mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: case h1.cons
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
b : α
ss : WSeq α
o : a = b ∨ ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
x✝ : α
s✝ : WSeq α
ej : (cons x✝ s✝).append S.join = cons b ss
m : a ∈ (cons x✝ s✝).append S.join
⊢ a ∈ cons x✝ s✝ ∨ ∃ s ∈ S, a ∈ s

Tactic:
skip

Goal state after:
case h1.cons
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
b : α
ss : WSeq α
o : a = b ∨ ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
x✝ : α
s✝ : WSeq α
ej : (cons x✝ s✝).append S.join = cons b ss
m : a ∈ (cons x✝ s✝).append S.join
⊢ a ∈ cons x✝ s✝ ∨ ∃ s ∈ S, a ∈ s
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: case h1.think
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
b : α
ss : WSeq α
o : a = b ∨ ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
s✝ : WSeq α
ej : s✝.think.append S.join = cons b ss
m : a ∈ s✝.think.append S.join
⊢ a ∈ s✝.think ∨ ∃ s ∈ S, a ∈ s

Tactic:
skip

Goal state after:
case h1.think
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
b : α
ss : WSeq α
o : a = b ∨ ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
s✝ : WSeq α
ej : s✝.think.append S.join = cons b ss
m : a ∈ s✝.think.append S.join
⊢ a ∈ s✝.think ∨ ∃ s ∈ S, a ∈ s
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: case h2.cons
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
ss : WSeq α
IH : ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
x✝ : α
s✝ : WSeq α
ej : (cons x✝ s✝).append S.join = ss.think
m : a ∈ (cons x✝ s✝).append S.join
⊢ a ∈ cons x✝ s✝ ∨ ∃ s ∈ S, a ∈ s

Tactic:
skip

Goal state after:
case h2.cons
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
ss : WSeq α
IH : ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
x✝ : α
s✝ : WSeq α
ej : (cons x✝ s✝).append S.join = ss.think
m : a ∈ (cons x✝ s✝).append S.join
⊢ a ∈ cons x✝ s✝ ∨ ∃ s ∈ S, a ∈ s
Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: case h2.think
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
ss : WSeq α
IH : ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
s✝ : WSeq α
ej : s✝.think.append S.join = ss.think
m : a ∈ s✝.think.append S.join
⊢ a ∈ s✝.think ∨ ∃ s ∈ S, a ∈ s

Tactic:
skip

Goal state after:
case h2.think
α : Type u
a : α
ss✝ : WSeq α
h : a ∈ ss✝
ss : WSeq α
IH : ∀ (s : WSeq α) (S : WSeq (WSeq α)), s.append S.join = ss → a ∈ s.append S.join → a ∈ s ∨ ∃ s ∈ S, a ∈ s
S : WSeq (WSeq α)
s✝ : WSeq α
ej : s✝.think.append S.join = ss.think
m : a ∈ s✝.think.append S.join
⊢ a ∈ s✝.think ∨ ∃ s ∈ S, a ∈ s
Module: Mathlib.Data.WSeq.Relation

Goal state before:
pp: α : Type u
SS : WSeq (WSeq (WSeq α))
s1 s2 : WSeq α
h✝ : ∃ s S SS, s1 = s.append (S.append SS.join).join ∧ s2 = s.append (S.join.append (map join SS).join)
c1 c2 : Computation (Option (α × WSeq α))
h :
  ∃ s S SS,
    c1 = (s.append (S.append SS.join).join).destruct ∧ c2 = (s.append (S.join.append (map join SS).join)).destruct
⊢ LiftRelAux
    (LiftRelO Eq fun s1 s2 ↦
      ∃ s S SS, s1 = s.append (S.append SS.join).join ∧ s2 = s.append (S.join.append (map join SS).join))
    (fun c1 c2 ↦
      ∃ s S SS,
        c1 = (s.append (S.append SS.join).join).destruct ∧ c2 = (s.append (S.join.append (map join SS).join)).destruct)
    c1.destruct c2.destruct

Tactic:
exact
    match c1, c2, h with
    | _, _, ⟨s, S, SS, rfl, rfl⟩ => by
      clear h
      induction s using WSeq.recOn with
      | nil =>
        induction S using WSeq.recOn with
        | nil =>
          simp only [nil_append, join_nil]
          induction SS using WSeq.recOn with
          | nil => simp
          | cons S SS => refine ⟨nil, S, SS, ?_, ?_⟩ <;> simp
          | think SS => refine ⟨nil, nil, SS, ?_, ?_⟩ <;> simp
        | cons s S => simpa using ⟨s, S, SS, rfl, rfl⟩
        | think S => refine ⟨nil, S, SS, ?_, ?_⟩ <;> simp
      | cons a s => simpa using ⟨s, S, SS, rfl, rfl⟩
      | think s => simpa using ⟨s, S, SS, rfl, rfl⟩

Goal state after:
Module: Mathlib.Data.Seq.Parallel

Goal state before:
pp: α : Type u
S T : WSeq (Computation α)
a : α
h1 : ∀ s ∈ S, s ~> a
H : WSeq.LiftRel Equiv S T
h2 : ∀ t ∈ T, t ~> a := (parallel_congr_lem H).mp h1
a' : α
h : a' ∈ parallel S
⊢ a' ∈ parallel T

Tactic:
have aa := parallel_promises h1 h

Goal state after:
α : Type u
S T : WSeq (Computation α)
a : α
h1 : ∀ s ∈ S, s ~> a
H : WSeq.LiftRel Equiv S T
h2 : ∀ t ∈ T, t ~> a := (parallel_congr_lem H).mp h1
a' : α
h : a' ∈ parallel S
aa : a = a'
⊢ a' ∈ parallel T
Module: Mathlib.Data.Vector.MapLemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
σ₁ : Type u_6
σ₂ : Type u_7
n : ℕ
xs✝ : Vector α n
f₁ : α → σ₁ → σ₁ × β
f₂ : α → σ₂ → σ₂ × β
R : σ₁ → σ₂ → Prop
hR✝ : ∀ {s : σ₁} {q : σ₂} (a : α), R s q → R (f₁ a s).1 (f₂ a q).1 ∧ (f₁ a s).2 = (f₂ a q).2
n✝ : ℕ
xs : Vector α n✝
x : α
ih :
  ∀ {s₁ : σ₁} {s₂ : σ₂},
    R s₁ s₂ → R (mapAccumr f₁ xs s₁).1 (mapAccumr f₂ xs s₂).1 ∧ (mapAccumr f₁ xs s₁).2 = (mapAccumr f₂ xs s₂).2
s₁ : σ₁
s₂ : σ₂
h₀ : R s₁ s₂
hR : R (f₁ x s₁).1 (f₂ x s₂).1
right✝ : (f₁ x s₁).2 = (f₂ x s₂).2
⊢ (mapAccumr f₂ xs (f₂ x s₂).1).2.snoc (f₁ x s₁).2 = (mapAccumr f₂ xs (f₂ x s₂).1).2.snoc (f₂ x s₂).2

