46 lines
1.2 KiB
Plaintext
46 lines
1.2 KiB
Plaintext
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 ℝ
|
||
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) : ℝ :=
|
||
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) : ℝ :=
|
||
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 ℝ Partition where
|
||
mem (x : ℝ) (p : Partition) := p.a ≤ x ∧ x ≤ p.b
|
||
|
||
end Partition
|
||
|
||
end Real
|