-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathtelescopes.v
176 lines (156 loc) · 6.79 KB
/
telescopes.v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
From stdpp Require Import coPset namespaces.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".
Section basic_tests.
Context {PROP : bi}.
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.
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.
Lemma test_pure_texist {TT : tele} (φ : TT → Prop) :
(∃.. x, ⌜ φ x ⌝) -∗ ∃.. x, ⌜ φ x ⌝ : PROP.
Proof. iIntros (H) "!%". done. Qed.
Lemma test_pure_tforall `{!BiPureForall PROP} {TT : tele} (φ : TT → Prop) :
(∀.. 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.
Proof. iIntros "#H !>" (x). done. Qed.
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.
Section accessor.
(* 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.
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.
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.
End accessor.
(* Robbert's tests *)
Section telescopes_and_tactics.
Definition test1 {PROP : bi} {X : tele} (α : X → PROP) : PROP :=
(∃.. 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).
Definition test2 {PROP : bi} {X : tele} (α : X → PROP) : PROP :=
(▷ ∃.. 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).
Definition test3 {PROP : bi} {X : tele} (α : X → PROP) : PROP :=
(◇ ∃.. 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".
Lemma test1_test {PROP : bi} :
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".
Lemma test2_test {PROP : bi} :
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".
Lemma test3_test {PROP : bi} :
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.