mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber

Goal state before:
pp: f : CircleDeg1Lift
n : ℕ
this : 0 < ↑n + 1
⊢ |((f ^ (n + 1)) 0 - (↑n + 1) * τ f) / (↑n + 1)| ≤ 1 / ↑(n + 1)

Tactic:
abs_div,

Goal state after:
f : CircleDeg1Lift
n : ℕ
this : 0 < ↑n + 1
⊢ |(f ^ (n + 1)) 0 - (↑n + 1) * τ f| / |↑n + 1| ≤ 1 / ↑(n + 1)
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
⊢ ∃ x, f x = x + τ f

Tactic:
exact intermediate_value_univ₂ hf (by fun_prop) ha hb

Goal state after:
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)
x : ℝ
g : G
⊢ τ (f₂ g⁻¹) = -τ (f₂ g)

Tactic:
← MonoidHom.coe_toHomUnits,

Goal state after:
G : Type u_1
inst✝ : Group G
f₁ f₂ : G →* CircleDeg1Lift
h : ∀ (g : G), τ (f₁ g) = τ (f₂ g)
x : ℝ
g : G
⊢ τ ↑(f₂.toHomUnits g⁻¹) = -τ (f₂ g)
Module: Mathlib.MeasureTheory.Measure.ContinuousPreimage

Goal state before:
pp: 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)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ∞
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen[inst✝⁵] s
hνs' : μ (⇑g ⁻¹' s) ≠ ∞
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed[inst✝⁹] K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε

Tactic:
← ENNReal.add_halves ε

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)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ∞
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen[inst✝⁵] s
hνs' : μ (⇑g ⁻¹' s) ≠ ∞
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed[inst✝⁹] K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε / 2 + ε / 2
Module: Mathlib.Topology.Instances.AddCircle.DenseSubgroup

Goal state before:
pp: case inr.mp.refine_2
a b : ℝ
hb : b ≠ 0
hd : Dense ↑(AddSubgroup.closure {a, b})
r : ℚ
hr : ↑r = a / b
⊢ (fun x ↦ x • (b / ↑r.den)) ↑r.den = b

Tactic:
· simp [field]

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℤ
f : AddCircle T → AddCircle T := fun y ↦ n • y + x
e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x (n - 1))
he : MeasurePreserving (⇑e) volume volume
h : n - 1 ≠ 0
⊢ n • DivisibleBy.div x (n - 1) = (n - 1) • DivisibleBy.div x (n - 1) + DivisibleBy.div x (n - 1)

Tactic:
sub_smul,

Goal state after:
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℤ
f : AddCircle T → AddCircle T := fun y ↦ n • y + x
e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x (n - 1))
he : MeasurePreserving (⇑e) volume volume
h : n - 1 ≠ 0
⊢ n • DivisibleBy.div x (n - 1) =
    n • DivisibleBy.div x (n - 1) - 1 • DivisibleBy.div x (n - 1) + DivisibleBy.div x (n - 1)
Module: Mathlib.Dynamics.SymbolicDynamics.Basic

Goal state before:
pp: A : Type u_1
inst✝² : Inhabited A
G : Type u_2
inst✝¹ : Monoid G
inst✝ : IsLeftCancelMul G
p : Pattern A G
v h : G
hmem : h ∈ Finset.image (fun x ↦ v * x) p.support
⊢ A

Tactic:
then

Goal state after:
A : Type u_1
inst✝² : Inhabited A
G : Type u_2
inst✝¹ : Monoid G
inst✝ : IsLeftCancelMul G
p : Pattern A G
v h : G
hmem : h ∈ Finset.image (fun x ↦ v * x) p.support
⊢ A
Module: Mathlib.Dynamics.SymbolicDynamics.Basic

Goal state before:
pp: A : Type u_3
G : Type u_4
inst✝¹ : Inhabited A
inst✝ : Monoid G
F : Set (Pattern A G)
h : G
⊢ MapsTo (mulShift h) (mulForbidden F) (mulForbidden F)

Tactic:
intro x hx p hp g
  specialize hx p hp (h * g)
  contrapose! hx
  simpa [mulOccursInAt_mulShift] using hx

