mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Differential.Basic

Goal state before:
pp: R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : Differential R
a b : R
F : Type u_2
inst✝⁴ : Field F
inst✝³ : Differential F
inst✝² : CharZero F
p : F[X]
inst✝¹ : Fact (Irreducible p)
inst✝ : Fact p.Monic
q : F[X]
⊢ (AdjoinRoot.mk p)
      ((implicitDeriv
          ((AdjoinRoot.modByMonicHom ⋯)
            (-(aeval (AdjoinRoot.root p)) (mapCoeffs p) / (aeval (AdjoinRoot.root p)) (derivative p))))
        p) =
    0

Tactic:
unfold implicitDeriv

Goal state after:
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : Differential R
a b : R
F : Type u_2
inst✝⁴ : Field F
inst✝³ : Differential F
inst✝² : CharZero F
p : F[X]
inst✝¹ : Fact (Irreducible p)
inst✝ : Fact p.Monic
q : F[X]
⊢ (AdjoinRoot.mk p)
      ((mapCoeffs +
          (AdjoinRoot.modByMonicHom ⋯)
              (-(aeval (AdjoinRoot.root p)) (mapCoeffs p) / (aeval (AdjoinRoot.root p)) (derivative p)) •
            Derivation.restrictScalars ℤ derivative')
        p) =
    0
Module: Mathlib.FieldTheory.Differential.Basic

Goal state before:
pp: R : Type u_1
inst✝⁷ : Field R
inst✝⁶ : Differential R
a✝ b : R
F : Type u_2
inst✝⁵ : Field F
inst✝⁴ : Differential F
inst✝³ : CharZero F
K : Type u_3
inst✝² : Field K
inst✝¹ : Algebra F K
inst✝ : FiniteDimensional F K
default : { _a // DifferentialAlgebra F K } := ⟨differentialFiniteDimensional F K, ⋯⟩
x✝ : { _a // DifferentialAlgebra F K }
a : Differential K
ha : DifferentialAlgebra F K
⊢ ⟨a, ha⟩ = Inhabited.default

Tactic:
ext x

Goal state after:
R : Type u_1
inst✝⁷ : Field R
inst✝⁶ : Differential R
a✝ b : R
F : Type u_2
inst✝⁵ : Field F
inst✝⁴ : Differential F
inst✝³ : CharZero F
K : Type u_3
inst✝² : Field K
inst✝¹ : Algebra F K
inst✝ : FiniteDimensional F K
default : { _a // DifferentialAlgebra F K } := ⟨differentialFiniteDimensional F K, ⋯⟩
x✝ : { _a // DifferentialAlgebra F K }
a : Differential K
ha : DifferentialAlgebra F K
x : K
⊢ x′ = x′
Module: Mathlib.FieldTheory.Finite.Extension

Goal state before:
pp: k : Type u_1
inst✝ : Field k
f : k[X]
hi : Irreducible f
n : ℕ
h : f ∣ X ^ Nat.card k ^ n - X
⊢ f.natDegree ∣ n

Tactic:
rcases eq_or_ne n 0 with rfl | hn

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
f : k[X]
hi : Irreducible f
h : f ∣ X ^ Nat.card k ^ 0 - X
⊢ f.natDegree ∣ 0

case inr
k : Type u_1
inst✝ : Field k
f : k[X]
hi : Irreducible f
n : ℕ
h : f ∣ X ^ Nat.card k ^ n - X
hn : n ≠ 0
⊢ f.natDegree ∣ n
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: K : Type u_1
σ : Type u_2
inst✝ : CommSemiring K
p q : MvPolynomial σ K
⊢ (fun e ↦ (eval e) (p + q)) = (fun e ↦ (eval e) p) + fun e ↦ (eval e) q

