mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
K : Type u_3
inst✝³ : CommRing K
inst✝² : Algebra R K
inst✝¹ : Algebra K A
inst✝ : IsScalarTower R K A
hinj : Injective ⇑(algebraMap R K)
ai : AlgebraicIndependent K x
this : (aeval x).comp (MvPolynomial.map (algebraMap R K)) = (aeval x).toRingHom
⊢ Injective (⇑(aeval x).toRingHom ∘ ⇑(MvPolynomial.map (algebraMap R K)))

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
K : Type u_3
inst✝³ : CommRing K
inst✝² : Algebra R K
inst✝¹ : Algebra K A
inst✝ : IsScalarTower R K A
hinj : Injective ⇑(algebraMap R K)
ai : AlgebraicIndependent K x
this : (aeval x).comp (MvPolynomial.map (algebraMap R K)) = (aeval x).toRingHom
⊢ Injective (⇑(aeval x).toRingHom ∘ ⇑(MvPolynomial.map (algebraMap R K)))
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
s : Set ι
⊢ AlgebraicIndependent R x ↔
    (AlgebraicIndependent R fun i ↦ x ↑i) ∧ AlgebraicIndepOn (↥(adjoin R (range fun i ↦ x ↑i))) x sᶜ

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
s : Set ι
⊢ AlgebraicIndependent R x ↔
    (AlgebraicIndependent R fun i ↦ x ↑i) ∧ AlgebraicIndepOn (↥(adjoin R (range fun i ↦ x ↑i))) x sᶜ
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
I : Set α
hI : I ⊆ M.E
h : ∀ ⦃J : Set α⦄, J ⊂ I → M.closure J ⊂ M.closure I
e : α
heI : e ∈ I
hecl : e ∈ M.closure (I \ {e})
⊢ M.closure (insert e (I \ {e}) \ {e}) = M.closure (M.closure (I \ {e}))

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
I : Set α
hI : I ⊆ M.E
h : ∀ ⦃J : Set α⦄, J ⊂ I → M.closure J ⊂ M.closure I
e : α
heI : e ∈ I
hecl : e ∈ M.closure (I \ {e})
⊢ M.closure (insert e (I \ {e}) \ {e}) = M.closure (M.closure (I \ {e}))
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X I : Set α
hI : M.Indep I
hX : (I ∩ X).Nonempty
h : I ∩ X = ∅
⊢ False

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X I : Set α
hI : M.Indep I
hX : (I ∩ X).Nonempty
h : I ∩ X = ∅
⊢ False
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f X ∩ M.E)
⊢ f ∈ M.closure (insert e X ∩ M.E)

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f X ∩ M.E)
⊢ f ∈ M.closure (insert e X ∩ M.E)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (X ∩ M.E)
hfE : f ∉ M.E
⊢ False

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (X ∩ M.E)
hfE : f ∉ M.E
⊢ False
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e X ∩ M.E)

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e X ∩ M.E)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e (X ∩ M.E))

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∉ M.closure (X ∩ M.E)
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
⊢ f ∈ M.closure (insert e (X ∩ M.E))
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ f ∈ M.closure (insert e (X ∩ M.E))

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
hef : e ∈ M.closure (insert f (X ∩ M.E))
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ f ∈ M.closure (insert e (X ∩ M.E))
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
p : MvPolynomial ι R
hp : (aeval x) p = 0
this : (aeval fun i ↦ ↑g (x i)) ((MvPolynomial.map ↑f) p) = 0
⊢ (aeval (⇑g ∘ x)) ((MvPolynomial.map ↑f) p) = 0

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
p : MvPolynomial ι R
hp : (aeval x) p = 0
this : (aeval fun i ↦ ↑g (x i)) ((MvPolynomial.map ↑f) p) = 0
⊢ (aeval (⇑g ∘ x)) ((MvPolynomial.map ↑f) p) = 0
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
s : Set ι
i : ι
hi : i ∉ s
⊢ Transcendental (↥(adjoin R (x '' s))) (x ↑⟨i, ⋯⟩) ↔ Transcendental (↥(adjoin R (x '' s))) (x i)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ M.Dep (insert f (insert e I)) ∨ f ∈ insert e I

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ M.Dep (insert f (insert e I)) ∨ f ∈ insert e I
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
hx : AlgebraicIndependent R x
inst✝ : Nontrivial A
s : Set ι
i : ι
⊢ Transcendental (↥(adjoin R (x '' s))) (x i) ↔ Disjoint s {i}

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
hx : AlgebraicIndependent R x
inst✝ : Nontrivial A
s : Set ι
i : ι
⊢ Transcendental (↥(adjoin R (x '' s))) (x i) ↔ Disjoint s {i}
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ M.Indep (insert e (insert f I)) → f ∈ insert e I

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e ∈ M.closure (insert f I)
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
⊢ M.Indep (insert e (insert f I)) → f ∈ insert e I
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: case e'_1
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
hx : AlgebraicIndependent R x
inst✝ : Nontrivial A
s : Set ι
i : ι
⊢ Transcendental (↥(adjoin R (x '' s))) (x ↑⟨i, ⋯⟩) ↔ Transcendental (↥(adjoin R (x '' s))) (x i)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e = f
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
h : M.Indep (insert e (insert f I))
⊢ f ∈ insert e I

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
hfE : f ∈ M.E
heE : e ∈ M.E
I : Set α
hef : e = f
he : M.Indep (insert e I) ∧ e ∉ I
hI : M.IsBasis I (X ∩ M.E)
h : M.Indep (insert e (insert f I))
⊢ f ∈ insert e I
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
p : MvPolynomial ι R
hp : (aeval x) p = 0
this : (MvPolynomial.map ↑f) p = 0
⊢ (MvPolynomial.map ↑f) p = 0

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
hf : Injective ⇑f
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
p : MvPolynomial ι R
hp : (aeval x) p = 0
this : (MvPolynomial.map ↑f) p = 0
⊢ (MvPolynomial.map ↑f) p = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : False
hf : f ∈ M.closure X
⊢ False

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
X : Set α
e f : α
he : False
hf : f ∈ M.closure X
⊢ False
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
e f : α
he : e ∈ M.closure (insert f X) \ M.closure X
hf : f ∈ M.closure (insert e X) \ M.closure X
⊢ M.closure (insert e X) = M.closure (insert e X)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
⊢ ∀ (p : MvPolynomial ι S), (aeval (⇑g ∘ x)) p = 0 → p = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X Y : Set α
h : Y ⊆ M.closure (X \ Y)
⊢ M.closure (X \ Y) = M.closure (X \ Y)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ M.closure S = M.E ↔ M.closure S = M.E

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: R : Type u_3
S : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Nontrivial R
inst✝¹ : FaithfulSMul R S
inst✝ : FaithfulSMul S A
x✝¹ : { s // AlgebraicIndepOn R _root_.id s }
s : Set S
hs : AlgebraicIndepOn R _root_.id s
x✝ : { s // AlgebraicIndepOn S _root_.id s }
t : Set A
ht : AlgebraicIndepOn S _root_.id t
this : AlgebraicIndependent R (Sum.elim (fun i ↦ _root_.id ↑i) (⇑(algebraMap S A) ∘ fun i ↦ _root_.id ↑i))
⊢ #(↑↑⟨t, ht⟩ ⊕ ↑↑⟨s, hs⟩) ≤ #(↑t ⊕ ↑s)

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ M.Spanning S ↔ M.Spanning S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: R : Type u_3
S : Type u
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : Nontrivial R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : FaithfulSMul R S
inst✝¹ : FaithfulSMul S A
inst✝ : IsScalarTower R S A
⊢ Cardinal.lift.{u, u} (trdeg R S) + Cardinal.lift.{u, u} (trdeg S A) ≤ Cardinal.lift.{u, u} (trdeg R A)

Tactic:
]

Goal state after:
R : Type u_3
S : Type u
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : Nontrivial R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : FaithfulSMul R S
inst✝¹ : FaithfulSMul S A
inst✝ : IsScalarTower R S A
⊢ Cardinal.lift.{u, u} (trdeg R S) + Cardinal.lift.{u, u} (trdeg S A) ≤ Cardinal.lift.{u, u} (trdeg R A)
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
inst✝ : CommRing R
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x : ι → MvPolynomial ι R := fun i ↦ (Polynomial.aeval (X i)) (f i)
t : Set ι
x✝¹ : t.Finite
x✝ : AlgebraicIndependent R fun i ↦ x ↑i
i : ι
hi : i ∉ t
⊢ t ⊆ x ⁻¹' ↑(supported R t)

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_3
inst✝ : CommRing R
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x : ι → MvPolynomial ι R := fun i ↦ (Polynomial.aeval (X i)) (f i)
t : Set ι
x✝¹ : t.Finite
x✝ : AlgebraicIndependent R fun i ↦ x ↑i
i : ι
hi : i ∉ t
⊢ t ⊆ x ⁻¹' ↑(supported R t)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ M.E ⊆ M.closure S ↔ M.E ⊆ M.closure S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
inst✝ : CommRing R
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x : ι → MvPolynomial ι R := fun i ↦ (Polynomial.aeval (X i)) (f i)
t : Set ι
x✝¹ : t.Finite
x✝ : AlgebraicIndependent R fun i ↦ x ↑i
i : ι
hi : i ∉ t
a✝ : ι
h : a✝ ∈ t
⊢ x a✝ ∈ ↑(supported R t)

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_3
inst✝ : CommRing R
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x : ι → MvPolynomial ι R := fun i ↦ (Polynomial.aeval (X i)) (f i)
t : Set ι
x✝¹ : t.Finite
x✝ : AlgebraicIndependent R fun i ↦ x ↑i
i : ι
hi : i ∉ t
a✝ : ι
h : a✝ ∈ t
⊢ x a✝ ∈ ↑(supported R t)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ ¬M.closure S = M.E → True

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ ¬M.closure S = M.E → True
Module: Mathlib.RingTheory.AlgebraicIndependent.Transcendental

Goal state before:
pp: ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x✝ : ι
⊢ (Polynomial.aeval (x x✝)) (f x✝) = ((aeval x).comp (Polynomial.aeval (X x✝))) (f x✝)

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
f : ι → Polynomial R
hf : ∀ (i : ι), Transcendental R (f i)
x✝ : ι
⊢ (Polynomial.aeval (x x✝)) (f x✝) = ((aeval x).comp (Polynomial.aeval (X x✝))) (f x✝)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S T : Set α
hS : M.Spanning S
hST : S ⊆ T
hT : T ⊆ M.E
⊢ M.closure S ⊆ M.closure T

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S T : Set α
hS : M.Spanning S
hST : S ⊆ T
hT : T ⊆ M.E
⊢ M.closure S ⊆ M.closure T
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S T : Set α
hS : M.Spanning S
hST : S ⊆ T
⊢ M.Spanning (T ∩ M.E)

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S T : Set α
hS : M.Spanning S
hST : S ⊆ T
⊢ M.Spanning (T ∩ M.E)
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
h : M.Spanning S
B : Set α
hB : M.IsBasis B S
hB' : M.IsBase B
⊢ ∃ B, M.IsBase B ∧ B ⊆ S

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S : Set α
h : M.Spanning S
B : Set α
hB : M.IsBasis B S
hB' : M.IsBase B
⊢ ∃ B, M.IsBase B ∧ B ⊆ S
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : S ⊆ M.E
⊢ (∃ B, M.IsBase B ∧ B ⊆ S) ↔ ∃ B, M.IsBase B ∧ B ⊆ S

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S : Set α
hS : ∃ B, M.IsBase B ∧ B ⊆ S
⊢ ∃ B, M.IsBase B ∧ B ⊆ S

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S : Set α
hS : ∃ B, M.IsBase B ∧ B ⊆ S
⊢ ∃ B, M.IsBase B ∧ B ⊆ S
Module: Mathlib.RingTheory.Polynomial.SeparableDegree

Goal state before:
pp: case h
F : Type u_1
inst✝ : CommSemiring F
q : ℕ
g : F[X]
m : ℕ
hf : IsSeparableContraction q ((expand F (q ^ m)) g) g
⊢ g.natDegree * q ^ m = g.natDegree * q ^ m

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.SeparableDegree

Goal state before:
pp: F : Type u_1
inst✝ : CommSemiring F
f : F[X]
hf : HasSeparableContraction 1 f
a : ℕ
ha : hf.degree * 1 ^ a = f.natDegree
⊢ hf.degree = hf.degree

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.SeparableDegree

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
f : F[X]
irred : Irreducible f
inst✝ : CharZero F
⊢ f = f

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
I : Set α
hI : I ⊆ M.E
⊢ (∃ B, M.IsBase B ∧ B ⊆ M.E \ I) ↔ ∃ B, M.IsBase B ∧ B ⊆ M.E \ I

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
q : MvPolynomial ι R
hp : (aeval (⇑g ∘ x)) ((MvPolynomial.map ↑f) q) = 0
⊢ (aeval fun i ↦ ↑g (x i)) ((MvPolynomial.map ↑f) q) = 0

Tactic:
]

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
q : MvPolynomial ι R
hp : (aeval (⇑g ∘ x)) ((MvPolynomial.map ↑f) q) = 0
⊢ (aeval fun i ↦ ↑g (x i)) ((MvPolynomial.map ↑f) q) = 0
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X : Set α
hK : X ⊆ M.E
⊢ M.closure (M.E \ X) = M.E ↔ M.closure (M.E \ X) = M.E

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
I : Set α
hI : M.Indep I
hIs : M.Spanning I
B : Set α
hB : M.IsBase B
hBI : B ⊆ I
⊢ M.IsBase B

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
I : Set α
hI : M.Indep I
hIs : M.Spanning I
B : Set α
hB : M.IsBase B
hBI : B ⊆ I
⊢ M.IsBase B
Module: Mathlib.RingTheory.Polynomial.SeparableDegree

Goal state before:
pp: F✝ : Type u_1
inst✝² : Field F✝
q✝ : ℕ
F : Type u_1
inst✝¹ : Field F
q : ℕ
hq : NeZero q
inst✝ : CharP F q
g g' : F[X]
m : ℕ
hg : g.Separable
hg' : g'.Separable
s : ℕ
h_expand : g = (expand F (q ^ s)) g'
hm : m ≤ m + s
⊢ g.natDegree = g'.natDegree

Tactic:
]

Goal state after:
F✝ : Type u_1
inst✝² : Field F✝
q✝ : ℕ
F : Type u_1
inst✝¹ : Field F
q : ℕ
hq : NeZero q
inst✝ : CharP F q
g g' : F[X]
m : ℕ
hg : g.Separable
hg' : g'.Separable
s : ℕ
h_expand : g = (expand F (q ^ s)) g'
hm : m ≤ m + s
⊢ g.natDegree = g'.natDegree
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
X I : Set α
hIX : M.IsBasis I X
⊢ M.closure X = M.E ↔ M.closure X = M.E

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S B : Set α
hS : M.Spanning S
⊢ M.IsBasis B S ↔ M.IsBase B ∧ B ⊆ S

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S B : Set α
hS : M.Spanning S
⊢ M.IsBasis B S ↔ M.IsBase B ∧ B ⊆ S
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
S : Type u_3
B : Type u_4
FRS : Type u_5
FAB : Type u_6
inst✝⁶ : CommRing S
inst✝⁵ : CommRing B
inst✝⁴ : Algebra S B
inst✝³ : FunLike FRS R S
inst✝² : RingHomClass FRS R S
inst✝¹ : FunLike FAB A B
inst✝ : RingHomClass FAB A B
f : FRS
g : FAB
H : ∀ (p : MvPolynomial ι R), (aeval x) p = 0 → p = 0
hf : Surjective ⇑f
hg : Injective ⇑g
h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)
q : MvPolynomial ι R
hp : (aeval (⇑g ∘ x)) ((MvPolynomial.map ↑f) q) = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Combinatorics.Matroid.Closure

Goal state before:
pp: α : Type u_2
M : Matroid α
S B : Set α
hS : M.Spanning S
h : M.IsBasis B S
⊢ M.Spanning S

Tactic:
]

Goal state after:
α : Type u_2
M : Matroid α
S B : Set α
hS : M.Spanning S
h : M.IsBasis B S
⊢ M.Spanning S