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stdpp
Commits
cafae3e7
Commit
cafae3e7
authored
Feb 05, 2021
by
Robbert Krebbers
Browse files
More WIP.
parent
d16ab1aa
Changes
3
Hide whitespace changes
Inline
Sidebyside
tests/solve_proper.v
View file @
cafae3e7
...
...
@@ 44,16 +44,26 @@ Section setoid_tests.
Definition
setoid_test1
(
rec
:
myfunS
nat
A
)
:
myfunS
nat
A
:
=
λ
n
,
h
(
f
(
rec
n
))
(
rec
n
).
Goal
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test1
.
#[
local
]
Instance
:
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test1
.
Proof
.
solve_proper
.
Qed
.
Definition
setoid_test2
(
rec
:
myfunS
nat
(
myfunS
nat
A
))
:
myfunS
nat
A
:
=
λ
n
,
h
(
f
(
rec
n
n
))
(
rec
n
n
).
Goal
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test2
.
#[
local
]
Instance
:
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test2
.
Proof
.
solve_proper
.
Qed
.
Definition
setoid_test3
(
rec
:
myfunS
nat
A
)
:
myfunS
nat
(
myfunS
nat
A
)
:
=
λ
n
m
,
h
(
f
(
rec
n
))
(
rec
m
).
Goal
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test3
.
#[
local
]
Instance
:
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test3
.
Proof
.
solve_proper
.
Qed
.
Definition
setoid_test4
(
rec
:
myfunS
nat
A
)
:
myfunS
nat
(
myfunS
nat
A
)
:
=
λ
n
,
setoid_test2
(
λ
m
,
setoid_test1
rec
).
#[
local
]
Instance
:
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test4
.
Proof
.
solve_proper
.
Qed
.
Definition
setoid_test5
(
rec
:
myfunS
nat
A
)
:
myfunS
nat
A
:
=
setoid_test2
(
setoid_test3
rec
).
#[
local
]
Instance
:
Proper
(
setoid_equiv
==>
setoid_equiv
)
setoid_test4
.
Proof
.
solve_proper
.
Qed
.
End
setoid_tests
.
theories/sets.v
View file @
cafae3e7
...
...
@@ 1044,6 +1044,14 @@ Section pred_finite_infinite.
pred_infinite
P
→
(
∀
x
,
P
x
→
Q
x
)
→
pred_infinite
Q
.
Proof
.
unfold
pred_infinite
.
set_solver
.
Qed
.
Lemma
pred_infinite_surj
{
A
B
}
(
P
:
B
→
Prop
)
(
f
:
A
→
B
)
:
(
∀
x
,
P
x
→
∃
y
,
f
y
=
x
)
→
pred_infinite
P
→
pred_infinite
(
P
∘
f
).
Proof
.
intros
Hf
HP
xs
.
generalize
(
HP
(
f
<$>
xs
)).
setoid_rewrite
elem_of_list_fmap
.
naive_solver
.
Qed
.
Lemma
pred_not_infinite_finite
{
A
}
(
P
:
A
→
Prop
)
:
pred_infinite
P
→
pred_finite
P
→
False
.
Proof
.
intros
Hinf
[
xs
?].
destruct
(
Hinf
xs
).
set_solver
.
Qed
.
...
...
theories/tactics.v
View file @
cafae3e7
...
...
@@ 369,18 +369,24 @@ Ltac f_equiv :=


(
?R
_
_
)
(
?f
_
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
R
_
_
==>
R
_
_
==>
R
_
_
==>
R
_
_
==>
_
)
f
)


(
?R
_
_
_
)
(
?f
_
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
R
_
_
_
==>
R
_
_
_
R
_
_
_
==>
R
_
_
_
==>
_
)
f
)
(* If we cannot find a [Proper] instance that involves the relation [R],
check if [R] is convertable to a [pointwise_relation], i.e., [R] is a
check if [R] is
a relation on functions. is
convertable to a [pointwise_relation], i.e., [R] is a
pointwise relation on functions. In this case, we introduce the function
argument, and [simpl]ify the resulting goal. *)


?R
_
_
=>
eunify
R
(
pointwise_relation
_
_
)
;
intros
?
;
csimpl
(* Next, try to infer the relation by searching for an arbitrary [Proper]
instance. Unfortunately, very often, it will turn the goal into a Leibniz
equality so we get stuck. *)
(* TODO: Can we exclude that instance? *)


?R
(
?f
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
R
)
f
)


?R
(
?f
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
R
)
f
)


?R
(
?f
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
_
==>
R
)
f
)


?R
(
?f
_
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
_
==>
_
==>
R
)
f
)
argument, and [simpl]ify the resulting goal.
Deal with other cases where we have an equivalence relation on functions
(e.g. a [pointwise_relation] that is hidden in some form in [R]). We do
this by checking if the arguments of the relation are actually functions,
and then forcefully introduce one ∀ and introduce the remaining ∀s that
show up in the goal.
*)


?R
?f
_
=>
(* To check that we actually have an equivalence relation
on functions, we try to eta expand [f], which will only succeed if [f] is
actually a function. *)
let
f'
:
=
constr
:
(
λ
x
,
f
x
)
in
(* Now forcefully introduce the first ∀ and other ∀s that show up in the
goal afterwards. *)
intros
?
;
csimpl
(* In case the function symbol differs, but the arguments are the same,
maybe we have a [pointwise_relation] in our context. *)
(* TODO: If only some of the arguments are the same, we could also
...
...
@@ 390,6 +396,14 @@ Ltac f_equiv :=

H
:
?R'
?f
?g

?R
(
?f
_
_
)
(
?g
_
_
)
=>
eunify
R'
(
pointwise_relation
_
_
)
;
simple
apply
H

H
:
?R'
?f
?g

?R
(
?f
_
_
_
)
(
?g
_
_
_
)
=>
eunify
R'
(
pointwise_relation
_
_
)
;
simple
apply
H

H
:
?R'
?f
?g

?R
(
?f
_
_
_
_
)
(
?g
_
_
_
_
)
=>
eunify
R'
(
pointwise_relation
_
_
)
;
simple
apply
H
(* Next, if all fails, try to infer the relation by searching for an arbitrary
[Proper] instance. Unfortunately, very often, it will turn the goal into a
Leibniz equality so we get stuck. *)
(* TODO: Can we exclude that instance? *)


?R
(
?f
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
R
)
f
)


?R
(
?f
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
R
)
f
)


?R
(
?f
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
_
==>
R
)
f
)


?R
(
?f
_
_
_
_
)
_
=>
simple
apply
(
_
:
Proper
(
_
==>
_
==>
_
==>
_
==>
R
)
f
)
end
;
try
simple
apply
reflexivity
.
Tactic
Notation
"f_equiv"
"/="
:
=
csimpl
in
*
;
f_equiv
.
...
...
@@ 410,6 +424,7 @@ Ltac solve_proper_unfold :=


?R
(
?f
_
_
)
(
?f
_
_
)
=>
unfold
f


?R
(
?f
_
)
(
?f
_
)
=>
unfold
f
end
.
(** [solve_proper_prepare] does some preparation work before the main
[solve_proper] loop. Having this as a separate tactic is useful for debugging
[solve_proper] failure. *)
...
...
@@ 419,23 +434,11 @@ Ltac solve_proper_prepare :=
repeat
lazymatch
goal
with


Proper
_
_
=>
intros
???


(
_
==>
_
)%
signature
_
_
=>
intros
???


pointwise_relation
_
_
_
_
=>
intros
?


?R
?f
_
=>
(* Deal with other cases where we have an equivalence relation on functions
(e.g. a [pointwise_relation] that is hidden in some form in [R]). We do
this by checking if the arguments of the relation are actually functions,
and then forcefully introduce one ∀ and introduce the remaining ∀s that
show up in the goal. To check that we actually have an equivalence relation
on functions, we try to eta expand [f], which will only succeed if [f] is
actually a function. *)
let
f'
:
=
constr
:
(
λ
x
,
f
x
)
in
(* Now forcefully introduce the first ∀ and other ∀s that show up in the
goal afterwards. *)
intros
?
;
intros
end
;
simplify_eq
;
(* We try with and without unfolding. We have to backtrack on
that because unfolding may succeed, but then the proof may fail. *)
(
solve_proper_unfold
+
idtac
)
;
simpl
.
(** The tactic [solve_proper_core tac] solves goals of the form "Proper (R1 ==> R2)", for
any number of relations. The actual work is done by repeatedly applying
[tac]. *)
...
...
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