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

Goal state before:
pp: case cons
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
ι : Type u_2
f : ι → R
a✝ : ι
s✝ : Multiset ι
h₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum
h : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0
⊢ logDeriv (f a✝ * (Multiset.map f s✝).prod) = logDeriv (f a✝) + (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum

Tactic:
rw [← h₂]

Goal state after:
case cons
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
ι : Type u_2
f : ι → R
a✝ : ι
s✝ : Multiset ι
h₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum
h : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0
⊢ logDeriv (f a✝ * (Multiset.map f s✝).prod) = logDeriv (f a✝) + logDeriv (Multiset.map f s✝).prod

case cons
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
ι : Type u_2
f : ι → R
a✝ : ι
s✝ : Multiset ι
h₂ : (∀ x ∈ s✝, f x ≠ 0) → logDeriv (Multiset.map f s✝).prod = (Multiset.map (fun x ↦ logDeriv (f x)) s✝).sum
h : ∀ x ∈ a✝ ::ₘ s✝, f x ≠ 0
⊢ ∀ x ∈ s✝, f x ≠ 0
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.Finite K (R σ K)

Tactic:
← IsNoetherian.iff_fg,

Goal state after:
case intro
σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
val✝ : Fintype σ
⊢ IsNoetherian K (R σ K)
Module: Mathlib.FieldTheory.Finite.Polynomial

Goal state before:
pp: σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
p : MvPolynomial σ K
h : ∀ (v : σ → K), (eval v) p = 0
hp : p ∈ restrictDegree σ K (Fintype.card K - 1)
p' : R σ K := ⟨p, hp⟩
this : p' ∈ ⊥
⊢ p' = 0

Tactic:
mem_bot

Goal state after:
σ K : Type u
inst✝² : Fintype K
inst✝¹ : Field K
inst✝ : Finite σ
p : MvPolynomial σ K
h : ∀ (v : σ → K), (eval v) p = 0
hp : p ∈ restrictDegree σ K (Fintype.card K - 1)
p' : R σ K := ⟨p, hp⟩
this : p' = 0
⊢ p' = 0
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
c₀ : ι → F := fun i ↦ c i / ↑(Fintype.card Gal(K/F))
u₁ : ι → K := fun i ↦ ∏ x, x (u i)
ffb : fixedField ⊤ = ⊥
u₀ : ι → F
hu₀ : ∀ (i : ι), (algebraMap F K) (u₀ i) = u₁ i
v₁ : K := (∑ x, x v) / ↑(Fintype.card Gal(K/F))
v₀ : F
hv₀ : (algebraMap F K) v₀ = v₁
⊢ Function.Injective ⇑(algebraMap F K)

Tactic:
case =>

Goal state after:
Module: Mathlib.FieldTheory.Differential.Liouville

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁹ : Field F
inst✝⁸ : Field K
inst✝⁷ : Differential F
inst✝⁶ : Differential K
inst✝⁵ : Algebra F K
inst✝⁴ : DifferentialAlgebra F K
inst✝³ : CharZero F
inst✝² : FiniteDimensional F K
inst✝¹ : IsGalois F K
a : F
ι : Type
inst✝ : Fintype ι
c : ι → F
hc : ∀ (x : ι), (c x)′ = 0
u : ι → K
v : K
h : ↑a = ∑ x, ↑(c x) * logDeriv (u x) + v′
this : CharZero K
ffb : fixedField ⊤ = ⊥
⊢ (↑(Fintype.card Gal(K/F)))′ = 0

Tactic:
case =>

Goal state after:
Module: Mathlib.FieldTheory.Galois.NormalBasis

Goal state before:
pp: case e'_4
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
this✝ : Finite K
this : Fintype K
x : L
hx :
  Ideal.span {X ^ finrank K L - 1} =
    (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
        ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x)).ker
e_1✝ : Fin (X ^ finrank K L - C 1).natDegree = Fin (AdjoinRoot.powerBasis ⋯).dim
i : Fin (X ^ finrank K L - C 1).natDegree
⊢ (⇑(frobeniusAlgEquivOfAlgebraic K L))^[↑(Fin.cast ⋯ i)] x =
    (↑(AEval'.of (frobeniusAlgHom K L).toLinearMap).symm ∘ₗ
        ↑K
          (liftQ (Ideal.span {X ^ finrank K L - 1})
            (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
              ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x))
            ⋯))
      ((AdjoinRoot.powerBasis ⋯).basis i)