Tactic:
congr 1

Goal state after:
Module: Mathlib.Data.Vector.MapLemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
σ₁ : Type u_6
σ₂ : Type u_7
n : ℕ
xs✝ : Vector α n
f₁ : α → β → σ₁ → σ₁ × γ
f₂ : α → β → σ₂ → σ₂ × γ
R : σ₁ → σ₂ → Prop
hR✝ : ∀ {s : σ₁} {q : σ₂} (a : α) (b : β), R s q → R (f₁ a b s).1 (f₂ a b q).1 ∧ (f₁ a b s).2 = (f₂ a b q).2
n✝ : ℕ
xs : Vector α n✝
ys : Vector β n✝
x : α
y : β
ih :
  ∀ {s₁ : σ₁} {s₂ : σ₂},
    R s₁ s₂ →
      R (mapAccumr₂ f₁ xs ys s₁).1 (mapAccumr₂ f₂ xs ys s₂).1 ∧ (mapAccumr₂ f₁ xs ys s₁).2 = (mapAccumr₂ f₂ xs ys s₂).2
s₁ : σ₁
s₂ : σ₂
h₀ : R s₁ s₂
hR : R (f₁ x y s₁).1 (f₂ x y s₂).1
right✝ : (f₁ x y s₁).2 = (f₂ x y s₂).2
⊢ (mapAccumr₂ f₂ xs ys (f₂ x y s₂).1).2.snoc (f₁ x y s₁).2 = (mapAccumr₂ f₂ xs ys (f₂ x y s₂).1).2.snoc (f₂ x y s₂).2

Tactic:
congr 1

Goal state after:
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: f : CircleDeg1Lift
x : ℝ
⊢ f (x + 1) = 1 + f x

Tactic:
map_add_one,

Goal state after:
f : CircleDeg1Lift
x : ℝ
⊢ f x + 1 = 1 + f x
Module: Mathlib.MeasureTheory.Measure.ContinuousPreimage

Goal state before:
pp: case inr
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
hs : NullMeasurableSet s ν
hνs : ν s ≠ ∞
this✝ : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
this :
  ∀ {s : Set Y},
    NullMeasurableSet s ν → ν s ≠ ∞ → ∀ ε > 0, IsOpen[inst✝⁵] s → ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
hso : ¬IsOpen[inst✝⁵] s
H : 0 < ε / 3
U : Set Y
hUo : IsOpen[inst✝⁵] U
hU : ν U < ∞
hmU : NullMeasurableSet U ν
hUs : ν (U ∆ s) ≤ ε / 3
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε

Tactic:
filter_upwards [hf, this hmU hU.ne _ H hUo] with a hfa ha

Goal state after:
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
hs : NullMeasurableSet s ν
hνs : ν s ≠ ∞
this✝ : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
this :
  ∀ {s : Set Y},
    NullMeasurableSet s ν → ν s ≠ ∞ → ∀ ε > 0, IsOpen[inst✝⁵] s → ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
hso : ¬IsOpen[inst✝⁵] s
H : 0 < ε / 3
U : Set Y
hUo : IsOpen[inst✝⁵] U
hU : ν U < ∞
hmU : NullMeasurableSet U ν
hUs : ν (U ∆ s) ≤ ε / 3
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : μ ((⇑(f a) ⁻¹' U) ∆ (⇑g ⁻¹' U)) ≤ ε / 3
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: f : CircleDeg1Lift
x : ℝ
⊢ x + ↑⌊τ f⌋ + 1 ≤ x + τ f + 1

Tactic:
gcongr; apply floor_le

Goal state after:
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: f : CircleDeg1Lift
hf : Continuous ⇑f
a : ℝ
ha : f a ≤ a + τ f
b : ℝ
hb : b + τ f ≤ f b
⊢ Continuous fun x ↦ x + τ f

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: f : CircleDeg1Lift
hf : Continuous ⇑f
m : ℤ
n : ℕ
hn : 0 < n
⊢ τ (f ^ n) = ↑m ↔ ∃ x, (f ^ n) x = x + ↑m

Tactic:
skip

Goal state after:
f : CircleDeg1Lift
hf : Continuous ⇑f
m : ℤ
n : ℕ
hn : 0 < n
⊢ τ (f ^ n) = ↑m ↔ ∃ x, (f ^ n) x = x + ↑m
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: G : Type u_1
inst✝ : Group G
f₁ f₂ : G →* CircleDeg1Lift
h : ∀ (g : G), τ (f₁ g) = τ (f₂ g)
this : ∀ (x : ℝ), BddAbove (range fun g ↦ (f₂ g⁻¹) ((f₁ g) x))
F₁ : G →* ℝ ≃o ℝ := toOrderIso.comp f₁.toHomUnits
F₂ : G →* ℝ ≃o ℝ := toOrderIso.comp f₂.toHomUnits
hF₁ : ∀ (g : G), ⇑(F₁ g) = ⇑(f₁ g)
hF₂ : ∀ (g : G), ⇑(F₂ g) = ⇑(f₂ g)
x : ℝ
⊢ ContinuousAt (fun x ↦ id x + 1) (⨆ g, (f₂ g⁻¹) ((f₁ g) x))

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Ergodic.Action.OfMinimal

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : MeasurableSpace G
inst✝¹ : OpensMeasurableSpace G
μ : Measure G
inst✝ : μ.IsOpenPosMeasure
g : G
hg : Ergodic (fun x ↦ g * x) μ
a : G
h : (range fun x ↦ g ^ x)ᶜ ∈ 𝓝 a
⊢ Continuous[instTopologicalSpaceProd, inst✝⁴] fun x ↦
    match x with
    | (x, y) => a * x / y

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Ergodic.RadonNikodym

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ ν : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : SigmaFinite ν
f : X → X
hfμ : MeasurePreserving f μ μ
hfν : MeasurePreserving f ν ν
s : Set X
hsm : MeasurableSet s
hνs : ν s = 0
hμs : (μ.singularPart ν) sᶜ = 0
⊢ (μ.singularPart ν) s = (μ.singularPart ν) s + 0

Tactic:
add_zero

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ ν : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : SigmaFinite ν
f : X → X
hfμ : MeasurePreserving f μ μ
hfν : MeasurePreserving f ν ν
s : Set X
hsm : MeasurableSet s
hνs : ν s = 0
hμs : (μ.singularPart ν) sᶜ = 0
⊢ (μ.singularPart ν) s = (μ.singularPart ν) s
Module: Mathlib.Dynamics.Ergodic.RadonNikodym

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ✝ ν : Measure X
inst✝¹ : IsFiniteMeasure ν
f : X → X
hfν : MeasurePreserving f ν ν
μ : Measure X
inst✝ : IsFiniteMeasure μ
hfμ : MeasurePreserving f μ μ
hμν : μ ≪ ν
c : ENNReal
s : Set X := {a | μ.rnDeriv ν a < c}
hsm : MeasurableSet s
hν_sdiff : ν (f ⁻¹' s \ s) = ν (s \ f ⁻¹' s)
h₀ : ν (f ⁻¹' s \ s) ≠ 0
⊢ Measurable (μ.rnDeriv ν)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Flow

Goal state before:
pp: τ : Type u_1
α : Type u_2
inst✝² : TopologicalSpace τ
inst✝¹ : TopologicalSpace α
inst✝ : AddZero τ
ϕ : Flow τ α
s : Set α
h : IsInvariant ϕ.toFun s
⊢ Continuous fun x ↦ ϕ.toFun x.1 ↑x.2

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Flow

Goal state before:
pp: τ : Type u_1
α : Type u_2
inst✝² : TopologicalSpace τ
inst✝¹ : TopologicalSpace α
inst✝ : AddMonoid τ
ϕ : Flow τ α
S : AddSubmonoid τ
⊢ Continuous (uncurry fun t x ↦ ϕ.toFun (↑t) x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Dynamics.Flow

Goal state before:
pp: τ : Type u_1
α : Type u_2
inst✝³ : TopologicalSpace τ
inst✝² : TopologicalSpace α
inst✝¹ : SubtractionCommMonoid τ
inst✝ : ContinuousNeg τ
ϕ : Flow τ α
⊢ Continuous (uncurry fun t ↦ ϕ.toFun (-t))

Tactic:
by fun_prop

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
hμν : μ ≪ ν
⊢ (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ

Tactic:
by
  suffices (μ.rnDeriv ν)⁻¹ =ᵐ[μ] (μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)))⁻¹
      ∧ ν.rnDeriv μ =ᵐ[μ] (μ.withDensity (ν.rnDeriv μ)).rnDeriv μ by
    refine (this.1.trans (Filter.EventuallyEq.trans ?_ this.2.symm))
    exact Measure.inv_rnDeriv_aux (absolutelyContinuous_withDensity_rnDeriv hμν)
      (withDensity_absolutelyContinuous _ _)
  constructor
  · filter_upwards [rnDeriv_withDensity_rnDeriv hμν] with x hx
    simp only [Pi.inv_apply, inv_inj]
    exact hx.symm
  · exact (Measure.rnDeriv_withDensity μ (Measure.measurable_rnDeriv ν μ)).symm

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
hμs : μ s ≠ ∞
t : Set α := toMeasurable μ s
ht : t = toMeasurable μ s
ht_m : MeasurableSet t
⊢ μ t ≠ ∞

Tactic:
rwa [ht, measure_toMeasurable s]

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

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
⊢ coverEntropyEntourage T ∅ U = ⊥

Tactic:
simp only [coverEntropyEntourage, coverMincard_empty]

Goal state after:
X : Type u_1
T : X → X
U : SetRel X X
⊢ (expGrowthSup fun n ↦ ↑0) = ⊥
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 ≤ coverMincard T F U n + coverMincard T G U n

Tactic:
rw [hF, top_add]; exact le_top

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

Goal state before:
pp: G : Type u_3
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
inst✝² : SFinite μ
ν₁ ν₂ : Measure G
inst✝¹ : ν₁.HaveLebesgueDecomposition μ
inst✝ : ν₂.HaveLebesgueDecomposition μ
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
⊢ Measurable (ν₁.rnDeriv μ)

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
inst✝² : SFinite μ
ν₁ ν₂ : Measure G
inst✝¹ : ν₁.HaveLebesgueDecomposition μ
inst✝ : ν₂.HaveLebesgueDecomposition μ
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
⊢ Measurable (ν₂.rnDeriv μ)

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
inst✝² : SFinite μ
ν₁ ν₂ : Measure G
inst✝¹ : ν₁.HaveLebesgueDecomposition μ
inst✝ : ν₂.HaveLebesgueDecomposition μ
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
⊢ Measurable (0, ν₁.rnDeriv μ ⋆ₘₗ[μ] ν₂.rnDeriv μ).2

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁸ : Group G
mG : MeasurableSpace G
inst✝⁷ : MeasurableMul₂ G
inst✝⁶ : MeasurableInv G
μ : Measure G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
ν₁ ν₂ : Measure G
inst✝³ : IsFiniteMeasure ν₁
inst✝² : IsFiniteMeasure ν₂
inst✝¹ : ν₁.HaveLebesgueDecomposition μ
inst✝ : ν₂.HaveLebesgueDecomposition μ
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
this : (ν₁ ∗ₘ ν₂).HaveLebesgueDecomposition μ
⊢ AEMeasurable ((ν₁ ∗ₘ ν₂).rnDeriv μ) μ

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁸ : Group G
mG : MeasurableSpace G
inst✝⁷ : MeasurableMul₂ G
inst✝⁶ : MeasurableInv G
μ : Measure G
inst✝⁵ : μ.IsMulLeftInvariant
inst✝⁴ : SFinite μ
ν₁ ν₂ : Measure G
inst✝³ : IsFiniteMeasure ν₁
inst✝² : IsFiniteMeasure ν₂
inst✝¹ : ν₁.HaveLebesgueDecomposition μ
inst✝ : ν₂.HaveLebesgueDecomposition μ
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
this : (ν₁ ∗ₘ ν₂).HaveLebesgueDecomposition μ
⊢ AEMeasurable (ν₁.rnDeriv μ ⋆ₘₗ[μ] ν₂.rnDeriv μ) μ

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
inst✝² : SigmaFinite μ
ν₁ ν₂ : Measure G
inst✝¹ : SigmaFinite ν₁
inst✝ : SigmaFinite ν₂
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
⊢ AEMeasurable ((ν₁ ∗ₘ ν₂).rnDeriv μ) μ

Tactic:
by fun_prop

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

Goal state before:
pp: G : Type u_3
inst✝⁶ : Group G
mG : MeasurableSpace G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ : Measure G
inst✝³ : μ.IsMulLeftInvariant
inst✝² : SigmaFinite μ
ν₁ ν₂ : Measure G
inst✝¹ : SigmaFinite ν₁
inst✝ : SigmaFinite ν₂
hν₁ : ν₁ ≪ μ
hν₂ : ν₂ ≪ μ
⊢ AEMeasurable (ν₁.rnDeriv μ ⋆ₘₗ[μ] ν₂.rnDeriv μ) μ

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : (map i (X ^ n - C a)).Splits
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (map i (X ^ n - C a)).roots
hs : map i (X ^ n - C a) = (Multiset.map (fun x ↦ X - C x) s).prod
hs' : s.card = n
C_mul_C : C (i a⁻¹) * C (i a) = 1
⊢ X ^ n - 1 = C (i a⁻¹) * (C (i a) * X ^ n) - C (i a⁻¹) * C (i a)

Tactic:
←
      mul_assoc,

Goal state after:
F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : (map i (X ^ n - C a)).Splits
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (map i (X ^ n - C a)).roots
hs : map i (X ^ n - C a) = (Multiset.map (fun x ↦ X - C x) s).prod
hs' : s.card = n
C_mul_C : C (i a⁻¹) * C (i a) = 1
⊢ X ^ n - 1 = C (i a⁻¹) * C (i a) * X ^ n - C (i a⁻¹) * C (i a)
Module: Mathlib.ModelTheory.Basic

Goal state before:
pp: L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ≃[L] N
x✝ : N
⊢ (f.comp f.symm) x✝ = (refl L N) x✝

Tactic:
rw [comp_apply, apply_symm_apply, refl_apply]

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
hx : x ∈ solvableByRad F E
n : ℕ
hn : n ≠ 0
hα : Group.IsSolvable (minpoly F (x ^ n)).Gal
p : F[X] := minpoly F (x ^ n)
h : p.comp (X ^ n) = 0
h' : eval ((X ^ n).coeff 0) p = 0 ∧ X ^ n = C ((X ^ n).coeff 0)
⊢ n = (C ?a).natDegree

pp: case a
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
hx : x ∈ solvableByRad F E
n : ℕ
hn : n ≠ 0
hα : Group.IsSolvable (minpoly F (x ^ n)).Gal
p : F[X] := ⋯
h : p.comp (X ^ n) = 0
h' : eval ((X ^ n).coeff 0) p = 0 ∧ X ^ n = C ((X ^ n).coeff 0)
⊢ F

Tactic:
← h'.2,

Goal state after:
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
hx : x ∈ solvableByRad F E
n : ℕ
hn : n ≠ 0
hα : Group.IsSolvable (minpoly F (x ^ n)).Gal
p : F[X] := minpoly F (x ^ n)
h : p.comp (X ^ n) = 0
h' : eval ((X ^ n).coeff 0) p = 0 ∧ X ^ n = C ((X ^ n).coeff 0)
⊢ n = (X ^ n).natDegree
Module: Mathlib.ModelTheory.Algebra.Ring.Basic

Goal state before:
pp: this : Fintype.card ring.Symbols = 5
⊢ ring.card = 5

Tactic:
simp [Language.card, this]

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
α : Type u'
n n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
h : n✝ ≤ n✝
⊢ castLE h (rel R✝ ts✝) = rel R✝ ts✝

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
α : Type u'
k n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ ∀ {m n : ℕ} (km : n✝ ≤ m) (mn : m ≤ n), castLE mn (castLE km (rel R✝ ts✝)) = castLE ⋯ (rel R✝ ts✝)

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
L' : Language
α : Type u'
β : Type v'
γ : Type u_1
L'' : Language
ft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin n)
fr : (n : ℕ) → L.Relations n → L'.Relations n
ft' : (n : ℕ) → L'.Term (β ⊕ Fin n) → L''.Term (γ ⊕ Fin n)
fr' : (n : ℕ) → L'.Relations n → L''.Relations n
n n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ mapTermRel ft' fr' (fun x ↦ id) (mapTermRel ft fr (fun x ↦ id) (rel R✝ ts✝)) =
    mapTermRel (fun x ↦ ft' x ∘ ft x) (fun x ↦ fr' x ∘ fr x) (fun x ↦ id) (rel R✝ ts✝)

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
α : Type u'
n n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ mapTermRel (fun x ↦ id) (fun x ↦ id) (fun x ↦ id) (rel R✝ ts✝) = rel R✝ ts✝

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
α : Type u'
n n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ mapTermRel (fun x t ↦ Term.relabel (Sum.map id (natAdd 0)) t) (fun x ↦ id) (fun x ↦ castLE ⋯) (rel R✝ ts✝) =
    castLE ⋯ (rel R✝ ts✝)

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
α : Type u'
n n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ (LHom.id L).onBoundedFormula (rel R✝ ts✝) = id (rel R✝ ts✝)

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case rel
L : Language
L' : Language
α : Type u'
n : ℕ
L'' : Language
φ : L' →ᴸ L''
ψ : L →ᴸ L'
n✝ l✝ : ℕ
R✝ : L.Relations l✝
ts✝ : Fin l✝ → L.Term (α ⊕ Fin n✝)
⊢ (φ.comp ψ).onBoundedFormula (rel R✝ ts✝) = (φ.onBoundedFormula ∘ ψ.onBoundedFormula) (rel R✝ ts✝)

Tactic:
| rel =>

Goal state after:
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: L : Language
M : Type w
inst✝² : L.Structure M
α : Type u'
β : Type v'
inst✝¹ : L[[α]].Structure M
inst✝ : (L.lhomWithConstants α).IsExpansionOn M
n : ℕ
t : L[[α]].Term (β ⊕ Fin n)
v : β → M
xs : Fin n → M
⊢ realize (Sum.elim (Sum.elim (fun a ↦ ↑(L.con a)) v) xs ∘ ⇑(Equiv.sumAssoc α β (Fin n)).symm) t.constantsToVars =
    realize (Sum.elim (fun a ↦ ↑(L.con a)) (Sum.elim v xs)) t.constantsToVars

Tactic:
congr 1

Goal state after:
case e_v
L : Language
M : Type w
inst✝² : L.Structure M
α : Type u'
β : Type v'
inst✝¹ : L[[α]].Structure M
inst✝ : (L.lhomWithConstants α).IsExpansionOn M
n : ℕ
t : L[[α]].Term (β ⊕ Fin n)
v : β → M
xs : Fin n → M
⊢ Sum.elim (Sum.elim (fun a ↦ ↑(L.con a)) v) xs ∘ ⇑(Equiv.sumAssoc α β (Fin n)).symm =
    Sum.elim (fun a ↦ ↑(L.con a)) (Sum.elim v xs)
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
l : ℕ
φ ψ : L.BoundedFormula α l
v : α → M
xs : Fin l → M
⊢ (φ ⊓ ψ).Realize v xs ↔ φ.Realize v xs ∧ ψ.Realize v xs

Tactic:
by
  simp [Realize, Min.min]

Goal state after:
Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
n : ℕ
l : List (L.BoundedFormula α n)
v : α → M
xs : Fin n → M
⊢ (List.foldr (fun x1 x2 ↦ x1 ⊔ x2) ⊥ l).Realize v xs ↔ ∃ φ ∈ l, φ.Realize v xs

Tactic:
by
  induction l with
  | nil => simp
  | cons φ l ih =>
    simp_rw [List.foldr_cons, realize_sup, ih, List.mem_cons, or_and_right, exists_or,
      exists_eq_left]

Goal state after: