ⋅ core a)) with "[Ha]" as "[Ha #Hac]".
    { by rewrite cmra_core_r. }
    iIntros "{$Hac $Ha}".
    iExists (S j + z1), z2.
    iNext.
    iApply ("H3" $! _ 0 with "[$]").
    - iSplit; first done. iApply "H2". iLeft. iApply "H2". by iRight.
    - done.
  Qed.

Lemma test_iFrame_pure(x y z : M) :
    ✓ x -> ⌜y ≡ z⌝ |-@{uPredI M} ✓ x /\ ✓ x /\ y ≡ z.
  Proof. iIntros (Hv) "Hxy". by iFrame (Hv) "Hxy". Qed.

Lemma test_iAssert_modalityP : (|==> False) -* |==> P.
  Proof. iIntros. iAssert False%I with "[> - //]" as %[]. Qed.

Lemma test_iStartProof_1P : P -* P.
  Proof. iStartProof. iStartProof. iIntros "$". Qed.

Lemma test_iStartProof_2P : P -* P.
  Proof. iStartProof (uPred _). iStartProof (uPredI _). iIntros "$". Qed.

Lemma test_iStartProof_3P : P -* P.
  Proof. iStartProof (uPredI _). iStartProof (uPredI _). iIntros "$". Qed.

Lemma test_iStartProof_4P : P -* P.
  Proof. iStartProof (uPredI _). iStartProof (uPred _). iIntros "$". Qed.

Lemma test_masksN E P Q R :
    ↑N ⊆ E ->
    (True -* P -* inv N Q -* True -* R) -* P -* |> Q ={E}=* R.
  Proof.
    iIntros (?) "H HP HQ".
    iApply ("H" with "[% //] [$] [> HQ] [> //]").
    by iApply inv_alloc.
  Qed.

Lemma test_iInv_0N P: inv N (<pers> P) ={⊤}=* |> P.
  Proof.
    iIntros "#H".
    iInv N as "#H2". Show.
    iModIntro. iSplit; auto.
  Qed.

Lemma test_iInv_0_with_closeN P: inv N (<pers> P) ={⊤}=* |> P.
  Proof.
    iIntros "#H".
    iInv N as "#H2" "Hclose". Show.
    iMod ("Hclose" with "H2").
    iModIntro. by iNext.
  Qed.

Lemma test_iInv_1N E P:
    ↑N ⊆ E ->
    inv N (<pers> P) ={E}=* |> P.
  Proof.
    iIntros (?) "#H".
    iInv N as "#H2".
    iModIntro. iSplit; auto.
  Qed.

Lemma test_iInv_2γ p N P:
    cinv N γ (<pers> P) ** cinv_own γ p ={⊤}=* cinv_own γ p ** |> P.
  Proof.
    iIntros "(#?&?)".
    iInv N as "(#HP&Hown)". Show.
    iModIntro. iSplit; auto with iFrame.
  Qed.

Lemma test_iInv_2_with_closeγ p N P:
    cinv N γ (<pers> P) ** cinv_own γ p ={⊤}=* cinv_own γ p ** |> P.
  Proof.
    iIntros "(#?&?)".
    iInv N as "(#HP&Hown)" "Hclose". Show.
    iMod ("Hclose" with "HP").
    iModIntro. iFrame. by iNext.
  Qed.

Lemma test_iInv_3γ p1 p2 N P:
    cinv N γ (<pers> P) ** cinv_own γ p1 ** cinv_own γ p2
      ={⊤}=* cinv_own γ p1 ** cinv_own γ p2  ** |> P.
  Proof.
    iIntros "(#?&Hown1&Hown2)".
    iInv N with "[Hown2 //]" as "(#HP&Hown2)".
    iModIntro. iSplit; auto with iFrame.
  Qed.

Lemma test_iInv_4t N E1 E2 P:
    ↑N ⊆ E2 ->
    na_inv t N (<pers> P) ** na_own t E1 ** na_own t E2
         |- |={⊤}=> na_own t E1 ** na_own t E2  ** |> P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
    iInv N as "(#HP&Hown2)". Show.
    iModIntro. iSplitL "Hown2"; auto with iFrame.
  Qed.

Lemma test_iInv_4_with_closet N E1 E2 P:
    ↑N ⊆ E2 ->
    na_inv t N (<pers> P) ** na_own t E1 ** na_own t E2
         |- |={⊤}=> na_own t E1 ** na_own t E2  ** |> P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
    iInv N as "(#HP&Hown2)" "Hclose". Show.
    iMod ("Hclose" with "[HP Hown2]").
    { iFrame. done. }
    iModIntro. iFrame. by iNext.
  Qed.

Lemma test_iInv_5t N E1 E2 P:
    ↑N ⊆ E2 ->
    na_inv t N (<pers> P) ** na_own t E1 ** na_own t E2
      ={⊤}=* na_own t E1 ** na_own t E2  ** |> P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
    iInv N with "Hown2" as "(#HP&Hown2)".
    iModIntro. iSplitL "Hown2"; auto with iFrame.
  Qed.

Lemma test_iInv_6t N E1 E2 P:
    ↑N ⊆ E1 ->
    na_inv t N (<pers> P) ** na_own t E1 ** na_own t E2
      ={⊤}=* na_own t E1 ** na_own t E2  ** |> P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
    iInv N with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
  Qed.

Lemma test_iInv_7t N1 N2 N3 E1 E2 P:
    ↑N3 ⊆ E1 ->
    inv N1 P ** na_inv t N3 (<pers> P) ** inv N2 P ** na_own t E1 ** na_own t E2
      ={⊤}=* na_own t E1 ** na_own t E2 ** |> P.
  Proof.
    iIntros (?) "(#?&#?&#?&Hown1&Hown2)".
    iInv N3 with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
  Qed.

Lemma test_iInv_8N : ∃ P, inv N P ={⊤}=* P ≡ True /\ inv N P.
  Proof.
    eexists. iIntros "#H".
    iInv N as "HP". iFrame "HP". auto.
  Qed.

Lemma test_iInv_9t N1 N2 N3 E1 E2 P:
    ↑N3 ⊆ E1 ->
    inv N1 P ** na_inv t N3 (<pers> P) ** inv N2 P ** na_own t E1 ** na_own t E2
      ={⊤}=* na_own t E1 ** na_own t E2 ** |> P.
  Proof.
    iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)".
    iInv "HInv" with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
  Qed.

Lemma test_iInv_10t N1 N2 N3 E1 E2 P:
    ↑N3 ⊆ E1 ->
    inv N1 P ** na_inv t N3 (<pers> P) ** inv N2 P ** na_own t E1 ** na_own t E2
      ={⊤}=* na_own t E1 ** na_own t E2 ** |> P.
  Proof.
    iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)".
    iInv "HInv" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
  Qed.

Lemma test_iInv_11N P: inv N (<pers> P) ={⊤}=* |> P.
  Proof.
    let H := iFresh in
    (iIntros H; iInv H as "#H2"). auto.
  Qed.

Lemma test_iInv_12N P: inv N (<pers> P) ={⊤}=* True.
  Proof.
    iIntros "H".
    Fail iInv 34 as "#H2".
    Fail iInv nroot as "#H2".
    Fail iInv "H2" as "#H2".
    done.
  Qed.

Lemma test_iInv_13N:
    inv N (∃ (v1 v2 v3 : nat), emp ** emp ** emp) ={⊤}=* |> emp.
  Proof.
    iIntros "H"; iInv "H" as (v1 v2 v3) "(?&?&_)".
    eauto.
  Qed.

Theorem test_iApply_inG`{!inG Σ A} γ (x x' : A) :
    x' ≼ x ->
    own γ x -* own γ x'.
  Proof. intros. by iApply own_mono. Qed.

Local NotationmonPred := (monPred I (iPropI Σ)).

Local NotationmonPredI := (monPredI I (iPropI Σ)).
  Implicit Types P Q R : monPred.
  Implicit Types 𝓟 𝓠 𝓡 : iProp Σ.
  Check "test_iInv".

Lemma test_iInvN E 𝓟 :
    ↑N ⊆ E ->
    ⎡inv N 𝓟⎤ |-@{monPredI} |={E}=> emp.
  Proof.
    iIntros (?) "Hinv".
    iInv N as "HP". Show.
    iFrame "HP". auto.
  Qed.
  Check "test_iInv_with_close".

Lemma test_iInv_with_closeN E 𝓟 :
    ↑N ⊆ E ->
    ⎡inv N 𝓟⎤ |-@{monPredI} |={E}=> emp.
  Proof.
    iIntros (?) "Hinv".
    iInv N as "HP" "Hclose". Show.
    iMod ("Hclose" with "HP"). auto.
  Qed.

Lemma test_bi_emp_valid: |-@{PROP} True.
Proof. naive_solver. Qed.

Lemma test_bi_emp_valid_parens: (|-@{PROP} True) /\ ((|-@{PROP} True)).
Proof. naive_solver. Qed.

Lemma test_bi_emp_valid_parens_space_open: ( |-@{PROP} True).
Proof. naive_solver. Qed.

Lemma test_bi_emp_valid_parens_space_close: (|-@{PROP} True ).
Proof. naive_solver. Qed.

Lemma test_entails_annot_sectionsP :
  (P |-@{PROP} P) /\ (|-@{PROP}) P P /\
  (P -|-@{PROP} P) /\ (-|-@{PROP}) P P.
Proof. naive_solver. Qed.

Lemma test_entails_annot_sections_parensP :
  ((P |-@{PROP} P)) /\ ((|-@{PROP})) P P /\
  ((P -|-@{PROP} P)) /\ ((-|-@{PROP})) P P.
Proof. naive_solver. Qed.

Lemma test_entails_annot_sections_space_openP :
  ( P |-@{PROP} P) /\
  ( P -|-@{PROP} P).
Proof. naive_solver. Qed.

Lemma test_entails_annot_sections_space_closeP :
  (P |-@{PROP} P ) /\ (|-@{PROP} ) P P /\
  (P -|-@{PROP} P ) /\ (-|-@{PROP} ) P P.
Proof. naive_solver. Qed.

Lemma p1: forall P, True -> P |- P.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p2".

Lemma p2: forall P, True /\ (P |- P).
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p3".

Lemma p3: exists P, P |- P.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p4".

Lemma p4: |-@{PROP} exists (x : nat), ⌜x = 0⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p5".

Lemma p5: |-@{PROP} exists (x : nat), ⌜forall y : nat, y = y⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p6".

Lemma p6: exists! (z : nat), |-@{PROP} exists (x : nat), ⌜forall y : nat, y = y⌝ ** ⌜z = 0⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p7".

Lemma p7: forall (a : nat), a = 0 -> forall y, True |-@{PROP} ⌜y >= 0⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p8".

Lemma p8: forall (a : nat), a = 0 -> forall y, |-@{PROP} ⌜y >= 0⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.
Check "p9".

Lemma p9: forall (a : nat), a = 0 -> forall y : nat, |-@{PROP} forall z : nat, ⌜z >= 0⌝.
Proof.
  Unset Printing Notations. Set Printing Raw Literals. Show. Set Printing Notations. Unset Printing Raw Literals.
Abort.