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From iris.proofmode Require Import tactics monpred.
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From iris.base_logic Require Import base_logic.
From iris.base_logic.lib Require Import invariants cancelable_invariants na_invariants.
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Set Ltac Backtrace.
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Section base_logic_tests.
  Context {M : ucmraT}.
  Implicit Types P Q R : uPred M.

  Lemma test_random_stuff (P1 P2 P3 : nat  uPred M) :
    ( (x y : nat) a b,
      x  y 
       (uPred_ownM (a  b) -
      ( y1 y2 c, P1 ((x + y1) + y2)  True   uPred_ownM c) -
        ( z, P2 z  True  P2 z) -
       ( n m : nat, P1 n   ((True  P2 n)   (n = n  P3 n))) -
       x = 0   x z,  P3 (x + z)  uPred_ownM b  uPred_ownM (core b)))%I.
  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 - ( x   x  y  z : uPred M).
  Proof. iIntros (Hv) "Hxy". by iFrame (Hv) "Hxy". Qed.

  Lemma test_iAssert_modality P : (|==> False) - |==> P.
  Proof. iIntros. iAssert False%I with "[> - //]" as %[]. Qed.
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  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 (uPredSI _). iIntros "$". Qed.
  Lemma test_iStartProof_4 P : P - P.
  Proof. iStartProof (uPredSI _). iStartProof (uPred _). iIntros "$". Qed.
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End base_logic_tests.

Section iris_tests.
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  Context `{!invG Σ, !cinvG Σ, !na_invG Σ}.
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  Implicit Types P Q R : iProp Σ.
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  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.
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  Lemma test_iInv_0 N P: inv N (<pers> P) ={}=  P.
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  Proof.
    iIntros "#H".
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    iInv N as "#H2". Show.
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    iModIntro. iSplit; auto.
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  Qed.

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  Lemma test_iInv_0_with_close N P: inv N (<pers> P) ={}=  P.
  Proof.
    iIntros "#H".
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    iInv N as "#H2" "Hclose". Show.
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    iMod ("Hclose" with "H2").
    iModIntro. by iNext.
  Qed.

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  Lemma test_iInv_1 N E P:
    N  E 
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    inv N (<pers> P) ={E}=  P.
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  Proof.
    iIntros (?) "#H".
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    iInv N as "#H2".
    iModIntro. iSplit; auto.
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  Qed.

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  Lemma test_iInv_2 γ p N P:
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    cinv N γ (<pers> P)  cinv_own γ p ={}= cinv_own γ p   P.
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  Proof.
    iIntros "(#?&?)".
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    iInv N as "(#HP&Hown)". Show.
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    iModIntro. iSplit; auto with iFrame.
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  Qed.

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  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 "(#?&?)".
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    iInv N as "(#HP&Hown)" "Hclose". Show.
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    iMod ("Hclose" with "HP").
    iModIntro. iFrame. by iNext.
  Qed.

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  Lemma test_iInv_3 γ p1 p2 N P:
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    cinv N γ (<pers> P)  cinv_own γ p1  cinv_own γ p2
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      ={}= cinv_own γ p1  cinv_own γ p2    P.
  Proof.
    iIntros "(#?&Hown1&Hown2)".
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    iInv N with "[Hown2 //]" as "(#HP&Hown2)".
    iModIntro. iSplit; auto with iFrame.
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  Qed.

  Lemma test_iInv_4 t N E1 E2 P:
    N  E2 
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    na_inv t N (<pers> P)  na_own t E1  na_own t E2
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          |={}=> na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
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    iInv N as "(#HP&Hown2)". Show.
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    iModIntro. iSplitL "Hown2"; auto with iFrame.
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  Qed.

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  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)".
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    iInv N as "(#HP&Hown2)" "Hclose". Show.
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    iMod ("Hclose" with "[HP Hown2]").
    { iFrame. done. }
    iModIntro. iFrame. by iNext.
  Qed.

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  (* test named selection of which na_own to use *)
  Lemma test_iInv_5 t N E1 E2 P:
    N  E2 
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    na_inv t N (<pers> P)  na_own t E1  na_own t E2
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      ={}= na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
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    iInv N with "Hown2" as "(#HP&Hown2)".
    iModIntro. iSplitL "Hown2"; auto with iFrame.
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  Qed.

  Lemma test_iInv_6 t N E1 E2 P:
    N  E1 
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    na_inv t N (<pers> P)  na_own t E1  na_own t E2
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      ={}= na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&Hown1&Hown2)".
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    iInv N with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
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  Qed.

  (* test robustness in presence of other invariants *)
  Lemma test_iInv_7 t N1 N2 N3 E1 E2 P:
    N3  E1 
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    inv N1 P  na_inv t N3 (<pers> P)  inv N2 P   na_own t E1  na_own t E2
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      ={}= na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&#?&#?&Hown1&Hown2)".
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    iInv N3 with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
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  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".
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    iInv N as "HP". iFrame "HP". auto.
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  Qed.
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  (* test selection by hypothesis name instead of namespace *)
  Lemma test_iInv_9 t N1 N2 N3 E1 E2 P:
    N3  E1 
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    inv N1 P  na_inv t N3 (<pers> P)  inv N2 P   na_own t E1  na_own t E2
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      ={}= na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)".
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    iInv "HInv" with "Hown1" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
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  Qed.

  (* test selection by hypothesis name instead of namespace *)
  Lemma test_iInv_10 t N1 N2 N3 E1 E2 P:
    N3  E1 
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    inv N1 P  na_inv t N3 (<pers> P)  inv N2 P   na_own t E1  na_own t E2
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      ={}= na_own t E1  na_own t E2    P.
  Proof.
    iIntros (?) "(#?&#HInv&#?&Hown1&Hown2)".
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    iInv "HInv" as "(#HP&Hown1)".
    iModIntro. iSplitL "Hown1"; auto with iFrame.
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  Qed.

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

  (* error messages *)
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  Check "test_iInv_12".
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  Lemma test_iInv_12 N P: inv N (<pers> P) ={}= True.
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  Proof.
    iIntros "H".
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    Fail iInv 34 as "#H2".
    Fail iInv nroot as "#H2".
    Fail iInv "H2" as "#H2".
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    done.
  Qed.
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  (* test destruction of existentials when opening an invariant *)
  Lemma test_iInv_13 N:
    inv N ( (v1 v2 v3 : nat), emp  emp  emp) ={}=  emp.
  Proof.
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    iIntros "H"; iInv "H" as (v1 v2 v3) "(?&?&_)".
    eauto.
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  Qed.
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End iris_tests.
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Section monpred_tests.
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  Context `{!invG Σ}.
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  Context {I : biIndex}.
  Local Notation monPred := (monPred I (iPropI Σ)).
  Local Notation monPredI := (monPredI I (iPropI Σ)).
  Local Notation monPredSI := (monPredSI I (iPropSI Σ)).
  Implicit Types P Q R : monPred.
  Implicit Types 𝓟 𝓠 𝓡 : iProp Σ.

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  Check "test_iInv".
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  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.

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  Check "test_iInv_with_close".
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  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.