Goal state after:
Module: Mathlib.Dynamics.SymbolicDynamics.Basic

Goal state before:
pp: A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Inhabited A
G : Type u_2
inst✝² : Monoid G
inst✝¹ : IsLeftCancelMul G
inst✝ : DiscreteTopology A
F : Set (Pattern A G)
h_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}
⊢ IsClosed[Pi.topologicalSpace] {x | ∀ p ∈ F, ∀ (g : G), ¬p.mulOccursInAt x g}

Tactic:
rw [h_eq]

Goal state after:
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Inhabited A
G : Type u_2
inst✝² : Monoid G
inst✝¹ : IsLeftCancelMul G
inst✝ : DiscreteTopology A
F : Set (Pattern A G)
h_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}
⊢ IsClosed[Pi.topologicalSpace] (⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v})
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'
⊢ μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν

Tactic:
haveLebesgueDecomposition_add μ ν'

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'
⊢ (μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') =ᵐ[ν]
    (μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν
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 < ⊤ ↔ ∃ s, IsDynNetIn T F U n ↑s ∧ ↑s.card = netMaxcard T F U n

Tactic:
by
  apply Iff.intro <;> intro h
  · obtain ⟨k, k_max⟩ := ENat.ne_top_iff_exists.mp h.ne
    rw [← k_max]
    simp only [Nat.cast_inj]
    -- The criterion we want to use is `Nat.sSup_mem`. We rewrite `netMaxcard` with an `sSup`,
    -- then check its `BddAbove` and `Nonempty` hypotheses.
    have : netMaxcard T F U n
      = sSup (WithTop.some '' Finset.card '' {s : Finset X | IsDynNetIn T F U n s}) := by
      rw [netMaxcard, ← image_comp, sSup_image]
      simp only [mem_ofPred_eq, ENat.some_eq_natCast, Function.comp_apply]
      exact biSup_congr (fun _ _ ↦ rfl)
    rw [this] at k_max
    have h_bdda : BddAbove (Finset.card '' {s : Finset X | IsDynNetIn T F U n s}) := by
      refine ⟨k, mem_upperBounds.2 ?_⟩
      simp only [mem_image, mem_ofPred_eq, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
      intro s h
      rw [← ENat.natCast_le_natCast, k_max]
      apply le_sSup
      exact Filter.frequently_principal.mp fun a ↦ a (by simpa using ⟨_, h, rfl⟩) rfl
    have h_nemp : (Finset.card '' {s : Finset X | IsDynNetIn T F U n s}).Nonempty := by
      refine ⟨0, ?_⟩
      simp only [mem_image, mem_ofPred_eq, Finset.card_eq_zero, exists_eq_right, Finset.coe_empty]
      exact isDynNetIn_empty
    rw [← WithTop.coe_sSup' h_bdda] at k_max
    have key := Nat.sSup_mem h_nemp h_bdda
    rw [← Nat.cast_inj.mp k_max, mem_image] at key
    simp only [mem_ofPred_eq] at key
    exact key
  · obtain ⟨s, _, s_card⟩ := h
    rw [← s_card]
    exact WithTop.coe_lt_top s.card

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
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 < ∞
⊢ μ.rnDeriv (μ + ν) a = μ.rnDeriv ν a / (μ.rnDeriv ν a + 1)

Tactic:
rw [ha, div_eq_mul_inv]

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 refine_2
X : Type u_1
T : X → X
F G : Set X
U : SetRel X X
⊢ max (coverEntropyEntourage T F U) (coverEntropyEntourage T G U) ≤ coverEntropyEntourage T (F ∪ G) U

Tactic:
exact max_le (coverEntropyEntourage_monotone T U subset_union_left)
      (coverEntropyEntourage_monotone T U subset_union_right)

