mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_in_q : Weight.toLinear K (↥H) L χ ∈ q
⊢ ⁅x_χ, m_α⁆ ∈ ⨆ α, sl2SubmoduleOfRoot ⋯

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : P
n : ℕ
⊢ spanSingleton S x ^ n = spanSingleton S (x ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
γ : Weight K (↥H) L
hγ : γ.IsNonZero
⊢ γ ∈ LieSubalgebra.root

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
⊢ IsDedekindDomain A ↔ IsDedekindDomainInv A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ ∀ (a b : FractionalIdeal A⁰ K), a / b = a * b⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ ∀ (a b : FractionalIdeal A⁰ K), a / b = a * b⁻¹

Tactic:
by
    let := (isDedekindDomain_iff_isDedekindDomainInv.mp ‹_›).commGroupWithZero K
    exact div_eq_mul_inv

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x : R
⊢ ↑(Ideal.span {x}) = spanSingleton S ((algebraMap R P) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K✝
inst✝ : IsFractionRing A K✝
I : { x // 0 < x }
J K : FractionalIdeal A⁰ K✝
hJK : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((fun x y ↦ ↑x * y) I K)
⊢ (fun x1 x2 ↦ x1 ≤ x2) J K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K✝
inst✝ : IsFractionRing A K✝
I : { x // 0 < x }
J K : FractionalIdeal A⁰ K✝
hJK : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ y * ↑x) I J) ((fun x y ↦ y * ↑x) I K)
⊢ (fun x1 x2 ↦ x1 ≤ x2) J K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K✝
inst✝ : IsFractionRing A K✝
I : { x // 0 < x }
J K : Ideal A
e : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((fun x y ↦ ↑x * y) I K)
⊢ 0 < ↑↑I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K✝ : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K✝
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K✝
inst✝ : IsFractionRing A K✝
I : { x // 0 < x }
J K : Ideal A
e : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) I J) ((fun x y ↦ ↑x * y) I K)
⊢ (fun x1 x2 ↦ x1 ≤ x2) J K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
hI : I = ⊥
⊢ J = ⊥

Tactic:
rwa [hI, ← eq_bot_iff] at h

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
hI : I = ⊥
⊢ J = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case mp
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
y : P'
h : y ∈ (canonicalEquiv S P P') (spanSingleton S x)
⊢ y ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)

Tactic:
rw [mem_spanSingleton]
    obtain ⟨x', hx', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h
    obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp hx'
    use z
    rw [IsLocalization.map_smul, RingHom.id_apply]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
⊢ (↑I)⁻¹ * ↑J ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
⊢ I ∣ J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra R A
p : Ideal R
P : Ideal A
hP : P ≠ ⊤
inst✝ : p.IsMaximal
⊢ IsCoatom p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra R A
p : Ideal R
P : Ideal A
hP : P ≠ ⊤
inst✝ : p.IsMaximal
⊢ P.LiesOver p ↔ P ∣ map (algebraMap R A) p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
x✝ : DvdNotUnit I J
hI : I ≠ 0
H : Ideal A
hunit : ¬IsUnit H
hmul : J = I * H
h : J = I
⊢ I * H = I * 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ WellFounded DvdNotUnit

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
inst✝³ : IsLocalization S P
P' : Type u_5
inst✝² : CommRing P'
inst✝¹ : Algebra R P'
inst✝ : IsLocalization S P'
x : P
⊢ (canonicalEquiv S P P') (spanSingleton S x) = spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
⊢ P.IsMaximal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hirr : Irreducible P
I J : Ideal A
⊢ P ∣ I * J → P ∣ I ∨ P ∣ J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ ∀ {a : Ideal A}, Irreducible a ↔ Prime a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) + ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_plus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
⊢ γ ∈ LieSubalgebra.root

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
x y : P
I : FractionalIdeal S P
⊢ y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) + ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_plus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
⊢ S.root i + S.root j ∈ Set.range ⇑S.root

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
⊢ spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.DedekindDomain

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
hI : I ≠ ⊥
hM : Module.IsTorsionBySet R M ↑I
P : Multiset (Ideal R) := factors I
prime_of_mem : ∀ p ∈ P.toFinset, Prime p
p : Ideal R
hp : p ∈ ↑(factors I).toFinset
q : Ideal R
hq : q ∈ ↑(factors I).toFinset
pq : p ≠ q
this : Multiset.count p (normalizedFactors ?m.252) = 0
⊢ p ^ min (Multiset.count p (normalizedFactors (q ^ Multiset.count q P))) (Multiset.count p P) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.DedekindDomain

