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

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc : c⁻¹ * (a * c) * c⁻¹ = c⁻¹ * (c * a) * c⁻¹
⊢ c⁻¹ * a = a * c⁻¹

Tactic:
rw [mul_assoc, mul_assoc, mul_inv_cancel₀ hb.1, mul_one, ← mul_assoc,
      inv_mul_cancel₀ hb.1, one_mul] at hc

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc : c⁻¹ * a = a * c⁻¹
⊢ c⁻¹ * a = a * c⁻¹
Module: Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality

Goal state before:

Tactic:
_ ≤ Cardinal.lift.{v} #F ⊔ Cardinal.lift.{u} (#A ⊔ #B) ⊔ ℵ₀ := by
      gcongr
      rw [Cardinal.lift_le]
      exact (mk_union_le _ _).trans_eq (by simp)
    _ = _ := by
      simp [lift_mk_le_lift_mk_of_injective (algebraMap F A).injective]

Goal state after:
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
⊢ a * c⁻¹ * ((⇑((ad (↥k) D) a))^[n - 1] b * a) = a * (c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a)

Tactic:
← mul_assoc,

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
⊢ a * c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = a * (c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a)
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
⊢ a * c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = a * (c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a)

Tactic:
← mul_assoc,

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
⊢ a * c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = a * (c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b) * a
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : a⁻¹ * (a * d) - a⁻¹ * (d * a) = a⁻¹ * a
⊢ False

Tactic:
← mul_assoc,

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : a⁻¹ * a * d - a⁻¹ * (d * a) = a⁻¹ * a
⊢ False
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : d - a⁻¹ * (d * a) = 1
⊢ False

Tactic:
← mul_assoc,

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : d - a⁻¹ * d * a = 1
⊢ False
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: 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
hcovers' : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
⊢ ⋃ i ∈ {x | K x = K k}, f i • ↑(K k) = Set.univ

Tactic:
Set.iUnion₂_congr fun i hi => by rw [← (Finset.mem_filter.mp hi).2]

Goal state after:
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
hcovers' : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
⊢ ⋃ i ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
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
fa : FR →ₐ[R] FA := IsFractionRing.liftAlgHom ⋯
fb : FR →ₐ[R] FB := IsFractionRing.liftAlgHom ⋯
algInst✝¹ : Algebra FR FA := fa.toAlgebra
algInst✝ : Algebra FR FB := fb.toAlgebra
⊢ Nontrivial (A ⊗[R] B)

Tactic:
let : CompatibleSMul FR R FA FB := CompatibleSMul.isScalarTower

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 ⋯
fb : FR →ₐ[R] FB := IsFractionRing.liftAlgHom ⋯
algInst✝¹ : Algebra FR FA := fa.toAlgebra
algInst✝ : Algebra FR FB := fb.toAlgebra
this : CompatibleSMul FR R FA FB := CompatibleSMul.isScalarTower
⊢ Nontrivial (A ⊗[R] B)
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝⁴ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
inst✝ : NeZero n
σ : Gal(L/K)
⊢ (autEquivRootsOfUnity hζ H L) σ • α = σ α

Tactic:
have ⟨ζ, hζ'⟩ := hζ

Goal state after:
K : Type u
inst✝⁴ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
inst✝ : NeZero n
σ : Gal(L/K)
ζ : K
hζ' : ζ ∈ primitiveRoots n K
⊢ (autEquivRootsOfUnity ⋯ H L) σ • α = σ α
Module: Mathlib.FieldTheory.KummerExtension

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✝ : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
⊢ [IsGalois K L ∧ IsCyclic Gal(L/K),
      ∃ a, Irreducible (X ^ finrank K L - C a) ∧ IsSplittingField K L (X ^ finrank K L - C a),
      ∃ α, α ^ finrank K L ∈ Set.range ⇑(algebraMap K L) ∧ K⟮α⟯ = ⊤].TFAE

Tactic:
by
  have : NeZero (Module.finrank K L) := NeZero.of_pos finrank_pos
  tfae_have 1 → 3
  | ⟨inst₁, inst₂⟩ => exists_root_adjoin_eq_top_of_isCyclic K L hK
  tfae_have 3 → 2
  | ⟨α, ⟨a, ha⟩, hα⟩ => ⟨a, irreducible_X_pow_sub_C_of_root_adjoin_eq_top ha.symm hα,
      isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top hK ha.symm hα⟩
  tfae_have 2 → 1
  | ⟨a, H, inst⟩ => ⟨isGalois_of_isSplittingField_X_pow_sub_C hK H L,
      isCyclic_of_isSplittingField_X_pow_sub_C hK H L⟩
  tfae_finish

Goal state after:
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this : Field (A ⊗[R] B) := H.toField
a : A
hta : Transcendental R a
b : B
htb : Transcendental R b
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
⊢ False

Tactic:
let fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight

Goal state after:
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this : Field (A ⊗[R] B) := H.toField
a : A
hta : Transcendental R a
b : B
htb : Transcendental R b
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
⊢ False
Module: Mathlib.FieldTheory.NormalizedTrace

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
a : F
⊢ normalizedTraceAux F F a = a

Tactic:
normalizedTraceAux_eq_of_finiteDimensional F a,

Goal state after:
F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
a : F
⊢ (↑(Module.finrank F F))⁻¹ • (trace F F) a = a
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 = ⊥
⊢ A.LinearDisjoint ↥B

Tactic:
apply LinearDisjoint.of_finrank_sup

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 F ↥(A ⊔ B) = finrank F ↥A * finrank F ↥B
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝⁴ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝³ : IsDomain A
this✝² : IsDomain B
this✝¹ : IsDomain R
this✝ : Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R R A B
this : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
⊢ a ⊗ₜ[R] 1 ∈ ↑fa.range ∧ a ⊗ₜ[R] 1 ∈ ↑fb.range

Tactic:
AlgHom.coe_range,

Goal state after:
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝⁴ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝³ : IsDomain A
this✝² : IsDomain B
this✝¹ : IsDomain R
this✝ : Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R R A B
this : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
⊢ a ⊗ₜ[R] 1 ∈ Set.range ⇑fa ∧ a ⊗ₜ[R] 1 ∈ Set.range ⇑fb
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✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
hf : ∀ (i : ι), l i ^ q ^ f i ∈ (algebraMap F E).range
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
i : ι
hs : i ∉ l.support
⊢ 0 = l i ^ q ^ n

Tactic:
Finsupp.notMem_support_iff.1 hs,

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✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
hf : ∀ (i : ι), l i ^ q ^ f i ∈ (algebraMap F E).range
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
i : ι
hs : i ∉ l.support
⊢ 0 = 0 ^ q ^ n
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
K : Type u_1
inst✝² : Field K
inst✝¹ : Algebra F K
inst✝ : IsDomain (E ⊗[F] K)
halg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F K
L : Type (max v u_1)
w✝¹ : Field L
w✝ : Algebra F L
fa : E →ₐ[F] L
fb : K →ₐ[F] L
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
H : fa.range.LinearDisjoint fb.range
⊢ Algebra.IsAlgebraic F ↥fa.fieldRange ∨ Algebra.IsAlgebraic F ↥fb.fieldRange

Tactic:
by
        rwa [(AlgEquiv.ofInjective fa hfa).isAlgebraic_iff,
          (AlgEquiv.ofInjective fb hfb).isAlgebraic_iff] at halg

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✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
lF : ι → F
hlF : ∀ (i : ι), (algebraMap F E) (lF i) = l i ^ q ^ n
i : ι
hs : i ∉ l.support
⊢ l i ^ q ^ n = 0

Tactic:
Finsupp.notMem_support_iff.1 hs,

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✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
lF : ι → F
hlF : ∀ (i : ι), (algebraMap F E) (lF i) = l i ^ q ^ n
i : ι
hs : i ∉ l.support
⊢ 0 ^ q ^ n = 0
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
h : PerfectField E
eq : perfectClosure F E = ⊥
p : ℕ := ringExpChar F
this : ExpChar E p
x : F
⊢ ∃ a, (frobenius F p) a = x

Tactic:
by
    obtain ⟨y, h⟩ := surjective_frobenius E p (algebraMap F E x)
    have : y ∈ perfectClosure F E := ⟨1, x, by rw [← h, pow_one, frobenius_def, ringExpChar.eq F p]⟩
    obtain ⟨z, rfl⟩ := eq ▸ this
    simp only [Algebra.ofId] at h
    exact ⟨z, (algebraMap F E).injective (by rw [RingHom.map_frobenius]; rw [h])⟩

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
f : K⟮X⟯
hf : ¬∃ c, f = C c
f_ne_zero : f ≠ 0
H : f⁻¹ * C f.num.leadingCoeff = f⁻¹ * (f * C (f.denom.coeff f.num.natDegree))
⊢ False

Tactic:
← mul_assoc,

Goal state after:
K : Type u_1
inst✝ : Field K
f : K⟮X⟯
hf : ¬∃ c, f = C c
f_ne_zero : f ≠ 0
H : f⁻¹ * C f.num.leadingCoeff = f⁻¹ * f * C (f.denom.coeff f.num.natDegree)
⊢ False
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝¹ : Field E
L : Type w
inst✝ : Field L
A : Subfield E
f : L →+* E
B : Subfield L
⊢ A ⊓ f.fieldRange ⊓ map f B = A ⊓ map f B

Tactic:
f.fieldRange_eq_map,

Goal state after:
E : Type v
inst✝¹ : Field E
L : Type w
inst✝ : Field L
A : Subfield E
f : L →+* E
B : Subfield L
⊢ A ⊓ map f ⊤ ⊓ map f B = A ⊓ map f B
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ Polynomial.C ((algebraMap K[X] K⟮X⟯) (Φ' E).content)⁻¹ *
      (Polynomial.C ((algebraMap K[X] K⟮X⟯) (Φ' E).content) * Polynomial.map (algebraMap K[X] K⟮X⟯) (Φ' E).primPart) =
    Polynomial.map (algebraMap K[X] K⟮X⟯) (Φ E)

Tactic:
← mul_assoc,

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ Polynomial.C ((algebraMap K[X] K⟮X⟯) (Φ' E).content)⁻¹ * Polynomial.C ((algebraMap K[X] K⟮X⟯) (Φ' E).content) *
      Polynomial.map (algebraMap K[X] K⟮X⟯) (Φ' E).primPart =
    Polynomial.map (algebraMap K[X] K⟮X⟯) (Φ E)
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ (algebraMap K[X] K⟮X⟯) ((Φ E).coeff (φ E).natDegree) = c E

Tactic:
by
  have := congr($(C_c_mul_φ h).coeff (φ E).natDegree)
  rw [coeff_C_mul, coeff_map, coeff_map, coeff_natDegree, IntermediateField.algebraMap_apply,
    φ_monic h, OneMemClass.coe_one, mul_one] at this
  exact this.symm

Goal state after: