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

Goal state before:
pp: case mp
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n < ⊤
k : ℕ
k_max : ↑k = ↑(sSup (Finset.card '' {s | IsDynNetIn T F U n ↑s}))
this : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})
h_bdda : BddAbove (Finset.card '' {s | IsDynNetIn T F U n ↑s})
h_nemp : (Finset.card '' {s | IsDynNetIn T F U n ↑s}).Nonempty
key : ∃ x ∈ {s | IsDynNetIn T F U n ↑s}, x.card = k
⊢ ∃ s, IsDynNetIn T F U n ↑s ∧ s.card = k

Tactic:
]

Goal state after:
case mp
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n < ⊤
k : ℕ
k_max : ↑k = ↑(sSup (Finset.card '' {s | IsDynNetIn T F U n ↑s}))
this : netMaxcard T F U n = sSup (WithTop.some '' Finset.card '' {s | IsDynNetIn T F U n ↑s})
h_bdda : BddAbove (Finset.card '' {s | IsDynNetIn T F U n ↑s})
h_nemp : (Finset.card '' {s | IsDynNetIn T F U n ↑s}).Nonempty
key : ∃ x ∈ {s | IsDynNetIn T F U n ↑s}, x.card = k
⊢ ∃ s, IsDynNetIn T F U n ↑s ∧ s.card = k
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = coverMincard T F U n
h : ¬Nonempty { s // IsDynCoverOf T F U n ↑s }
⊢ ∀ (i : Finset X), IsDynCoverOf T F U n ↑i → ↑(#i) = ⊤

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = coverMincard T F U n
h : ¬Nonempty { s // IsDynCoverOf T F U n ↑s }
⊢ ∀ (i : Finset X), IsDynCoverOf T F U n ↑i → ↑(#i) = ⊤
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: case mpr
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
s : Finset X
left✝ : IsDynNetIn T F U n ↑s
s_card : ↑s.card = netMaxcard T F U n
⊢ ↑s.card < ⊤

Tactic:
]

Goal state after:
case mpr
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
s : Finset X
left✝ : IsDynNetIn T F U n ↑s
s_card : ↑s.card = netMaxcard T F U n
⊢ ↑s.card < ⊤
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: case mpr
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : ∃ s, IsDynNetIn T F U n ↑s ∧ ↑s.card = netMaxcard T F U n
⊢ netMaxcard T F U n < ⊤

Tactic:
obtain ⟨s, _, s_card⟩ := h
    rw [← s_card]
    exact WithTop.coe_lt_top s.card

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = coverMincard T F U n
h : ∀ (x : Finset X), ¬IsDynCoverOf T F U n ↑x
⊢ ∀ (i : Finset X), ¬IsDynCoverOf T F U n ↑i

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = coverMincard T F U n
h : ∀ (x : Finset X), ¬IsDynCoverOf T F U n ↑x
⊢ ∀ (i : Finset X), ¬IsDynCoverOf T F U n ↑i
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
n : ℕ
⊢ ∀ (i : Finset X), IsDynNetIn T ∅ U n ↑i → ↑i.card = ⊥

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
U : SetRel X X
n : ℕ
⊢ ∀ (i : Finset X), IsDynNetIn T ∅ U n ↑i → ↑i.card = ⊥
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
n : ℕ
s : Finset X
s_net : s = ∅
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
  (μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') =ᵐ[ν + ν']
    (μ.singularPart ν').rnDeriv (ν + ν') + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν')
h₂ :
  (μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν =ᵐ[ν]
    (μ.singularPart ν').rnDeriv ν + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν
h₃ : (μ.singularPart ν').rnDeriv (ν + ν') =ᵐ[ν] (μ.singularPart ν').rnDeriv ν
h₄ : (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') =ᵐ[ν] 0
h₅ : (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν =ᵐ[ν] 0
x : α
hx₃ : (μ.singularPart ν').rnDeriv (ν + ν') x = (μ.singularPart ν').rnDeriv ν x
hx₄ : (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') x = 0 x
hx₅ : (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν x = 0 x
⊢ (μ.singularPart ν').rnDeriv ν x + 0 x = (μ.singularPart ν').rnDeriv ν x + 0 x

