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Tej Chajed
stdpp
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be4eb648
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be4eb648
authored
Nov 26, 2018
by
Robbert Krebbers
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Versions of `elem_of_list_split` that give first or last element.
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theories/list.v
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be4eb648
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@@ -647,6 +647,28 @@ Proof.
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@@ -647,6 +647,28 @@ Proof.
induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
induction 1 as [x l|x y l ? [l1 [l2 ->]]]; [by eexists [], l|].
by exists (y :: l1), l2.
by exists (y :: l1), l2.
Qed.
Qed.
Lemma elem_of_list_split_l `{EqDecision A} l x :
x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2 ∧ x ∉ l1.
Proof.
induction 1 as [x l|x y l ? IH].
{ exists [], l. rewrite elem_of_nil. naive_solver. }
destruct (decide (x = y)) as [->|?].
- exists [], l. rewrite elem_of_nil. naive_solver.
- destruct IH as (l1 & l2 & -> & ?).
exists (y :: l1), l2. rewrite elem_of_cons. naive_solver.
Qed.
Lemma elem_of_list_split_r `{EqDecision A} l x :
x ∈ l → ∃ l1 l2, l = l1 ++ x :: l2 ∧ x ∉ l2.
Proof.
induction l as [|y l IH] using rev_ind.
{ by rewrite elem_of_nil. }
destruct (decide (x = y)) as [->|].
- exists l, []. rewrite elem_of_nil. naive_solver.
- rewrite elem_of_app, elem_of_list_singleton. intros [?| ->]; try done.
destruct IH as (l1 & l2 & -> & ?); auto.
exists l1, (l2 ++ [y]).
rewrite elem_of_app, elem_of_list_singleton, <-(assoc_L (++)). naive_solver.
Qed.
Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x.
Lemma elem_of_list_lookup_1 l x : x ∈ l → ∃ i, l !! i = Some x.
Proof.
Proof.
induction 1 as [|???? IH]; [by exists 0 |].
induction 1 as [|???? IH]; [by exists 0 |].
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