mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: case mp
a b c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b → a = b ∨ c ∉ [[a, b]]

pp: case mpr
a b c : ℝ
⊢ a = b ∨ c ∉ [[a, b]] → IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b

Tactic:
·

Goal state after:
case mp
a b c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b → a = b ∨ c ∉ [[a, b]]
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: case mpr
a b c : ℝ
⊢ a = b ∨ c ∉ [[a, b]] → IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b

Tactic:
·

Goal state after:
case mpr
a b c : ℝ
⊢ a = b ∨ c ∉ [[a, b]] → IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b
Module: Mathlib.Analysis.SpecialFunctions.NonIntegrable

Goal state before:
pp: case mpr.inl
a c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a a

pp: case mpr.inr
a b c : ℝ
h₀ : c ∉ [[a, b]]
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b

Tactic:
·

Goal state after:
case mpr.inl
a c : ℝ
⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a a
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕))

pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)) = ν (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)

Tactic:
·

Goal state after:
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕))
Module: Mathlib.Order.Filter.AtTopBot.Floor

Goal state before:
pp: case calc_1
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ |a * c ^ n| ≤ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n

pp: case calc_2
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n)

Tactic:
·

Goal state after:
case calc_1
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ |a * c ^ n| ≤ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n
Module: Mathlib.Order.Filter.AtTopBot.Floor

Goal state before:
pp: case calc_2
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n)

Tactic:
·

Goal state after:
case calc_2
K : Type u_1
inst✝³ : Ring K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a c : K
d n : ℕ
h : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!
⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∫ (u : ℝ) in Ioo a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioo a b, g u ∂μ

Tactic:
apply setIntegral_mono_on

Goal state after:
case hf
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn f (Ioo a b) μ

case hg
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn g (Ioo a b) μ

case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)

case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case hf
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn f (Ioo a b) μ

pp: case hg
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn g (Ioo a b) μ

pp: case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)

pp: case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x

Tactic:
·

Goal state after:
case hf
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn f (Ioo a b) μ
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ ε < a * c ^ n / ↑(n - d)!

pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε

Tactic:
·

Goal state after:
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε < 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ ε < a * c ^ n / ↑(n - d)!
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case hg
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn g (Ioo a b) μ

pp: case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)

pp: case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x

Tactic:
·

Goal state after:
case hg
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ IntegrableOn g (Ioo a b) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)

pp: case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x

Tactic:
·

Goal state after:
case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case hs
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ MeasurableSet (Ioo a b)

Tactic:
exact measurableSet_Ioo

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε

Tactic:
·

Goal state after:
K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
ε : K
hε : ε > 0
n : ℕ
h : a * c ^ n / ε < ↑(n - d)!
⊢ a * c ^ n / ↑(n - d)! < ε
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x

Tactic:
·

Goal state after:
case h
f g : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
inst✝ : NullSingletonClass μ
h : ∀ x ∈ Ioo a b, f x ≤ g x
⊢ ∀ x ∈ Ioo a b, f x ≤ g x
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 ↦ x * x_1) μ) A = μ ((fun x ↦ x₁ • x) ⁻¹' A)

pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ SMulInvariantMeasure G (G ⧸ Γ) μ

Tactic:
·

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
inst✝⁵ : PolishSpace G
Γ : Subgroup G
inst✝⁴ : Γ.Normal
inst✝³ : T2Space (G ⧸ Γ)
inst✝² : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝¹ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 ↦ x * x_1) μ) A = μ ((fun x ↦ x₁ • x) ⁻¹' A)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case hfi
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ Integrable (fun t ↦ ‖f p t - f q t‖) (μ.restrict (Icc a b))

pp: case hs
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ Ι a₀ b₀ ⊆ Icc a b

Tactic:
·

Goal state after:
case hfi
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ Integrable (fun t ↦ ‖f p t - f q t‖) (μ.restrict (Icc a b))
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, ↑n)

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
t : α
ht : ¬∃ n, t = ↑n
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, t)

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, ↑n)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case hs
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ Ι a₀ b₀ ⊆ Icc a b

Tactic:
·

Goal state after:
case hs
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this✝ : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ Ι a₀ b₀ ⊆ Icc a b
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[<] ↑n) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))

pp: case inl.right
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[≥] ↑n) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))

Tactic:
·

Goal state after:
case inl.left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[<] ↑n) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.left.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 s ×ˢ 𝓝[<] 1 ≤ 𝓝[univ ×ˢ Icc 0 1] (s, 1)

pp: case inl.left.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 (uncurry f (s, 1)) = 𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n))

Tactic:
·

Goal state after:
case inl.left.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 s ×ˢ 𝓝[<] 1 ≤ 𝓝[univ ×ˢ Icc 0 1] (s, 1)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.left.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 (uncurry f (s, 1)) = 𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n))

Tactic:
·

Goal state after:
case inl.left.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 (uncurry f (s, 1)) = 𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n))
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.right
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[≥] ↑n) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))

Tactic:
·

Goal state after:
case inl.right
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ Tendsto (uncurry f ∘ Prod.map id fract) (𝓝 s ×ˢ 𝓝[≥] ↑n) (𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n)))