36 lines
1.1 KiB
Plaintext
36 lines
1.1 KiB
Plaintext
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import Common.List.Basic
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import Common.Set.Intervals.Partition
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/-! # Common.Set.Intervals.StepFunction
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Characterization of step functions.
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-/
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namespace Set.Intervals
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/--
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A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
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if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
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constant on each open subinterval of `P`.
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-/
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structure StepFunction (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
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/- A partition of some closed interval `[a, b]`. -/
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partition : Partition α
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/-- A function whose domain is a closed interval `[a, b]`. -/
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function : ∀ x ∈ Icc partition.a partition.b, α
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/-- Ensure the function is constant on each open subinterval of `p`. -/
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const_open_subintervals :
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∀ (hI : I ∈ partition.openSubintervals), ∃ c : α, ∀ (hy : y ∈ I),
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function y (Partition.mem_open_subinterval_mem_closed_interval hI hy) = c
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namespace StepFunction
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/--
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The locus of points between the `x`-axis and the function.
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-/
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def toSet [Preorder α] [@DecidableRel α LT.lt]
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(s : StepFunction α) : Set (α × α) := sorry
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end StepFunction
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end Set.Intervals
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