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

Goal state before:
pp: case h₂
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 ((⇑g ⁻¹' U) ∆ (⇑g ⁻¹' s)) ≤ ε / 3

Tactic:
rwa [← preimage_symmDiff, hg.measure_preimage (hmU.symmDiff hs)]

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

Goal state before:
pp: M : Type u_1
inst✝⁹ : TopologicalSpace M
X : Type u_2
inst✝⁸ : TopologicalSpace X
inst✝⁷ : R1Space X
inst✝⁶ : MeasurableSpace X
inst✝⁵ : BorelSpace X
inst✝⁴ : SMul M X
inst✝³ : ContinuousSMul M X
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegular
inst✝ : ErgodicSMul M X μ
s : Set X
hsm : NullMeasurableSet s μ
hd : closure[inst✝⁹] {g | (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s} = univ
this✝¹ : MeasurableSpace M := borel M
this✝ : BorelSpace M
⊢ IsClosed[inst✝⁹] {g | (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s}

Tactic:
let f : C(M × X, X) := ⟨(· • ·).uncurry, continuous_smul⟩

Goal state after:
M : Type u_1
inst✝⁹ : TopologicalSpace M
X : Type u_2
inst✝⁸ : TopologicalSpace X
inst✝⁷ : R1Space X
inst✝⁶ : MeasurableSpace X
inst✝⁵ : BorelSpace X
inst✝⁴ : SMul M X
inst✝³ : ContinuousSMul M X
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegular
inst✝ : ErgodicSMul M X μ
s : Set X
hsm : NullMeasurableSet s μ
hd : closure[inst✝⁹] {g | (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s} = univ
this✝¹ : MeasurableSpace M := borel M
this✝ : BorelSpace M
f : C(M × X, X) := { toFun := uncurry fun x1 x2 ↦ x1 • x2, continuous_toFun := ⋯ }
⊢ IsClosed[inst✝⁹] {g | (fun x ↦ g • x) ⁻¹' s =ᵐ[μ] s}
Module: Mathlib.Topology.Instances.AddCircle.DenseSubgroup

Goal state before:
pp: case inr.mp
a b : ℝ
hb : b ≠ 0
hd : Dense ↑(AddSubgroup.closure {a, b})
r : ℚ
hr : ↑r = a / b
⊢ False

Tactic:
refine not_denseRange_zsmul (a := b / r.den) <| hd.mono ?_

Goal state after:
case inr.mp
a b : ℝ
hb : b ≠ 0
hd : Dense ↑(AddSubgroup.closure {a, b})
r : ℚ
hr : ↑r = a / b
⊢ ↑(AddSubgroup.closure {a, b}) ⊆ range fun x ↦ x • (b / ↑r.den)
Module: Mathlib.Dynamics.Ergodic.Conservative

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hs : NullMeasurableSet s μ
n : ℕ
H : ¬μ {x | x ∈ s ∧ ∀ m ≥ n, f^[m] x ∉ s} = 0
⊢ False

Tactic:
have : NullMeasurableSet (s ∩ { x | ∀ m ≥ n, f^[m] x ∉ s }) μ := by
    simp only [ofPred_forall, ← compl_ofPred]
    exact hs.inter <| .biInter (to_countable _) fun m _ ↦
      (hs.preimage <| hf.toQuasiMeasurePreserving.iterate m).compl

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hs : NullMeasurableSet s μ
n : ℕ
H : ¬μ {x | x ∈ s ∧ ∀ m ≥ n, f^[m] x ∉ s} = 0
this : NullMeasurableSet (s ∩ {x | ∀ m ≥ n, f^[m] x ∉ s}) μ
⊢ False
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'
hμν' : μ ⟂ₘ 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 : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᵐ[ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᵐ[ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0
x : α
⊢ (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x →
    (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x →
      (μ.singularPart ν).rnDeriv ν x = 0 x →
        ((μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')) x =
          ((μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν) x

Tactic:
hx₃

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hμν' : μ ⟂ₘ 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 : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᵐ[ν] (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν
h₄ : (μ.singularPart ν).rnDeriv (ν + ν') =ᵐ[ν] 0
h₅ : (μ.singularPart ν).rnDeriv ν =ᵐ[ν] 0
x : α
hx₃ : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') x = (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x
⊢ (μ.singularPart ν).rnDeriv (ν + ν') x = 0 x →
    (μ.singularPart ν).rnDeriv ν x = 0 x →
      ((μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')) x =
        ((μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν) x
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
⊢ (μ.singularPart ν').rnDeriv (ν + ν') x = (μ.singularPart ν').rnDeriv ν x →
    (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') x = 0 x →
      (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν x = 0 x →
        ((μ.singularPart ν').rnDeriv (ν + ν') + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν')) x =
          ((μ.singularPart ν').rnDeriv ν + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν) x

Tactic:
hx₃

Goal state after:
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
⊢ (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') x = 0 x →
    (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν x = 0 x →
      ((μ.singularPart ν').rnDeriv (ν + ν') + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν')) x =
        ((μ.singularPart ν').rnDeriv ν + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν) x
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
F : Set X
T : X → X
h : F.Nonempty
U : SetRel X X
⊢ 0 ≤ netEntropyInfEntourage T F U

Tactic:
apply Monotone.expGrowthInf_nonneg
  · exact fun _ _ m_n ↦ ENat.toENNReal_mono (netMaxcard_monotone_time T F U m_n)
  · rw [ne_eq, funext_iff.not, not_forall]
    use 0
    rw [netMaxcard_zero T h U, Pi.zero_apply, ENat.toENNReal_one]
    exact one_ne_zero

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
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
⊢ μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ μ.rnDeriv ν x / (μ.rnDeriv ν x + 1)

Tactic:
have : (μ.rnDeriv (μ + ν)) =ᵐ[ν] fun x ↦ 1 - (μ.rnDeriv ν x + 1)⁻¹ := by
    filter_upwards [h_add, h_one_add, rnDeriv_add_self ν μ] with a h4 h5 h6
    rw [h5, Pi.add_apply] at h4
    nth_rw 1 [h4, h6]
    simp

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
⊢ μ.rnDeriv (μ + ν) =ᵐ[ν] fun x ↦ μ.rnDeriv ν x / (μ.rnDeriv ν x + 1)
Module: Mathlib.FieldTheory.IsAlgClosed.Classification

Goal state before:
pp: R : Type u
K : Type v
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsAlgClosed K
ι : Type w
v : ι → K
inst✝ : Nontrivial R
hv : IsTranscendenceBasis R v
hR : #R ≤ ℵ₀
hK : ℵ₀ < #K
this : ℵ₀ ≤ Cardinal.lift.{max u v, w} #ι
⊢ max (max (Cardinal.lift.{max v w, u} #R) (Cardinal.lift.{max u v, w} #ι)) ℵ₀ = Cardinal.lift.{max u v, w} #ι

Tactic:
rw [max_eq_left, max_eq_right]
        · exact le_trans (by simpa using hR) this
        · exact le_max_of_le_right this

Goal state after:
Module: Mathlib.ModelTheory.LanguageMap

Goal state before:
pp: L : Language
L' : Language
M : Type w
inst✝ : L.Structure M
L'' : Language
e'✝ : L' ≃ᴸ L''
e✝ e : L ≃ᴸ L'
e' : L' ≃ᴸ L''
⊢ (e.invLHom.comp e'.invLHom).comp (e'.toLHom.comp e.toLHom) = LHom.id L

Tactic:
rw [LHom.comp_assoc, ← LHom.comp_assoc e'.invLHom, e'.left_inv, LHom.id_comp, e.left_inv]

Goal state after:
Module: Mathlib.ModelTheory.LanguageMap

Goal state before:
pp: case refine_2
L : Language
L' : Language
M : Type w
inst✝² : L.Structure M
α : Type u_1
inst✝¹ : (constantsOn α).Structure M
A B : Set M
h : A ⊆ B
N : Type w'
inst✝ : L.Structure N
f : M ↪[L] N
n : ℕ
R : L[[↑A]].Relations n
x : Fin n → M
⊢ RelMap R (f.toFun ∘ x) ↔ RelMap R x

Tactic:
cases R with

Goal state after:
Module: Mathlib.FieldTheory.AbelRuffini

Goal state before:
pp: case hq.refine_2
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] := ⋯
hp : p.comp (X ^ n) ≠ 0
s : Multiset p.SplittingField
hs :
  map (algebraMap F p.SplittingField) p =
    C (map (algebraMap F p.SplittingField) p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod
⊢ Group.IsSolvable ((Multiset.map (fun x ↦ X - C x) s).prod.comp (X ^ n)).Gal

Tactic:
rw [multiset_prod_comp]

Goal state after:
case hq.refine_2
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)
hp : p.comp (X ^ n) ≠ 0
s : Multiset p.SplittingField
hs :
  map (algebraMap F p.SplittingField) p =
    C (map (algebraMap F p.SplittingField) p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod
⊢ Group.IsSolvable (Multiset.map (fun p_1 ↦ p_1.comp (X ^ n)) (Multiset.map (fun x ↦ X - C x) s)).prod.Gal
Module: Mathlib.ModelTheory.Ultraproducts

Goal state before:
pp: case refine_1
α : Type u_1
M : α → Type u_2
u : Ultrafilter α
L : Language
inst✝¹ : (a : α) → L.Structure (M a)
inst✝ : ∀ (a : α), Nonempty (M a)
β : Type u_3
n : ℕ
x : β → (a : α) → M a
k : ℕ
φ : L.BoundedFormula β (k + 1)
ih :
  ∀ (v : Fin (k + 1) → (a : α) → M a),
    (φ.Realize (fun i ↦ Quotient.mk' (x i)) fun i ↦ Quotient.mk' (v i)) ↔
      ∀ᶠ (a : α) in ↑u, φ.Realize (fun i ↦ x i a) fun i ↦ v i a
v : Fin k → (a : α) → M a
m : (a : α) → M a
a : α
⊢ Fin.snoc v m (Fin.last k) a = Fin.snoc (fun i ↦ v i a) (m a) (Fin.last k)

Tactic:
simp only [Fin.snoc_last]

Goal state after:
Module: Mathlib.ModelTheory.Encoding

Goal state before:
pp: case refine_1
L : Language
α : Type u'
⊢ ⨆ i, #(L.Term (α ⊕ Fin i)) ≠ 0

Tactic:
rw [← Cardinal.one_le_iff_ne_zero]
      refine _root_.trans ?_ (le_ciSup bddAbove_of_small 1)
      rw [Cardinal.one_le_iff_ne_zero, mk_ne_zero_iff]
      exact ⟨var (Sum.inr 0)⟩

Goal state after:
Module: Mathlib.ModelTheory.ElementaryMaps

Goal state before:
pp: L : Language
M : Type u_1
N : Type u_2
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ↪ₑ[L] N
α : Type u_5
φ : L.Formula α
x : α → M
⊢ BoundedFormula.Realize φ (⇑f ∘ x) default ↔ BoundedFormula.Realize φ (⇑f ∘ x) (⇑f ∘ default)

Tactic:
Unique.eq_default (f ∘ default)

Goal state after:
L : Language
M : Type u_1
N : Type u_2
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ↪ₑ[L] N
α : Type u_5
φ : L.Formula α
x : α → M
⊢ BoundedFormula.Realize φ (⇑f ∘ x) default ↔ BoundedFormula.Realize φ (⇑f ∘ x) default
Module: Mathlib.ModelTheory.ElementaryMaps

Goal state before:
pp: L : Language
M : Type u_1
N : Type u_2
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ↪ₑ[L] N
φ : L.Sentence
⊢ Formula.Realize φ (⇑f ∘ default) ↔ Formula.Realize φ default

Tactic:
Unique.eq_default (f ∘ default)

Goal state after:
L : Language
M : Type u_1
N : Type u_2
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M ↪ₑ[L] N
φ : L.Sentence
⊢ Formula.Realize φ default ↔ Formula.Realize φ default
Module: Mathlib.ModelTheory.Encoding

Goal state before:
pp: L : Language
α : Type u'
l : List ((n : ℕ) × L.BoundedFormula α n)
h :
  ∀ (φ : (n : ℕ) × L.BoundedFormula α n) (l' : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ)),
    listDecode (φ.snd.listEncode ++ l') = φ :: listDecode l'
⊢ listDecode (flatMap (fun φ ↦ φ.snd.listEncode) l) = l

Tactic:
by
    induction l with
    | nil =>
      simp [listDecode]
    | cons φ l ih => rw [flatMap_cons, h φ _, ih]

Goal state after:
Module: Mathlib.ModelTheory.Substructures

Goal state before:
pp: case refine_2
L : Language
M : Type w
inst✝ : L.Structure M
s : Set M
h : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f s
⊢ s ∈ (closure L).closed

Tactic:
have h' : closure L s = ⟨s, h⟩ := closure_eq_of_le (refl _) subset_closure

Goal state after:
case refine_2
L : Language
M : Type w
inst✝ : L.Structure M
s : Set M
h : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f s
h' : (closure L).toFun s = { carrier := s, fun_mem := ⋯ }
⊢ s ∈ (closure L).closed
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: case mp
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
s : Set (α → M)
⊢ (∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}) →
    ∃ A0, ↑A0 ⊆ A ∧ ∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}

Tactic:
rintro ⟨φ, rfl⟩

Goal state after:
case mp
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
φ : L.Formula (↑A ⊕ α)
⊢ ∃ A0, ↑A0 ⊆ A ∧ ∃ φ_1, {v | φ.Realize (Sum.elim Subtype.val v)} = {v | φ_1.Realize (Sum.elim Subtype.val v)}
Module: Mathlib.ModelTheory.Algebra.Field.IsAlgClosed

Goal state before:
pp: p : ℕ
κ : Cardinal.{u_2}
hκ : ℵ₀ < κ
⊢ κ.Categorical (Theory.ACF p)

Tactic:
by
  rintro ⟨M⟩ ⟨N⟩ hM hN
  let _ := fieldOfModelACF p M
  have := modelField_of_modelACF p M
  let _ := compatibleRingOfModelField M
  have := isAlgClosed_of_model_ACF p M
  have := charP_of_model_fieldOfChar p M
  let _ := fieldOfModelACF p N
  have := modelField_of_modelACF p N
  let _ := compatibleRingOfModelField N
  have := isAlgClosed_of_model_ACF p N
  have := charP_of_model_fieldOfChar p N
  constructor
  refine languageEquivEquivRingEquiv.symm ?_
  apply Classical.choice
  refine IsAlgClosed.ringEquiv_of_equiv_of_char_eq p ?_ ?_
  · rw [hM]; exact hκ
  · rw [← Cardinal.eq, hM, hN]

Goal state after:
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
T' : L.Theory
h : ∀ φ ∈ T', T ⊨ᵇ φ
φ : L.Formula α
hφ : T' ⊨ᵇ φ
M : T.ModelType
⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs

Tactic:
have hM : M ⊨ T' := T'.model_iff.2 (fun ψ hψ => (h ψ hψ).realize_sentence M)
  let M' : ModelType T' := ⟨M⟩
  exact hφ M'

Goal state after:
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: case inr
L : Language
T : L.Theory
h : T.IsComplete
φ : L.Sentence
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
hφn : T ⊨ᵇ Formula.not φ
⊢ M ⊨ φ ↔ T ⊨ᵇ φ

Tactic:
exact
      iff_of_false ((Sentence.realize_not M).1 (hφn.realize_sentence M))
        ((h.models_not_iff φ).1 hφn)

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Divisibility

Goal state before:
pp: case refine_2.inl
a : Cardinal.{u}
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c = 0
⊢ a ∣ b ∨ a ∣ c

pp: case refine_2.inr
a : Cardinal.{u}
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
⊢ a ∣ b ∨ a ∣ c

Tactic:
· rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp

Goal state after:
case refine_2.inr
a : Cardinal.{u}
ha : ℵ₀ ≤ a
b c : Cardinal.{u}
hbc : a ∣ b * c
hz : b * c ≠ 0
⊢ a ∣ b ∨ a ∣ c
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁷ : Field M
inst✝⁶ : Algebra K M
inst✝⁵ : Algebra L M
inst✝⁴ : IsScalarTower K L M
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
⊢ (algebraMap S M) s ∈ F.extendRight M

Tactic:
rw [IsScalarTower.algebraMap_apply S F M, IsScalarTower.algebraMap_apply F L M]

Goal state after:
K : Type u_1
L : Type u_2
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝⁷ : Field M
inst✝⁶ : Algebra K M
inst✝⁵ : Algebra L M
inst✝⁴ : IsScalarTower K L M
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra S ↥F
inst✝¹ : Algebra S M
inst✝ : IsScalarTower S (↥F) M
s : S
⊢ (algebraMap L M) ((algebraMap (↥F) L) ((algebraMap S ↥F) s)) ∈ F.extendRight M
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: k✝ : Type u_1
K✝ : Type u_2
inst✝⁶ : Field k✝
inst✝⁵ : Field K✝
inst✝⁴ : Algebra k✝ K✝
k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
g : ↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop
x : K
⊢ x ∈ {x}

Tactic:
simp only [Set.mem_singleton_iff]

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : Set.SurjOn (fun x ↦ ↑(g x)) (↑s) Set.univ
⊢ H.FiniteIndex

Tactic:
have := Set.finite_univ_iff.mp <| Set.Finite.of_surjOn _ hcovers s.finite_toSet

Goal state after:
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : Set.SurjOn (fun x ↦ ↑(g x)) (↑s) Set.univ
this : Finite (G ⧸ H)
⊢ H.FiniteIndex
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
h' : i ≠ j
c : Set G
x : G
hj' : x ∈ (fun x ↦ g x • ↑H) j
hi' : x ∈ (fun x ↦ g x • ↑H) i
hx : x ∈ c
⊢ x ∈ ⊥

Tactic:
rw [mem_leftCoset_iff, SetLike.mem_coe, ← QuotientGroup.eq] at hi' hj'

Goal state after:
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
h' : i ≠ j
c : Set G
x : G
hj' : ↑(g j) = ↑x
hi' : ↑(g i) = ↑x
hx : x ∈ c
⊢ x ∈ ⊥
Module: Mathlib.RingTheory.TensorProduct.Nontrivial

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra R B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
inst✝¹ : IsDomain A
inst✝ : IsDomain B
this : IsDomain R
FR : Type u_1 := FractionRing R
FA : Type u_2 := FractionRing A
FB : Type u_3 := FractionRing B
⊢ Nontrivial (A ⊗[R] B)

Tactic:
let fa : FR →ₐ[R] FA := IsFractionRing.liftAlgHom (g := Algebra.ofId R FA)
    ((IsFractionRing.injective A FA).comp ha)

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra R B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
inst✝¹ : IsDomain A
inst✝ : IsDomain B
this : IsDomain R
FR : Type u_1 := FractionRing R
FA : Type u_2 := FractionRing A
FB : Type u_3 := FractionRing B
fa : FR →ₐ[R] FA := IsFractionRing.liftAlgHom ⋯
⊢ Nontrivial (A ⊗[R] B)
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case pos
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this : Fact (Irreducible (X ^ n - C a))
h : root (X ^ n - C a) = 0
⊢ root (X ^ n - C a) = σ (root (X ^ n - C a))

Tactic:
rw [h, map_zero]

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ (ζ ^ n • v) ^ finrank K L = v ^ finrank K L

Tactic:
smul_pow,

Goal state after:
case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ (ζ ^ n) ^ finrank K L • v ^ finrank K L = v ^ finrank K L
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case inr
G : Type u_1
inst✝ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
h : ∀ i ∈ s, (H i).FiniteIndex → s.card < (H i).index
hs : s.Nonempty
⊢ ∑ i ∈ s, (↑(H i).index)⁻¹ < 1

Tactic:
have hs' : 0 < s.card := hs.card_pos
  have hlt : ∀ i ∈ s, ((H i).index : ℚ)⁻¹ < (s.card : ℚ)⁻¹ := fun i hi ↦ by
    cases eq_or_ne (H i).index 0 with
    | inl hindex =>
      rwa [hindex, Nat.cast_zero, inv_zero, inv_pos, Nat.cast_pos]
    | inr hindex =>
      exact inv_strictAnti₀ (by exact_mod_cast hs') (by exact_mod_cast h i hi ⟨hindex⟩)
  apply (Finset.sum_lt_sum_of_nonempty hs hlt).trans_eq
  rw [Finset.sum_const, nsmul_eq_mul, mul_inv_cancel₀ (Nat.cast_ne_zero.mpr hs'.ne')]

Goal state after: