Documentation

Mathlib.Topology.UnitInterval

The unit interval, as a topological space #

Use open unitInterval to turn on the notation I := Set.Icc (0 : ℝ) (1 : ℝ).

We provide basic instances, as well as a custom tactic for discharging 0 ≤ ↑x, 0 ≤ 1 - ↑x, ↑x ≤ 1, and 1 - ↑x ≤ 1 when x : I.

The unit interval #

@[reducible, inline]

abbrev unitInterval :

The unit interval [0,1] in ℝ.

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def unitInterval.termI :

Lean.ParserDescr

The unit interval [0,1] in ℝ.

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theorem unitInterval.zero_mem :

0 ∈ unitInterval

theorem unitInterval.one_mem :

1 ∈ unitInterval

theorem unitInterval.mul_mem {x y : ℝ} (hx : x ∈ unitInterval) (hy : y ∈ unitInterval) :

x * y ∈ unitInterval

theorem unitInterval.div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) :

x / y ∈ unitInterval

theorem unitInterval.fract_mem (x : ℝ) :

Int.fract x ∈ unitInterval

theorem unitInterval.mem_iff_one_sub_mem {t : ℝ} :

t ∈ unitInterval ↔ 1 - t ∈ unitInterval

theorem unitInterval.univ_eq_Icc :

Set.univ = Set.Icc 0 1

theorem unitInterval.coe_ne_zero {x : ↑unitInterval} :

↑x ≠ 0 ↔ x ≠ 0

theorem unitInterval.coe_ne_one {x : ↑unitInterval} :

↑x ≠ 1 ↔ x ≠ 1

@[simp]

theorem unitInterval.coe_pos {x : ↑unitInterval} :

0 < ↑x ↔ 0 < x

@[simp]

theorem unitInterval.coe_lt_one {x : ↑unitInterval} :

↑x < 1 ↔ x < 1

theorem unitInterval.mul_le_left {x y : ↑unitInterval} :

x * y ≤ x

theorem unitInterval.mul_le_right {x y : ↑unitInterval} :

x * y ≤ y

def unitInterval.symm :

↑unitInterval → ↑unitInterval

Unit interval central symmetry.

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def unitInterval.termσ :

Lean.ParserDescr

Unit interval central symmetry.

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@[simp]

theorem unitInterval.symm_zero :

symm 0 = 1

@[simp]

theorem unitInterval.symm_one :

symm 1 = 0

@[simp]

theorem unitInterval.symm_symm (x : ↑unitInterval) :

symm (symm x) = x

theorem unitInterval.symm_involutive :

Function.Involutive symm

theorem unitInterval.symm_bijective :

Function.Bijective symm

@[simp]

theorem unitInterval.coe_symm_eq (x : ↑unitInterval) :

↑(symm x) = 1 - ↑x

@[simp]

theorem unitInterval.symm_projIcc (x : ℝ) :

symm (Set.projIcc 0 1 ⋯ x) = Set.projIcc 0 1 ⋯ (1 - x)

theorem unitInterval.continuous_symm :

Continuous symm

def unitInterval.symmHomeomorph :

↑unitInterval ≃ₜ ↑unitInterval

unitInterval.symm as a Homeomorph.

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@[simp]

theorem unitInterval.symmHomeomorph_symm_apply (a✝ : ↑unitInterval) :

symmHomeomorph.symm a✝ = symm a✝

@[simp]

theorem unitInterval.symmHomeomorph_apply (a✝ : ↑unitInterval) :

symmHomeomorph a✝ = symm a✝

theorem unitInterval.strictAnti_symm :

StrictAnti symm

@[simp]

theorem unitInterval.symm_inj {i j : ↑unitInterval} :

symm i = symm j ↔ i = j

theorem unitInterval.half_le_symm_iff (t : ↑unitInterval) :

