mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: case inr
X : Type u_1
T : X → X
V : SetRel X X
inst✝ : UniformSpace X
h : Continuous T
F : Set X
U : SetRel X X
V_uni : V ∈ 𝓤 X
n : ℕ
h' : coverMincard T F U n < ⊤
⊢ coverMincard T (closure F) (V ○ U) n ≤ coverMincard T F U n

Tactic:
obtain ⟨s, s_cover, s_coverMincard⟩ := (coverMincard_finite_iff T F U n).1 h'

Goal state after:
case inr
X : Type u_1
T : X → X
V : SetRel X X
inst✝ : UniformSpace X
h : Continuous T
F : Set X
U : SetRel X X
V_uni : V ∈ 𝓤 X
n : ℕ
h' : coverMincard T F U n < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U n ↑s
s_coverMincard : ↑s.card = coverMincard T F U n
⊢ coverMincard T (closure F) (V ○ U) n ≤ coverMincard T F U n
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
a : α
h4 : 1 = μ.rnDeriv (μ + ν) a + ν.rnDeriv (μ + ν) a
h5 : (μ + ν).rnDeriv (μ + ν) a = 1
h6 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
⊢ μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
a : α
h4 : 1 = μ.rnDeriv (μ + ν) a + ν.rnDeriv (μ + ν) a
h5 : (μ + ν).rnDeriv (μ + ν) a = 1
h6 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
⊢ μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
F : Set X
inst✝ : UniformSpace X
F_comp : IsCompact F
F_inv : MapsTo T F F
U_uni : U ∈ 𝓤 X
V : Set (X × X)
V_uni : V ∈ 𝓤 X
V_symm : SetRel.IsSymm V
V_U : V ○ V ⊆ U
s : Finset X
s_cover : IsDynCoverOf T F V 1 ↑s
⊢ ↑(#s) < ↑⊤

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
inst✝ : UniformSpace X
F_comp : IsCompact F
F_inv : MapsTo T F F
U_uni : U ∈ 𝓤 X
V : Set (X × X)
V_uni : V ∈ 𝓤 X
V_symm : SetRel.IsSymm V
V_U : V ○ V ⊆ U
s : Finset X
s_cover : IsDynCoverOf T F V 1 ↑s
⊢ ↑(#s) < ↑⊤
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
a : α
h4 : 1 = μ.rnDeriv (μ + ν) a + ν.rnDeriv (μ + ν) a
h5 : (μ + ν).rnDeriv (μ + ν) a = 1
h6 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
⊢ μ.rnDeriv (μ + ν) a = μ.rnDeriv (μ + ν) a + (μ.rnDeriv ν a + 1)⁻¹ - (μ.rnDeriv ν a + 1)⁻¹

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
a : α
h4 : 1 = μ.rnDeriv (μ + ν) a + ν.rnDeriv (μ + ν) a
h5 : (μ + ν).rnDeriv (μ + ν) a = 1
h6 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
⊢ μ.rnDeriv (μ + ν) a = μ.rnDeriv (μ + ν) a + (μ.rnDeriv ν a + 1)⁻¹ - (μ.rnDeriv ν a + 1)⁻¹
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: case inl
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n = ⊤
⊢ coverMincard T (F ∪ G) U n ≤ ⊤

Tactic:
]

Goal state after:
case inl
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n = ⊤
⊢ coverMincard T (F ∪ G) U n ≤ ⊤
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: case inr.inl
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n = ⊤
⊢ coverMincard T (F ∪ G) U n ≤ ⊤

Tactic:
]

Goal state after:
case inr.inl
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n = ⊤
⊢ coverMincard T (F ∪ G) U n ≤ ⊤
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
this : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹
a : α
ha : μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ 1 - (μ.rnDeriv ν a + 1)⁻¹ = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
this : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹
a : α
ha : μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ 1 - (μ.rnDeriv ν a + 1)⁻¹ = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: case inr.inr
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U n ↑s
s_coverMincard : ↑s.card = coverMincard T F U n
t : Finset X
t_cover : IsDynCoverOf T G U n ↑t
t_coverMincard : ↑t.card = coverMincard T G U n
⊢ coverMincard T (F ∪ G) U n ≤ ↑(s.card + t.card)

Tactic:
]

Goal state after:
case inr.inr
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U n ↑s
s_coverMincard : ↑s.card = coverMincard T F U n
t : Finset X
t_cover : IsDynCoverOf T G U n ↑t
t_coverMincard : ↑t.card = coverMincard T G U n
⊢ coverMincard T (F ∪ G) U n ≤ ↑(s.card + t.card)
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U n ↑s
s_coverMincard : ↑s.card = coverMincard T F U n
t : Finset X
t_cover : IsDynCoverOf T G U n ↑t
t_coverMincard : ↑t.card = coverMincard T G U n
⊢ IsDynCoverOf T (F ∪ G) U n (↑s ∪ ↑t)

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
hF : coverMincard T F U n < ⊤
hG : coverMincard T G U n < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U n ↑s
s_coverMincard : ↑s.card = coverMincard T F U n
t : Finset X
t_cover : IsDynCoverOf T G U n ↑t
t_coverMincard : ↑t.card = coverMincard T G U n
⊢ IsDynCoverOf T (F ∪ G) U n (↑s ∪ ↑t)
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
this : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹
a : α
ha : μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹
ha_lt_top : μ.rnDeriv ν a < ∞
h : 1 = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹ + 1 * (μ.rnDeriv ν a + 1)⁻¹
⊢ 1 = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹ + 1 * (μ.rnDeriv ν a + 1)⁻¹

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
h_add : (μ + ν).rnDeriv (μ + ν) =ᵐ[ν] μ.rnDeriv (μ + ν) + ν.rnDeriv (μ + ν)
h_one_add : ∀ᵐ (x : α) ∂ν, (μ + ν).rnDeriv (μ + ν) x = (fun x ↦ 1) x
this : μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹
a : α
ha : μ.rnDeriv (μ + ν) a = 1 - (μ.rnDeriv ν a + 1)⁻¹
ha_lt_top : μ.rnDeriv ν a < ∞
h : 1 = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹ + 1 * (μ.rnDeriv ν a + 1)⁻¹
⊢ 1 = μ.rnDeriv ν a * (μ.rnDeriv ν a + 1)⁻¹ + 1 * (μ.rnDeriv ν a + 1)⁻¹
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
⊢ ↑(coverMincard T (F ∪ G) U n) ≤ ↑(coverMincard T F U n + coverMincard T G U n)

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
n : ℕ
⊢ ↑(coverMincard T (F ∪ G) U n) ≤ ↑(coverMincard T F U n + coverMincard T G U n)
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
a : α
ha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
ha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ μ.rnDeriv ν a = μ.rnDeriv ν a

pp: case h0
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
a : α
ha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
ha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ μ.rnDeriv ν a + 1 ≠ 0

pp: case ht
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
a : α
ha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
ha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ μ.rnDeriv ν a + 1 ≠ ∞

Tactic:
]

Goal state after:
case h0
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
a : α
ha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
ha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ μ.rnDeriv ν a + 1 ≠ 0

case ht
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
a : α
ha1 : ν.rnDeriv (μ + ν) a = (μ.rnDeriv ν a + 1)⁻¹
ha2 : μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)
ha_lt_top : μ.rnDeriv ν a < ∞
⊢ μ.rnDeriv ν a + 1 ≠ ∞
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: X : Type u_1
ι : Type u_2
inst✝ : UniformSpace X
T : X → X
F : ι → Set X
s : Finset ι
this : coverEntropy T (⋃ x ∈ s, F x) = ⨆ a ∈ s, (coverEntropy T ∘ F) a
⊢ coverEntropy T (⋃ i ∈ s, F i) = ⨆ i ∈ s, coverEntropy T (F i)

Tactic:
]

Goal state after:
X : Type u_1
ι : Type u_2
inst✝ : UniformSpace X
T : X → X
F : ι → Set X
s : Finset ι
this : coverEntropy T (⋃ x ∈ s, F x) = ⨆ a ∈ s, (coverEntropy T ∘ F) a
⊢ coverEntropy T (⋃ i ∈ s, F i) = ⨆ i ∈ s, coverEntropy T (F i)
Module: Mathlib.Dynamics.TopologicalEntropy.Subset

Goal state before:
pp: X : Type u_1
ι : Type u_2
inst✝ : UniformSpace X
T : X → X
F : ι → Set X
s : Finset ι
this : coverEntropy T (⋃ x ∈ s, F x) = ⨆ a ∈ s, (coverEntropy T ∘ F) a
⊢ ⨆ a ∈ s, (coverEntropy T ∘ F) a = ⨆ i ∈ s, coverEntropy T (F i)

Tactic:
]

