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Add hd_error_skipn_iff_In #803

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Dec 8, 2022
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Add hd_error_skipn_iff_In #803

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14 changes: 14 additions & 0 deletions template-coq/theories/utils/MCList.v
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Original file line number Diff line number Diff line change
Expand Up @@ -1454,3 +1454,17 @@ Proof.
case=> [|l' z];
[exact: (snoc_view_snoc nil) | exact: (snoc_view_snoc (a :: l'))].
Qed.

Lemma hd_error_skipn_iff_In {A v ls}
: (exists n, hd_error (@skipn A n ls) = Some v) <-> In v ls.
Proof.
move: ls; elim => //=.
1: setoid_rewrite skipn_nil; cbn; firstorder congruence.
move => ?? <-.
split.
{ move => [[|?]]; rewrite ?skipn_0 ?skipn_S => //=.
1: move => [->].
all: eauto. }
{ move => [->|[n H]]; [ exists 0 | exists (S n) ];
rewrite ?skipn_0 ?skipn_S => //=. }
Qed.