1 / 2 ≤ ↑(symm t) ↔ ↑t ≤ 1 / 2

@[simp]

theorem unitInterval.symm_eq_one {i : ↑unitInterval} :

symm i = 1 ↔ i = 0

@[simp]

theorem unitInterval.symm_eq_zero {i : ↑unitInterval} :

symm i = 0 ↔ i = 1

@[simp]

theorem unitInterval.symm_le_symm {i j : ↑unitInterval} :

symm i ≤ symm j ↔ j ≤ i

theorem unitInterval.le_symm_comm {i j : ↑unitInterval} :

i ≤ symm j ↔ j ≤ symm i

theorem unitInterval.symm_le_comm {i j : ↑unitInterval} :

symm i ≤ j ↔ symm j ≤ i

@[simp]

theorem unitInterval.symm_lt_symm {i j : ↑unitInterval} :

symm i < symm j ↔ j < i

theorem unitInterval.lt_symm_comm {i j : ↑unitInterval} :

i < symm j ↔ j < symm i

theorem unitInterval.symm_lt_comm {i j : ↑unitInterval} :

symm i < j ↔ symm j < i

instance unitInterval.instConnectedSpaceElemReal :

ConnectedSpace ↑unitInterval

theorem unitInterval.nonneg (x : ↑unitInterval) :

0 ≤ ↑x

theorem unitInterval.one_minus_nonneg (x : ↑unitInterval) :

0 ≤ 1 - ↑x

theorem unitInterval.le_one (x : ↑unitInterval) :

↑x ≤ 1

theorem unitInterval.one_minus_le_one (x : ↑unitInterval) :

1 - ↑x ≤ 1

theorem unitInterval.add_pos {t : ↑unitInterval} {x : ℝ} (hx : 0 < x) :

0 < x + ↑t

theorem unitInterval.nonneg' {t : ↑unitInterval} :

0 ≤ t

like unitInterval.nonneg, but with the inequality in I.

theorem unitInterval.le_one' {t : ↑unitInterval} :

t ≤ 1

like unitInterval.le_one, but with the inequality in I.

theorem unitInterval.pos_iff_ne_zero {x : ↑unitInterval} :

0 < x ↔ x ≠ 0

theorem unitInterval.lt_one_iff_ne_one {x : ↑unitInterval} :

x < 1 ↔ x ≠ 1

theorem unitInterval.eq_one_or_eq_zero_of_le_mul {i j : ↑unitInterval} (h : i ≤ j * i) :

i = 0 ∨ j = 1

instance unitInterval.instNontrivialElemReal :

Nontrivial ↑unitInterval

theorem unitInterval.mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) :

a * t ∈ unitInterval ↔ t ∈ Set.Icc 0 (1 / a)

theorem unitInterval.two_mul_sub_one_mem_iff {t : ℝ} :

2 * t - 1 ∈ unitInterval ↔ t ∈ Set.Icc (1 / 2) 1

def unitInterval.submonoid :

The unit interval as a submonoid of ℝ.

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@[simp]

theorem unitInterval.coe_unitIntervalSubmonoid :

↑submonoid = unitInterval

@[simp]

theorem unitInterval.mem_unitIntervalSubmonoid {x : ℝ} :

x ∈ submonoid ↔ x ∈ unitInterval

theorem unitInterval.prod_mem {ι : Type u_1} {t : Finset ι} {f : ι → ℝ} (h : ∀ c ∈ t, f c ∈ unitInterval) :

∏ c ∈ t, f c ∈ unitInterval

instance unitInterval.instLinearOrderedCommMonoidWithZeroElemReal :

LinearOrderedCommMonoidWithZero ↑unitInterval

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theorem Set.abs_projIcc_sub_projIcc {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] {a b c d : α} (h : a ≤ b) :

|↑(projIcc a b h c) - ↑(projIcc a b h d)| ≤ |c - d|

Set.projIcc is a contraction.

def Set.Icc.addNSMul {α : Type u_1} [AddCommGroup α] [LinearOrder α] {a b : α} (h : a ≤ b) (δ : α) (n : ℕ) :

↑(Icc a b)

When h : a ≤ b and δ > 0, addNSMul h δ is a sequence of points in the closed interval [a,b], which is initially equally spaced but eventually stays at the right endpoint b.