Tactic:
ext x

Goal state after:
K : Type u_1
σ : Type u_2
inst✝ : CommSemiring K
p q : MvPolynomial σ K
x : σ → K
⊢ (eval x) (p + q) = ((fun e ↦ (eval e) p) + fun e ↦ (eval e) q) x
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
⊢ (evalᵢ σ K).range = ⊤

Tactic:
evalᵢ,

Goal state after:
σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
⊢ (evalₗ K σ ∘ₗ (restrictDegree σ K (Fintype.card K - 1)).subtype).range = ⊤
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: case intro
σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
val✝ : Fintype σ
⊢ Module.finrank K (R σ K) = Module.finrank K ((σ → K) → K)

Tactic:
rw [Module.finrank_fintype_fun_eq_card, finrank_R]

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.ExtendRight

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝¹⁵ : Field K
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
F : IntermediateField K L
M : Type u_3
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : Algebra L M
inst✝⁹ : IsScalarTower K L M
R : Type u_4
S : Type u_5
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra S ↥F
inst✝⁵ : Algebra S M
inst✝⁴ : IsScalarTower S (↥F) M
inst✝³ : Algebra R ↥F
inst✝² : Algebra R M
inst✝¹ : IsScalarTower R (↥F) M
inst✝ : IsIntegralClosure S R ↥F
⊢ IsIntegralClosure S R ↥(F.extendRight M)

Tactic:
refine .of_algEquiv S (F.extendRightEquiv' M R) fun x ↦ ?_
  rw [Subtype.ext_iff, ← algebraMap_apply (F.extendRight M), ← algebraMap_apply (F.extendRight M),
    algebraMap_extendRightEquiv', ← IsScalarTower.algebraMap_apply,
    ← IsScalarTower.algebraMap_apply]

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
x y1 y2 : ℕ × K
H : R K p y1 y2
n : ℕ
y : K
⊢ R K p (x.1 + (n, y).1, (⇑(frobenius K p))^[(n, y).1] x.2 + (⇑(frobenius K p))^[x.1] (n, y).2)
    (x.1 + (n + 1, (frobenius K p) y).1,
      (⇑(frobenius K p))^[(n + 1, (frobenius K p) y).1] x.2 + (⇑(frobenius K p))^[x.1] (n + 1, (frobenius K p) y).2)

Tactic:
by
      rw [← iterate_succ_apply, iterate_succ_apply', iterate_succ_apply', ← map_add]
      apply R.intro

Goal state after:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁴ : CommRing K
inst✝³ : CommRing L
inst✝² : CommRing M
i : K →+* L
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : IsPRadical i p
h : pNilradical M p = ⊥
f g : L →+* M
heq : (fun f ↦ f.comp i) f = (fun f ↦ f.comp i) g
⊢ f = g

Tactic:
ext x

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁴ : CommRing K
inst✝³ : CommRing L
inst✝² : CommRing M
i : K →+* L
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : IsPRadical i p
h : pNilradical M p = ⊥
f g : L →+* M
heq : (fun f ↦ f.comp i) f = (fun f ↦ f.comp i) g
x : L
⊢ f x = g x
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
L : (FiniteGaloisIntermediateField k K)ᵒᵖ
⊢ (limitToAlgEquiv g).restrictNormal ↥(unop L).toIntermediateField = ↑g L

Tactic:
ext x

Goal state after:
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
L : (FiniteGaloisIntermediateField k K)ᵒᵖ
x : ↥(unop L).toIntermediateField
⊢ ↑(((limitToAlgEquiv g).restrictNormal ↥(unop L).toIntermediateField) x) = ↑((↑g L) x)
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
⊢ (algEquivToLimit k K) (limitToAlgEquiv g) = g

Tactic:
by
    apply Subtype.val_injective
    ext L
    change (limitToAlgEquiv g).restrictNormal _ = _
    ext x
    have : ((limitToAlgEquiv g).restrictNormal L.unop) x = (limitToAlgEquiv g) x.1 := by
      exact AlgEquiv.restrictNormal_commutes (limitToAlgEquiv g) L.unop x
    simp_rw [this]
    exact proj_adjoin_singleton_val _ _ _ _ x.2

Goal state after:
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
L : FiniteGaloisIntermediateField k K
fix1 : Set
  ((L : (FiniteGaloisIntermediateField k K)ᵒᵖ) → ↑((asProfiniteGaloisGroupFunctor k K).obj L).toProfinite.toTop) :=
  {f | f (op L) = 1}
⊢ ⇑(mulEquivToLimit k K) '' ↑L.fixingSubgroup = Subtype.val ⁻¹' fix1

Tactic:
ext x

Goal state after:
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
L : FiniteGaloisIntermediateField k K
fix1 : Set
  ((L : (FiniteGaloisIntermediateField k K)ᵒᵖ) → ↑((asProfiniteGaloisGroupFunctor k K).obj L).toProfinite.toTop) :=
  {f | f (op L) = 1}
x : ↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop
⊢ x ∈ ⇑(mulEquivToLimit k K) '' ↑L.fixingSubgroup ↔ x ∈ Subtype.val ⁻¹' fix1
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
D H : Subgroup G
inst✝ : D.FiniteIndex
hD_le_H : D ≤ H
t : Set ↥H
ht : IsComplement t ↑(D.subgroupOf H) ∧ 1 ∈ t
hf : t.Finite
⊢ ⋃ g ∈ hf.toFinset, ↑g • ↑D = ↑H

Tactic:
ext x

Goal state after:
G : Type u_1
inst✝¹ : Group G
D H : Subgroup G
inst✝ : D.FiniteIndex
hD_le_H : D ≤ H
t : Set ↥H
ht : IsComplement t ↑(D.subgroupOf H) ∧ 1 ∈ t
hf : t.Finite
x : G
⊢ x ∈ ⋃ g ∈ hf.toFinset, ↑g • ↑D ↔ x ∈ ↑H
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
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
⊢ (⇑((ad (↥k) D) a))^[p ^ m] = 0

Tactic:
ext x

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
x : D
⊢ (⇑((ad (↥k) D) a))^[p ^ m] x = 0 x
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case neg
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
j : ι
hj : j ∈ s
hcovers' : ¬⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) = Set.univ
⊢ ∃ k ∈ s, (H k).FiniteIndex

Tactic:
· have ⟨i, hi, _, hfi⟩ :=
      exists_finiteIndex_of_leftCoset_cover_aux hcovers j hj hcovers'
    exact ⟨i, hi, hfi⟩

Goal state after:
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
⊢ {x | K x = D}.card = D.index → {x | x ∈ s ∧ (H x).FiniteIndex}.PairwiseDisjoint fun i ↦ g i • ↑(H i)

Tactic:
intro h i hi j hj hij c hi' hj' x hx

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 ∈ {x | x ∈ s ∧ (H x).FiniteIndex}
j : ι
hj : j ∈ {x | x ∈ s ∧ (H x).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
⊢ x ∈ ⊥
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
inst✝⁴ : CommSemiring R
A : Type v
inst✝³ : Semiring A
B : Type w
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
⊢ Algebra.TensorProduct.includeLeft.range.LinearDisjoint Algebra.TensorProduct.includeRight.range

Tactic:
by
  rw [Subalgebra.LinearDisjoint, Submodule.linearDisjoint_iff]
  change Function.Injective <|
    Submodule.mulMap (LinearMap.range Algebra.TensorProduct.includeLeft.toLinearMap)
      (LinearMap.range Algebra.TensorProduct.includeRight.toLinearMap)
  rw [← Algebra.TensorProduct.linearEquivIncludeRange_symm_toLinearMap]
  exact LinearEquiv.injective _

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsAlgebraic K L
c : ConjRootClass K L
⊢ c.minpoly.rootSet L = c.carrier

Tactic:
ext x

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsAlgebraic K L
c : ConjRootClass K L
x : L
⊢ x ∈ c.minpoly.rootSet L ↔ x ∈ c.carrier
Module: Mathlib.FieldTheory.NormalizedTrace

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : CharZero F
inst✝ : Algebra.IsIntegral F K
a b : K
E : IntermediateField F K := F⟮a⟯ ⊔ F⟮b⟯
this✝ : FiniteDimensional F ↥F⟮a⟯
this : FiniteDimensional F ↥F⟮b⟯
ha : a ∈ E
hb : b ∈ E
hab : a + b ∈ E
a' : ↥E := ⟨a, ha⟩
b' : ↥E := ⟨b, hb⟩
ab' : ↥E := ⟨a + b, hab⟩
⊢ normalizedTraceAux F (↥E) ab' =
    (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) a' + (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) b'

Tactic:
normalizedTraceAux_eq_of_finiteDimensional F ab',

Goal state after:
F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : CharZero F
inst✝ : Algebra.IsIntegral F K
a b : K
E : IntermediateField F K := F⟮a⟯ ⊔ F⟮b⟯
this✝ : FiniteDimensional F ↥F⟮a⟯
this : FiniteDimensional F ↥F⟮b⟯
ha : a ∈ E
hb : b ∈ E
hab : a + b ∈ E
a' : ↥E := ⟨a, ha⟩
b' : ↥E := ⟨b, hb⟩
ab' : ↥E := ⟨a + b, hab⟩
⊢ (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) ab' =
    (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) a' + (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) b'
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: case e_a
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
L' : Subalgebra F E := (IsScalarTower.toAlgHom F L E).range
i : L ≃ₐ[F] ↥L' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)
⊢ Set.range ⇑(algebraMap L E) = Set.range ⇑(algebraMap (↥L') E)

Tactic:
ext x

Goal state after:
case e_a
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
L' : Subalgebra F E := (IsScalarTower.toAlgHom F L E).range
i : L ≃ₐ[F] ↥L' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)
x : E
⊢ x ∈ Set.range ⇑(algebraMap L E) ↔ x ∈ Set.range ⇑(algebraMap (↥L') E)
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: case e_a
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
L' : Subalgebra F E := (IsScalarTower.toAlgHom F L E).range
⊢ ↑A = Set.range ⇑(algebraMap (↥A) E)

Tactic:
ext x

Goal state after:
case e_a
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
L' : Subalgebra F E := (IsScalarTower.toAlgHom F L E).range
x : E
⊢ x ∈ ↑A ↔ x ∈ Set.range ⇑(algebraMap (↥A) E)
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
⊢ comap i (perfectClosure F K) = perfectClosure F E

Tactic:
ext x

Goal state after:
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
⊢ x ∈ comap i (perfectClosure F K) ↔ x ∈ perfectClosure F E
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 : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
⊢ Cardinal.lift.{v, w} (Module.rank F L) * Cardinal.lift.{w, v} (Module.rank (↥(adjoin L ↑A)) E) =
    Cardinal.lift.{w, v} (Module.rank (↥A) E)

Tactic:
rw [← H.lift_adjoin_rank_eq_lift_rank_right_of_isAlgebraic halg, ← Cardinal.lift_mul,
    Cardinal.lift_inj]
  exact rank_mul_rank A (extendScalars
    (show A ≤ (adjoin L (A : Set E)).restrictScalars F from subset_adjoin L (A : Set E))) E

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 ⊥

Tactic:
exact bot_right _

Goal state after:
Module: Mathlib.FieldTheory.PurelyInseparable.Tower

Goal state before:
pp: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
⊢ Cardinal.lift.{w, v} (insepDegree F E) * Cardinal.lift.{v, w} (insepDegree E K) =
    Cardinal.lift.{v, w} (insepDegree F K)

Tactic:
have h := lift_rank_mul_lift_insepDegree_of_isPurelyInseparable (separableClosure F E) E K