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : F = ∅
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: X : Type u_1
Y : Type u_2
s F : Set X
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
n : ℕ
h : Semiconj φ S T
h' : IsDynCoverOf S F (map φ φ ⁻¹' V) n s
x : X
hx : x ∈ F
y : X
hy : y ∈ s
hxy : (x, y) ∈ dynEntourage S (map φ φ ⁻¹' V) n
⊢ (x, y) ∈ dynEntourage S (map φ φ ⁻¹' V) n

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
s F : Set X
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
n : ℕ
h : Semiconj φ S T
h' : IsDynCoverOf S F (map φ φ ⁻¹' V) n s
x : X
hx : x ∈ F
y : X
hy : y ∈ s
hxy : (x, y) ∈ dynEntourage S (map φ φ ⁻¹' V) n
⊢ (x, y) ∈ dynEntourage S (map φ φ ⁻¹' V) n
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
F : Set X
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
n : ℕ
h : Semiconj φ S T
inst✝ : V.IsSymm
t : Finset Y
h' : IsDynCoverOf T (φ '' F) V n ↑t
h✝¹ : Nonempty X
s : Finset Y
s_cover : IsDynCoverOf T (φ '' F) V n ↑s
s_card : s.card ≤ t.card
s_inter : ∀ x ∈ s, (ball x (dynEntourage T V n) ∩ φ '' F).Nonempty
g : Y → Y
g_rel : ∀ x ∈ s, g x ∈ ball x (dynEntourage T V n)
g_mem : ∀ x ∈ s, g x ∈ φ '' F
f : Y → X
h✝ : ∀ y ∈ φ '' F, f y ∈ F
φ_f : ∀ y ∈ φ '' F, φ (f y) = y
x : X
hx : x ∈ F
y : Y
hy : y ∈ ↑s
hxy : (φ x, y) ∈ dynEntourage T V n
⊢ (y, g y) ∈ dynEntourage T V n

Tactic:
]

Goal state after:
case inr
X : Type u_1
Y : Type u_2
F : Set X
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
n : ℕ
h : Semiconj φ S T
inst✝ : V.IsSymm
t : Finset Y
h' : IsDynCoverOf T (φ '' F) V n ↑t
h✝¹ : Nonempty X
s : Finset Y
s_cover : IsDynCoverOf T (φ '' F) V n ↑s
s_card : s.card ≤ t.card
s_inter : ∀ x ∈ s, (ball x (dynEntourage T V n) ∩ φ '' F).Nonempty
g : Y → Y
g_rel : ∀ x ∈ s, g x ∈ ball x (dynEntourage T V n)
g_mem : ∀ x ∈ s, g x ∈ φ '' F
f : Y → X
h✝ : ∀ y ∈ φ '' F, f y ∈ F
φ_f : ∀ y ∈ φ '' F, φ (f y) = y
x : X
hx : x ∈ F
y : Y
hy : y ∈ ↑s
hxy : (φ x, y) ∈ dynEntourage T V n
⊢ (y, g y) ∈ dynEntourage T V n
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
⊢ ∀ (x : X), x ∉ F

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
⊢ ∀ (x : X), x ∉ F
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = ⨅ x, ↑(#↑x)
this : Nonempty { s // IsDynCoverOf T F U n ↑s }
key : ⨅ s, ↑(#↑s) ∈ range fun s ↦ ↑(#↑s)
⊢ ∃ s, IsDynCoverOf T F U n ↑s ∧ #s = k

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = ⨅ x, ↑(#↑x)
this : Nonempty { s // IsDynCoverOf T F U n ↑s }
key : ⨅ s, ↑(#↑s) ∈ range fun s ↦ ↑(#↑s)
⊢ ∃ s, IsDynCoverOf T F U n ↑s ∧ #s = k
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
x : X
x_F : x ∈ F
key : IsDynNetIn T F U n ↑{x}
⊢ False

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
x : X
x_F : x ∈ F
key : IsDynNetIn T F U n ↑{x}
⊢ False
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
x : X
x_F : x ∈ F
key : 1 ≤ 0
⊢ False

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : netMaxcard T F U n = 0
x : X
x_F : x ∈ F
key : 1 ≤ 0
⊢ False
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
⊢ netMaxcard T F U n ≠ 0 ↔ F ≠ ∅

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
⊢ netMaxcard T F U n ≠ 0 ↔ F ≠ ∅
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
inst✝ : V.IsSymm
n : ℕ
h' : coverMincard T (φ '' F) V n < ⊤
t : Finset Y
t_cover : IsDynCoverOf T (φ '' F) V n ↑t
t_card : ↑t.card = coverMincard T (φ '' F) V n
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' (V ○ V)) n ↑s
s_card : s.card ≤ t.card
⊢ coverMincard S F (map φ φ ⁻¹' (V ○ V)) n ≤ ↑t.card

Tactic:
]

Goal state after:
case inr
X : Type u_1
Y : Type u_2
V : SetRel Y Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
inst✝ : V.IsSymm
n : ℕ
h' : coverMincard T (φ '' F) V n < ⊤
t : Finset Y
t_cover : IsDynCoverOf T (φ '' F) V n ↑t
t_card : ↑t.card = coverMincard T (φ '' F) V n
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' (V ○ V)) n ↑s
s_card : s.card ≤ t.card
⊢ coverMincard S F (map φ φ ⁻¹' (V ○ V)) n ≤ ↑t.card
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
hμν : μ ≪ ν
⊢ μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν

Tactic:
conv_rhs => rw [ν.haveLebesgueDecomposition_add μ, add_comm]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
⊢ μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv (μ.withDensity (ν.rnDeriv μ) + ν.singularPart μ)
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = ⨅ x, ↑(#↑x)
this : Nonempty { s // IsDynCoverOf T F U n ↑s }
key : ∃ a, ∃ (b : IsDynCoverOf T F U n ↑a), ↑(#↑⟨a, b⟩) = ↑k
⊢ ∃ s, IsDynCoverOf T F U n ↑s ∧ #s = k

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h_fin : coverMincard T F U n < ⊤
k : ℕ
k_min : ↑k = ⨅ x, ↑(#↑x)
this : Nonempty { s // IsDynCoverOf T F U n ↑s }
key : ∃ a, ∃ (b : IsDynCoverOf T F U n ↑a), ↑(#↑⟨a, b⟩) = ↑k
⊢ ∃ s, IsDynCoverOf T F U n ↑s ∧ #s = k
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝² : μ.HaveLebesgueDecomposition ν
inst✝¹ : ν.HaveLebesgueDecomposition μ
inst✝ : SigmaFinite μ
hμν : μ ≪ ν
hνμ : ν ≪ μ
this : μ.withDensity (μ.rnDeriv ν)⁻¹ = μ.withDensity (ν.rnDeriv μ)
⊢ (μ.withDensity (ν.rnDeriv μ)).rnDeriv μ = (μ.withDensity (ν.rnDeriv μ)).rnDeriv μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝² : μ.HaveLebesgueDecomposition ν
inst✝¹ : ν.HaveLebesgueDecomposition μ
inst✝ : SigmaFinite μ
hμν : μ ≪ ν
hνμ : ν ≪ μ
⊢ (ν.withDensity (μ.rnDeriv ν)).withDensity ((ν.withDensity (μ.rnDeriv ν)).rnDeriv ν)⁻¹ = ν

