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

Goal state before:
pp: X : Type u_1
T : X → X
U V : SetRel X X
n k : ℕ
k_n : k < n
xy : X × X
xy_comp : xy ∈ (⋂ k, ⋂ (_ : k < n), map T^[k] T^[k] ⁻¹' U) ○ ⋂ k, ⋂ (_ : k < n), map T^[k] T^[k] ⁻¹' V
⊢ xy ∈ map T^[k] T^[k] ⁻¹' (U ○ V)

Tactic:
simp only [SetRel.comp, mem_iInter, mem_preimage, map_apply, mem_ofPred_eq] at xy_comp ⊢

Goal state after:
X : Type u_1
T : X → X
U V : SetRel X X
n k : ℕ
k_n : k < n
xy : X × X
xy_comp : ∃ b, (∀ i < n, (T^[i] xy.1, T^[i] b) ∈ U) ∧ ∀ i < n, (T^[i] b, T^[i] xy.2) ∈ V
⊢ ∃ b, ((map T^[k] T^[k] xy).1, b) ∈ U ∧ (b, (map T^[k] T^[k] xy).2) ∈ V
Module: Mathlib.Dynamics.OmegaLimit

Goal state before:
pp: τ : Type u_1
inst✝² : TopologicalSpace τ
inst✝¹ : AddCommGroup τ
α : Type u_2
inst✝ : TopologicalSpace α
f : Filter τ
ϕ : Flow τ α
s : Set α
hf : ∀ (t : τ), Tendsto (fun x ↦ t + x) f f
x✝ : α
h : ∀ n ∈ 𝓝 x✝, ∀ {U : Set τ}, U ∈ f → ∃ x ∈ U, (ϕ.toFun x '' ω f ϕ.toFun s ∩ n).Nonempty
n : Set α
hn : n ∈ 𝓝 x✝
u : Set τ
hu : u ∈ f
o : Set α
ho₁ : o ⊆ n
ho₂ : IsOpen[inst✝] o
ho₃ : x✝ ∈ o
⊢ ∃ x ∈ u, (ϕ.toFun x '' s ∩ n).Nonempty

Tactic:
rcases h o (IsOpen.mem_nhds ho₂ ho₃) hu with ⟨t, _ht₁, ht₂⟩

Goal state after:
τ : Type u_1
inst✝² : TopologicalSpace τ
inst✝¹ : AddCommGroup τ
α : Type u_2
inst✝ : TopologicalSpace α
f : Filter τ
ϕ : Flow τ α
s : Set α
hf : ∀ (t : τ), Tendsto (fun x ↦ t + x) f f
x✝ : α
h : ∀ n ∈ 𝓝 x✝, ∀ {U : Set τ}, U ∈ f → ∃ x ∈ U, (ϕ.toFun x '' ω f ϕ.toFun s ∩ n).Nonempty
n : Set α
hn : n ∈ 𝓝 x✝
u : Set τ
hu : u ∈ f
o : Set α
ho₁ : o ⊆ n
ho₂ : IsOpen[inst✝] o
ho₃ : x✝ ∈ o
t : τ
_ht₁ : t ∈ u
ht₂ : (ϕ.toFun t '' ω f ϕ.toFun s ∩ o).Nonempty
⊢ ∃ x ∈ u, (ϕ.toFun x '' s ∩ n).Nonempty
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
U : SetRel X X
T : X → X
F : Set X
n : ℕ
⊢ netMaxcard T F U n ≤ coverMincard T F U n

Tactic:
by
  rcases eq_top_or_lt_top (coverMincard T F U n) with h | h
  · exact h ▸ le_top
  · obtain ⟨t, t_cover, t_mincard⟩ := (coverMincard_finite_iff T F U n).1 h
    rw [← t_mincard]
    exact iSup₂_le fun s s_net ↦ Nat.cast_le.2 (s_net.card_le_card_of_isDynCoverOf t_cover)

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

Goal state before:
pp: X : Type u_1
T : X → X
U : SetRel X X
⊢ (expGrowthSup fun n ↦ ↑0) = ⊥

Tactic:
ENat.toENNReal_zero,

Goal state after:
X : Type u_1
T : X → X
U : SetRel X X
⊢ (expGrowthSup fun n ↦ 0) = ⊥
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
⊢ netEntropyEntourage T F univ = 0

Tactic:
rw [← expGrowthSup_const one_ne_zero one_ne_top, netEntropyEntourage]
  simp only [netMaxcard_univ T h, ENat.toENNReal_one]

