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
h_h_containment : ⁅x_χ, ↑y⁆ ∈ genWeightSpace L ⇑χ
h_bracket_decomp :
  ⁅x_χ, m_α⁆ ∈
    genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ⊔
        genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ) - ⇑(Weight.toLinear K (↥H) L α)) ⊔
      genWeightSpace L ⇑χ
I : LieSubmodule K (↥H) L := ⨆ β, sl2SubmoduleOfRoot ⋯
genWeightSpace_le_I : ∀ β_lin ∈ q, β_lin ≠ 0 → genWeightSpace L ⇑β_lin ≤ I
h_plus_contain : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ≤ I
this : -Weight.toLinear K (↥H) L α = -1 • Weight.toLinear K (↥H) L α
⊢ Weight.toLinear K (↥H) L χ + -1 • Weight.toLinear K (↥H) L α ∈ q

Tactic:
]

Goal state after:
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
h_h_containment : ⁅x_χ, ↑y⁆ ∈ genWeightSpace L ⇑χ
h_bracket_decomp :
  ⁅x_χ, m_α⁆ ∈
    genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ⊔
        genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ) - ⇑(Weight.toLinear K (↥H) L α)) ⊔
      genWeightSpace L ⇑χ
I : LieSubmodule K (↥H) L := ⨆ β, sl2SubmoduleOfRoot ⋯
genWeightSpace_le_I : ∀ β_lin ∈ q, β_lin ≠ 0 → genWeightSpace L ⇑β_lin ≤ I
h_plus_contain : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ≤ I
this : -Weight.toLinear K (↥H) L α = -1 • Weight.toLinear K (↥H) L α
⊢ Weight.toLinear K (↥H) L χ + -1 • Weight.toLinear K (↥H) L α ∈ q
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
S : Submonoid R
P : Type u_2
inst✝³ : CommRing P
inst✝² : Algebra R P
inst✝¹ : IsLocalization S P
I : FractionalIdeal S P
inst✝ : (↑I).IsPrincipal
⊢ ↑I = ↑I

Tactic:
]

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

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := commGroupWithZero (FractionRing A) h
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (↑M)⁻¹ * ↑P
z : A
hzM : z ∈ M
hzp : z ∉ P
zy_mem : (algebraMap A (FractionRing A)) (z * y) ∈ ↑P
⊢ (algebraMap A (FractionRing A)) y ∈ ↑P

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := commGroupWithZero (FractionRing A) h
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (↑M)⁻¹ * ↑P
z : A
hzM : z ∈ M
hzp : z ∉ P
zy_mem : (algebraMap A (FractionRing A)) (z * y) ∈ ↑P
⊢ (algebraMap A (FractionRing A)) y ∈ ↑P
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := commGroupWithZero (FractionRing A) h
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (↑M)⁻¹ * ↑P
z : A
hzM : z ∈ M
hzp : z ∉ P
zy_mem : (algebraMap A (FractionRing A)) (z * y) ∈ ↑P
zy : A
hzy : z * y ∈ P
zy_eq : (algebraMap A (FractionRing A)) zy = (algebraMap A (FractionRing A)) (z * y)
⊢ (algebraMap A (FractionRing A)) y ∈ ↑P

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal A⁰ (FractionRing A)) := commGroupWithZero (FractionRing A) h
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : ↑P ≠ 0
M'_ne : ↑M ≠ 0
le_one : (↑M)⁻¹ * ↑P ≤ 1
y : A
_hy : y ∈ ⊤
hx : (algebraMap A (FractionRing A)) y ∈ (↑M)⁻¹ * ↑P
z : A
hzM : z ∈ M
hzp : z ∉ P
zy_mem : (algebraMap A (FractionRing A)) (z * y) ∈ ↑P
zy : A
hzy : z * y ∈ P
zy_eq : (algebraMap A (FractionRing A)) zy = (algebraMap A (FractionRing A)) (z * y)
⊢ (algebraMap A (FractionRing A)) y ∈ ↑P
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
I : FractionalIdeal A⁰ (FractionRing A)
hI : I ≠ ⊥
a : A
J : Ideal A
ha : a ≠ 0
hJ : I = spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J
h₂ : I * (spanSingleton A⁰ ((algebraMap A (FractionRing A)) a) * (↑J)⁻¹) = 1
⊢ ∃ J, I * J = 1

