mathlib-initiative/mathlib-tactics huggingface dataset:
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
ι : Type u_1
a : Basis ι F ↥A
H : LinearIndependent L (⇑A.val ∘ ⇑a)
x✝¹ : L
x✝ : E
⊢ (AddMonoidHom.id E) ((algebraMap L E) x✝¹ * x✝) =
    (algebraMap (↥(IsScalarTower.toAlgHom F L E).range) E)
        ((AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)) x✝¹) *
      (AddMonoidHom.id E) x✝

Tactic:
;

Goal state after:
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
ι : Type u_1
a : Basis ι F ↥A
H : LinearIndependent L (⇑A.val ∘ ⇑a)
x✝¹ : L
x✝ : E
⊢ (AddMonoidHom.id E) ((algebraMap L E) x✝¹ * x✝) =
    (algebraMap (↥(IsScalarTower.toAlgHom F L E).range) E)
        ((AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)) x✝¹) *
      (AddMonoidHom.id E) x✝
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝¹³ : Field F
inst✝¹² : Field E
inst✝¹¹ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝¹⁰ : Field L
inst✝⁹ : Algebra F L
inst✝⁸ : Algebra L E
inst✝⁷ : IsScalarTower F L E
H : A.LinearDisjoint L
L' : Type u_1
inst✝⁶ : Field L'
inst✝⁵ : Algebra F L'
inst✝⁴ : Algebra L' L
inst✝³ : IsScalarTower F L' L
inst✝² : Algebra L' E
inst✝¹ : IsScalarTower F L' E
inst✝ : IsScalarTower L' L E
x✝ : L'
⊢ (IsScalarTower.toAlgHom F L' E) x✝ = ((IsScalarTower.toAlgHom F L E).comp (IsScalarTower.toAlgHom F L' L)) x✝

Tactic:
;

Goal state after:
F : Type u
E : Type v
inst✝¹³ : Field F
inst✝¹² : Field E
inst✝¹¹ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝¹⁰ : Field L
inst✝⁹ : Algebra F L
inst✝⁸ : Algebra L E
inst✝⁷ : IsScalarTower F L E
H : A.LinearDisjoint L
L' : Type u_1
inst✝⁶ : Field L'
inst✝⁵ : Algebra F L'
inst✝⁴ : Algebra L' L
inst✝³ : IsScalarTower F L' L
inst✝² : Algebra L' E
inst✝¹ : IsScalarTower F L' E
inst✝ : IsScalarTower L' L E
x✝ : L'
⊢ (IsScalarTower.toAlgHom F L' E) x✝ = ((IsScalarTower.toAlgHom F L E).comp (IsScalarTower.toAlgHom F L' L)) x✝
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: case e'_2
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
⊢ (algebraMap R[X] B) X = b

Tactic:
;

Goal state after:
case e'_2
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
⊢ (algebraMap R[X] B) X = b
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: case e'_3
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
⊢ (algebraMap R[X] A) X = a

Tactic:
;

Goal state after:
case e'_3
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
⊢ (algebraMap R[X] A) X = a
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: case h
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
x : E
n : ℕ
hx : x ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS
⊢ x⁻¹ ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS

Tactic:
;

Goal state after:
case h
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
x : E
n : ℕ
hx : x ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS
⊢ x⁻¹ ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
inst✝⁶ : Field F
A : Type v
inst✝⁵ : Field A
B : Type w
inst✝⁴ : Field B
inst✝³ : Algebra F A
inst✝² : Algebra F B
H : IsField (A ⊗[F] B)
K : Type u_1
inst✝¹ : Field K
inst✝ : Algebra F K
fa : A →ₐ[F] K
fb : B →ₐ[F] K
⊢ fa.fieldRange.LinearDisjoint ↥fb.fieldRange

Tactic:
rw [linearDisjoint_iff']

Goal state after:
F : Type u
inst✝⁶ : Field F
A : Type v
inst✝⁵ : Field A
B : Type w
inst✝⁴ : Field B
inst✝³ : Algebra F A
inst✝² : Algebra F B
H : IsField (A ⊗[F] B)
K : Type u_1
inst✝¹ : Field K
inst✝ : Algebra F K
fa : A →ₐ[F] K
fb : B →ₐ[F] K
⊢ fa.fieldRange.LinearDisjoint fb.fieldRange.toSubalgebra
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
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
E' : IntermediateField F E := adjoin F (Set.range v)
y : ι
⊢ IsSeparable F (v y)

