File: proofmode_monpred.v

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From iris.bi.lib Require Import fractional.
From iris.proofmode Require Import tactics monpred.
From iris.base_logic.lib Require Import invariants ghost_var.
From iris.prelude Require Import options.

Unset Mangle Names.

Section tests.
  Context {I : biIndex} {PROP : bi}.
  Local Notation monPred := (monPred I PROP).
  Local Notation monPredI := (monPredI I PROP).
  Implicit Types P Q R : monPred.
  Implicit Types π“Ÿ 𝓠 𝓑 : PROP.
  Implicit Types i j : I.

  Lemma test0 P : P -βˆ— P.
  Proof. iIntros "$". Qed.

  Lemma test_iStartProof_1 P : P -βˆ— P.
  Proof. iStartProof. iStartProof. iIntros "$". Qed.
  Lemma test_iStartProof_2 P : P -βˆ— P.
  Proof. iStartProof monPred. iStartProof monPredI. iIntros "$". Qed.
  Lemma test_iStartProof_3 P : P -βˆ— P.
  Proof. iStartProof monPredI. iStartProof monPredI. iIntros "$". Qed.
  Lemma test_iStartProof_4 P : P -βˆ— P.
  Proof. iStartProof monPredI. iStartProof monPred. iIntros "$". Qed.
  Lemma test_iStartProof_5 P : P -βˆ— P.
  Proof. iStartProof PROP. iIntros (i) "$". Qed.
  Lemma test_iStartProof_6 P : P ⊣⊒ P.
  Proof. iStartProof PROP. iIntros (i). iSplit; iIntros "$". Qed.
  Lemma test_iStartProof_7 `{!BiInternalEq PROP} P : ⊒@{monPredI} P ≑ P.
  Proof. iStartProof PROP. done. Qed.

  Lemma test_intowand_1 P Q : (P -βˆ— Q) -βˆ— P -βˆ— Q.
  Proof.
    iStartProof PROP. iIntros (i) "HW". Show.
    iIntros (j ->) "HP". Show. by iApply "HW".
  Qed.
  Lemma test_intowand_2 P Q : (P -βˆ— Q) -βˆ— P -βˆ— Q.
  Proof.
    iStartProof PROP. iIntros (i) "HW". iIntros (j ->) "HP".
    iSpecialize ("HW" with "[HP //]"). done.
  Qed.
  Lemma test_intowand_3 P Q : (P -βˆ— Q) -βˆ— P -βˆ— Q.
  Proof.
    iStartProof PROP. iIntros (i) "HW". iIntros (j ->) "HP".
    iSpecialize ("HW" with "HP"). done.
  Qed.
  Lemma test_intowand_4 P Q : (P -βˆ— Q) -βˆ— β–· P -βˆ— β–· Q.
  Proof.
    iStartProof PROP. iIntros (i) "HW". iIntros (j ->) "HP". by iApply "HW".
  Qed.
  Lemma test_intowand_5 P Q : (P -βˆ— Q) -βˆ— β–· P -βˆ— β–· Q.
  Proof.
    iStartProof PROP. iIntros (i) "HW". iIntros (j ->) "HP".
    iSpecialize ("HW" with "HP"). done.
  Qed.

  Lemma test_apply_in_elim (P : monPredI) (i : I) : monPred_in i -βˆ— ⎑ P i ⎀ β†’ P.
  Proof. iIntros. by iApply monPred_in_elim. Qed.

  Lemma test_iStartProof_forall_1 (Ξ¦ : nat β†’ monPredI) : βˆ€ n, Ξ¦ n -βˆ— Ξ¦ n.
  Proof.
    iStartProof PROP. iIntros (n i) "$".
  Qed.
  Lemma test_iStartProof_forall_2 (Ξ¦ : nat β†’ monPredI) : βˆ€ n, Ξ¦ n -βˆ— Ξ¦ n.
  Proof.
    iStartProof. iIntros (n) "$".
  Qed.

  Lemma test_embed_wand (P Q : PROP) : (⎑P⎀ -βˆ— ⎑Q⎀) ⊒@{monPredI} ⎑P -βˆ— Q⎀.
  Proof.
    iIntros "H HP". by iApply "H".
  Qed.

  Lemma test_objectively `{!BiPersistentlyForall PROP} P Q :
    <obj> emp -βˆ— <obj> P -βˆ— <obj> Q -βˆ— <obj> (P βˆ— Q).
  Proof. iIntros "#? HP HQ". iModIntro. by iSplitL "HP". Qed.

