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Commits
546ed060
Commit
546ed060
authored
Nov 23, 2016
by
Robbert Krebbers
Browse files
Make some arguments explicit that could not be infered.
parent
b81fb420
Pipeline
#3044
passed with stage
in 10 minutes and 23 seconds
Changes
2
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2 changed files
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10 additions
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12 deletions
+10
12
prelude/fin_collections.v
prelude/fin_collections.v
+5
5
prelude/list.v
prelude/list.v
+5
7
No files found.
prelude/fin_collections.v
View file @
546ed060
...
...
@@ 254,21 +254,21 @@ Proof. rewrite Forall_forall. by setoid_rewrite elem_of_elements. Qed.
Lemma
set_Exists_elements
P
X
:
set_Exists
P
X
↔
Exists
P
(
elements
X
).
Proof
.
rewrite
Exists_exists
.
by
setoid_rewrite
elem_of_elements
.
Qed
.
Lemma
set_Forall_Exists_dec
{
P
Q
:
A
→
Prop
}
(
dec
:
∀
x
,
{
P
x
}
+
{
Q
x
})
X
:
Lemma
set_Forall_Exists_dec
(
P
Q
:
A
→
Prop
)
(
dec
:
∀
x
,
{
P
x
}
+
{
Q
x
})
X
:
{
set_Forall
P
X
}
+
{
set_Exists
Q
X
}.
Proof
.
refine
(
cast_if
(
Forall_Exists_dec
dec
(
elements
X
)))
;
refine
(
cast_if
(
Forall_Exists_dec
P
Q
dec
(
elements
X
)))
;
[
by
apply
set_Forall_elements

by
apply
set_Exists_elements
].
Defined
.
Lemma
not_set_Forall_Exists
P
`
{
dec
:
∀
x
,
Decision
(
P
x
)}
X
:
¬
set_Forall
P
X
→
set_Exists
(
not
∘
P
)
X
.
Proof
.
intro
.
by
destruct
(
set_Forall_Exists_dec
dec
X
).
Qed
.
Proof
.
intro
.
by
destruct
(
set_Forall_Exists_dec
P
(
not
∘
P
)
dec
X
).
Qed
.
Lemma
not_set_Exists_Forall
P
`
{
dec
:
∀
x
,
Decision
(
P
x
)}
X
:
¬
set_Exists
P
X
→
set_Forall
(
not
∘
P
)
X
.
Proof
.
by
destruct
(
@
set_Forall_Exists_dec
(
not
∘
P
)
_
(
λ
x
,
swap_if
(
decide
(
P
x
)))
X
).
by
destruct
(
set_Forall_Exists_dec
(
not
∘
P
)
P
(
λ
x
,
swap_if
(
decide
(
P
x
)))
X
).
Qed
.
Global
Instance
set_Forall_dec
(
P
:
A
→
Prop
)
`
{
∀
x
,
Decision
(
P
x
)}
X
:
...
...
prelude/list.v
View file @
546ed060
...
...
@@ 2051,7 +2051,7 @@ Lemma list_subseteq_nil l : [] ⊆ l.
Proof. intros x. by rewrite elem_of_nil. Qed.
(** ** Properties of the [Forall] and [Exists] predicate *)
Lemma Forall_Exists_dec
{
P Q : A → Prop
}
(dec : ∀ x, {P x} + {Q x}) :
Lemma Forall_Exists_dec
(
P Q : A → Prop
)
(dec : ∀ x, {P x} + {Q x}) :
∀ l, {Forall P l} + {Exists Q l}.
Proof.
refine (
...
...
@@ 2232,21 +2232,19 @@ Section Forall_Exists.
Context {dec : ∀ x, Decision (P x)}.
Lemma not_Forall_Exists l : ¬Forall P l → Exists (not ∘ P) l.
Proof. intro. destruct (Forall_Exists_dec
dec l); intuition
. Qed.
Proof. intro.
by
destruct (Forall_Exists_dec
P (not ∘ P) dec l)
. Qed.
Lemma not_Exists_Forall l : ¬Exists P l → Forall (not ∘ P) l.
Proof.
(* TODO: Coq 8.6 needs type annotation here, Coq 8.5 did not.
Should we report this? *)
by destruct (@Forall_Exists_dec (not ∘ P) _
by destruct (Forall_Exists_dec (not ∘ P) P
(λ x : A, swap_if (decide (P x))) l).
Qed.
Global Instance Forall_dec l : Decision (Forall P l) :=
match Forall_Exists_dec dec l with
match Forall_Exists_dec
P (not ∘ P)
dec l with
 left H => left H
 right H => right (Exists_not_Forall _ H)
end.
Global Instance Exists_dec l : Decision (Exists P l) :=
match Forall_Exists_dec (λ x, swap_if (decide (P x))) l with
match Forall_Exists_dec
(not ∘ P) P
(λ x, swap_if (decide (P x))) l with
 left H => right (Forall_not_Exists _ H)
 right H => left H
end.
...
...
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