telescopes.v 6.81 KB
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From stdpp Require Import coPset namespaces.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".

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Unset Mangle Names.

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Section basic_tests.
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  Context {PROP : bi}.
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  Implicit Types P Q R : PROP.

  Lemma test_iIntros_tforall {TT : tele} (Φ : TT  PROP) :
     .. x, Φ x - Φ x.
  Proof. iIntros (x) "H". done. Qed.
  Lemma test_iPoseProof_tforall {TT : tele} P (Φ : TT  PROP) :
    (.. x, P  Φ x)  P - .. x, Φ x.
  Proof.
    iIntros (H1) "H2"; iIntros (x).
    iPoseProof (H1) as "H1". by iApply "H1".
  Qed.
  Lemma test_iApply_tforall {TT : tele} P (Φ : TT  PROP) :
    (.. x, P - Φ x) - P - .. x, Φ x.
  Proof. iIntros "H1 H2" (x). by iApply "H1". Qed.
  Lemma test_iAssumption_tforall {TT : tele} (Φ : TT  PROP) :
    (.. x, Φ x) - .. x, Φ x.
  Proof. iIntros "H" (x). iAssumption. Qed.
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  Lemma test_exist_texist_auto_name {TT : tele} (Φ : TT  PROP) :
    (.. y, Φ y) - .. x, Φ x.
  Proof. iDestruct 1 as (?) "H". by iExists y. Qed.
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  Lemma test_pure_texist {TT : tele} (φ : TT  Prop) :
    (.. x,  φ x ) - .. x,  φ x  : PROP.
  Proof. iIntros (H) "!%". done. Qed.
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  Lemma test_pure_tforall `{!BiPureForall PROP} {TT : tele} (φ : TT  Prop) :
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    (.. x,  φ x ) - .. x,  φ x  : PROP.
  Proof. iIntros (H) "!%". done. Qed.
  Lemma test_pure_tforall_persistent {TT : tele} (Φ : TT  PROP) :
    (.. x, <pers> (Φ x)) - <pers> .. x, Φ x.
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  Proof. iIntros "#H !>" (x). done. Qed.
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  Lemma test_pure_texists_intuitionistic {TT : tele} (Φ : TT  PROP) :
    (.. x,  (Φ x)) -  .. x, Φ x.
  Proof. iDestruct 1 as (x) "#H". iExists (x). done. Qed.
  Lemma test_iMod_tforall {TT : tele} P (Φ : TT  PROP) :
     P - (.. x, Φ x) - .. x,  (P  Φ x).
  Proof. iIntros ">H1 H2" (x) "!> {$H1}". done. Qed.
  Lemma test_timeless_tforall {TT : tele} (φ : TT  Prop) :
     (.. x,  φ x ) -  .. x,  φ x  : PROP.
  Proof. iIntros ">H1 !>". done. Qed.
  Lemma test_timeless_texist {TT : tele} (φ : TT  Prop) :
     (.. x,  φ x ) -  .. x,  φ x  : PROP.
  Proof. iIntros ">H1 !>". done. Qed.
  Lemma test_add_model_texist `{!BiBUpd PROP} {TT : tele} P Q (Φ : TT  PROP) :
    (|==> P) - (P - Q) - .. x, |==> Q  (Φ x - Φ x).
  Proof. iIntros "H1 H2". iDestruct ("H2" with "[> $H1]") as "$". auto. Qed.
  Lemma test_iFrame_tforall {TT : tele} P (Φ : TT  PROP) :
    P - .. x, P  (Φ x - Φ x).
  Proof. iIntros "$". auto. Qed.
  Lemma test_iFrame_texist {TT : tele} P (Φ : TT  PROP) :
    P - (.. x, Φ x) - .. x, P  Φ x.
  Proof. iIntros "$". auto. Qed.
End basic_tests.

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Section accessor.
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(* Just playing around a bit with a telescope version
   of accessors with just one binder list. *)
Definition accessor `{!BiFUpd PROP} {X : tele} (E1 E2 : coPset)
           (α β γ : X  PROP) : PROP :=
  (|={E1,E2}=> .. x, α x  (β x - |={E2,E1}=> (γ x)))%I.

Notation "'ACC' @ E1 , E2 {{ ∃ x1 .. xn , α | β | γ } }" :=
  (accessor (X:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. ))
            E1 E2
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => α%I) ..)
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => β%I) ..)
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => γ%I) ..))
  (at level 20, α, β, γ at level 200, x1 binder, xn binder, only parsing).

(* Working with abstract telescopes. *)
Section tests.
Context `{!BiFUpd PROP} {X : tele}.
Implicit Types α β γ : X  PROP.

Lemma acc_mono E1 E2 α β γ1 γ2 :
  (.. x, γ1 x - γ2 x) -
  accessor E1 E2 α β γ1 - accessor E1 E2 α β γ2.
Proof.
  iIntros "Hγ12 >Hacc". iDestruct "Hacc" as (x') "[Hα Hclose]". Show.
  iModIntro. iExists x'. iFrame. iIntros "Hβ".
  iMod ("Hclose" with "Hβ") as "Hγ". iApply "Hγ12". auto.
Qed.
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Lemma acc_mono_disj E1 E2 α β γ1 γ2 :
  accessor E1 E2 α β γ1 - accessor E1 E2 α β (λ.. x, γ1 x  γ2 x).
Proof.
  Show.
  iApply acc_mono. iIntros (x) "Hγ1". Show.
  rewrite ->tele_app_bind. Show.
  iLeft. done.
Qed.
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End tests.

Section printing_tests.
Context `{!BiFUpd PROP}.

(* Working with concrete telescopes: Testing the reduction into normal quantifiers. *)
Lemma acc_elim_test_1 E1 E2 :
  ACC @ E1, E2 {{  a b : nat, <affine> a = b | True | <affine> a  b }}
    @{PROP} |={E1}=> False.
Proof.
  iIntros ">H". Show.
  iDestruct "H" as (a b) "[% Hclose]". iMod ("Hclose" with "[//]") as "%".
  done.
Qed.
End printing_tests.
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End accessor.

(* Robbert's tests *)
Section telescopes_and_tactics.

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Definition test1 {PROP : bi} {X : tele} (α : X  PROP) : PROP :=
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  (.. x, α x)%I.

Notation "'TEST1' {{ ∃ x1 .. xn , α } }" :=
  (test1 (X:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. ))
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => α%I) ..))
  (at level 20, α at level 200, x1 binder, xn binder, only parsing).

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Definition test2 {PROP : bi} {X : tele} (α : X  PROP) : PROP :=
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  ( .. x, α x)%I.

Notation "'TEST2' {{ ∃ x1 .. xn , α } }" :=
  (test2 (X:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. ))
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => α%I) ..))
  (at level 20, α at level 200, x1 binder, xn binder, only parsing).

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Definition test3 {PROP : bi} {X : tele} (α : X  PROP) : PROP :=
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  ( .. x, α x)%I.

Notation "'TEST3' {{ ∃ x1 .. xn , α } }" :=
  (test3 (X:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. ))
            (tele_app (TT:=TeleS (fun x1 => .. (TeleS (fun xn => TeleO)) .. )) $
                      fun x1 => .. (fun xn => α%I) ..))
  (at level 20, α at level 200, x1 binder, xn binder, only parsing).

Check "test1_test".
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Lemma test1_test {PROP : bi}  :
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  TEST1 {{  a b : nat, <affine> a = b }} @{PROP}  False.
Proof.
  iIntros "H". iDestruct "H" as (x) "H". Show.
Restart.
  iIntros "H". unfold test1. iDestruct "H" as (x) "H". Show.
Abort.

Check "test2_test".
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Lemma test2_test {PROP : bi}  :
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  TEST2 {{  a b : nat, <affine> a = b }} @{PROP}  False.
Proof.
  iIntros "H". iModIntro. Show.
  iDestruct "H" as (x) "H". Show.
Restart.
  iIntros "H". iDestruct "H" as (x) "H". Show.
Abort.

Check "test3_test".
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Lemma test3_test {PROP : bi}  :
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  TEST3 {{  a b : nat, <affine> a = b }} @{PROP}  False.
Proof.
  iIntros "H". iMod "H".
  iDestruct "H" as (x) "H".
  Show.
Restart.
  iIntros "H". iDestruct "H" as (x) "H". Show.
Abort.

End telescopes_and_tactics.