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

Goal state before:
pp: case a
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
V : Set (X × X)
V_uni : V ∈ 𝓤 X
V_symm : SetRel.IsSymm V
V_U : V ○ V ⊆ U
this : SetRel.IsRefl V
⊢ coverEntropyInfEntourage T F U ≤ ⨆ U ∈ 𝓤 X, netEntropyInfEntourage T F U

Tactic:
apply (coverEntropyInfEntourage_antitone T F V_U).trans (le_iSup₂_of_le V V_uni _)

Goal state after:
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
V : Set (X × X)
V_uni : V ∈ 𝓤 X
V_symm : SetRel.IsSymm V
V_U : V ○ V ⊆ U
this : SetRel.IsRefl V
⊢ (fun U ↦ coverEntropyInfEntourage T F U) (V ○ V) ≤ netEntropyInfEntourage T F V
Module: Mathlib.Dynamics.TopologicalEntropy.NetEntropy

Goal state before:
pp: X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
⊢ coverEntropy T F = ⨆ U ∈ 𝓤 X, netEntropyEntourage T F U

Tactic:
apply le_antisymm <;> refine iSup₂_le fun U U_uni ↦ ?_
  · obtain ⟨V, V_uni, V_symm, V_comp_U⟩ := comp_symm_mem_uniformity_sets U_uni
    apply (coverEntropyEntourage_antitone T F V_comp_U).trans (le_iSup₂_of_le V V_uni _)
    have := isRefl_of_mem_uniformity V_uni
    exact coverEntropyEntourage_le_netEntropyEntourage T F
  · apply (netEntropyEntourage_antitone T F SetRel.symmetrize_subset_self).trans
    apply (le_iSup₂ (SetRel.symmetrize U) (symmetrize_mem_uniformity U_uni)).trans'
    exact netEntropyEntourage_le_coverEntropyEntourage T F

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ Measurable fun x ↦ (map f μ).rnDeriv (map f ν) (f x)

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ Measurable (μ.rnDeriv ν)

pp: case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
⊢ ∫⁻ (x : α) in s, (map f μ).rnDeriv (map f ν) (f x) ∂ν = ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν

Tactic:
· exact (Measure.measurable_rnDeriv _ _).comp hf.measurable

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ Measurable (μ.rnDeriv ν)

case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ∞
⊢ ∫⁻ (x : α) in s, (map f μ).rnDeriv (map f ν) (f x) ∂ν = ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν
Module: Mathlib.FieldTheory.IsAlgClosed.Classification

Goal state before:
pp: K : Type u
L : Type v
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : IsAlgClosed K
inst✝² : IsAlgClosed L
inst✝¹ : CharZero K
inst✝ : CharZero L
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
s : Set K
hs : IsTranscendenceBasis ℤ Subtype.val
t : Set L
ht : IsTranscendenceBasis ℤ Subtype.val
⊢ ℵ₀ < Cardinal.lift.{v, u} #K

Tactic:
aleph0_lt_lift

Goal state after:
K : Type u
L : Type v
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : IsAlgClosed K
inst✝² : IsAlgClosed L
inst✝¹ : CharZero K
inst✝ : CharZero L
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
s : Set K
hs : IsTranscendenceBasis ℤ Subtype.val
t : Set L
ht : IsTranscendenceBasis ℤ Subtype.val
⊢ ℵ₀ < #K
Module: Mathlib.FieldTheory.IsAlgClosed.Classification

Goal state before:
pp: K : Type u
L : Type v
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : IsAlgClosed K
inst✝³ : IsAlgClosed L
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : CharP K p
inst✝ : CharP L p
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
this✝ : Algebra (ZMod p) K := ZMod.algebra K p
this : Algebra (ZMod p) L := ZMod.algebra L p
s : Set K
hs : IsTranscendenceBasis (ZMod p) Subtype.val
t : Set L
ht : IsTranscendenceBasis (ZMod p) Subtype.val
⊢ ℵ₀ < Cardinal.lift.{v, u} #K

Tactic:
aleph0_lt_lift

Goal state after:
K : Type u
L : Type v
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : IsAlgClosed K
inst✝³ : IsAlgClosed L
p : ℕ
inst✝² : Fact (Nat.Prime p)
inst✝¹ : CharP K p
inst✝ : CharP L p
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
this✝ : Algebra (ZMod p) K := ZMod.algebra K p
this : Algebra (ZMod p) L := ZMod.algebra L p
s : Set K
hs : IsTranscendenceBasis (ZMod p) Subtype.val
t : Set L
ht : IsTranscendenceBasis (ZMod p) Subtype.val
⊢ ℵ₀ < #K
Module: Mathlib.ModelTheory.Syntax