Goal state after:
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
h :
  Cardinal.lift.{w, v} (Module.rank (↥(separableClosure F E)) E) * Cardinal.lift.{v, w} (insepDegree E K) =
    Cardinal.lift.{v, w} (insepDegree (↥(separableClosure F E)) K)
⊢ Cardinal.lift.{w, v} (insepDegree F E) * Cardinal.lift.{v, w} (insepDegree E K) =
    Cardinal.lift.{v, w} (insepDegree F K)
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
C F : PointedCone R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
sub : map f F ≤ map f C
hF : ∀ {x y : N} {a : R}, x ∈ map f C → y ∈ map f C → 0 < a → a • x + y ∈ map f F → x ∈ map f F
x : M
xf : x ∈ F
y : M
yC : y ∈ C
hy : f y = (↑{ c // 0 ≤ c } f) x
⊢ x ∈ C

Tactic:
hf hy

Goal state after:
case refine_1
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
C F : PointedCone R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
sub : map f F ≤ map f C
hF : ∀ {x y : N} {a : R}, x ∈ map f C → y ∈ map f C → 0 < a → a • x + y ∈ map f F → x ∈ map f F
x : M
xf : x ∈ F
y : M
yC : x ∈ C
hy : f y = (↑{ c // 0 ≤ c } f) x
⊢ x ∈ C
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
eq :
  (Polynomial.mapRingHom (algebraMap K F)) (g E) * Polynomial.C ((aeval α) (f E)) =
    (Polynomial.mapRingHom (algebraMap K F)) (f E) * Polynomial.C ((aeval α) (g E))
⊢ IsUnit ((aeval α) (f E)) ∧ IsUnit (Polynomial.C ((aeval α) (g E)))

Tactic:
isUnit_iff_ne_zero,

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
eq :
  (Polynomial.mapRingHom (algebraMap K F)) (g E) * Polynomial.C ((aeval α) (f E)) =
    (Polynomial.mapRingHom (algebraMap K F)) (f E) * Polynomial.C ((aeval α) (g E))
⊢ (aeval α) (f E) ≠ 0 ∧ IsUnit (Polynomial.C ((aeval α) (g E)))
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
⊢ (o.rotationAux θ).toLinearMap ∘ₗ (θ.cos • LinearMap.id - θ.sin • ↑o.rightAngleRotation.toLinearEquiv) = LinearMap.id

Tactic:
ext x

Goal state after:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
x : V
⊢ ((o.rotationAux θ).toLinearMap ∘ₗ (θ.cos • LinearMap.id - θ.sin • ↑o.rightAngleRotation.toLinearEquiv)) x =
    LinearMap.id x
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
⊢ (θ.cos • LinearMap.id - θ.sin • ↑o.rightAngleRotation.toLinearEquiv) ∘ₗ (o.rotationAux θ).toLinearMap = LinearMap.id

Tactic:
ext x

Goal state after:
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
x : V
⊢ ((θ.cos • LinearMap.id - θ.sin • ↑o.rightAngleRotation.toLinearEquiv) ∘ₗ (o.rotationAux θ).toLinearMap) x =
    LinearMap.id x
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

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)
⊢ o.rotation ↑π = LinearIsometryEquiv.neg ℝ

Tactic:
ext x

Goal state after:
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
⊢ (o.rotation ↑π) x = (LinearIsometryEquiv.neg ℝ) x
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

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)
⊢ o.rotation ↑(π / 2) = o.rightAngleRotation

Tactic:
ext x

Goal state after:
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
⊢ (o.rotation ↑(π / 2)) x = o.rightAngleRotation x
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

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
⊢ (o.rotation ↑(π + -(π / 2))) x = (o.rotation ↑(π / 2)) x

Tactic:
← sub_eq_add_neg,

Goal state after:
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
⊢ (o.rotation ↑(π - π / 2)) x = (o.rotation ↑(π / 2)) x