bookshelf/OneVariableCalculus/Real/Set/Partition.lean

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import Bookshelf.List.Basic
import Bookshelf.Real.Set.Interval
namespace Real
/--
A `Partition` is some finite subset of `[a, b]` containing points `a` and `b`.
It is assumed that the points of the `Partition` are distinct and sorted. The
use of a `List` ensures finite-ness.
-/
structure Partition where
xs : List
has_min_length : xs.length ≥ 2
sorted : ∀ x ∈ xs.pairwise (fun x₁ x₂ => x₁ < x₂), x
namespace Partition
lemma length_partition_gt_zero (p : Partition) : p.xs.length > 0 :=
calc p.xs.length
_ ≥ 2 := p.has_min_length
_ > 0 := by simp
/--
The left-most subdivision point of the `Partition`.
-/
def a (p : Partition) : :=
p.xs.head (List.length_gt_zero_imp_not_nil (length_partition_gt_zero p))
/--
The right-most subdivision point of the `Partition`.
-/
def b (p : Partition) : :=
p.xs.getLast (List.length_gt_zero_imp_not_nil (length_partition_gt_zero p))
/--
Define `∈` syntax for a `Partition`. We say a real is a member of a partition
provided it lies somewhere in closed interval `[a, b]`.
-/
instance : Membership Partition where
mem (x : ) (p : Partition) := p.a ≤ x ∧ x ≤ p.b
end Partition
end Real