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝² : μ.HaveLebesgueDecomposition ν
inst✝¹ : ν.HaveLebesgueDecomposition μ
inst✝ : SigmaFinite μ
hμν : μ ≪ ν
hνμ : ν ≪ μ
⊢ (ν.withDensity (μ.rnDeriv ν)).withDensity ((ν.withDensity (μ.rnDeriv ν)).rnDeriv ν)⁻¹ = ν
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
n : ℕ
⊢ coverMincard T ∅ U n ≤ 0

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
U : SetRel X X
n : ℕ
⊢ coverMincard T ∅ U n ≤ 0
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝² : μ.HaveLebesgueDecomposition ν
inst✝¹ : ν.HaveLebesgueDecomposition μ
inst✝ : SigmaFinite μ
hμν : μ ≪ ν
hνμ : ν ≪ μ
⊢ ν = ν

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
V : SetRel Y Y
n : ℕ
h' : coverMincard S F (map φ φ ⁻¹' V) n < ⊤
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' V) n ↑s
s_card : ↑s.card = coverMincard S F (map φ φ ⁻¹' V) n
⊢ coverMincard T (φ '' F) V n ≤ ↑s.card

Tactic:
]

Goal state after:
case inr
X : Type u_1
Y : Type u_2
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
V : SetRel Y Y
n : ℕ
h' : coverMincard S F (map φ φ ⁻¹' V) n < ⊤
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' V) n ↑s
s_card : ↑s.card = coverMincard S F (map φ φ ⁻¹' V) n
⊢ coverMincard T (φ '' F) V n ≤ ↑s.card
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
⊢ coverMincard T F U n = 0 ↔ F = ∅

Tactic:
]

Goal state after:
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
⊢ coverMincard T F U n = 0 ↔ F = ∅
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
F : Set X
T : X → X
U : SetRel X X
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F U 0 ↑{x}
⊢ coverMincard T F U 0 ≤ 1

Tactic:
]

Goal state after:
X : Type u_1
F : Set X
T : X → X
U : SetRel X X
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F U 0 ↑{x}
⊢ coverMincard T F U 0 ≤ 1
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: case inr
X : Type u_1
Y : Type u_2
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
V : SetRel Y Y
n : ℕ
h' : coverMincard S F (map φ φ ⁻¹' V) n < ⊤
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' V) n ↑s
s_card : ↑s.card = coverMincard S F (map φ φ ⁻¹' V) n
this : IsDynCoverOf T (φ '' F) V n ↑(Finset.image φ s)
⊢ coverMincard T (φ '' F) V n ≤ ↑s.card

Tactic:
]

Goal state after:
case inr
X : Type u_1
Y : Type u_2
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
V : SetRel Y Y
n : ℕ
h' : coverMincard S F (map φ φ ⁻¹' V) n < ⊤
s : Finset X
s_cover : IsDynCoverOf S F (map φ φ ⁻¹' V) n ↑s
s_card : ↑s.card = coverMincard S F (map φ φ ⁻¹' V) n
this : IsDynCoverOf T (φ '' F) V n ↑(Finset.image φ s)
⊢ coverMincard T (φ '' F) V n ≤ ↑s.card
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
F : Set X
T : X → X
U : SetRel X X
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F U 0 ↑{x}
⊢ 1 = 1

Tactic:
]

Goal state after:
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
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
⊢ (∀ᵐ (x : α) ∂ν.singularPart μ, μ.rnDeriv (ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)) x ≠ 0 → p x) ∧
    ∀ᵐ (x : α) ∂μ.withDensity (ν.rnDeriv μ), μ.rnDeriv (ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)) x ≠ 0 → p x

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
⊢ (∀ᵐ (x : α) ∂ν.singularPart μ, μ.rnDeriv (ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)) x ≠ 0 → p x) ∧
    ∀ᵐ (x : α) ∂μ.withDensity (ν.rnDeriv μ), μ.rnDeriv (ν.singularPart μ + μ.withDensity (ν.rnDeriv μ)) x ≠ 0 → p x
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
F : Set X
T : X → X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n ↑{x}
⊢ coverMincard T F univ n ≤ 1

Tactic:
]

Goal state after:
X : Type u_1
F : Set X
T : X → X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n ↑{x}
⊢ coverMincard T F univ n ≤ 1
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: case left
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
⊢ ∀ᵐ (x : α) ∂ν.singularPart μ, μ.rnDeriv ν x ≠ 0 → p x

Tactic:
]

Goal state after:
case left
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
⊢ ∀ᵐ (x : α) ∂ν.singularPart μ, μ.rnDeriv ν x ≠ 0 → p x
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: X : Type u_1
F : Set X
T : X → X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n ↑{x}
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: case mp
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : ∀ b < ⊤, ∃ i, b < ↑(↑i).card
k : ℕ
⊢ ∃ s, IsDynNetIn T F U n ↑s ∧ k ≤ s.card

Tactic:
]

Goal state after:
case mp
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n : ℕ
h : ∀ b < ⊤, ∃ i, b < ↑(↑i).card
k : ℕ
⊢ ∃ s, IsDynNetIn T F U n ↑s ∧ k ≤ s.card
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: case right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
h_ac : μ.withDensity (ν.rnDeriv μ) ≪ μ
⊢ ∀ᵐ (x : α) ∂μ.withDensity (ν.rnDeriv μ), μ.rnDeriv ν x ≠ 0 → p x

Tactic:
]

Goal state after:
case right
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
p : α → Prop
h : ∀ᵐ (a : α) ∂μ, p a
h_ac : μ.withDensity (ν.rnDeriv μ) ≪ μ
⊢ ∀ᵐ (x : α) ∂μ.withDensity (ν.rnDeriv μ), μ.rnDeriv ν x ≠ 0 → p x
Module: Mathlib.Dynamics.TopologicalEntropy.Semiconj

Goal state before:
pp: case a
X : Type u_1
Y : Type u_2
u : UniformSpace Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
this : UniformSpace X := UniformSpace.comap φ u
V : Set (Y × Y)
V_uni : V ∈ 𝓤 Y
⊢ ∃ t ∈ 𝓤 Y, Prod.map φ φ ⁻¹' t ⊆ Prod.map φ φ ⁻¹' SetRel.symmetrize V

Tactic:
]

Goal state after:
case a
X : Type u_1
Y : Type u_2
u : UniformSpace Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
this : UniformSpace X := UniformSpace.comap φ u
V : Set (Y × Y)
V_uni : V ∈ 𝓤 Y
⊢ ∃ t ∈ 𝓤 Y, Prod.map φ φ ⁻¹' t ⊆ Prod.map φ φ ⁻¹' SetRel.symmetrize V
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: case mpr
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n k : ℕ
s : Finset X
s_net : IsDynNetIn T F U n ↑s
s_card : k + 1 ≤ s.card
⊢ k < s.card

Tactic:
]

Goal state after:
case mpr
X : Type u_1
T : X → X
F : Set X
U : SetRel X X
n k : ℕ
s : Finset X
s_net : IsDynNetIn T F U n ↑s
s_card : k + 1 ≤ s.card
⊢ k < s.card
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
s : Set α
hs : MeasurableSet s
⊢ μ s = μ s

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: case inr.inl
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
F_inv : MapsTo T F F
inst✝ : U.IsSymm
m : ℕ
F_nonempty : F.Nonempty
⊢ 1 ≤ 1

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : μ.HaveLebesgueDecomposition ν
inst✝ : SFinite ν
hμν : μ ≪ ν
s : Set α
⊢ μ s = μ s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
⊢ μ Set.univ = μ Set.univ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ (∫⁻ (a : α) in s, μ.rnDeriv ν a ∂ν).toReal = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ ∀ᵐ (x : α) ∂ν.restrict s, μ.rnDeriv ν x < ∞

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ (∫⁻ (a : α) in s, μ.rnDeriv ν a ∂ν).toReal = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ ∀ᵐ (x : α) ∂ν.restrict s, μ.rnDeriv ν x < ∞
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: case inr
X : Type u_1
U : SetRel X X
T : X → X
F : Set X
n : ℕ
h : coverMincard T F U n < ⊤
t : Finset X
t_cover : IsDynCoverOf T F U n ↑t
t_mincard : ↑t.card = coverMincard T F U n
⊢ netMaxcard T F U n ≤ ↑t.card

Tactic:
]

Goal state after:
case inr
X : Type u_1
U : SetRel X X
T : X → X
F : Set X
n : ℕ
h : coverMincard T F U n < ⊤
t : Finset X
t_cover : IsDynCoverOf T F U n ↑t
t_mincard : ↑t.card = coverMincard T F U n
⊢ netMaxcard T F U n ≤ ↑t.card
Module: Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ (ν.withDensity (μ.rnDeriv ν)) s = (ν.withDensity (μ.rnDeriv ν)) s ∨
    (ν.withDensity (μ.rnDeriv ν)) s = 0 ∧ (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∨
      (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∧ (ν.withDensity (μ.rnDeriv ν)) s = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
⊢ (ν.withDensity (μ.rnDeriv ν)) s = (ν.withDensity (μ.rnDeriv ν)) s ∨
    (ν.withDensity (μ.rnDeriv ν)) s = 0 ∧ (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∨
      (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∧ (ν.withDensity (μ.rnDeriv ν)) s = 0
Module: Mathlib.Dynamics.TopologicalEntropy.CoverEntropy

Goal state before:
pp: case inr.inr.inr
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
F_inv : MapsTo T F F
inst✝ : U.IsSymm
m n : ℕ
F_nonempty : F.Nonempty
hn : n ≠ 0
h : coverMincard T F U m < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U m ↑s
s_coverMincard : ↑(#s) = coverMincard T F U m
t : Finset X
t_cover : IsDynCoverOf T F (U ○ U) (m * n) ↑t
t_sn : #t ≤ #s ^ n
⊢ coverMincard T F (U ○ U) (m * n) ≤ ↑(#s) ^ n

Tactic:
]

Goal state after:
case inr.inr.inr
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
F_inv : MapsTo T F F
inst✝ : U.IsSymm
m n : ℕ
F_nonempty : F.Nonempty
hn : n ≠ 0
h : coverMincard T F U m < ⊤
s : Finset X
s_cover : IsDynCoverOf T F U m ↑s
s_coverMincard : ↑(#s) = coverMincard T F U m
t : Finset X
t_cover : IsDynCoverOf T F (U ○ U) (m * n) ↑t
t_sn : #t ≤ #s ^ n
⊢ coverMincard T F (U ○ U) (m * n) ≤ ↑(#s) ^ 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✝ : SFinite ν
s : Set α
⊢ (∫⁻ (a : α) in s, μ.rnDeriv ν a ∂ν).toReal = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
s : Set α
⊢ ∀ᵐ (x : α) ∂ν.restrict s, μ.rnDeriv ν x < ∞

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
s : Set α
⊢ (∫⁻ (a : α) in s, μ.rnDeriv ν a ∂ν).toReal = ((ν.withDensity (μ.rnDeriv ν)) s).toReal

α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
s : Set α
⊢ ∀ᵐ (x : α) ∂ν.restrict s, μ.rnDeriv ν x < ∞
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✝ : SFinite ν
s : Set α
⊢ (ν.withDensity (μ.rnDeriv ν)) s = (ν.withDensity (μ.rnDeriv ν)) s ∨
    (ν.withDensity (μ.rnDeriv ν)) s = 0 ∧ (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∨
      (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∧ (ν.withDensity (μ.rnDeriv ν)) s = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SFinite ν
s : Set α
⊢ (ν.withDensity (μ.rnDeriv ν)) s = (ν.withDensity (μ.rnDeriv ν)) s ∨
    (ν.withDensity (μ.rnDeriv ν)) s = 0 ∧ (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∨
      (ν.withDensity (μ.rnDeriv ν)) s = ∞ ∧ (ν.withDensity (μ.rnDeriv ν)) s = 0