81 lines
1.8 KiB
Plaintext
81 lines
1.8 KiB
Plaintext
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import Mathlib.Algebra.Group.Defs
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import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic.NormNum
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/-! # Common.List.NonEmpty
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A `List` with at least one member.
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-/
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namespace List
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structure NonEmpty (α : Type _) : Type _ where
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head : α
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tail : List α
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instance : Coe (NonEmpty α) (List α) where
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coe (xs : NonEmpty α) := xs.head :: xs.tail
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instance : CoeDep (List α) (cons x xs : List α) (NonEmpty α) where
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coe := { head := x, tail := xs }
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namespace NonEmpty
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/--
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The length of a `List.NonEmpty`.
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-/
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def length (xs : NonEmpty α) : Nat := 1 + xs.tail.length
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/--
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The length of a `List.NonEmpty` is always one plus the length of its tail.
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-/
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theorem length_self_eq_one_add_length_tail (xs : NonEmpty α)
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: xs.length = 1 + xs.tail.length := rfl
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/--
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A proof that an index is within the bounds of the `List.NonEmpty`.
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-/
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abbrev inBounds (xs : NonEmpty α) (i : Nat) : Prop :=
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i < xs.length
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/--
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Retrieves the member of the `List.NonEmpty` at the specified index.
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-/
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def get (xs : NonEmpty α) : (i : Nat) → (h : xs.inBounds i) → α
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| 0, _ => xs.head
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| n + 1, h =>
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have : n < xs.tail.length := by
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unfold inBounds at h
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rw [length_self_eq_one_add_length_tail, add_comm] at h
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norm_num at h
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exact h
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xs.tail[n]
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/--
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Variant of `get` that returns an `Option α` in the case of an invalid index.
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-/
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def get? : NonEmpty α → Nat → Option α
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| xs, 0 => some xs.head
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| xs, n + 1 => xs.tail.get? n
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/--
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Type class instance for allowing direct indexing notation.
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-/
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instance : GetElem (NonEmpty α) Nat α inBounds where
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getElem := get
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/--
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Convert a `List.NonEmpty` into a plain `List`.
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-/
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def toList (xs : NonEmpty α) : List α := xs
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/--
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Retrieve the last member of the `List.NonEmpty`.
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-/
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def last : NonEmpty α → α
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| ⟨head, []⟩ => head
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| ⟨_, cons x xs⟩ => last (cons x xs)
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end NonEmpty
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end List
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