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
I : FractionalIdeal A⁰ (FractionRing A)
hI : I ≠ ⊥
a : A
J : Ideal A
ha : a ≠ 0
hJ : I = spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J
h₂ : I * (spanSingleton A⁰ ((algebraMap A (FractionRing A)) a) * (↑J)⁻¹) = 1
⊢ ∃ J, I * J = 1
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
x' : R
⊢ (algebraMap R P) x' = x✝ ↔ (algebraMap R P) x' = x✝

Tactic:
]

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

Goal state before:
pp: case zero
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
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: case succ
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
hn : spanSingleton S x ^ n = spanSingleton S (x ^ n)
⊢ spanSingleton S (x ^ n * x) = spanSingleton S (x ^ n * x)

Tactic:
]

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

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
a : A
J : Ideal A
ha : a ≠ 0
hI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥
⊢ 1 = 1

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
a : A
J : Ideal A
ha : a ≠ 0
hI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥
⊢ (algebraMap A (FractionRing A)) a ≠ 0

pp: case hI0
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
a : A
J : Ideal A
ha : a ≠ 0
hI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥
⊢ J ≠ ⊥

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
a : A
J : Ideal A
ha : a ≠ 0
hI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥
⊢ (algebraMap A (FractionRing A)) a ≠ 0

case hI0
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDomain A
x✝ : IsDedekindDomain A
a : A
J : Ideal A
ha : a ≠ 0
hI : spanSingleton A⁰ ((algebraMap A (FractionRing A)) a)⁻¹ * ↑J ≠ ⊥
⊢ J ≠ ⊥
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case h
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 x' : R
hy' : x' • x ∈ Ideal.span {x}
⊢ (algebraMap R P) x' * (algebraMap R P) x = (algebraMap R P) x' * (algebraMap R P) x

Tactic:
]

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

Goal state before:
pp: case mpr
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' : R
⊢ (algebraMap R P) y' * (algebraMap R P) x = (algebraMap R P) y' * (algebraMap R P) x

Tactic:
]

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)
⊢ ∃ z, z • (IsLocalization.map P' (RingHom.id R) ⋯) x = y

Tactic:
]

Goal state after:
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)
⊢ ∃ z, z • (IsLocalization.map P' (RingHom.id R) ⋯) x = y
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)
⊢ ↑I * J ≤ ↑I * K

Tactic:
]

Goal state after:
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)
⊢ ↑I * J ≤ ↑I * K
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 = ⊥
hI : I = ⊥
⊢ J = ⊥

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J = ⊥
hI : I = ⊥
⊢ J = ⊥
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

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

Tactic:
]

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

Goal state before:
pp: case h
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
z : R
hx' : z • x ∈ spanSingleton S x
h : (IsLocalization.map P' (RingHom.id R) ⋯) (z • x) ∈ (canonicalEquiv S P P') (spanSingleton S x)
⊢ z • (IsLocalization.map P' (RingHom.id R) ⋯) x = z • (IsLocalization.map P' (RingHom.id R) ⋯) x

Tactic:
]

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 ≤ (↑I)⁻¹ * ↑I

Tactic:
]

Goal state after:
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 ≤ (↑I)⁻¹ * ↑I
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case mpr
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 ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)
⊢ ∃ y_1 ∈ spanSingleton S x, (IsLocalization.map P' (RingHom.id R) ⋯) y_1 = y

Tactic:
]

Goal state after:
case mpr
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 ∈ spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)
⊢ ∃ y_1 ∈ spanSingleton S x, (IsLocalization.map P' (RingHom.id R) ⋯) y_1 = y
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Basic

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

Tactic:
]

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.IsMaximal

Tactic:
]

Goal state after:
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.IsMaximal
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
⊢ comap (algebraMap R A) P = p ↔ comap (algebraMap R A) P = p

Tactic:
]

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 = I

Tactic:
]

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
this : P.IsMaximal
⊢ (∀ ⦃x : A⦄, x ∈ I * J → x ∈ P) → (∀ ⦃x : A⦄, x ∈ I → x ∈ P) ∨ ∀ ⦃x : A⦄, x ∈ J → x ∈ P

Tactic:
]

Goal state after:
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
this : P.IsMaximal
⊢ (∀ ⦃x : A⦄, x ∈ I * J → x ∈ P) → (∀ ⦃x : A⦄, x ∈ I → x ∈ P) ∨ ∀ ⦃x : A⦄, x ∈ J → x ∈ P
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
⊢ Associated (factors I).prod I

pp: case prime
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
⊢ ∀ i ∈ (factors I).toFinset, Prime i

pp: case coprime
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
⊢ ∀ i ∈ (factors I).toFinset, ∀ j ∈ (factors I).toFinset, i ≠ j → i ≠ j

Tactic:
]

Goal state after:
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
⊢ Associated (factors I).prod I

case prime
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
⊢ ∀ i ∈ (factors I).toFinset, Prime i

case coprime
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
⊢ ∀ i ∈ (factors I).toFinset, ∀ j ∈ (factors I).toFinset, i ≠ j → i ≠ j
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: case right
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
h : y ∈ spanSingleton S x * I
x' : P
hx' : x' ∈ spanSingleton S x
y' : P
hy' : y' ∈ I
a : R
ha : a • x = x'
⊢ a • (x * y') = a • (x * y')

Tactic:
]

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
⊢ 1 = 1

Tactic:
]

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
⊢ p ^ min (Multiset.count p (normalizedFactors (q ^ Multiset.count q P))) (Multiset.count p P) = ⊤

pp: case hI
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
⊢ q ^ Multiset.count q P ≠ ⊥

pp: case hJ
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
⊢ Irreducible p

Tactic:
]

Goal state after:
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
⊢ p ^ min (Multiset.count p (normalizedFactors (q ^ Multiset.count q P))) (Multiset.count p P) = ⊤

case hI
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
⊢ q ^ Multiset.count q P ≠ ⊥

case hJ
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
⊢ Irreducible p
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 (q ^ Multiset.count q P)) = 0
⊢ ⊤ = ⊤

Tactic:
]

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
⊢ ¬(Multiset.count q P ≠ 0 ∧ p = normalize q)

Tactic:
]

Goal state after:
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
⊢ ¬(Multiset.count q P ≠ 0 ∧ p = normalize q)
Module: Mathlib.Algebra.Module.DedekindDomain

Goal state before:
pp: case hI
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
⊢ q ^ Multiset.count q P ≠ 0

Tactic:
]

Goal state after:
case hI
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
⊢ q ^ Multiset.count q P ≠ 0
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
hM' : Module.IsTorsionBySet R M ↑⊤.annihilator
hI : (↑⊤.annihilator ∩ ↑(nonZeroDivisors R)).Nonempty
⊢ ∃ x ∈ ⊤.annihilator, x ≠ 0

Tactic:
]

Goal state after:
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
hM' : Module.IsTorsionBySet R M ↑⊤.annihilator
hI : (↑⊤.annihilator ∩ ↑(nonZeroDivisors R)).Nonempty
⊢ ∃ x ∈ ⊤.annihilator, x ≠ 0
Module: Mathlib.LinearAlgebra.LeftExact

Goal state before:
pp: case refine_2
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
h : M2 →ₗ[R] N
x✝ : ∃ y, y ∘ₗ g = h
y : M3 →ₗ[R] N
hy : y ∘ₗ g = h
⊢ 0 = 0

Tactic:
]

Goal state after: