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From iris.proofmode Require Import tactics.
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From iris.proofmode Require Import pviewshifts.
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Lemma demo_0 {M : cmraT} (P Q : uPred M) :
   (P  Q)  (( x, x = 0  x = 1)  (Q  P)).
Proof.
  iIntros "#H #H2".
  (* should remove the disjunction "H" *)
  iDestruct "H" as "[?|?]"; last by iLeft.
  (* should keep the disjunction "H" because it is instantiated *)
  iDestruct ("H2" $! 10) as "[%|%]". done. done.
Qed.

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Lemma demo_1 (M : cmraT) (P1 P2 P3 : nat  uPred M) :
  True  ( (x y : nat) a b,
    x  y 
     (uPred_ownM (a  b) -
    ( y1 y2 c, P1 ((x + y1) + y2)  True   uPred_ownM c) -
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      ( z, P2 z  True  P2 z) -
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     ( 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))).
Proof.
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  iIntros {i [|j] a b ?} "! [Ha Hb] H1 #H2 H3"; setoid_subst.
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  { iLeft. by iNext. }
  iRight.
  iDestruct "H1" as {z1 z2 c} "(H1&_&#Hc)".
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  iPoseProof "Hc" as "foo".
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  iRevert {a b} "Ha Hb". iIntros {b a} "Hb {foo} Ha".
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  iAssert (uPred_ownM (a  core a))%I as "[Ha #Hac]" with "[Ha]".
  { by rewrite cmra_core_r. }
  iFrame "Ha Hac".
  iExists (S j + z1), z2.
  iNext.
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  iApply ("H3" $! _ 0 with "H1 ! [] !").
  - iSplit. done. iApply "H2". iLeft. iApply "H2". by iRight.
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  - done.
Qed.

Lemma demo_2 (M : cmraT) (P1 P2 P3 P4 Q : uPred M) (P5 : nat  uPredC M):
    (P2  (P3  Q)  True  P1  P2  (P4  ( x:nat, P5 x  P3))  True)
   (P1 - (True  True) - (((P2  False  P2  0 = 0)  P3)  Q  P1  True) 
     (P2  False)  (False  P5 0)).
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".
  * iFrame "H3". by iRight.
  * iSplitL "HQ". iAssumption. by iSplitL "H1".
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Qed.

Lemma demo_3 (M : cmraT) (P1 P2 P3 : uPred M) :
  (P1  P2  P3)  ( P1   (P2   x, (P3  x = 0)  P3)).
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Proof. iIntros "($ & $ & H)". iFrame "H". iNext. by iExists 0. Qed.

Definition foo {M} (P : uPred M) := (P  P)%I.
Definition bar {M} : uPred M := ( P, foo P)%I.

Lemma demo_4 (M : cmraT) : True  @bar M.
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Proof. iIntros. iIntros {P} "HP". done. Qed.
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Lemma demo_5 (M : cmraT) (x y : M) (P : uPred M) :
  ( z, P  z  y)  (P - (x,x)  (y,x)).
Proof.
  iIntros "H1 H2".
  iRewrite (uPred.eq_sym x x with "- !"). iApply uPred.eq_refl.
  iRewrite -("H1" $! _ with "- !"); first done.
  iApply uPred.eq_refl.
Qed.

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Lemma demo_6 (M : cmraT) (P Q : uPred M) :
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  True  ( x y z : nat,
    x = plus 0 x  y = 0  z = 0  P   Q  foo (x  x)).
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Proof.
  iIntros {a} "*".
  iIntros "#Hfoo **".
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  by iIntros "# _".
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Qed.
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Section iris.
  Context {Λ : language} {Σ : iFunctor}.

  Lemma demo_7 (E1 E2 E : coPset) (P : iProp Λ Σ) :
    E1  E2  E  E1 
    (|={E1,E}=>  P)  (|={E2,E  E2  E1}=>  P).
  Proof.
    iIntros {? ?} "Hpvs".
    iPvs "Hpvs"; first (split_and?; set_solver).
    done.
  Qed.
End iris.