92 lines
2.2 KiB
Plaintext
92 lines
2.2 KiB
Plaintext
import Common.Finset
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import Common.Geometry.Rectangle.Orthogonal
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import Common.List.Basic
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import Common.List.NonEmpty
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/-! # Common.Geometry.StepFunction
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Characterization of step functions.
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-/
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namespace Geometry
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/--
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An interval defines a range of values, characterized by a "left" value and a
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"right" value. We require these values to be distinct; we do not support the
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notion of an empty interval.
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-/
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structure Interval (α : Type _) [LT α] where
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left : α
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right : α
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h : left < right
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namespace Interval
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/--
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Computes the size of the interval.
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-/
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def size [LT α] [Sub α] (i : Interval α) : α := i.right - i.left
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/--
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Computes the midpoint of the interval.
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-/
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def midpoint [LT α] [Add α] [HDiv α ℝ α] (i : Interval α) : α :=
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(i.left + i.right) / (2 : ℝ)
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/--
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Convert an `Interval` into a `Set.Ico`.
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-/
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def toIco [Preorder α] (i : Interval α) : Set α := Set.Ico i.left i.right
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/--
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Convert an `Interval` into a `Set.Ioc`.
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-/
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def toIoc [Preorder α] (i : Interval α) : Set α := Set.Ioc i.left i.right
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/--
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Convert an `Interval` into a `Set.Icc`.
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-/
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def toIcc [Preorder α] (i : Interval α) : Set α := Set.Icc i.left i.right
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/--
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Convert an `Interval` into a `Set.Ioo`.
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-/
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def toIoo [Preorder α] (i : Interval α) : Set α := Set.Ioo i.left i.right
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end Interval
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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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Instead of maintaining a function from `[a, b]` to `ℝ`, we instead maintain a
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function that maps each partition index to some constant value.
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-/
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structure StepFunction where
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ivls : List.NonEmpty (Interval ℝ)
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connected : ∀ I ∈ ivls.toList.pairwise (·.right = ·.left), I
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toFun : Fin ivls.length → ℝ
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namespace StepFunction
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/--
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The ordinate set of the `StepFunction`.
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-/
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def toSet (sf : StepFunction) : Set Point :=
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⋃ i ∈ Finset.finRange sf.ivls.length,
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let I := sf.ivls[i]
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Rectangle.Orthogonal.toSet
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⟨
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{
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tl := ⟨I.left, sf.toFun i⟩,
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bl := ⟨I.left, 0⟩,
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br := ⟨I.right, 0⟩,
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has_right_angle := sorry
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},
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by simp
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⟩
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end StepFunction
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end Geometry |