Projection of a line onto a closed interval #

Given a linearly ordered type α, in this file we define

Set.projIcc (a b : α) (h : a ≤ b) to be the map α → [a, b] sending (-∞, a] to a, [b, ∞) to b, and each point x ∈ [a, b] to itself;
Set.IccExtend {a b : α} (h : a ≤ b) (f : Icc a b → β) to be the extension of f to α defined as f ∘ projIcc a b h.

We also prove some trivial properties of these maps.

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def Set.projIcc {α : Type u_1} [inst : LinearOrder α] (a : α) (b : α) (h : a ≤ b) (x : α) :

↑(Set.Icc a b)

Projection of α to the closed interval [a, b].

Equations

Set.projIcc a b h x = { val := max a (min b x), property := (_ : a ≤ max a (min b x) ∧ max a (min b x) ≤ b) }

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theorem Set.projIcc_of_le_left {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (hx : x ≤ a) :

Set.projIcc a b h x = { val := a, property := (_ : a ∈ Set.Icc a b) }

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theorem Set.projIcc_left {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Set.projIcc a b h a = { val := a, property := (_ : a ∈ Set.Icc a b) }

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theorem Set.projIcc_of_right_le {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (hx : b ≤ x) :

Set.projIcc a b h x = { val := b, property := (_ : b ∈ Set.Icc a b) }

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theorem Set.projIcc_right {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Set.projIcc a b h b = { val := b, property := (_ : b ∈ Set.Icc a b) }

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theorem Set.projIcc_eq_left {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} {x : α} (h : a < b) :

Set.projIcc a b (_ : a ≤ b) x = { val := a, property := (_ : a ∈ Set.Icc a b) } ↔ x ≤ a

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theorem Set.projIcc_eq_right {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} {x : α} (h : a < b) :

Set.projIcc a b (_ : a ≤ b) x = { val := b, property := (_ : b ∈ Set.Icc a b) } ↔ b ≤ x

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theorem Set.projIcc_of_mem {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (hx : x ∈ Set.Icc a b) :

Set.projIcc a b h x = { val := x, property := hx }

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theorem Set.projIcc_val {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (x : ↑(Set.Icc a b)) :

Set.projIcc a b h ↑x = x

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theorem Set.projIcc_surjOn {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Set.SurjOn (Set.projIcc a b h) (Set.Icc a b) Set.univ

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theorem Set.projIcc_surjective {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Function.Surjective (Set.projIcc a b h)

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theorem Set.range_projIcc {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Set.range (Set.projIcc a b h) = Set.univ

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theorem Set.monotone_projIcc {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

Monotone (Set.projIcc a b h)

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theorem Set.strictMonoOn_projIcc {α : Type u_1} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) :

StrictMonoOn (Set.projIcc a b h) (Set.Icc a b)

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def Set.IccExtend {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) :

α → β

Extend a function [a, b] → β to a map α → β.

Equations

Set.IccExtend h f = f ∘ Set.projIcc a b h

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theorem Set.IccExtend_range {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) :

Set.range (Set.IccExtend h f) = Set.range f

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theorem Set.IccExtend_of_le_left {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : x ≤ a) :

Set.IccExtend h f x = f { val := a, property := (_ : a ∈ Set.Icc a b) }

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theorem Set.IccExtend_left {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) :

Set.IccExtend h f a = f { val := a, property := (_ : a ∈ Set.Icc a b) }

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theorem Set.IccExtend_of_right_le {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : b ≤ x) :

Set.IccExtend h f x = f { val := b, property := (_ : b ∈ Set.Icc a b) }

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theorem Set.IccExtend_right {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) :

Set.IccExtend h f b = f { val := b, property := (_ : b ∈ Set.Icc a b) }

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theorem Set.IccExtend_of_mem {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : x ∈ Set.Icc a b) :

Set.IccExtend h f x = f { val := x, property := hx }

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theorem Set.Icc_extend_coe {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {a : α} {b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) (x : ↑(Set.Icc a b)) :

Set.IccExtend h f ↑x = f x

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theorem Monotone.IccExtend {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] {a : α} {b : α} (h : a ≤ b) {f : ↑(Set.Icc a b) → β} (hf : Monotone f) :

Monotone (Set.IccExtend h f)

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theorem StrictMono.strictMonoOn_IccExtend {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] {a : α} {b : α} (h : a ≤ b) {f : ↑(Set.Icc a b) → β} (hf : StrictMono f) :

StrictMonoOn (Set.IccExtend h f) (Set.Icc a b)