Goal state after:
X : Type u_1
ι : Type u_2
inst✝ : UniformSpace X
T : X → X
F : ι → Set X
s : Finset ι
this : coverEntropy T (⋃ x ∈ s, F x) = ⨆ a ∈ s, (coverEntropy T ∘ F) a
⊢ ⨆ a ∈ s, (coverEntropy T ∘ F) a = ⨆ i ∈ s, coverEntropy T (F i)
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case h₂
F : Type u_1
inst✝ : Field F
s : Multiset F[X]
hs : ∀ p ∈ s, Group.IsSolvable p.Gal
p : F[X]
t : Multiset F[X]
hps : p ∈ s
a✝ : t ⊆ s
ht : Group.IsSolvable t.prod.Gal
⊢ Group.IsSolvable (p * t.prod).Gal

Tactic:
]

Goal state after:
case h₂
F : Type u_1
inst✝ : Field F
s : Multiset F[X]
hs : ∀ p ∈ s, Group.IsSolvable p.Gal
p : F[X]
t : Multiset F[X]
hps : p ∈ s
a✝ : t ⊆ s
ht : Group.IsSolvable t.prod.Gal
⊢ Group.IsSolvable (p * t.prod).Gal
Module: Mathlib.Topology.Algebra.Group.TopologicalAbelianization

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a b : G
h : (a * b)⁻¹ * (b * a) = ⁅b⁻¹, a⁻¹⁆
⊢ ⁅b⁻¹, a⁻¹⁆ ∈ (commutator G).topologicalClosure

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a b : G
h : (a * b)⁻¹ * (b * a) = ⁅b⁻¹, a⁻¹⁆
⊢ ⁅b⁻¹, a⁻¹⁆ ∈ (commutator G).topologicalClosure
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case pos
F : Type u_1
inst✝ : Field F
n : ℕ
hn : n = 0
⊢ Group.IsSolvable (Gal 0)

Tactic:
]

Goal state after:
case pos
F : Type u_1
inst✝ : Field F
n : ℕ
hn : n = 0
⊢ Group.IsSolvable (Gal 0)
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ξ : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ξ
hμ : μ ≪ ξ
hν : ν ≪ ξ
h1✝ : μ.rnDeriv (μ + ν) * (μ + ν).rnDeriv ξ =ᵐ[ξ] μ.rnDeriv ξ
h2✝ : ν.rnDeriv (μ + ν) * (μ + ν).rnDeriv ξ =ᵐ[ξ] ν.rnDeriv ξ
h_ac : μ + ν ≪ ξ
a : α
h1 : (μ.rnDeriv (μ + ν) * (μ + ν).rnDeriv ξ) a = μ.rnDeriv ξ a
h2 : (ν.rnDeriv (μ + ν) * (μ + ν).rnDeriv ξ) a = ν.rnDeriv ξ a
h_lt_top1 : (μ + ν).rnDeriv ξ a < ∞
h_lt_top2 : ν.rnDeriv (μ + ν) a < ∞
h_pos : 0 < (μ + ν).rnDeriv ξ a
⊢ μ.rnDeriv (μ + ν) a * (ν.rnDeriv (μ + ν) a)⁻¹ = μ.rnDeriv (μ + ν) a * (ν.rnDeriv (μ + ν) a)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
n : ℕ
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - 1 ≠ 0
σ τ : (X ^ n - 1).Gal
a : (X ^ n - 1).SplittingField
ha : a ^ n = 1
key : ∀ (σ : (X ^ n - 1).Gal), ∃ m, σ a = a ^ m
c : ℕ
hc : σ a = a ^ c
d : ℕ
hd : τ a = a ^ d
⊢ (a ^ d) ^ c = (a ^ d) ^ c

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case pos
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : a = 0
⊢ Group.IsSolvable (X ^ n).Gal

Tactic:
]

Goal state after:
case pos
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : a = 0
⊢ Group.IsSolvable (X ^ n).Gal
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
⊢ ∫⁻ (a : β), (map f μ).rnDeriv (map f ν) a ∂map f (ν.restrict s) = μ s

Tactic:
]

Goal state after:
case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
⊢ ∫⁻ (a : β), (map f μ).rnDeriv (map f ν) a ∂map f (ν.restrict s) = μ s
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case pos
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : n = 0
⊢ Group.IsSolvable (C (1 - a)).Gal

Tactic:
]

Goal state after:
case pos
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : n = 0
⊢ Group.IsSolvable (C (1 - a)).Gal
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
hs_eq : s = f ⁻¹' f '' s
this : SigmaFinite (map f ν)
⊢ μ (f ⁻¹' f '' s) = μ (f ⁻¹' f '' s)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
c : (X ^ n - C a).SplittingField
hc : c ^ n = 1
⊢ c ^ n = 1

