mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (a : α), (f * fun x ↦ ∫⁻ (y : β), φ (x, y) ∂ν) a ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((fun z ↦ f z.1) * φ) (x, y) ∂ν ∂μ

pp: case hf
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable ((fun z ↦ f z.1) * φ) (μ.prod ν)

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ f z.1) (μ.prod ν)

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun x ↦ ∫⁻ (y : β), φ (x, y) ∂ν) μ

Tactic:
·

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
hf : AEMeasurable f μ
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (a : α), (f * fun x ↦ ∫⁻ (y : β), φ (x, y) ∂ν) a ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((fun z ↦ f z.1) * φ) (x, y) ∂ν ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (x : α), ∫⁻ (y : β), φ (x, y) ∂ν.withDensity g ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((fun z ↦ g z.2) * φ) (x, y) ∂ν ∂μ

pp: case hf
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable ((fun z ↦ g z.2) * φ) (μ.prod ν)

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ g z.2) (μ.prod ν)

Tactic:
·

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
g : β → ℝ≥0∞
hg : AEMeasurable g ν
φ : α × β → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (x : α), ∫⁻ (y : β), φ (x, y) ∂ν.withDensity g ∂μ = ∫⁻ (x : α), ∫⁻ (y : β), ((fun z ↦ g z.2) * φ) (x, y) ∂ν ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ (μ.prod ν).withDensity ((fun z ↦ f z.1) * fun z ↦ g z.2) = (μ.prod ν).withDensity fun z ↦ f z.1 * g z.2

pp: case hf
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ AEMeasurable (fun z ↦ g z.2) (μ.prod ν)

pp: case hg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ AEMeasurable (fun z ↦ f z.1) (μ.prod ν)

Tactic:
·

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
mβ : MeasurableSpace β
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ (μ.prod ν).withDensity ((fun z ↦ f z.1) * fun z ↦ g z.2) = (μ.prod ν).withDensity fun z ↦ f z.1 * g z.2
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: case inl
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
h_zero : m 0 = 0
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : ∀ i ∈ ∅, p (f i)
⊢ m (𝔼 i ∈ ∅, f i) ≤ 𝔼 i ∈ ∅, m (f i)

pp: case inr
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
s : Finset ι
h_zero : m 0 = 0
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : ∀ i ∈ s, p (f i)
hs_nonempty : s.Nonempty
⊢ m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
h_zero : m 0 = 0
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : ∀ i ∈ ∅, p (f i)
⊢ m (𝔼 i ∈ ∅, f i) ≤ 𝔼 i ∈ ∅, m (f i)
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: case inr
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
s : Finset ι
h_zero : m 0 = 0
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : ∀ i ∈ s, p (f i)
hs_nonempty : s.Nonempty
⊢ m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)

Tactic:
·

Goal state after:
case inr
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
s : Finset ι
h_zero : m 0 = 0
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : ∀ i ∈ s, p (f i)
hs_nonempty : s.Nonempty
⊢ m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (y : G), ∫⁻ (x : G), ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (x, x⁻¹ * y) ∂μ ∂μ =
    ∫⁻ (a : G), ((f ⋆ₘₗ[μ] g) * φ) a ∂μ

pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (Function.uncurry fun a x ↦ ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (a, a⁻¹ * x)) (μ.prod μ)

pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (μ.prod μ)

pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (f ⋆ₘₗ[μ] g) μ

pp: case hg
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable φ μ

pp: case hf
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ f z.1 * g z.2) (μ.prod μ)

pp: case hg
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ AEMeasurable (fun z ↦ φ (z.1 * z.2)) (μ.prod μ)

Tactic:
·

Goal state after:
G : Type u_3
inst✝⁴ : Group G
mG : MeasurableSpace G
inst✝³ : MeasurableMul₂ G
inst✝² : MeasurableInv G
μ : Measure G
inst✝¹ : SFinite μ
inst✝ : μ.IsMulLeftInvariant
f g : G → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
φ : G → ℝ≥0∞
hφ : Measurable φ
⊢ ∫⁻ (y : G), ∫⁻ (x : G), ((fun z ↦ f z.1 * g z.2) * fun z ↦ φ (z.1 * z.2)) (x, x⁻¹ * y) ∂μ ∂μ =
    ∫⁻ (a : G), ((f ⋆ₘₗ[μ] g) * φ) a ∂μ
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖

pp: case neg
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ¬‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖

Tactic:
·

Goal state after:
case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case neg
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ¬‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖

Tactic:
·

Goal state after:
case neg
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ¬‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_1
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ |C| + 1 > 0

pp: case refine_2
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ f.SurjectiveOnWith K (|C| + 1)

Tactic:
·

Goal state after:
case refine_1
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ |C| + 1 > 0
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_2
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ f.SurjectiveOnWith K (|C| + 1)

Tactic:
·

Goal state after:
case refine_2
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ f.SurjectiveOnWith K (|C| + 1)
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
x : V₁
C : ℝ
_Cpos : 0 < C
hC : ‖f x‖ ≤ C * ‖x‖
h : ‖x‖ = 0
⊢ ‖f x‖ ≤ ‖f‖ * ‖x‖

pp: case neg
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
x : V₁
C : ℝ
_Cpos : 0 < C
hC : ‖f x‖ ≤ C * ‖x‖
h : ¬‖x‖ = 0
⊢ ‖f x‖ ≤ ‖f‖ * ‖x‖

Tactic:
·

Goal state after:
case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
x : V₁
C : ℝ
_Cpos : 0 < C
hC : ‖f x‖ ≤ C * ‖x‖
h : ‖x‖ = 0
⊢ ‖f x‖ ≤ ‖f‖ * ‖x‖
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ star (U * B) * (U * B) = 1

pp: case refine_2
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ U * B * star (U * B) = 1

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ star (U * B) * (U * B) = 1
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : x⋆ = 0
⊢ ‖x⋆‖ ≤ ‖x‖

pp: case inr
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖

Tactic:
·

Goal state after:
case inl
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : x⋆ = 0
⊢ ‖x⋆‖ ≤ ‖x‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖

Tactic:
·

Goal state after:
case inr
E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ U * B * star (U * B) = 1

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ U * B * star (U * B) = 1
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : x⋆ = 0
⊢ ‖x⋆‖ ≤ ‖x‖

pp: case inr
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖

Tactic:
·

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : x⋆ = 0
⊢ ‖x⋆‖ ≤ ‖x‖
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖

Tactic:
·

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CStarRing E
x : E
hx : 0 < ‖x⋆‖
⊢ ‖x⋆‖ ≤ ‖x‖
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U B : R
x✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}
x✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}
hA₁ : star U * U = 1
hA₂ : U * star U = 1
hB₁ : star B * B = 1
hB₂ : B * star B = 1
⊢ U * B ∈ {U | star U * U = 1 ∧ U * star U = 1}

Tactic:
refine ⟨?_, ?_⟩
    · calc
        star (U * B) * (U * B) = star B * star U * U * B := by simp only [mul_assoc, star_mul]
        _ = star B * (star U * U) * B := by rw [← mul_assoc]
        _ = 1 := by rw [hA₁, mul_one, hB₁]
    · calc
        U * B * star (U * B) = U * B * (star B * star U) := by rw [star_mul]
        _ = U * (B * star B) * star U := by simp_rw [← mul_assoc]
        _ = 1 := by rw [hB₂, mul_one, hA₂]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_1
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : f.NormNoninc
⊢ ‖f‖ ≤ 1

pp: case refine_2
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : ‖f‖ ≤ 1
v : V
⊢ ‖f v‖ ≤ ‖v‖

Tactic:
·

Goal state after:
case refine_1
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : f.NormNoninc
⊢ ‖f‖ ≤ 1
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: case refine_2
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : ‖f‖ ≤ 1
v : V
⊢ ‖f v‖ ≤ ‖v‖

Tactic:
·

Goal state after:
case refine_2
V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f : NormedAddGroupHom V W
h : ‖f‖ ≤ 1
v : V
⊢ ‖f v‖ ≤ ‖v‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝¹ : SeminormedAddCommGroup V
inst✝ : SeminormedAddCommGroup W
f g : NormedAddGroupHom V W
⊢ f.comp (ι f g) = g.comp (ι f g)

Tactic:
ext x
  rw [comp_apply, comp_apply, ← sub_eq_zero, ← NormedAddGroupHom.sub_apply]
  exact x.2

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: case ofNat
R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
a✝ : ℕ
⊢ ↑(U ^ Int.ofNat a✝) = ↑U ^ Int.ofNat a✝

pp: case negSucc
R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
a✝ : ℕ
⊢ ↑(U ^ Int.negSucc a✝) = ↑U ^ Int.negSucc a✝

Tactic:
·

Goal state after:
case ofNat
R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
a✝ : ℕ
⊢ ↑(U ^ Int.ofNat a✝) = ↑U ^ Int.ofNat a✝
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: case negSucc
R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
a✝ : ℕ
⊢ ↑(U ^ Int.negSucc a✝) = ↑U ^ Int.negSucc a✝

Tactic:
·

Goal state after:
case negSucc
R : Type u_1
inst✝¹ : GroupWithZero R
inst✝ : StarMul R
U : ↥(unitary R)
a✝ : ℕ
⊢ ↑(U ^ Int.negSucc a✝) = ↑U ^ Int.negSucc a✝
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
⊢ ∑'[L] (i : β), 0 • f i = 0 • ∑'[L] (i : β), f i

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
g : γ
hg : g ≠ 0
⊢ ∑'[L] (i : β), g • f i = g • ∑'[L] (i : β), f i

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
⊢ ∑'[L] (i : β), 0 • f i = 0 • ∑'[L] (i : β), f i
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
g : γ
hg : g ≠ 0
⊢ ∑'[L] (i : β), g • f i = g • ∑'[L] (i : β), f i

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : AddCommMonoid α
f : β → α
L : SummationFilter β
γ : Type u_5
inst✝³ : DivisionSemiring γ
inst✝² : Module γ α
inst✝¹ : ContinuousConstSMul γ α
inst✝ : T2Space α
g : γ
hg : g ≠ 0
⊢ ∑'[L] (i : β), g • f i = g • ∑'[L] (i : β), f i
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case pos
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
hf : Summable f L
⊢ ∑'[L] (z : ι), e (f z) = y ↔ ∑'[L] (z : ι), f z = e.symm y

pp: case neg
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
hf : ¬Summable f L
⊢ ∑'[L] (z : ι), e (f z) = y ↔ ∑'[L] (z : ι), f z = e.symm y

Tactic:
·

Goal state after:
case pos
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
hf : Summable f L
⊢ ∑'[L] (z : ι), e (f z) = y ↔ ∑'[L] (z : ι), f z = e.symm y