Tactic:
AdjoinRoot.powerBasis,

Goal state after:
case e'_4
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Finite L
this✝ : Finite K
this : Fintype K
x : L
hx :
  Ideal.span {X ^ finrank K L - 1} =
    (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
        ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x)).ker
e_1✝ : Fin (X ^ finrank K L - C 1).natDegree = Fin (AdjoinRoot.powerBasis ⋯).dim
i : Fin (X ^ finrank K L - C 1).natDegree
⊢ (⇑(frobeniusAlgEquivOfAlgebraic K L))^[↑(Fin.cast ⋯ i)] x =
    (↑(AEval'.of (frobeniusAlgHom K L).toLinearMap).symm ∘ₗ
        ↑K
          (liftQ (Ideal.span {X ^ finrank K L - 1})
            (toSpanSingleton K[X] (AEval' (frobeniusAlgHom K L).toLinearMap)
              ((AEval'.of (frobeniusAlgHom K L).toLinearMap) x))
            ⋯))
      ((AdjoinRoot.powerBasisAux ⋯) i)
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
hp : ¬1 < p
h : x = 0
⊢ x ^ p ^ 0 = 0

Tactic:
rw [pow_zero, pow_one, h]

Goal state after:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
p : ℕ
inst✝ : ExpChar R p
x y : R
⊢ (∃ n, (x - y) ^ p ^ n = 0) ↔ ∃ n, x ^ p ^ n = y ^ p ^ n

Tactic:
sub_pow_expChar_pow,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
p : ℕ
inst✝ : ExpChar R p
x y : R
⊢ (∃ n, x ^ p ^ n - y ^ p ^ n = 0) ↔ ∃ n, x ^ p ^ n = y ^ p ^ n
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: case refine_1
K : Type u_1
L : Type u_2
inst✝² : CommSemiring K
inst✝¹ : CommSemiring L
i : K →+* L
p : ℕ
inst✝ : IsPRadical i p
x : K
h : x ∈ Ideal.comap i (pNilradical L p)
⊢ ∃ n, x ^ p ^ n = 0

pp: case refine_2
K : Type u_1
L : Type u_2
inst✝² : CommSemiring K
inst✝¹ : CommSemiring L
i : K →+* L
p : ℕ
inst✝ : IsPRadical i p
x : K
h : x ∈ pNilradical K p
⊢ x ∈ Ideal.comap i (pNilradical L p)

Tactic:
· obtain ⟨n, h⟩ := mem_pNilradical.1 <| Ideal.mem_comap.1 h
    obtain ⟨m, h⟩ := mem_pNilradical.1 <| ker_le i p ((map_pow i x _).symm ▸ h)
    exact ⟨n + m, by rwa [pow_add, pow_mul]⟩

Goal state after:
case refine_2
K : Type u_1
L : Type u_2
inst✝² : CommSemiring K
inst✝¹ : CommSemiring L
i : K →+* L
p : ℕ
inst✝ : IsPRadical i p
x : K
h : x ∈ pNilradical K p
⊢ x ∈ Ideal.comap i (pNilradical L p)
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
x1 x2 : L
⊢ liftAux i j p (x1 + x2) = liftAux i j p x1 + liftAux i j p x2

Tactic:
obtain ⟨n1, y1, h1⟩ := IsPRadical.pow_mem i p x1
    obtain ⟨n2, y2, h2⟩ := IsPRadical.pow_mem i p x2
    rw [liftAux_apply i j p _ _ _ h1, liftAux_apply i j p _ _ _ h2,
      liftAux_apply i j p (x1 + x2) (n1 + n2) (y1 ^ p ^ n2 + y2 ^ p ^ n1) (by rw [map_add,
        map_pow, map_pow, h1, h2, ← pow_mul, ← pow_add, ← pow_mul, ← pow_add,
        add_comm n2, add_pow_expChar_pow]),
      map_add, map_pow, map_pow, map_add, ← iterateFrobeniusEquiv_def]
    nth_rw 1 [iterateFrobeniusEquiv_symm_add_apply]
    rw [RingEquiv.symm_apply_apply, add_comm n1, iterateFrobeniusEquiv_symm_add_apply,
      ← iterateFrobeniusEquiv_def, RingEquiv.symm_apply_apply]

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case mp
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
y : G
hy : y ∈ H
d : G
h : d ∈ D
⊢ ↑⟨y, hy⟩ * d ∈ H

Tactic:
exact H.mul_mem hy (hD_le_H h)

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
h : H.index = 0
⊢ False

Tactic:
rw [hind, Finset.card_eq_zero] at h
    rw [h, ← Finset.set_biUnion_coe, Finset.coe_empty, Set.biUnion_empty] at hcovers
    exact Set.empty_ne_univ hcovers

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
⊢ i ≠ j → Function.onFun Disjoint (fun x ↦ g x • ↑H) i j

Tactic:
h'

Goal state after:
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
i : ι
hi : i ∈ ↑s
j : ι
hj : j ∈ ↑s
h' : i ≠ j
⊢ Function.onFun Disjoint (fun x ↦ g x • ↑H) i j
Module: Mathlib.RingTheory.MvPolynomial

Goal state before:
pp: case inl
σ : Type u
K : Type v
inst✝¹ : CommRing K
inst✝ : IsEmpty σ
h✝ : Subsingleton K
⊢ Module.rank K (MvPolynomial σ K) = 1

Tactic:
simp [rank_eq_lift]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial

Goal state before:
pp: case inr
σ : Type u
K : Type v
inst✝¹ : CommRing K
inst✝ : IsEmpty σ
h✝ : Nontrivial K
⊢ Module.rank K (MvPolynomial σ K) = 1

Tactic:
simp [rank_eq_lift]

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝³ : DivisionRing k
inst✝² : Infinite k
inst✝¹ : AddCommGroup E
inst✝ : Module k E
s : Finset (Subspace k E)
hcovers : ⊤ ∉ s
⊢ ⋃ p ∈ s, ↑p ≠ Set.univ

Tactic:
exact Subspace.biUnion_ne_univ_of_top_notMem hcovers

Goal state after:
Module: Mathlib.FieldTheory.NormalizedTrace

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

Tactic:
normalizedTrace_def E K,

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

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

Tactic:
eq_bot_of_finrank_one H

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

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

Tactic:
eq_bot_of_finrank_one H

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

Goal state before:
pp: case refine_1
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
S : Set E
inst✝¹ : Algebra.IsSeparable F ↥(adjoin F S)
q : ℕ
inst✝ : ExpChar F q
n : ℕ
M : IntermediateField F E := adjoin F ((fun x ↦ x ^ q ^ n) '' S)
this✝¹ : ExpChar (↥M) q
x : E
hx : x ∈ S
this✝ : Algebra.IsSeparable ↥M ↥(↥M)⟮x⟯
this : IsPurelyInseparable ↥M ↥(↥M)⟮x⟯
hx' : x ∈ ⊥
⊢ x ∈ ↑M

Tactic:
mem_bot

Goal state after:
case refine_1
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
S : Set E
inst✝¹ : Algebra.IsSeparable F ↥(adjoin F S)
q : ℕ
inst✝ : ExpChar F q
n : ℕ
M : IntermediateField F E := adjoin F ((fun x ↦ x ^ q ^ n) '' S)
this✝¹ : ExpChar (↥M) q
x : E
hx : x ∈ S
this✝ : Algebra.IsSeparable ↥M ↥(↥M)⟮x⟯
this : IsPurelyInseparable ↥M ↥(↥M)⟮x⟯
hx' : x ∈ Set.range ⇑(algebraMap (↥M) E)
⊢ x ∈ ↑M
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
this : PerfectRing F p
⊢ PerfectField F

Tactic:
exact PerfectRing.toPerfectField F p

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A : Subfield E
⊢ A.relrank ⊤ = Module.rank (↥A) E

Tactic:
rw [relrank_eq_rank_of_le (show A ≤ ⊤ from le_top), extendScalars_top,
    IntermediateField.topEquiv.toLinearEquiv.rank_eq]

Goal state after:
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 ⊓ map f B).toSubring

Tactic:
rw [← map_inf]; rfl

Goal state after:
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 : L →+* E
h : B ≤ f.fieldRange
⊢ (comap f A).relfinrank (comap f B) = A.relfinrank B

Tactic:
simpa using! congr(toNat $(lift_relrank_comap_comap_eq_lift_relrank_of_le A B f h))

Goal state after:
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 : PointedCone R M
F : Set (PointedCone R M)
h : ∀ f ∈ F, f.IsFaceOf C
x✝ y✝ : M
a : R
xc : x✝ ∈ C
yc : y✝ ∈ C
a0 : 0 < a
a✝ : a • x✝ + y✝ ∈ C
⊢ (∀ p ∈ F, a • x✝ + y✝ ∈ p) → x✝ ∈ C ∧ ∀ p ∈ F, x✝ ∈ p

Tactic:
h'

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 : PointedCone R M
F : Set (PointedCone R M)
h : ∀ f ∈ F, f.IsFaceOf C
x✝ y✝ : M
a : R
xc : x✝ ∈ C
yc : y✝ ∈ C
a0 : 0 < a
a✝ : a • x✝ + y✝ ∈ C
h' : ∀ p ∈ F, a • x✝ + y✝ ∈ p
⊢ x✝ ∈ C ∧ ∀ p ∈ F, x✝ ∈ p
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
L : Type w
inst✝¹ : Field L
inst✝ : Algebra F L
A B : IntermediateField F E
f : L →ₐ[F] E
h : B ≤ f.fieldRange
⊢ (comap f A).relfinrank (comap f B) = A.relfinrank B

Tactic:
simpa using! congr(toNat $(lift_relrank_comap_comap_eq_lift_relrank_of_le A B f h))

Goal state after:
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))
⊢ (generator E).num.natDegree = 0 ∧ (generator E).denom.natDegree = 0

Tactic:
obtain ⟨isUnit₁, isUnit₂⟩ :
      IsUnit (Polynomial.C <| aeval α (f E)) ∧ IsUnit (Polynomial.C <| aeval α (g E)) := by
    rw [Polynomial.isUnit_C, isUnit_iff_ne_zero, Polynomial.isUnit_C, isUnit_iff_ne_zero]
    obtain (H | H) := aeval_ne_zero_of_isCoprime (generator E).isCoprime_num_denom α
    · refine ⟨H, Polynomial.C_injective.ne_iff.mp ?_⟩
      rw [map_zero, ← mul_ne_zero_iff_left <| Polynomial.map_ne_zero <|
        num_ne_zero (generator_ne_zero h)]
      exact eq ▸ mul_ne_zero (Polynomial.map_ne_zero (generator E).denom_ne_zero) <|
        Polynomial.C_ne_zero.mpr H
    · refine ⟨Polynomial.C_injective.ne_iff.mp ?_, H⟩
      rw [map_zero, ← mul_ne_zero_iff_left <| Polynomial.map_ne_zero (generator E).denom_ne_zero]
      exact eq ▸ mul_ne_zero (Polynomial.map_ne_zero (num_ne_zero (generator_ne_zero h))) <|
        Polynomial.C_ne_zero.mpr H

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))
isUnit₁ : IsUnit (Polynomial.C ((aeval α) (f E)))
isUnit₂ : IsUnit (Polynomial.C ((aeval α) (g E)))
⊢ (generator E).num.natDegree = 0 ∧ (generator E).denom.natDegree = 0
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
p : P
hp : p ∈ s
v : V
hv : v ∈ s.directionᗮ
⊢ ↑⟨p, hp⟩ ∈ s ∧ (v +ᵥ p) -ᵥ ↑⟨p, hp⟩ ∈ s.directionᗮ

Tactic:
by simp [*]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: case mp
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
hp₂ : p₂ ∈ s
this : Nonempty ↥s
h : dist p₁ ↑((orthogonalProjection s) p₁) * dist p₁ ↑((orthogonalProjection s) p₁) = dist p₁ p₂ * dist p₁ p₂
⊢ ↑((orthogonalProjection s) p₁) = p₂

Tactic:
dist_comm _ p₂,

Goal state after:
case mp
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
hp₂ : p₂ ∈ s
this : Nonempty ↥s
h : dist p₁ ↑((orthogonalProjection s) p₁) * dist p₁ ↑((orthogonalProjection s) p₁) = dist p₂ p₁ * dist p₂ p₁
⊢ ↑((orthogonalProjection s) p₁) = p₂
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: case mp.left
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
p : P
h : vectorSpan ℝ {p, s.points i} = (s.altitude i).direction
heq : p = s.points i
⊢ False

Tactic:
heq,

Goal state after:
case mp.left
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
p : P
h : vectorSpan ℝ {s.points i, s.points i} = (s.altitude i).direction
heq : p = s.points i
⊢ False
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: case mpr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
p : P
hne : p ≠ s.points i
h : p -ᵥ s.points i ∈ (s.altitude i).direction
⊢ Submodule.span ℝ ((fun x ↦ p -ᵥ x) '' ({p, s.points i} \ {p})) = (s.altitude i).direction

Tactic:
Set.insert_sdiff_of_mem _ (Set.mem_singleton _),

Goal state after:
case mpr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
p : P
hne : p ≠ s.points i
h : p -ᵥ s.points i ∈ (s.altitude i).direction
⊢ Submodule.span ℝ ((fun x ↦ p -ᵥ x) '' ({s.points i} \ {p})) = (s.altitude i).direction
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace 𝕜 V
V₂ : Type u_4
P₂ : Type u_5
inst✝⁸ : NormedAddCommGroup V₂
inst✝⁷ : InnerProductSpace 𝕜 V₂
inst✝⁶ : MetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : MetricSpace P₂
inst✝³ : NormedAddTorsor V₂ P₂
s : AffineSubspace 𝕜 P
inst✝² : Nonempty ↥s
inst✝¹ : s.direction.HasOrthogonalProjection
f : P →ᵃⁱ[𝕜] P₂
inst✝ : (map f.toAffineMap s).direction.HasOrthogonalProjection
p : P
v : V
hv : v ∈ s.direction
⊢ inner 𝕜 (f.linearIsometry.toLinearMap v) (f.linearIsometry.toLinearMap (p -ᵥ ↑((orthogonalProjection s) p))) = 0

Tactic:
LinearIsometry.coe_toLinearMap,

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace 𝕜 V
V₂ : Type u_4
P₂ : Type u_5
inst✝⁸ : NormedAddCommGroup V₂
inst✝⁷ : InnerProductSpace 𝕜 V₂
inst✝⁶ : MetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : MetricSpace P₂
inst✝³ : NormedAddTorsor V₂ P₂
s : AffineSubspace 𝕜 P
inst✝² : Nonempty ↥s
inst✝¹ : s.direction.HasOrthogonalProjection
f : P →ᵃⁱ[𝕜] P₂
inst✝ : (map f.toAffineMap s).direction.HasOrthogonalProjection
p : P
v : V
hv : v ∈ s.direction
⊢ inner 𝕜 (f.linearIsometry v) (f.linearIsometry (p -ᵥ ↑((orthogonalProjection s) p))) = 0
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: case a
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| ≤ s.height i * s.height j

pp: case a
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| ≠ s.height i * s.height j

Tactic:
· convert! abs_real_inner_le_norm _ _ using 1
    simp only [dist_eq_norm_vsub, height]

Goal state after:
case a
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
⊢ |⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫| ≠ s.height i * s.height j
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

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

Tactic:
rw [oangle, kahler_apply_apply, Complex.arg_coe_angle_eq_iff_eq_toReal, Real.Angle.toReal_zero,
    Complex.arg_eq_zero_iff]
  simpa using o.nonneg_inner_and_areaForm_eq_zero_iff_sameRay x y

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : ∠ p₁ p₂ p₃ = π
heq : p₁ = p₂
⊢ False

Tactic:
heq,

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : ∠ p₂ p₂ p₃ = π
heq : p₁ = p₂
⊢ False