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
M : Type v
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
hI : I ≠ ⊥
hM : Module.IsTorsionBySet R M ↑I
⊢ DirectSum.IsInternal fun p ↦ torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (factors I))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.DedekindDomain

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
hM : Module.IsTorsion R M
⊢ DirectSum.IsInternal fun p ↦ torsionBySet R M ↑(↑p ^ Multiset.count (↑p) (factors ⊤.annihilator))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.DedekindDomain

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module.Finite R M
hM : Module.IsTorsion R M
⊢ ∃ P x, ∃ (_ : ∀ p ∈ P, Prime p), ∃ e, DirectSum.IsInternal fun p ↦ torsionBySet R M ↑(↑p ^ e p)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.LeftExact

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
M1 : Type u_2
M2 : Type u_3
M3 : Type u_4
N : Type u_5
inst✝⁷ : AddCommGroup M1
inst✝⁶ : AddCommGroup M2
inst✝⁵ : AddCommGroup M3
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M1
inst✝² : Module R M2
inst✝¹ : Module R M3
inst✝ : Module R N
f : M1 →ₗ[R] M2
g : M2 →ₗ[R] M3
exac : Function.Exact ⇑f ⇑g
surj : Function.Surjective ⇑g
⊢ Function.Exact ⇑(lcomp R N g) ⇑(lcomp R N f)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) + ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_plus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
h_sum_in_range : S.root i + S.root j ∈ Set.range ⇑S.root
⊢ q ∈ S.invtRootSubmodule

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
x y : R₁
hy : y ∈ R₁⁰
⊢ spanSingleton R₁⁰ (mk' K x ⟨y, hy⟩) * ↑I = ↑J ↔ Ideal.span {x} * I = Ideal.span {y} * J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝³ : CommRing R₁
K : Type u_4
inst✝² : Field K
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I J : Ideal R₁
z : K
⊢ spanSingleton R₁⁰ z * ↑I = ↑J ↔ Ideal.span {(sec R₁⁰ z).1} * I = Ideal.span {↑(sec R₁⁰ z).2} * J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : K
h : x = 0
⊢ 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : K
h : ¬x = 0
⊢ spanSingleton R₁⁰ x * spanSingleton R₁⁰ x⁻¹ = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
x : K
⊢ 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
x : K
hx : x * d ∈ ↑J
⊢ x = d⁻¹ * (x * d)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) + ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_plus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
h_sum_in_range : S.root i + S.root j ∈ Set.range ⇑S.root
h_equiv : Weight.toLinear K (↥H) L χ ∈ ↑⟨q, ⋯⟩ ↔ S.root j ∈ ↑⟨q, ⋯⟩
⊢ S.root j ∈ ↑⟨q, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
⊢ genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
⊢ J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Flat.IsBaseChange

Goal state before:
pp: R : Type u_1
inst✝²¹ : CommRing R
S : Type u_2
inst✝²⁰ : CommRing S
inst✝¹⁹ : Algebra R S
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
inst✝¹⁸ : AddCommGroup M₁
inst✝¹⁷ : AddCommGroup M₂
inst✝¹⁶ : AddCommGroup M₃
inst✝¹⁵ : AddCommGroup N₁
inst✝¹⁴ : AddCommGroup N₂
inst✝¹³ : AddCommGroup N₃
inst✝¹² : Module R M₁
inst✝¹¹ : Module R M₂
inst✝¹⁰ : Module R M₃
inst✝⁹ : Module R N₁
inst✝⁸ : Module R N₂
inst✝⁷ : Module R N₃
inst✝⁶ : Module S N₁
inst✝⁵ : Module S N₂
inst✝⁴ : Module S N₃
inst✝³ : IsScalarTower R S N₁
inst✝² : IsScalarTower R S N₂
inst✝¹ : IsScalarTower R S N₃
h₁ : M₁ →ₗ[R] N₁
h₂ : M₂ →ₗ[R] N₂
h₃ : M₃ →ₗ[R] N₃
f : M₁ →ₗ[R] M₂
g : M₂ →ₗ[R] M₃
f' : N₁ →ₗ[S] N₂
g' : N₂ →ₗ[S] N₃
comm₁ : h₂ ∘ₗ f = ↑R f' ∘ₗ h₁
comm₂ : h₃ ∘ₗ g = ↑R g' ∘ₗ h₂
inst✝ : Module.Flat R S
isb₂ : IsBaseChange S h₂
isb₃ : IsBaseChange S h₃
exact₁ : Function.Exact ⇑f ⇑g
inj₁ : Function.Injective ⇑f
exact₂ : Function.Exact ⇑f' ⇑g'
inj₂ : Function.Injective ⇑f'
⊢ IsBaseChange S h₁

Tactic:
by

Goal state after: