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53a18cf8
Commit
53a18cf8
authored
Jul 13, 2019
by
Paolo G. Giarrusso
Committed by
Robbert
Jul 13, 2019
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Mark projections for sigTO as NonExpansive and Proper
parent
94533e33
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theories/algebra/ofe.v
theories/algebra/ofe.v
+37
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theories/algebra/ofe.v
View file @
53a18cf8
...
...
@@ -1301,7 +1301,7 @@ Section sigT.
on the first component.
*)
Instance
sigT_dist
:
Dist
(
sigT
P
)
:
=
λ
n
x1
x2
,
∃
eq
:
projT1
x1
=
projT1
x2
,
rew
eq
in
projT2
x1
≡
{
n
}
≡
projT2
x2
.
∃
Heq
:
projT1
x1
=
projT1
x2
,
rew
H
eq
in
projT2
x1
≡
{
n
}
≡
projT2
x2
.
(**
Usually we'd give a direct definition, and show it equivalent to
...
...
@@ -1319,11 +1319,11 @@ Section sigT.
reflexivity
_
.
Definition
sigT_dist_eq
x1
x2
n
:
(
x1
≡
{
n
}
≡
x2
)
↔
∃
eq
:
projT1
x1
=
projT1
x2
,
(
rew
eq
in
projT2
x1
)
≡
{
n
}
≡
projT2
x2
:
=
∃
Heq
:
projT1
x1
=
projT1
x2
,
(
rew
H
eq
in
projT2
x1
)
≡
{
n
}
≡
projT2
x2
:
=
reflexivity
_
.
Definition
sigT_dist_proj1
n
{
x
y
}
:
x
≡
{
n
}
≡
y
→
projT1
x
=
projT1
y
:
=
proj1_ex
.
Definition
sigT_equiv_proj1
x
y
:
x
≡
y
→
projT1
x
=
projT1
y
:
=
λ
H
,
proj1_ex
(
H
0
).
Definition
sigT_equiv_proj1
{
x
y
}
:
x
≡
y
→
projT1
x
=
projT1
y
:
=
λ
H
,
proj1_ex
(
H
0
).
Definition
sigT_ofe_mixin
:
OfeMixin
(
sigT
P
).
Proof
.
...
...
@@ -1342,16 +1342,47 @@ Section sigT.
Canonical
Structure
sigTO
:
ofeT
:
=
OfeT
(
sigT
P
)
sigT_ofe_mixin
.
Lemma
sigT_equiv_eq_alt
`
{!
∀
a
b
:
A
,
ProofIrrel
(
a
=
b
)}
x1
x2
:
x1
≡
x2
↔
∃
Heq
:
projT1
x1
=
projT1
x2
,
rew
Heq
in
projT2
x1
≡
projT2
x2
.
Proof
.
setoid_rewrite
equiv_dist
;
setoid_rewrite
sigT_dist_eq
;
split
=>
Heq
.
-
move
:
(
Heq
0
)
=>
[
H0eq1
_
].
exists
H0eq1
=>
n
.
move
:
(
Heq
n
)
=>
[]
Hneq1
.
by
rewrite
(
proof_irrel
H0eq1
Hneq1
).
-
move
:
Heq
=>
[
Heq1
Heqn2
]
n
.
by
exists
Heq1
.
Qed
.
(** [projT1] is non-expansive and proper. *)
Global
Instance
projT1_ne
:
NonExpansive
(
projT1
:
sigTO
→
leibnizO
A
).
Proof
.
solve_proper
.
Qed
.
Global
Instance
projT1_proper
:
Proper
((
≡
)
==>
(
≡
))
(
projT1
:
sigTO
→
leibnizO
A
).
Proof
.
apply
ne_proper
,
projT1_ne
.
Qed
.
(** [projT2] is "non-expansive"; the properness lemma [projT2_ne] requires UIP. *)
Lemma
projT2_ne
n
(
x1
x2
:
sigTO
)
(
Heq
:
x1
≡
{
n
}
≡
x2
)
:
rew
(
sigT_dist_proj1
n
Heq
)
in
projT2
x1
≡
{
n
}
≡
projT2
x2
.
Proof
.
by
destruct
Heq
.
Qed
.
Lemma
projT2_proper
`
{!
∀
a
b
:
A
,
ProofIrrel
(
a
=
b
)}
(
x1
x2
:
sigTO
)
(
Heqs
:
x1
≡
x2
)
:
rew
(
sigT_equiv_proj1
Heqs
)
in
projT2
x1
≡
projT2
x2
.
Proof
.
move
:
x1
x2
Heqs
=>
[
a1
x1
]
[
a2
x2
]
Heqs
.
case
:
(
proj1
(
sigT_equiv_eq_alt
_
_
)
Heqs
)
=>
/=.
intros
->.
rewrite
(
proof_irrel
(
sigT_equiv_proj1
Heqs
)
eq_refl
)
/=.
done
.
Qed
.
(** [existT] is "non-expansive" — general, dependently-typed statement. *)
Lemma
existT_ne
n
{
i1
i2
}
{
v1
:
P
i1
}
{
v2
:
P
i2
}
:
∀
(
eq
:
i1
=
i2
),
(
rew
f_equal
P
eq
in
v1
≡
{
n
}
≡
v2
)
→
∀
(
Heq
:
i1
=
i2
),
(
rew
f_equal
P
H
eq
in
v1
≡
{
n
}
≡
v2
)
→
existT
i1
v1
≡
{
n
}
≡
existT
i2
v2
.
Proof
.
intros
->
;
simpl
.
exists
eq_refl
=>
/=.
done
.
Qed
.
Lemma
existT_proper
{
i1
i2
}
{
v1
:
P
i1
}
{
v2
:
P
i2
}
:
∀
(
eq
:
i1
=
i2
),
(
rew
f_equal
P
eq
in
v1
≡
v2
)
→
∀
(
Heq
:
i1
=
i2
),
(
rew
f_equal
P
H
eq
in
v1
≡
v2
)
→
existT
i1
v1
≡
existT
i2
v2
.
Proof
.
intros
eq
Heq
n
.
apply
(
existT_ne
n
eq
),
equiv_dist
,
Heq
.
Qed
.
Proof
.
intros
Heq
Heqv
n
.
apply
(
existT_ne
n
Heq
),
equiv_dist
,
Heqv
.
Qed
.
(** [existT] is "non-expansive" — non-dependently-typed version. *)
Global
Instance
existT_ne_2
a
:
NonExpansive
(@
existT
A
P
a
).
...
...
@@ -1374,17 +1405,6 @@ Section sigT.
Lemma
sigT_chain_const_proj1
c
n
:
projT1
(
c
n
)
=
projT1
(
c
0
).
Proof
.
refine
(
sigT_dist_proj1
_
(
chain_cauchy
c
0
n
_
)).
lia
.
Qed
.
Lemma
sigT_equiv_eq_alt
`
{!
∀
a
b
:
A
,
ProofIrrel
(
a
=
b
)}
x1
x2
:
x1
≡
x2
↔
∃
eq
:
projT1
x1
=
projT1
x2
,
rew
eq
in
projT2
x1
≡
projT2
x2
.
Proof
.
setoid_rewrite
equiv_dist
;
setoid_rewrite
sigT_dist_eq
;
split
=>
Heq
.
-
move
:
(
Heq
0
)
=>
[
H0eq1
_
].
exists
H0eq1
=>
n
.
move
:
(
Heq
n
)
=>
[]
Hneq1
.
by
rewrite
(
proof_irrel
H0eq1
Hneq1
).
-
move
:
Heq
=>
[
Heq1
Heqn2
]
n
.
by
exists
Heq1
.
Qed
.
(* For this COFE construction we need UIP (Uniqueness of Identity Proofs)
on [A] (i.e. [∀ x y : A, ProofIrrel (x = y)]. UIP is most commonly obtained
from decidable equality (by Hedberg’s theorem, see
...
...
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