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[stdlib] several NoDup lemmas #18172

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Feb 18, 2024
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[stdlib] several NoDup lemmas #18172

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827a2e9
add NoDup_app, InjectiveOn_map_NoDup, ForallOrdPairs_NoDup, NoDup_concat
Oct 16, 2023
3d1c825
InjectiveOn_map_NoDup -> ForallPairs_inj_map_NoDup
Oct 19, 2023
a9dad4d
lemma names as suggested by Andres Erbsen, changelog
Feb 17, 2024
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4 changes: 4 additions & 0 deletions doc/changelog/11-standard-library/18172-NoDup_lemmas.rst
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Original file line number Diff line number Diff line change
@@ -0,0 +1,4 @@
- **Added:**
lemmas :g:`NoDup_app`, :g:`NoDup_iff_ForallOrdPairs`, :g:`NoDup_map_NoDup_ForallPairs` and :g:`NoDup_concat`
(`#18172 <https://github.com/coq/coq/pull/18172>`_,
by Stefan Haani and Andrej Dudenhefner).
57 changes: 57 additions & 0 deletions theories/Lists/List.v
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Expand Up @@ -2479,6 +2479,20 @@ Section ReDun.
+ now constructor.
Qed.

Lemma NoDup_app (l1 l2 : list A):
NoDup l1 -> NoDup l2 -> (forall a, In a l1 -> ~ In a l2) ->
NoDup (l1 ++ l2).
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Proof.
intros H1 H2 H. induction l1 as [|a l1 IHl1]; [assumption|].
apply NoDup_cons_iff in H1 as [].
cbn. constructor.
- intros H3%in_app_or. destruct H3.
+ contradiction.
+ apply (H a); [apply in_eq|assumption].
- apply IHl1; [assumption|].
intros. apply H, in_cons. assumption.
Qed.

Lemma NoDup_app_remove_l l l' : NoDup (l++l') -> NoDup l'.
Proof.
induction l as [|a l IHl]; intro H.
Expand Down Expand Up @@ -3310,6 +3324,49 @@ Section ForallPairs.
Qed.
End ForallPairs.

Lemma NoDup_iff_ForallOrdPairs [A] (l: list A):
NoDup l <-> ForallOrdPairs (fun a b => a <> b) l.
Proof.
split; intro H.
- induction H; constructor.
+ apply Forall_forall.
intros y Hy ->. contradiction.
+ assumption.
- induction H as [|a l H1 H2]; constructor.
+ rewrite Forall_forall in H1. intro E.
contradiction (H1 a E). reflexivity.
+ assumption.
Qed.

Lemma NoDup_map_NoDup_ForallPairs [A B] (f: A->B) (l: list A) :
ForallPairs (fun x y => f x = f y -> x = y) l -> NoDup l -> NoDup (map f l).
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Proof.
intros Hinj Hl.
induction Hl as [|x ?? _ IH]; cbn; constructor.
- intros [y [??]]%in_map_iff.
destruct (Hinj y x); cbn; auto.
- apply IH.
intros x' y' Hx' Hy'.
now apply Hinj; right.
Qed.

Lemma NoDup_concat [A] (L: list (list A)):
Forall (@NoDup A) L ->
ForallOrdPairs (fun l1 l2 => forall a, In a l1 -> ~ In a l2) L ->
NoDup (concat L).
Proof.
intros H1 H2. induction L as [|l1 L IHL]; [constructor|].
cbn. apply NoDup_app.
- apply Forall_inv in H1. assumption.
- apply IHL.
+ apply Forall_inv_tail in H1. assumption.
+ inversion H2. assumption.
- intros a aInl1 ainL%in_concat. destruct ainL as [l2 [l2inL ainL2]].
inversion H2 as [|l L' H3].
rewrite Forall_forall in H3.
apply (H3 _ l2inL _ aInl1). assumption.
Qed.

Section Repeat.

Variable A : Type.
Expand Down