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 - C a).Gal

Tactic:
rw [ha, C_0, sub_zero]
    exact gal_X_pow_isSolvable n

Goal state after:
Module: Mathlib.ModelTheory.Basic

Goal state before:
pp: L : Language
L' : Language
M : Type w
N : Type w'
inst✝³ : L.Structure M
inst✝² : L.Structure N
P : Type u_1
inst✝¹ : L.Structure P
Q : Type u_2
inst✝ : L.Structure Q
hnp : N ↪[L] P
hmn : M ↪[L] N
n✝ : ℕ
f✝ : L.Functions n✝
x✝ : Fin n✝ → M
⊢ funMap f✝ (⇑hnp ∘ ⇑hmn ∘ x✝) = funMap f✝ ((⇑hnp ∘ ⇑hmn) ∘ x✝)

Tactic:
trivial

Goal state after:
Module: Mathlib.ModelTheory.LanguageMap

Goal state before:
pp: L : Language
L' : Language
M : Type w
inst✝ : L.Structure M
α : Type w'
ie : IsEmpty α
⊢ ((LHom.id L).sumElim (LHom.ofIsEmpty (constantsOn α) L)).comp (L.lhomWithConstants α) = LHom.id L

Tactic:
by rw [lhomWithConstants, LHom.sumElim_comp_inl]

Goal state after:
Module: Mathlib.ModelTheory.LanguageMap

Goal state before:
pp: case refine_1.inl
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
x : Fin n → M
g : L.Functions n
⊢ f.toFun (funMap (Sum.inl g) x) = funMap (Sum.inl g) (f.toFun ∘ x)

Tactic:
| inl g =>

Goal state after:
Module: Mathlib.ModelTheory.Basic

Goal state before:
pp: L : Language
L' : Language
M : Type w
N : Type w'
inst✝³ : L.Structure M
inst✝² : L.Structure N
P : Type u_1
inst✝¹ : L.Structure P
Q : Type u_2
inst✝ : L.Structure Q
hnp : N ≃[L] P
hmn : M ≃[L] N
n✝ : ℕ
f✝ : L.Functions n✝
x✝ : Fin n✝ → M
⊢ funMap f✝ (⇑hnp ∘ ⇑hmn ∘ x✝) = funMap f✝ ((⇑hnp ∘ ⇑hmn) ∘ x✝)

Tactic:
trivial

Goal state after:
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case func
L : Language
α : Type u'
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
ih : ∀ (a : Fin l✝), (LHom.id L).onTerm (_ts✝ a) = id (_ts✝ a)
⊢ (LHom.id L).onTerm (func _f✝ _ts✝) = id (func _f✝ _ts✝)

Tactic:
simp_rw [onTerm, ih]

Goal state after:
case func
L : Language
α : Type u'
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
ih : ∀ (a : Fin l✝), (LHom.id L).onTerm (_ts✝ a) = id (_ts✝ a)
⊢ (func ((LHom.id L).onFunction _f✝) fun i ↦ id (_ts✝ i)) = id (func _f✝ _ts✝)
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case func
L : Language
L' : Language
α : Type u'
L'' : Language
φ : L' →ᴸ L''
ψ : L →ᴸ L'
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
ih : ∀ (a : Fin l✝), (φ.comp ψ).onTerm (_ts✝ a) = (φ.onTerm ∘ ψ.onTerm) (_ts✝ a)
⊢ (φ.comp ψ).onTerm (func _f✝ _ts✝) = (φ.onTerm ∘ ψ.onTerm) (func _f✝ _ts✝)

Tactic:
simp_rw [onTerm, ih]

Goal state after:
case func
L : Language
L' : Language
α : Type u'
L'' : Language
φ : L' →ᴸ L''
ψ : L →ᴸ L'
l✝ : ℕ
_f✝ : L.Functions l✝
_ts✝ : Fin l✝ → L.Term α
ih : ∀ (a : Fin l✝), (φ.comp ψ).onTerm (_ts✝ a) = (φ.onTerm ∘ ψ.onTerm) (_ts✝ a)
⊢ (func ((φ.comp ψ).onFunction _f✝) fun i ↦ (φ.onTerm ∘ ψ.onTerm) (_ts✝ i)) = (φ.onTerm ∘ ψ.onTerm) (func _f✝ _ts✝)
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: L : Language
α : Type u'
β : Type v'
n : ℕ
g : α → β ⊕ Fin n
k : ℕ
φ : L.BoundedFormula α k
⊢ relabel g φ.not = (relabel g φ).not

Tactic:
simp [BoundedFormula.not]

Goal state after:
Module: Mathlib.ModelTheory.Ultraproducts

Goal state before:
pp: α : Type u_1
M : α → Type u_2
u : Ultrafilter α
L : Language
inst✝ : (a : α) → L.Structure (M a)
n : ℕ
f : L.Functions n
x : Fin n → (a : α) → M a
⊢ (funMap f fun i ↦ Quotient.mk' (x i)) = Quotient.mk' fun a ↦ funMap f fun i ↦ x i a

Tactic:
apply funMap_quotient_mk'

Goal state after:
Module: Mathlib.ModelTheory.Encoding

Goal state before:
pp: case refine_2.refine_2
L : Language
α : Type u'
⊢ Nonempty (α ⊕ (i : ℕ) × L.Functions i ↪ (n : ℕ) × L.Term (α ⊕ Fin n))

Tactic:
refine ⟨⟨Sum.elim (fun i => ⟨0, var (Sum.inl i)⟩)
        fun F => ⟨1, func F.2 fun _ => var (Sum.inr 0)⟩, ?_⟩⟩

Goal state after:
case refine_2.refine_2
L : Language
α : Type u'
⊢ Function.Injective (Sum.elim (fun i ↦ ⟨0, var (Sum.inl i)⟩) fun F ↦ ⟨1, func F.snd fun x ↦ var (Sum.inr 0)⟩)
Module: Mathlib.ModelTheory.Substructures

Goal state before:
pp: L : Language
M : Type w
inst✝ : L.Structure M
s : Set M
⊢ lift.{max u w, w} #↑↑((closure L).toFun s) ≤ #(L.Term ↑s)

Tactic:
coe_closure_eq_range_term_realize

Goal state after:
L : Language
M : Type w
inst✝ : L.Structure M
s : Set M
⊢ lift.{max u w, w} #↑(range (Term.realize Subtype.val)) ≤ #(L.Term ↑s)
Module: Mathlib.ModelTheory.ElementarySubstructures

Goal state before:
pp: L : Language
M : Type u_1
inst✝ : L.Structure M
A : Set M
hA : MeetsDefinable A
⊢ ↑((closure L).toFun A) ⊆ A

Tactic:
coe_closure_eq_range_term_realize

Goal state after:
L : Language
M : Type u_1
inst✝ : L.Structure M
A : Set M
hA : MeetsDefinable A
⊢ range (Term.realize Subtype.val) ⊆ A
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
ι : Type u_2
f : ι → Set (α → M)
hf : ∀ (i : ι), A.Definable L (f i)
s : Finset ι
⊢ A.Definable L (s.inf f)

Tactic:
exact definable_finset_inf hf s

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.FreeCommRing

Goal state before:
pp: ι : Type u_1
κ : Type u_2
R : Type u_3
inst✝² : DecidableEq κ
inst✝¹ : CommRing R
inst✝ : DecidableEq R
monoms : ι → Finset (κ →₀ ℕ)
f : (i : ι) × ↥(monoms i) ⊕ κ → R
i : ι
⊢ (lift f) (genericPolyMap monoms i) =
    (eval (f ∘ Sum.inr)) (↑((mvPolynomialSupportLEEquiv monoms).symm (f ∘ Sum.inl)) i)

Tactic:
by
  simp only [genericPolyMap, map_sum, map_mul, lift_of, support,
    mvPolynomialSupportLEEquiv, coeff, Finset.sum_filter, MvPolynomial.eval_eq,
    ne_eq, Function.comp, Equiv.coe_fn_symm_mk, Finsupp.coe_mk]
  conv_rhs => rw [← Finset.sum_attach]
  refine Finset.sum_congr rfl ?_
  intro m _
  simp only [Finsupp.prod, map_prod, map_pow, lift_of, Subtype.coe_eta, Finset.coe_mem,
    exists_prop, true_and, dite_eq_ite, ite_true, ite_not]
  split_ifs with h0 <;> simp_all

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

Goal state before:
pp: a : Cardinal.{u}
ha : ℵ₀ ≤ a
⊢ ¬Irreducible a

Tactic:
by
  rw [irreducible_iff, not_and_or]
  refine Or.inr fun h => ?_
  simpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using
    @h a ℵ₀

Goal state after:
Module: Mathlib.FieldTheory.Finite.Extension

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
inst✝ : Finite k
f : k[X]
hi : Irreducible f
n : ℕ
hdvd : f.natDegree ∣ n
a : AdjoinRoot f := AdjoinRoot.root f
this✝ : NeZero f.natDegree
this : Fact (Irreducible f)
⊢ f ∣ X ^ Nat.card k ^ f.natDegree - X

Tactic:
← hi.dvd_iff_aeval_eq_zero (b := a) (by aesop)

Goal state after:
k : Type u_1
inst✝¹ : Field k
inst✝ : Finite k
f : k[X]
hi : Irreducible f
n : ℕ
hdvd : f.natDegree ∣ n
a : AdjoinRoot f := AdjoinRoot.root f
this✝ : NeZero f.natDegree
this : Fact (Irreducible f)
⊢ (aeval a) (X ^ Nat.card k ^ f.natDegree - X) = 0
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
m n : ℕ
x : K
⊢ mk K p (n, x) = mk K p (m + n, (⇑(frobenius K p))^[m] x)

Tactic:
induction m with

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
e f : PerfectClosure K p
x✝¹ : ℕ × K
m : ℕ
x : K
x✝ : ℕ × K
n : ℕ
y : K
⊢ ((m, x).1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] (m, x).2 + (⇑(frobenius K p))^[(m, x).1] (n, y).2) =
    ((n, y).1 + (m, x).1, (⇑(frobenius K p))^[(m, x).1] (n, y).2 + (⇑(frobenius K p))^[(n, y).1] (m, x).2)

Tactic:
by simp only [add_comm]

Goal state after:
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
this : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
pure : IsPurelyInseparable (↥k) D
n : ℕ
m : ↥k
hm : (algebraMap (↥k) D) m = a ^ p ^ n
⊢ ∃ n, a ^ p ^ n ∈ k

Tactic:
have := Subalgebra.range_subset (R := k) ⟨(k).toSubsemiring, fun r ↦ r.2⟩

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
this✝ : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
pure : IsPurelyInseparable (↥k) D
n : ℕ
m : ↥k
hm : (algebraMap (↥k) D) m = a ^ p ^ n
this : Set.range ⇑(algebraMap (↥k) D) ⊆ ↑{ toSubsemiring := k.toSubsemiring, algebraMap_mem' := ⋯ }
⊢ ∃ n, a ^ p ^ n ∈ k
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case refine_3
G : Type u_1
inst✝¹ : 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
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
hdensity : ∑ i ∈ s, (↑(H i).index)⁻¹ = ↑{x | K x = D}.card * (↑D.index)⁻¹
h : {x | K x = D}.card = D.index
i : ι
hi : i ∈ s ∧ (H i).FiniteIndex
j : ι
hj : j ∈ s ∧ (H j).FiniteIndex
hij : i ≠ j
c : Set G
hi' : c ⊆ (fun i ↦ g i • ↑(H i)) i
hj' : c ⊆ (fun i ↦ g i • ↑(H i)) j
x : G
hx : x ∈ c
hdisjoint : (↑{x | K x = D}).PairwiseDisjoint fun x ↦ f x • ↑D
hk' : ∀ (i : ι), i ∈ s ∧ (H i).FiniteIndex → c ⊆ g i • ↑(H i) → ∃ k, ↑k.fst = i ∧ K k = D ∧ x ∈ f k • ↑D
k₁ : κ
hik₁ : ↑k₁.fst = i
hk₁ : K k₁ = D
hxk₁ : x ∈ f k₁ • ↑D
⊢ x ∈ ⊥

Tactic:
have ⟨k₂, hjk₂, hk₂, hxk₂⟩ := hk' j hj hj'

Goal state after:
case refine_3
G : Type u_1
inst✝¹ : 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
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
hdensity : ∑ i ∈ s, (↑(H i).index)⁻¹ = ↑{x | K x = D}.card * (↑D.index)⁻¹
h : {x | K x = D}.card = D.index
i : ι
hi : i ∈ s ∧ (H i).FiniteIndex
j : ι
hj : j ∈ s ∧ (H j).FiniteIndex
hij : i ≠ j
c : Set G
hi' : c ⊆ (fun i ↦ g i • ↑(H i)) i
hj' : c ⊆ (fun i ↦ g i • ↑(H i)) j
x : G
hx : x ∈ c
hdisjoint : (↑{x | K x = D}).PairwiseDisjoint fun x ↦ f x • ↑D
hk' : ∀ (i : ι), i ∈ s ∧ (H i).FiniteIndex → c ⊆ g i • ↑(H i) → ∃ k, ↑k.fst = i ∧ K k = D ∧ x ∈ f k • ↑D
k₁ : κ
hik₁ : ↑k₁.fst = i
hk₁ : K k₁ = D
hxk₁ : x ∈ f k₁ • ↑D
k₂ : κ
hjk₂ : ↑k₂.fst = j
hk₂ : K k₂ = D
hxk₂ : x ∈ f k₂ • ↑D
⊢ x ∈ ⊥
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝²⁰ : Field F
inst✝¹⁹ : Field E
inst✝¹⁸ : Algebra F E
L : Type w
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra F L
inst✝¹⁵ : Algebra L E
inst✝¹⁴ : IsScalarTower F L E
L' : Type u_1
inst✝¹³ : Field L'
inst✝¹² : Algebra F L'
inst✝¹¹ : Algebra L' E
inst✝¹⁰ : IsScalarTower F L' E
H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint (IsScalarTower.toAlgHom F L' E).range
K : Type u_2
inst✝⁹ : Field K
inst✝⁸ : Algebra F K
inst✝⁷ : Algebra L K
inst✝⁶ : IsScalarTower F L K
inst✝⁵ : Algebra L' K
inst✝⁴ : IsScalarTower F L' K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
inst✝¹ : IsScalarTower L E K
inst✝ : IsScalarTower L' E K
this :
  ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L E)).range.LinearDisjoint
    ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L' E)).range
⊢ (IsScalarTower.toAlgHom F L K).fieldRange.LinearDisjoint (IsScalarTower.toAlgHom F L' K).range

Tactic:
rw [AlgHom.fieldRange_toSubalgebra]

Goal state after:
F : Type u
E : Type v
inst✝²⁰ : Field F
inst✝¹⁹ : Field E
inst✝¹⁸ : Algebra F E
L : Type w
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra F L
inst✝¹⁵ : Algebra L E
inst✝¹⁴ : IsScalarTower F L E
L' : Type u_1
inst✝¹³ : Field L'
inst✝¹² : Algebra F L'
inst✝¹¹ : Algebra L' E
inst✝¹⁰ : IsScalarTower F L' E
H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint (IsScalarTower.toAlgHom F L' E).range
K : Type u_2
inst✝⁹ : Field K
inst✝⁸ : Algebra F K
inst✝⁷ : Algebra L K
inst✝⁶ : IsScalarTower F L K
inst✝⁵ : Algebra L' K
inst✝⁴ : IsScalarTower F L' K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
inst✝¹ : IsScalarTower L E K
inst✝ : IsScalarTower L' E K
this :
  ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L E)).range.LinearDisjoint
    ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L' E)).range
⊢ (IsScalarTower.toAlgHom F L K).range.LinearDisjoint (IsScalarTower.toAlgHom F L' K).range
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
A B : IntermediateField F E
inst✝¹ : IsGalois F ↥A
inst✝ : FiniteDimensional F E
h₁ : A ⊔ B = ⊤
h₂ : A ⊓ B = ⊥
⊢ finrank F ↥(A ⊔ B) = finrank F ↥A * finrank F ↥B

Tactic:
rw [h₁, finrank_top', ← Module.finrank_mul_finrank F B E, mul_comm, mul_left_inj'
    Module.finrank_pos.ne']

Goal state after:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
A B : IntermediateField F E
inst✝¹ : IsGalois F ↥A
inst✝ : FiniteDimensional F E
h₁ : A ⊔ B = ⊤
h₂ : A ⊓ B = ⊥
⊢ finrank (↥B) E = finrank F ↥A