Tactic:
;

Goal state after:
F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
E' : IntermediateField F E := adjoin F (Set.range v)
y : ι
⊢ IsSeparable F (v y)
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM)).range
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
x✝ : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hx : x✝ ∈ {x, y}
⊢ ↑s ⊆ ↑(toSubmodule A')

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM)).range
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
x✝ : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hx : x✝ ∈ {x, y}
⊢ ↑s ⊆ ↑(toSubmodule A')
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 z : F
h : (frobenius E p) ((algebraMap F E) z) = (algebraMap F E) x
this : (Algebra.ofId F E).toRingHom z ∈ perfectClosure F E
⊢ (frobenius E p) ((algebraMap F E) z) = (algebraMap F E) x

Tactic:
;

Goal state after:
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 z : F
h : (frobenius E p) ((algebraMap F E) z) = (algebraMap F E) x
this : (Algebra.ofId F E).toRingHom z ∈ perfectClosure F E
⊢ (frobenius E p) ((algebraMap F E) z) = (algebraMap F E) x
Module: Mathlib.FieldTheory.RatFunc.IntermediateField

Goal state before:
pp: K : Type u_1
inst✝ : Field K
f : K⟮X⟯
hf : ↑((f.minpolyX ↥K⟮f⟯).coeff f.denom.natDegree) = 0
⊢ ∃ c, f = C c

Tactic:
use f.num.coeff f.denom.natDegree / f.denom.leadingCoeff
  rw [map_div₀, eq_div_iff ((_root_.map_ne_zero C).mpr
    (leadingCoeff_ne_zero.mpr f.denom_ne_zero)), eq_comm]
  simpa [sub_eq_zero] using hf

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
c : K
⊢ (C c).minpolyX ↥K⟮C c⟯ = 0

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
c : K
⊢ (C c).minpolyX ↥K⟮C c⟯ = 0
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.IsSeparable F E
⊢ Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Module.rank E ↥(separableClosure E K)) =
    Cardinal.lift.{v, w} (Module.rank F ↥(separableClosure F K))

Tactic:
separableClosure.eq_restrictScalars_of_isSeparable F 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.IsSeparable F E
⊢ Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Module.rank E ↥(separableClosure E K)) =
    Cardinal.lift.{v, w} (Module.rank F ↥(restrictScalars F (separableClosure E K)))
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)).range
x' : ↥(toSubmodule A') ⊗[R] ↥(toSubmodule B)
hx' : (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) x' = x
y' : ↥(toSubmodule A') ⊗[R] ↥(toSubmodule B)
hy' : (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) y' = y
⊢ (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) x' =
    (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) y'

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)).range
x' : ↥(toSubmodule A') ⊗[R] ↥(toSubmodule B)
hx' : (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) x' = x
y' : ↥(toSubmodule A') ⊗[R] ↥(toSubmodule B)
hy' : (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) y' = y
⊢ (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) x' =
    (LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)) y'
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.IsSeparable F E
⊢ Module.rank (↥(separableClosure F K)) K = Module.rank (↥(separableClosure E K)) K

Tactic:
separableClosure.eq_restrictScalars_of_isSeparable F 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.IsSeparable F E
⊢ Module.rank (↥(restrictScalars F (separableClosure E K))) K = Module.rank (↥(separableClosure E K)) K
Module: Mathlib.FieldTheory.PurelyInseparable.Tower

Goal state before:
pp: F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := this✝².toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree F ↥M * sepDegree ↥M ↥L = sepDegree F ↥M

Tactic:
IsPurelyInseparable.sepDegree_eq_one M L,

Goal state after:
F : Type u
E : Type v
inst✝⁸ : Field F
inst✝⁷ : Field E
inst✝⁶ : Algebra F E
K : Type w
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
S : Set K
inst✝¹ : IsPurelyInseparable F E
M : IntermediateField F K := adjoin F S
inst✝ : Algebra.IsAlgebraic F ↥M
L : IntermediateField E K := adjoin E S
E' : IntermediateField F K := (IsScalarTower.toAlgHom F E K).fieldRange
j : E ≃ₐ[F] ↥E' := AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K)
hi : M ≤ restrictScalars F L
i : ↥M →+* ↥L := Subsemiring.inclusion hi
this✝² : Algebra ↥M ↥L := i.toAlgebra
this✝¹ : SMul ↥M ↥L := this✝².toSMul
this✝ : IsScalarTower F ↥M ↥L
this : IsPurelyInseparable ↥M ↥L
h : Cardinal.lift.{v, w} (sepDegree E ↥L) = Cardinal.lift.{v, w} (sepDegree F ↥L)
⊢ sepDegree F ↥M * 1 = sepDegree F ↥M
Module: Mathlib.FieldTheory.RatFunc.IntermediateField

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
c : K
hf₁ : C c ∈ E
hf₂ : C c ∉ ⊥
⊢ False

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
c : K
hf₁ : C c ∈ E
hf₂ : C c ∉ ⊥
⊢ False
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
C : PointedCone R N
s : Finset M
hs : dual p ↑s = C
⊢ ∃ D, Submodule.FG D ∧ dual p ↑D = C

Tactic:
;

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
C : PointedCone R N
s : Finset M
hs : dual p ↑s = C
⊢ ∃ D, Submodule.FG D ∧ dual p ↑D = C
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
C : PointedCone R N
hC : DualFG p C
⊢ ∃ D, Submodule.FG D ∧ dual p ↑D = C

Tactic:
by
  obtain ⟨s, hs⟩ := hC; exact ⟨_, Submodule.fg_span s.finite_toSet, by simp [hs]⟩

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s : Finset M
⊢ dual p ↑s = dual p ↑(Submodule.span { c // 0 ≤ c } ↑s)

Tactic:
;

Goal state after:
case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s : Finset M
⊢ dual p ↑s = dual p ↑(Submodule.span { c // 0 ≤ c } ↑s)
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A B : Subfield E
⊢ (A ⊓ B).relrank A = B.relrank A

Tactic:
inf_comm,

Goal state after:
E : Type v
inst✝ : Field E
A B : Subfield E
⊢ (B ⊓ A).relrank A = B.relrank A
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
D : PointedCone R N
hD : DualFG p D
S : Finset M
⊢ DualFG p (dual p ↑S ⊓ D)

Tactic:
;

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
D : PointedCone R N
hD : DualFG p D
S : Finset M
⊢ DualFG p (dual p ↑S ⊓ D)
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
S T : Finset M
⊢ dual p ↑(S ∪ T) = dual p ↑S ⊓ dual p ↑T

Tactic:
;

Goal state after:
case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
S T : Finset M
⊢ dual p ↑(S ∪ T) = dual p ↑S ⊓ dual p ↑T
Module: Mathlib.FieldTheory.Relrank

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

Tactic:
;

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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
f : K
hf : (algebraMap K K⟮X⟯) f = ↑((φ E).coeff (generatorIndex h))
⊢ ↑((algebraMap K ↥E) f) = ↑((φ E).coeff ⋯.choose)

Tactic:
;

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
f : K
hf : (algebraMap K K⟮X⟯) f = ↑((φ E).coeff (generatorIndex h))
⊢ ↑((algebraMap K ↥E) f) = ↑((φ E).coeff ⋯.choose)
Module: Mathlib.FieldTheory.RatFunc.Luroth

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

Tactic:
rw [coeff_C_mul, coeff_map, coeff_map, coeff_natDegree, IntermediateField.algebraMap_apply,
    φ_monic h, OneMemClass.coe_one, mul_one] at this

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
this : c E = (algebraMap K[X] K⟮X⟯) ((Φ E).coeff (φ E).natDegree)
⊢ (algebraMap K[X] K⟮X⟯) ((Φ E).coeff (φ E).natDegree) = c E
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C F : PointedCone R M
x y : M
a b : R
hF : F.IsFaceOf C
hx : x ∈ C
hy : y ∈ C
ha : 0 < a
hb : 0 < b
h : a • x + b • y ∈ F
⊢ b • y + a • x ∈ F

Tactic:
;

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C F : PointedCone R M
x y : M
a b : R
hF : F.IsFaceOf C
hx : x ∈ C
hy : y ∈ C
ha : 0 < a
hb : 0 < b
h : a • x + b • y ∈ F
⊢ b • y + a • x ∈ F