proofmode_ascii.v

From iris.proofmode Require Import tactics monpred.
From iris.base_logic Require Import base_logic.
From iris.base_logic.lib Require Import invariants cancelable_invariants na_invariants.

From iris.bi Require Import ascii.

Set Default Proof Using "Type".
Unset Printing Use Implicit Types. (* FIXME: remove once we drop support for Coq <=8.11. *)

Section base_logic_tests.
  Context {M : ucmraT}.
  Implicit Types P Q R : uPred M.

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

  Lemma test_iStartProof_1 P : P -* P.
  Proof. iStartProof. iStartProof. iIntros "$". Qed.
  Lemma test_iStartProof_2 P : P -* P.
  Proof. iStartProof (uPred _). iStartProof (uPredI _). iIntros "$". Qed.
  Lemma test_iStartProof_3 P : P -* P.
  Proof. iStartProof (uPredI _). iStartProof (uPredI _). iIntros "$". Qed.
  Lemma test_iStartProof_4 P : P -* P.
  Proof. iStartProof (uPredI _). iStartProof (uPred _). iIntros "$". Qed.
End base_logic_tests.

Section iris_tests.
  Context `{!invG Σ, !cinvG Σ, !na_invG Σ}.
  Implicit Types P Q R : iProp Σ.

  Lemma test_masks  N 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_0 N P: inv N (<pers> P) ={⊤}=* |> P.
  Proof.
    iIntros "#H".
    iInv N as "#H2". Show.
    iModIntro. iSplit; auto.
  Qed.

  Lemma test_iInv_0_with_close N 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_1 N E P:
    ↑N ⊆ E ->
    inv N (<pers> P) ={E}=* |> P.
  Proof.
    iIntros (?) "#H".
    iInv N as "#H2".
    iModIntro. iSplit; auto.
  Qed.

  Lemma test_iInv_2 γ 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_close γ 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_3 γ 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_4 t 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_close t 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.

  (* test named selection of which na_own to use *)
  Lemma test_iInv_5 t 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_6 t 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.

  (* test robustness in presence of other invariants *)
  Lemma test_iInv_7 t 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.

  (* iInv should work even where we have "inv N P" in which P contains an evar *)
  Lemma test_iInv_8 N : ∃ P, inv N P ={⊤}=* P ≡ True /\ inv N P.
  Proof.
    eexists. iIntros "#H".
    iInv N as "HP". iFrame "HP". auto.
  Qed.

  (* test selection by hypothesis name instead of namespace *)
  Lemma test_iInv_9 t 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.

  (* test selection by hypothesis name instead of namespace *)
  Lemma test_iInv_10 t 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.

  (* test selection by ident name *)
  Lemma test_iInv_11 N P: inv N (<pers> P) ={⊤}=* |> P.
  Proof.
    let H := iFresh in
    (iIntros H; iInv H as "#H2"). auto.
  Qed.

  (* error messages *)
  Check "test_iInv_12".
  Lemma test_iInv_12 N 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.

  (* test destruction of existentials when opening an invariant *)
  Lemma test_iInv_13 N:
    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.
End iris_tests.

Section monpred_tests.
  Context `{!invG Σ}.
  Context {I : biIndex}.
  Local Notation monPred := (monPred I (iPropI Σ)).
  Local Notation monPredI := (monPredI I (iPropI Σ)).
  Implicit Types P Q R : monPred.
  Implicit Types 𝓟 𝓠 𝓡 : iProp Σ.

  Check "test_iInv".
  Lemma test_iInv N 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_close N 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.

End monpred_tests.

(** Test specifically if certain things parse correctly. *)
Section parsing_tests.
Context {PROP : bi}.
Implicit Types P : PROP.

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

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

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

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


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

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

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

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

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

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

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

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

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

Set Printing Notations.

End parsing_tests.