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
c : (X ^ n - C a).SplittingField
hc : c ^ n = 1
⊢ c ^ n = 1
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
  (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν] fun x ↦
    ((map f (μ.singularPart ν)).rnDeriv (map f ν) + (map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν)) (f x)
⊢ (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
  (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν] fun x ↦
    ((map f (μ.singularPart ν)).rnDeriv (map f ν) + (map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν)) (f x)
⊢ (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν]
    (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝² : SigmaFinite (map f ν)
this✝¹ : SigmaFinite (map f (μ.singularPart ν))
this✝ : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
  (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν] fun x ↦
    ((map f (μ.singularPart ν)).rnDeriv (map f ν) + (map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν)) (f x)
this : ∀ᵐ (x : α) ∂ν, (map f (μ.singularPart ν)).rnDeriv (map f ν) (f x) = 0 (f x)
⊢ (fun x ↦ (map f (μ.singularPart ν)).rnDeriv (map f ν) (f x)) =ᵐ[ν] 0

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝² : SigmaFinite (map f ν)
this✝¹ : SigmaFinite (map f (μ.singularPart ν))
this✝ : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_add :
  (fun x ↦ (map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν) (f x)) =ᵐ[ν] fun x ↦
    ((map f (μ.singularPart ν)).rnDeriv (map f ν) + (map f (ν.withDensity (μ.rnDeriv ν))).rnDeriv (map f ν)) (f x)
this : ∀ᵐ (x : α) ∂ν, (map f (μ.singularPart ν)).rnDeriv (map f ν) (f x) = 0 (f x)
⊢ (fun x ↦ (map f (μ.singularPart ν)).rnDeriv (map f ν) (f x)) =ᵐ[ν] 0
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case neg
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
mem_range : ∀ {c : (X ^ n - C a).SplittingField}, c ^ n = 1 → ∃ d, (algebraMap F (X ^ n - C a).SplittingField) d = c
σ τ : (X ^ n - C a).Gal
b : (X ^ n - C a).SplittingField
hb : b ^ n = (algebraMap F (X ^ n - C a).SplittingField) a
⊢ (σ * τ) b = (τ * σ) b

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
mem_range : ∀ {c : (X ^ n - C a).SplittingField}, c ^ n = 1 → ∃ d, (algebraMap F (X ^ n - C a).SplittingField) d = c
σ τ : (X ^ n - C a).Gal
b : (X ^ n - C a).SplittingField
hb : b ^ n = (algebraMap F (X ^ n - C a).SplittingField) a
⊢ (σ * τ) b = (τ * σ) b
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in f ⁻¹' s, μ.rnDeriv ν a ∂ν = ∫⁻ (x : α) in f ⁻¹' s, (map f μ).rnDeriv (map f ν) (f x) ∂ν

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in f ⁻¹' s, μ.rnDeriv ν a ∂ν = ∫⁻ (x : α) in f ⁻¹' s, (map f μ).rnDeriv (map f ν) (f x) ∂ν
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
mem_range : ∀ {c : (X ^ n - C a).SplittingField}, c ^ n = 1 → ∃ d, (algebraMap F (X ^ n - C a).SplittingField) d = c
σ τ : (X ^ n - C a).Gal
b : (X ^ n - C a).SplittingField
hb : 0 = (algebraMap F (X ^ n - C a).SplittingField) a
hb' : b = 0
⊢ False

Tactic:
]

Goal state after:
F : Type u_1
inst✝ : Field F
n : ℕ
a : F
h : (map (RingHom.id F) (X ^ n - 1)).Splits
ha : ¬a = 0
ha' : (algebraMap F (X ^ n - C a).SplittingField) a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : X ^ n - C a ≠ 0
hn''' : X ^ n - 1 ≠ 0
mem_range : ∀ {c : (X ^ n - C a).SplittingField}, c ^ n = 1 → ∃ d, (algebraMap F (X ^ n - C a).SplittingField) d = c
σ τ : (X ^ n - C a).Gal
b : (X ^ n - C a).SplittingField
hb : 0 = (algebraMap F (X ^ n - C a).SplittingField) a
hb' : b = 0
⊢ False
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν)) =
    map f (μ.singularPart ν) + map f (ν.withDensity (μ.rnDeriv ν))

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_3
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
f : α → ℝ≥0∞
hf : AEMeasurable f ν
⊢ ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂ν.withDensity (μ.rnDeriv ν)

Tactic:
]

Goal state after:
α : Type u_3
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
f : α → ℝ≥0∞
hf : AEMeasurable f ν
⊢ ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂ν.withDensity (μ.rnDeriv ν)