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