  Lemma test_objectively_absorbing `{!BiPersistentlyForall PROP} P Q R `{!Absorbing P} :
    <obj> emp -βˆ— <obj> P -βˆ— <obj> Q -βˆ— R -βˆ— <obj> (P βˆ— Q).
  Proof. iIntros "#? HP HQ HR". iModIntro. by iSplitL "HP". Qed.

  Lemma test_objectively_affine `{!BiPersistentlyForall PROP} P Q R `{!Affine R} :
    <obj> emp -βˆ— <obj> P -βˆ— <obj> Q -βˆ— R -βˆ— <obj> (P βˆ— Q).
  Proof. iIntros "#? HP HQ HR". iModIntro. by iSplitL "HP". Qed.

  Lemma test_iModIntro_embed P `{!Affine Q} π“Ÿ 𝓠 :
    β–‘ P -βˆ— Q -βˆ— βŽ‘π“ŸβŽ€ -βˆ— βŽ‘π“ βŽ€ -βˆ— ⎑ π“Ÿ βˆ— 𝓠 ⎀.
  Proof. iIntros "#H1 _ H2 H3". iModIntro. iFrame. Qed.

  Lemma test_iModIntro_embed_objective P `{!Objective Q} π“Ÿ 𝓠 :
    β–‘ P -βˆ— Q -βˆ— βŽ‘π“ŸβŽ€ -βˆ— βŽ‘π“ βŽ€ -βˆ— ⎑ βˆ€ i, π“Ÿ βˆ— 𝓠 βˆ— Q i ⎀.
  Proof. iIntros "#H1 H2 H3 H4". iModIntro. Show. iFrame. Qed.

  Lemma test_iModIntro_embed_nested P π“Ÿ 𝓠 :
    β–‘ P -βˆ— βŽ‘β—‡ π“ŸβŽ€ -βˆ— βŽ‘β—‡ π“ βŽ€ -βˆ— ⎑ β—‡ (π“Ÿ βˆ— 𝓠) ⎀.
  Proof. iIntros "#H1 H2 H3". iModIntro ⎑ _ ⎀%I. by iSplitL "H2". Qed.

  Lemma test_into_wand_embed π“Ÿ 𝓠 :
    (π“Ÿ -βˆ— β—‡ 𝓠) β†’
    βŽ‘π“ŸβŽ€ ⊒@{monPredI} β—‡ βŽ‘π“ βŽ€.
  Proof.
    iIntros (HPQ) "HP".
    iMod (HPQ with "[-]") as "$"; last by auto.
    iAssumption.
  Qed.

  Lemma test_monPred_at_and_pure P i j :
    (monPred_in j ∧ P) i ⊒ ⌜ j βŠ‘ i ⌝ ∧ P i.
  Proof.
    iDestruct 1 as "[% $]". done.
  Qed.
  Lemma test_monPred_at_sep_pure P i j :
    (monPred_in j βˆ— <absorb> P) i ⊒ ⌜ j βŠ‘ i ⌝ ∧ <absorb> P i.
  Proof.
    iDestruct 1 as "[% ?]". auto.
  Qed.

  Context (FU : BiFUpd PROP).

  Lemma test_apply_fupd_intro_mask_subseteq E1 E2 P :
    E2 βŠ† E1 β†’ P -βˆ— |={E1,E2}=> |={E2,E1}=> P.
  Proof. iIntros. by iApply @fupd_mask_intro_subseteq. Qed.
  Lemma test_apply_fupd_mask_subseteq E1 E2 P :
    E2 βŠ† E1 β†’ P -βˆ— |={E1,E2}=> |={E2,E1}=> P.
  Proof. iIntros. iFrame. by iApply @fupd_mask_subseteq. Qed.

  Lemma test_iFrame_embed_persistent (P : PROP) (Q: monPred) :
    Q βˆ— β–‘ ⎑P⎀ ⊒ Q βˆ— ⎑P βˆ— P⎀.
  Proof.
    iIntros "[$ #HP]". iFrame "HP".
  Qed.

  Lemma test_iNext_Bi P :
    β–· P ⊒@{monPredI} β–· P.
  Proof. iIntros "H". by iNext. Qed.

  (** Test monPred_at framing *)
  Lemma test_iFrame_monPred_at_wand (P Q : monPred) i :
    P i -βˆ— (Q -βˆ— P) i.
  Proof. iIntros "$". Show. Abort.

  Program Definition monPred_id (R : monPred) : monPred :=
    MonPred (Ξ» V, R V) _.
  Next Obligation. intros ? ???. eauto. Qed.

  Lemma test_iFrame_monPred_id (Q R : monPred) i :
    Q i βˆ— R i -βˆ— (Q βˆ— monPred_id R) i.
  Proof.
    iIntros "(HQ & HR)". iFrame "HR". iAssumption.
  Qed.

  Lemma test_iFrame_rel P i j ij :
    IsBiIndexRel i ij β†’ IsBiIndexRel j ij β†’
    P i -βˆ— P j -βˆ— P ij βˆ— P ij.
  Proof. iIntros (??) "HPi HPj". iFrame. Qed.

  Lemma test_iFrame_later_rel `{!BiAffine PROP} P i j :
    IsBiIndexRel i j β†’
    β–· (P i) -βˆ— (β–· P) j.
  Proof. iIntros (?) "?". iFrame. Qed.

  Lemma test_iFrame_laterN n P i :
    β–·^n (P i) -βˆ— (β–·^n P) i.
  Proof. iIntros "?". iFrame. Qed.

  Lemma test_iFrame_quantifiers P i :
    P i -βˆ— (βˆ€ _:(), βˆƒ _:(), P) i.
  Proof. iIntros "?". iFrame. Show. iIntros ([]). iExists (). iEmpIntro. Qed.

  Lemma test_iFrame_embed (P : PROP) i :
    P -βˆ— (embed (B:=monPredI) P) i.
  Proof. iIntros "$". Qed.

  (* Make sure search doesn't diverge on an evar *)
  Lemma test_iFrame_monPred_at_evar (P : monPred) i j :
    P i -βˆ— βˆƒ Q, (Q j).
  Proof. iIntros "HP". iExists _. Fail iFrame "HP". Abort.

End tests.

Section tests_iprop.
  Context {I : biIndex} `{!invGS_gen hlc Ξ£}.

  Local Notation monPred := (monPred I (iPropI Ξ£)).
  Local Notation monPredI := (monPredI I (iPropI Ξ£)).
  Implicit Types P Q R : monPred.
  Implicit Types π“Ÿ 𝓠 𝓑 : iProp Ξ£.

  Lemma test_iInv_0 N π“Ÿ :
    embed (B:=monPred) (inv N (<pers> π“Ÿ)) ={⊀}=βˆ— βŽ‘β–· π“ŸβŽ€.
  Proof.
    iIntros "#H".
    iInv N as "#H2". Show.
    iModIntro. iSplit=>//. iModIntro. iModIntro; auto.
  Qed.

  Lemma test_iInv_0_with_close N π“Ÿ :
    embed (B:=monPred) (inv N (<pers> π“Ÿ)) ={⊀}=βˆ— βŽ‘β–· π“ŸβŽ€.
  Proof.
    iIntros "#H".
    iInv N as "#H2" "Hclose". Show.
    iMod ("Hclose" with "H2").
    iModIntro. iModIntro. by iNext.
  Qed.

  Lemma test_iPoseProof `{inG Ξ£ A} P Ξ³ (x y : A) :
    x ~~> y β†’ P βˆ— ⎑own Ξ³ x⎀ ==βˆ— ⎑own Ξ³ y⎀.
  Proof.
    iIntros (?) "[_ HΞ³]".
    iPoseProof (own_update with "HΞ³") as "H"; first done.
    by iMod "H".
  Qed.

  Lemma test_embed_fractional `{!ghost_varG Ξ£ A} Ξ³ q (a : A) :
    ⎑ghost_var Ξ³ q a⎀ ⊒@{monPredI} ⎑ghost_var Ξ³ (q/2) a⎀ βˆ— ⎑ghost_var Ξ³ (q/2) a⎀.
  Proof. iIntros "[$ $]". Qed.

  Lemma test_embed_combine `{!ghost_varG Ξ£ A} Ξ³ q (a1 a2 : A) :
    β–· ⎑ghost_var Ξ³ (q/2) a1⎀ βˆ— β–· ⎑ghost_var Ξ³ (q/2) a2⎀ ⊒@{monPredI}
            β–·βŽ‘ghost_var Ξ³ q a1⎀ βˆ— β–· ⌜a1 = a2⌝.
  Proof.
    iIntros "[H1 H2]".
    iCombine "H1 H2" as "$" gives "#H". iNext.
    by iDestruct "H" as %[_ ->].
  Qed.
End tests_iprop.