Theorem: MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: induction' k with k ih
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
k : ℕ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ f^[k] ⁻¹' s =ᵐ[μ] s
state_after: case zero
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ f^[0] ⁻¹' s =ᵐ[μ] s

case succ
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
⊢ f^[k + 1] ⁻¹' s =ᵐ[μ] s

tactic: rw [iterate_succ, preimage_comp]
state_before: case succ
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
⊢ f^[k + 1] ⁻¹' s =ᵐ[μ] s
state_after: case succ
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
⊢ f ⁻¹' (f^[k] ⁻¹' s) =ᵐ[μ] s

tactic: exact EventuallyEq.trans (hf.preimage_ae_eq ih) hs
state_before: case succ
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
⊢ f ⁻¹' (f^[k] ⁻¹' s) =ᵐ[μ] s
state_after: no goals

tactic: rfl
state_before: case zero
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
⊢ f^[0] ⁻¹' s =ᵐ[μ] s
state_after: no goals