Commit 4fb85912 authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

Merge branch 'robbert/pigeon_hole' into 'master'

Add Pigeon Hole principle.

See merge request !373
parents f83560b2 6f9dea44
Pipeline #64673 passed with stage
in 5 minutes and 3 seconds
......@@ -29,6 +29,8 @@ lot to everyone involved!
- Add some more lemmas about `Finite` and `pred_finite`.
- Add lemmas about `last`: `last_app_cons`, `last_app`, `last_Some`, and
`last_Some_elem_of`.
- Add versions of Pigeonhole principle for Finite types, natural numbers, and
lists.
The following `sed` script should perform most of the renaming
(on macOS, replace `sed` by `gsed`, installed via e.g. `brew install gnu-sed`).
......
......@@ -419,3 +419,46 @@ Section sig_finite.
Lemma sig_card : card (sig P) = length (filter P (enum A)).
Proof. by rewrite <-list_filter_sig_filter, fmap_length. Qed.
End sig_finite.
Lemma finite_pigeonhole `{Finite A} `{Finite B} (f : A B) :
card B < card A x1 x2, x1 x2 f x1 = f x2.
Proof.
intros. apply dec_stable; intros Heq.
cut (Inj eq eq f); [intros ?%inj_card; lia|].
intros x1 x2 ?. apply dec_stable. naive_solver.
Qed.
Lemma nat_pigeonhole (f : nat nat) (n1 n2 : nat) :
n2 < n1
( i, i < n1 f i < n2)
i1 i2, i1 < i2 < n1 f i1 = f i2.
Proof.
intros Hn Hf. pose (f' (i : fin n1) := nat_to_fin (Hf _ (fin_to_nat_lt i))).
destruct (finite_pigeonhole f') as (i1&i2&Hi&Hf'); [by rewrite !fin_card|].
apply (not_inj (f:=fin_to_nat)) in Hi. apply (f_equal fin_to_nat) in Hf'.
unfold f' in Hf'. rewrite !fin_to_nat_to_fin in Hf'.
pose proof (fin_to_nat_lt i1); pose proof (fin_to_nat_lt i2).
destruct (decide (i1 < i2)); [exists i1, i2|exists i2, i1]; lia.
Qed.
Lemma list_pigeonhole {A} (l1 l2 : list A) :
l1 l2
length l2 < length l1
i1 i2 x, i1 < i2 l1 !! i1 = Some x l1 !! i2 = Some x.
Proof.
intros Hl Hlen.
assert ( i : fin (length l1), (j : fin (length l2)) x,
l1 !! (fin_to_nat i) = Some x
l2 !! (fin_to_nat j) = Some x) as [f Hf]%fin_choice.
{ intros i. destruct (lookup_lt_is_Some_2 l1 i)
as [x Hix]; [apply fin_to_nat_lt|].
assert (x l2) as [j Hjx]%elem_of_list_lookup_1
by (by eapply Hl, elem_of_list_lookup_2).
exists (nat_to_fin (lookup_lt_Some _ _ _ Hjx)), x.
by rewrite fin_to_nat_to_fin. }
destruct (finite_pigeonhole f) as (i1&i2&Hi&Hf'); [by rewrite !fin_card|].
destruct (Hf i1) as (x1&?&?), (Hf i2) as (x2&?&?).
assert (x1 = x2) as -> by congruence.
apply (not_inj (f:=fin_to_nat)) in Hi. apply (f_equal fin_to_nat) in Hf'.
destruct (decide (i1 < i2)); [exists i1, i2|exists i2, i1]; eauto with lia.
Qed.
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