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theorem Set.Icc.addNSMul_zero {α : Type u_1} [AddCommGroup α] [LinearOrder α] {a b : α} (h : a ≤ b) {δ : α} :

↑(addNSMul h δ 0) = a

theorem Set.Icc.addNSMul_eq_right {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] {a b : α} (h : a ≤ b) {δ : α} [Archimedean α] (hδ : 0 < δ) :

∃ (m : ℕ), ∀ n ≥ m, ↑(addNSMul h δ n) = b

theorem Set.Icc.monotone_addNSMul {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] {a b : α} (h : a ≤ b) {δ : α} (hδ : 0 ≤ δ) :

Monotone (addNSMul h δ)

theorem Set.Icc.abs_sub_addNSMul_le {α : Type u_1} [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α] {a b : α} (h : a ≤ b) {δ : α} (hδ : 0 ≤ δ) {t : ↑(Icc a b)} (n : ℕ) (ht : t ∈ Icc (addNSMul h δ n) (addNSMul h δ (n + 1))) :

|↑t - ↑(addNSMul h δ n)| ≤ δ

theorem exists_monotone_Icc_subset_open_cover_Icc {ι : Sort u_1} {a b : ℝ} (h : a ≤ b) {c : ι → Set ↑(Set.Icc a b)} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ (i : ι), c i) :

∃ (t : ℕ → ↑(Set.Icc a b)), ↑(t 0) = a ∧ Monotone t ∧ (∃ (m : ℕ), ∀ n ≥ m, ↑(t n) = b) ∧ ∀ (n : ℕ), ∃ (i : ι), Set.Icc (t n) (t (n + 1)) ⊆ c i

Any open cover c of a closed interval [a, b] in ℝ can be refined to a finite partition into subintervals.

theorem exists_monotone_Icc_subset_open_cover_unitInterval {ι : Sort u_1} {c : ι → Set ↑unitInterval} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ (i : ι), c i) :

∃ (t : ℕ → ↑unitInterval), t 0 = 0 ∧ Monotone t ∧ (∃ (n : ℕ), ∀ m ≥ n, t m = 1) ∧ ∀ (n : ℕ), ∃ (i : ι), Set.Icc (t n) (t (n + 1)) ⊆ c i

Any open cover of the unit interval can be refined to a finite partition into subintervals.

theorem exists_monotone_Icc_subset_open_cover_unitInterval_prod_self {ι : Sort u_1} {c : ι → Set (↑unitInterval × ↑unitInterval)} (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : Set.univ ⊆ ⋃ (i : ι), c i) :

∃ (t : ℕ → ↑unitInterval), t 0 = 0 ∧ Monotone t ∧ (∃ (n : ℕ), ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ (i : ι), Set.Icc (t n) (t (n + 1)) ×ˢ Set.Icc (t m) (t (m + 1)) ⊆ c i

@[simp]

theorem projIcc_eq_zero {x : ℝ} :

Set.projIcc 0 1 ⋯ x = 0 ↔ x ≤ 0

@[simp]

theorem projIcc_eq_one {x : ℝ} :

Set.projIcc 0 1 ⋯ x = 1 ↔ 1 ≤ x

def Tactic.Interactive.tacticUnit_interval :

Lean.ParserDescr

A tactic that solves 0 ≤ ↑x, 0 ≤ 1 - ↑x, ↑x ≤ 1, and 1 - ↑x ≤ 1 for x : I.

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theorem affineHomeomorph_image_I {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : 0 < a) :

⇑(affineHomeomorph a b ⋯) '' Set.Icc 0 1 = Set.Icc b (a + b)

The image of [0,1] under the homeomorphism fun x ↦ a * x + b is [b, a+b].

def iccHomeoI {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : a < b) :

↑(Set.Icc a b) ≃ₜ ↑(Set.Icc 0 1)

The affine homeomorphism from a nontrivial interval [a,b] to [0,1].

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@[simp]

theorem iccHomeoI_apply_coe {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : a < b) (x : ↑(Set.Icc a b)) :

↑((iccHomeoI a b h) x) = (↑x - a) / (b - a)

@[simp]

theorem iccHomeoI_symm_apply_coe {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] (a b : 𝕜) (h : a < b) (x : ↑(Set.Icc 0 1)) :

↑((iccHomeoI a b h).symm x) = (b - a) * ↑x + a

def unitInterval.toNNReal :

↑unitInterval → NNReal

The coercion from I to ℝ≥0.

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theorem unitInterval.toNNReal_continuous :

Continuous toNNReal

@[simp]

theorem unitInterval.coe_toNNReal (x : ↑unitInterval) :

↑(toNNReal x) = ↑x