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Adam
Iris
Commits
a9354385
Commit
a9354385
authored
Sep 03, 2021
by
Robbert Krebbers
Browse files
Add `iInduction` tests.
parent
6ab9c4e7
Changes
2
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tests/proofmode.ref
View file @
a9354385
...
...
@@ 781,3 +781,11 @@ Tactic failure: iPure: (φ n) not pure.
: string
The command has indeed failed with message:
Tactic failure: iIntuitionistic: Q not persistent.
The command has indeed failed with message:
Tactic failure: iInduction: cannot import IH
(my_Forall
(λ t : tree,
"H" : ∀ l : list tree, ([∗ list] x ∈ l, P x) ∗ P (Tree l)
□
P t
) l) into proof mode context.
tests/proofmode.v
View file @
a9354385
...
...
@@ 1618,3 +1618,62 @@ Proof.
Qed
.
End
tactic_tests
.
Section
mutual_induction
.
Context
{
PROP
:
bi
}.
Implicit
Types
P
Q
R
:
PROP
.
Implicit
Types
φ
:
nat
→
PROP
.
Implicit
Types
Ψ
:
nat
→
nat
→
PROP
.
Unset
Elimination
Schemes
.
Inductive
tree
:
=
Tree
:
list
tree
→
tree
.
(** The common induction principle for finitely branching trees. By default,
Coq generates a too weak induction principle, so we have to prove it by hand. *)
Lemma
tree_ind
(
φ
:
tree
→
Prop
)
:
(
∀
l
,
Forall
φ
l
→
φ
(
Tree
l
))
→
∀
t
,
φ
t
.
Proof
.
intros
Hrec
.
fix
REC
1
.
intros
[
l
].
apply
Hrec
.
clear
Hrec
.
induction
l
as
[
t
l
IH
]
;
constructor
;
auto
.
Qed
.
(** Now let's test that we can derive the internal induction principle for
finitely branching trees in separation logic. There are many variants of the
induction principle, but we pick the variant with the big op [[∗ list]] in
the induction hypothesis. This is also most interesting, since the proof mode
generates an induction hypothesis of the form [∀ x, ⌜ x ∈ l ⌝ → ...]. *)
Lemma
test_iInduction_Forall
(
P
:
tree
→
PROP
)
:
□
(
∀
l
,
([
∗
list
]
x
∈
l
,
P
x
)

∗
P
(
Tree
l
))

∗
∀
t
,
P
t
.
Proof
.
iIntros
"#H"
(
t
).
iInduction
t
as
[]
"IH"
.
iApply
"H"
.
iApply
big_sepL_intro
.
iIntros
"!#"
(
k
t'
?%
elem_of_list_lookup_2
).
by
iApply
(
"IH"
with
"[%]"
).
Qed
.
(** Now let's define a custom version of [Forall], called [my_Forall], and
use that in the variant [tree_ind_alt] of the induction principle. The proof
mode does not support [my_Forall], so we test if [iInduction] generates a
proper error message. *)
Inductive
my_Forall
{
A
}
(
φ
:
A
→
Prop
)
:
list
A
→
Prop
:
=

my_Forall_nil
:
my_Forall
φ
[]

my_Forall_cons
x
l
:
φ
x
→
my_Forall
φ
l
→
my_Forall
φ
(
x
::
l
).
Lemma
tree_ind_alt
(
φ
:
tree
→
Prop
)
:
(
∀
l
,
my_Forall
φ
l
→
φ
(
Tree
l
))
→
∀
t
,
φ
t
.
Proof
.
intros
Hrec
.
fix
REC
1
.
intros
[
l
].
apply
Hrec
.
clear
Hrec
.
induction
l
as
[
t
l
IH
]
;
constructor
;
auto
.
Qed
.
Lemma
test_iInduction_Forall_fail
(
P
:
tree
→
PROP
)
:
□
(
∀
l
,
([
∗
list
]
x
∈
l
,
P
x
)

∗
P
(
Tree
l
))

∗
∀
t
,
P
t
.
Proof
.
iIntros
"#H"
(
t
).
Fail
iInduction
t
as
[]
"IH"
using
tree_ind_alt
.
Abort
.
End
mutual_induction
.
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