Goal state before:
pp: case equal
L : Language
α : Type u'
n n✝ : ℕ
t₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)
⊢ ((LHom.id L).onTerm t₁✝).bdEqual ((LHom.id L).onTerm t₂✝) = id (equal t₁✝ t₂✝)

Tactic:
LHom.id_onTerm,

Goal state after:
case equal
L : Language
α : Type u'
n n✝ : ℕ
t₁✝ t₂✝ : L.Term (α ⊕ Fin n✝)
⊢ (id t₁✝).bdEqual (id t₂✝) = id (equal t₁✝ t₂✝)
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
β : Type u_1
f : α → β
s : Set (α → M)
h : A.Definable L s
⊢ A.Definable L ((fun g ↦ g ∘ f) ⁻¹' s)

Tactic:
obtain ⟨φ, rfl⟩ := h
  refine ⟨φ.relabel f, ?_⟩
  ext
  simp only [Set.preimage_ofPred_eq, mem_ofPred_eq, Formula.realize_relabel]

Goal state after:
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
m : ℕ
φ : L[[↑A]].Formula (α ⊕ Fin m)
y : α ⊕ Fin m → M
hy : φ.Realize y
⊢ ∃ xs, BoundedFormula.Realize φ (Sum.elim (y ∘ Sum.inl) xs) (xs ∘ Fin.natAdd m)

Tactic:
exact
      ⟨y ∘ Sum.inr, (congr (congr rfl (Sum.elim_comp_inl_inr y).symm) (funext finZeroElim)).mp hy⟩

Goal state after:
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: case intro.intro.mp
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
x : α → M
⊢ ((∃ a ∈ s,
        ((a ∘ ⇑(Equiv.Set.sumCompl (range f))) ∘
              ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ∘
            Sum.inl =
          x ∘ rangeSplitting f) ∧
      ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))) →
    ∃ x_1 ∈ s, x_1 ∘ f = x

pp: case intro.intro.mpr
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
x : α → M
⊢ (∃ x_1 ∈ s, x_1 ∘ f = x) →
    (∃ a ∈ s,
        ((a ∘ ⇑(Equiv.Set.sumCompl (range f))) ∘
              ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ∘
            Sum.inl =
          x ∘ rangeSplitting f) ∧
      ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))

Tactic:
· rintro ⟨⟨y, ys, hy⟩, hx⟩
      refine ⟨y, ys, ?_⟩
      ext a
      rw [hx a, ← Function.comp_apply (f := x), ← hy]
      simp

Goal state after:
case intro.intro.mpr
M : Type w
A : Set M
L : Language
inst✝² : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h✝ : A.Definable L s
f : α → β
inst✝¹ : Finite α
inst✝ : Finite β
val✝¹ : Fintype α
val✝ : Fintype β
h :
  A.Definable L
    ((fun g ↦ g ∘ rangeSplitting f) ⁻¹'
      (fun g ↦ g ∘ Sum.inl) ''
        (fun g ↦ g ∘ ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ''
          (fun g ↦ g ∘ ⇑(Equiv.Set.sumCompl (range f))) '' s)
h' : A.Definable L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}
x : α → M
⊢ (∃ x_1 ∈ s, x_1 ∘ f = x) →
    (∃ a ∈ s,
        ((a ∘ ⇑(Equiv.Set.sumCompl (range f))) ∘
              ⇑((Equiv.refl ↑(range f)).sumCongr (Fintype.equivFin ↑(range f)ᶜ).symm)) ∘
            Sum.inl =
          x ∘ rangeSplitting f) ∧
      ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))
Module: Mathlib.ModelTheory.Algebra.Field.IsAlgClosed

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : CompatibleRing K
n : ℕ
⊢ K ⊨ genericMonicPolyHasRoot n ↔ ∀ (p : { p // p.Monic ∧ p.natDegree = n }), ∃ x, eval x ↑p = 0

Tactic:
Equiv.forall_congr_left ((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv)

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : CompatibleRing K
n : ℕ
⊢ K ⊨ genericMonicPolyHasRoot n ↔
    ∀ (b : Fin n → K), ∃ x, eval x ↑(((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv).symm b) = 0
Module: Mathlib.FieldTheory.Differential.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
a b : R
ha : a ≠ 0
hb : b ≠ 0
⊢ (a / b)′ / (a / b) = a′ / a - b′ / b

Tactic:
simp [field, Derivation.leibniz_div]

Goal state after:
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
ih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType
s : Set E := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : Module.rank F ↥(adjoin F s) = Module.rank F ↥⊤
| Module.rank F ↥⊤

Tactic:
topEquiv.toLinearEquiv.rank_eq

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i : (Module.rank F E).ord.ToType
ih : (y : (Module.rank F E).ord.ToType) → y < i → (Module.rank F E).ord.ToType
s : Set E := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : Module.rank F ↥(adjoin F s) = Module.rank F ↥⊤
| Module.rank F E
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: case top
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
hi✝ : IsSuccPrelimit ⊤
hi : ¬Nonempty ↑(Iio ⊤)
this : Nonempty (Module.rank F E).ord.ToType
⊢ filtration ⊤ = ⊥

Tactic:
rw [← range_coe] at hi

Goal state after:
case top
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
hi✝ : IsSuccPrelimit ⊤
hi : ¬Nonempty ↑(range WithTop.some)
this : Nonempty (Module.rank F E).ord.ToType
⊢ filtration ⊤ = ⊥
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
| ℵ₀ ^ #(Module.rank F E).ord.ToType

Tactic:
← prod_const'

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
| Cardinal.prod fun x ↦ ℵ₀
Module: Mathlib.FieldTheory.CardinalEmb

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
| 2 ^ #(Module.rank F E).ord.ToType

Tactic:
← prod_const'

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
| Cardinal.prod fun x ↦ 2
Module: Mathlib.FieldTheory.Galois.Profinite

Goal state before:
pp: 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
L : FiniteGaloisIntermediateField k K
hx : x ∈ L.toIntermediateField
⊢ ↑(((proj L) g) ⟨x, hx⟩) = (((proj L) g).liftNormal K) x

Tactic:
exact (AlgEquiv.liftNormal_commutes (proj L g) _ ⟨x, hx⟩).symm

Goal state after:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
h : (of K p) x = 0
⊢ x ∈ pNilradical K p

Tactic:
of_apply,

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
h : mk K p (0, x) = 0
⊢ x ∈ pNilradical K p
Module: Mathlib.FieldTheory.MvRatFunc.Rank

Goal state before:
pp: case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))

Tactic:
have h1 := (IsScalarTower.toAlgHom F R K).toLinearMap.rank_le_of_injective hinj

Goal state after:
case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : Module.rank F R ≤ Module.rank F K
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
B : Subalgebra R S
⊢ ⊥.LinearDisjoint B

Tactic:
by
  rw [Subalgebra.LinearDisjoint, Algebra.toSubmodule_bot]
  exact Submodule.LinearDisjoint.one_left _

Goal state after:
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
H' : A ⊔ B = ⊤
inst✝¹ : Free R ↥B
inst✝ : Module.Finite R ↥B
x : ↥B
⊢ (Algebra.norm ↥A) ((algebraMap (↥B) S) x) = (algebraMap R ↥A) ((Algebra.norm R) x)

Tactic:
Algebra.norm_eq_matrix_det (Module.Free.chooseBasis R B),

Goal state after:
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
H' : A ⊔ B = ⊤
inst✝¹ : Free R ↥B
inst✝ : Module.Finite R ↥B
x : ↥B
⊢ (Algebra.norm ↥A) ((algebraMap (↥B) S) x) = (algebraMap R ↥A) ((Algebra.leftMulMatrix (Free.chooseBasis R ↥B)) x).det
Module: Mathlib.FieldTheory.NormalizedTrace

Goal state before:
pp: F : Type u_3
E : Type u_4
K : Type u_5
inst✝¹² : Field F
inst✝¹¹ : Field E
inst✝¹⁰ : Field K
inst✝⁹ : Algebra F E
inst✝⁸ : Algebra E K
inst✝⁷ : Algebra F K
inst✝⁶ : IsScalarTower F E K
inst✝⁵ : Algebra.IsIntegral F E
inst✝⁴ : Algebra.IsIntegral F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F E
inst✝¹ : Algebra.IsIntegral E K
inst✝ : CharZero E
a : K
this : FiniteDimensional E ↥E⟮a⟯
⊢ (↑(Module.finrank E ↥E⟮a⟯))⁻¹ • (normalizedTrace F E) ((trace E ↥E⟮a⟯) (AdjoinSimple.gen E a)) =
    (normalizedTrace F K) a

Tactic:
normalizedTrace_eq_of_finiteDimensional F E,

Goal state after:
F : Type u_3
E : Type u_4
K : Type u_5
inst✝¹² : Field F
inst✝¹¹ : Field E
inst✝¹⁰ : Field K
inst✝⁹ : Algebra F E
inst✝⁸ : Algebra E K
inst✝⁷ : Algebra F K
inst✝⁶ : IsScalarTower F E K
inst✝⁵ : Algebra.IsIntegral F E
inst✝⁴ : Algebra.IsIntegral F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F E
inst✝¹ : Algebra.IsIntegral E K
inst✝ : CharZero E
a : K
this : FiniteDimensional E ↥E⟮a⟯
⊢ (↑(Module.finrank E ↥E⟮a⟯))⁻¹ • ((↑(Module.finrank F E))⁻¹ • trace F E) ((trace E ↥E⟮a⟯) (AdjoinSimple.gen E a)) =
    (normalizedTrace F K) a
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : Type u_1
B : Type u_2
inst✝³ : Field A
inst✝² : Algebra F A
inst✝¹ : Field B
inst✝ : Algebra F B
fa : A →ₐ[F] E
fb : B →ₐ[F] E
H : fa.range.LinearDisjoint fb.range
⊢ IsDomain (A ⊗[F] B)

Tactic:
exact H.isDomain_of_injective fa.injective fb.injective

Goal state after:
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: case pos
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
inst✝³ : Free R ↥A
inst✝² : Free R ↥B
inst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)
inst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)
H : finrank R ↥B = 1
a✝ : Nontrivial R
h1 : finrank R ↥A = 0
⊢ A.LinearDisjoint B

Tactic:
rw [eq_bot_of_finrank_one H]

Goal state after:
case pos
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
inst✝³ : Free R ↥A
inst✝² : Free R ↥B
inst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)
inst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)
H : finrank R ↥B = 1
a✝ : Nontrivial R
h1 : finrank R ↥A = 0
⊢ A.LinearDisjoint ⊥
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: case pos
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
inst✝³ : Free R ↥A
inst✝² : Free R ↥B
inst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)
inst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)
H : finrank R ↥A = 1
a✝ : Nontrivial R
h1 : ¬finrank R ↥A = 0
h2 : finrank R ↥B = 0
⊢ A.LinearDisjoint B

Tactic:
rw [eq_bot_of_finrank_one H]

Goal state after:
case pos
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
inst✝³ : Free R ↥A
inst✝² : Free R ↥B
inst✝¹ : Free ↥A ↥(Algebra.adjoin ↥A ↑B)
inst✝ : Free ↥B ↥(Algebra.adjoin ↥B ↑A)
H : finrank R ↥A = 1
a✝ : Nontrivial R
h1 : ¬finrank R ↥A = 0
h2 : finrank R ↥B = 0
⊢ ⊥.LinearDisjoint B
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
| fun x ↦ Bivariate.swap ((monomial x) ((Φ E).coeff x))

Tactic:
ext
    rw [Bivariate.swap_monomial, mul_comm, ← Polynomial.smul_eq_C_mul,
      ← monomial_one_right_eq_X_pow, ← Polynomial.algebraMap_eq]

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
x✝ : ℕ
| (monomial x✝) 1 • Polynomial.map (algebraMap K K[X]) ((Φ E).coeff x✝)
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : DivisionRing R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C : PointedCone R M
x✝ y✝ : M
xc : x✝ ∈ C
yc : y✝ ∈ C
xyf : x✝ + y✝ ∈ ↑C.lineal
⊢ x✝ ∈ ↑C.lineal

Tactic:
simp [neg_add_rev, xc, true_and] at xyf ⊢

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : DivisionRing R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C : PointedCone R M
x✝ y✝ : M
xc : x✝ ∈ C
yc : y✝ ∈ C
xyf : x✝ + y✝ ∈ C ∧ -y✝ + -x✝ ∈ C
⊢ -x✝ ∈ C
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
⊢ dist p₁ ((reflection s) p₂) = dist ((reflection s) p₁) p₂

Tactic:
conv_lhs => rw [← reflection_reflection s p₁]

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
⊢ dist ((reflection s) ((reflection s) p₁)) ((reflection s) p₂) = dist ((reflection s) p₁) p₂
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
r₁ r₂ : ℝ
⊢ 2 • o.oangle (r₁ • x) (r₂ • x) = 0

Tactic:
by_cases h : r₁ = 0 <;> simp [h]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[ℝ] P₂
p₁ p₂ p₃ : P
⊢ ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃

Tactic:
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]

Goal state after: