mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
H : OpenNormalSubgroup ↑P.toProfinite.toTop
s : Set ↑(P.diagram.obj H).toProfinite.toTop
a✝ : IsOpen s
i : ↑P.toProfinite.toTop ⧸ ↑H.toOpenSubgroup
x✝ : i ∈ s
x : ↑P.toProfinite.toTop
⊢ (Quotient.out i)⁻¹ * x ∈ ↑H.toOpenSubgroup ↔ x ∈ Quotient.out i • (↑H.toOpenSubgroup).carrier

Tactic:
]

Goal state after:
P : ProfiniteGrp.{u}
H : OpenNormalSubgroup ↑P.toProfinite.toTop
s : Set ↑(P.diagram.obj H).toProfinite.toTop
a✝ : IsOpen s
i : ↑P.toProfinite.toTop ⧸ ↑H.toOpenSubgroup
x✝ : i ∈ s
x : ↑P.toProfinite.toTop
⊢ (Quotient.out i)⁻¹ * x ∈ ↑H.toOpenSubgroup ↔ x ∈ Quotient.out i • (↑H.toOpenSubgroup).carrier
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
⊢ IsOpen (⋂ i, ↑↑(↑i).toOpenSubgroup)

Tactic:
]

Goal state after:
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
⊢ IsOpen (⋂ i, ↑↑(↑i).toOpenSubgroup)
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
⊢ (Hom.hom P.toLimit) origin ∈ Subtype.val ⁻¹' s

Tactic:
]

Goal state after:
case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
⊢ (Hom.hom P.toLimit) origin ∈ Subtype.val ⁻¹' s
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
hI' : ∃ y < I, y.IsPrime
⊢ I.IsMaximal

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
hI' : ∃ y < I, y.IsPrime
⊢ I.IsMaximal
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
⊢ (Hom.hom P.toLimit) origin ∈ Subtype.val ⁻¹' s

Tactic:
apply hJ2

Goal state after:
case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⋯
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := ⋯
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
⊢ ↑((Hom.hom P.toLimit) origin) ∈ (↑J).pi fJ
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
P : Ideal R
hlt : P < I
hP : P.IsPrime
this✝ : IsPrincipalIdealRing (R ⧸ P)
this : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime
⊢ ¬I ≤ RingHom.ker (Ideal.Quotient.mk P)

Tactic:
Ideal.mk_ker

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
P : Ideal R
hlt : P < I
hP : P.IsPrime
this✝ : IsPrincipalIdealRing (R ⧸ P)
this : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime
⊢ ¬I ≤ P
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
P : Ideal R
hlt : P < I
hP : P.IsPrime
this✝ : IsPrincipalIdealRing (R ⧸ P)
this : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime
⊢ ¬I ≤ P

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsPrincipalIdealRing R
I : Ideal R
hI : I.IsPrime
P : Ideal R
hlt : P < I
hP : P.IsPrime
this✝ : IsPrincipalIdealRing (R ⧸ P)
this : (Ideal.map (Ideal.Quotient.mk P) I).IsPrime
⊢ ¬I ≤ P
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
⊢ Ring.KrullDimLE 1 R

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
⊢ Ring.KrullDimLE 1 R
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
h' : ringKrullDim R < 1
h'' : Ring.KrullDimLE 0 R
⊢ IsField R

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
h : ¬IsField R
h' : ringKrullDim R < 1
h'' : Ring.KrullDimLE 0 R
⊢ IsField R
Module: Mathlib.RingTheory.KrullDimension.PID

Goal state before:
pp: case refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
m : Ideal R
inst✝ : m.IsMaximal
h : ¬IsField R
⊢ ↑m.height ≤ ringKrullDim R

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
m : Ideal R
inst✝ : m.IsMaximal
h : ¬IsField R
⊢ ↑m.height ≤ ringKrullDim R
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
H :
  ∀ m < n + 1,
    ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),
      p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m
w✝ : IsLocalRing R
this✝ : (maximalIdeal R).IsPrime
q : Ideal R
hq : q.IsPrime
hpq : q < maximalIdeal R
hq' : ∀ (q' : Ideal R), q'.IsPrime → q < q' → ¬q' < maximalIdeal R
x : R
s' : Finset R
hxs' : x ∉ s'
hxq : x ∉ q
hn : (insert x s').card ≤ n + 1
hp : maximalIdeal R ∈ (span (insert x ↑s')).minimalPrimes
this : maximalIdeal R ≤ (q ⊔ span {x}).radical
h : ↑s' ⊆ ↑(q ⊔ span {x}).radical
t✝ : ↑↑s' → R
ht : Set.range t✝ ⊆ ↑q
hspan : ↑s' ⊆ ↑(span (Set.range t✝) ⊔ span {x}).radical
t : Finset R := Finset.image t✝ Finset.univ
⊢ q ∈ (span ↑t).minimalPrimes

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
H :
  ∀ m < n + 1,
    ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),
      p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m
w✝ : IsLocalRing R
this✝ : (maximalIdeal R).IsPrime
q : Ideal R
hq : q.IsPrime
hpq : q < maximalIdeal R
hq' : ∀ (q' : Ideal R), q'.IsPrime → q < q' → ¬q' < maximalIdeal R
x : R
s' : Finset R
hxs' : x ∉ s'
hxq : x ∉ q
hn : (insert x s').card ≤ n + 1
hp : maximalIdeal R ∈ (span (insert x ↑s')).minimalPrimes
this : maximalIdeal R ≤ (q ⊔ span {x}).radical
h : ↑s' ⊆ ↑(q ⊔ span {x}).radical
t✝ : ↑↑s' → R
ht : Set.range t✝ ⊆ ↑q
hspan : ↑s' ⊆ ↑(span (Set.range t✝) ⊔ span {x}).radical
t : Finset R := Finset.image t✝ Finset.univ
⊢ q ∈ (span ↑t).minimalPrimes
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (Hom.hom (P.diagram.map M_to_Na)) (↑((Hom.hom P.toLimit) origin) m) ∈ fJ a

Tactic:
]

Goal state after:
case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (Hom.hom (P.diagram.map M_to_Na)) (↑((Hom.hom P.toLimit) origin) m) ∈ fJ a
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (ConcreteCategory.hom (P.toFiniteQuotientFunctor.map M_to_Na)) (spc m) ∈ fJ a

Tactic:
]

Goal state after:
case h
P : ProfiniteGrp.{u}
U : Set ↑(limit P.diagram).toProfinite.toTop
s : Set ((j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : Subtype.val ⁻¹' s = U
spc : (j : OpenNormalSubgroup ↑P.toProfinite.toTop) → ↑(P.diagram.obj j).toProfinite.toTop
hspc : spc ∈ limitConePtAux P.diagram
uDefaultSpec : spc ∈ s
J : Finset (OpenNormalSubgroup ↑P.toProfinite.toTop)
fJ : (a : OpenNormalSubgroup ↑P.toProfinite.toTop) → Set ↑(P.diagram.obj a).toProfinite.toTop
hJ1 : ∀ a ∈ J, IsOpen (fJ a) ∧ spc a ∈ fJ a
hJ2 : (↑J).pi fJ ⊆ s
M : Subgroup ↑P.toProfinite.toTop := ⨅ j, ↑(↑j).toOpenSubgroup
hM : M.Normal
hMOpen : IsOpen ↑M
m : OpenNormalSubgroup ↑P.toProfinite.toTop := { toSubgroup := M, isOpen' := hMOpen, isNormal' := ⋯ }
origin : ↑P.toProfinite.toTop
horigin : (QuotientGroup.mk' M) origin = spc m
a : OpenNormalSubgroup ↑P.toProfinite.toTop
a_in_J : a ∈ ↑J
M_to_Na : m ⟶ a := ⋯.hom
⊢ (ConcreteCategory.hom (P.toFiniteQuotientFunctor.map M_to_Na)) (spc m) ∈ fJ a
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
this : IsClosed (Set.range ⇑(Hom.hom P.toLimit))
⊢ Set.univ = Set.univ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
⊢ ∀ x ∈ (Hom.hom P.toLimit).ker, x = 1

Tactic:
]

Goal state after:
P : ProfiniteGrp.{u}
⊢ ∀ x ∈ (Hom.hom P.toLimit).ker, x = 1
Module: Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits

Goal state before:
pp: P : ProfiniteGrp.{u}
⊢ Function.Bijective ⇑(ConcreteCategory.hom P.toLimit)

Tactic:
]

Goal state after:
P : ProfiniteGrp.{u}
⊢ Function.Bijective ⇑(ConcreteCategory.hom P.toLimit)
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
n : ℕ
H :
  ∀ m < n + 1,
    ∀ {R : Type u_1} [inst : CommRing R] [IsNoetherianRing R] (p : Ideal R) (s : Finset R),
      p ∈ (span ↑s).minimalPrimes → s.card = m → p.height ≤ ↑m
w✝ : IsLocalRing R
this✝ : (maximalIdeal R).IsPrime
q : Ideal R
hq : q.IsPrime
hpq : q < maximalIdeal R
hq' : ∀ (q' : Ideal R), q'.IsPrime → q < q' → ¬q' < maximalIdeal R
x : R
s' : Finset R
hxs' : x ∉ s'
hxq : x ∉ q
hn : (insert x s').card ≤ n + 1
hp : maximalIdeal R ∈ (span (insert x ↑s')).minimalPrimes
this : maximalIdeal R ≤ (q ⊔ span {x}).radical
h : ↑s' ⊆ ↑(q ⊔ span {x}).radical
t✝ : ↑↑s' → R
ht : Set.range t✝ ⊆ ↑q
hspan : ↑s' ⊆ ↑(span (Set.range t✝) ⊔ span {x}).radical
t : Finset R := Finset.image t✝ Finset.univ
⊢ span (Set.range t✝) ⊔ span {x} ≤ span (Set.range t✝) ⊔ span {x}

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
p : Ideal R
s : Set R
hs : s.Finite
hI : p ∈ (span s).minimalPrimes
⊢ p.height ≤ ↑hs.toFinset.card

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
p : Ideal R
s : Set R
hs : s.Finite
hI : p ∈ (span s).minimalPrimes
⊢ p.height ≤ ↑hs.toFinset.card
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
hJ : J ∈ I.minimalPrimes
⊢ ⨅ J ∈ I.minimalPrimes, J.height ≤ Cardinal.toENat (Submodule.spanRank I)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
hJ : J ∈ I.minimalPrimes
⊢ ⨅ J ∈ I.minimalPrimes, J.height ≤ Cardinal.toENat (Submodule.spanRank I)
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
hI : I ≠ ⊤
⊢ ↑(Submodule.spanFinrank I) = ↑(Submodule.spanFinrank I)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
⊢ ↑(IsLocalRing.maximalIdeal R).height ≠ ⊥ ∧ ↑(IsLocalRing.maximalIdeal R).height ≠ ⊤

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
⊢ ↑(IsLocalRing.maximalIdeal R).height ≠ ⊥ ∧ ↑(IsLocalRing.maximalIdeal R).height ≠ ⊤
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: case right
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
⊢ ¬(IsLocalRing.maximalIdeal R).height = ⊤

Tactic:
]

Goal state after:
case right
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
⊢ ¬(IsLocalRing.maximalIdeal R).height = ⊤
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
hJ₁ : J ≤ I
hJ₂ : Submodule.spanRank J ≤ ↑I.height.toNat
hJ₃ : I.height ≤ J.height
⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsNoetherianRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
hJ₁ : J ≤ I
hJ₂ : Submodule.spanRank J ≤ ↑I.height.toNat
hJ₃ : I.height ≤ J.height
⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ Order.krullDim (PrimeSpectrum R[X]) ≤ 2 * Order.krullDim (PrimeSpectrum R) + 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ Order.krullDim (PrimeSpectrum R[X]) ≤ 2 * Order.krullDim (PrimeSpectrum R) + 1
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : PrimeSpectrum R
⊢ Ring.KrullDimLE 1 p.asIdeal.ResidueField[X]

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
p : PrimeSpectrum R
⊢ Ring.KrullDimLE 1 p.asIdeal.ResidueField[X]
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
p : Ideal R
inst✝² : p.IsMaximal
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
x✝ : Field (R ⧸ p) := Quotient.field p
h : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P).height = 1
⊢ p.height + 1 = p.height + 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ p ∈ I.minimalPrimes

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ p ∈ I.minimalPrimes
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑(Submodule.spanFinrank I) = p.height

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : Submodule.spanRank I ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑(Submodule.spanFinrank I) = p.height
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: case refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : ↑(Submodule.spanFinrank I) ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑(Submodule.spanFinrank I) ≤ p.height

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
p : Ideal R
inst✝ : p.IsPrime
I : Ideal R
hI : p ∈ I.minimalPrimes
hr : ↑(Submodule.spanFinrank I) ≤ ↑p.height
hs : (Submodule.generators I).Finite
⊢ ↑(Submodule.spanFinrank I) ≤ p.height
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
p : Ideal R
inst✝² : p.IsMaximal
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
x✝ : Field (R ⧸ p) := Quotient.field p
e : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial.symm
P' : Ideal (R ⧸ p)[X] := Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)
⊢ Ideal.map (algebraMap R R[X]) (under R P) ≤ P

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
p : Ideal R
inst✝² : p.IsMaximal
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
x✝ : Field (R ⧸ p) := Quotient.field p
e : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial.symm
P' : Ideal (R ⧸ p)[X] := Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)
⊢ Ideal.map (algebraMap R R[X]) (under R P) ≤ P
Module: Mathlib.RingTheory.KrullDimension.Polynomial

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
p : Ideal R
inst✝² : p.IsMaximal
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
x✝ : Field (R ⧸ p) := Quotient.field p
e : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial.symm
P' : Ideal (R ⧸ p)[X] := Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)
this✝¹ : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P).IsMaximal
this✝ : P'.IsMaximal
this : P'.height = 1
⊢ (Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)).height = 1

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
p : Ideal R
inst✝² : p.IsMaximal
P : Ideal R[X]
inst✝¹ : P.IsMaximal
inst✝ : P.LiesOver p
x✝ : Field (R ⧸ p) := Quotient.field p
e : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial.symm
P' : Ideal (R ⧸ p)[X] := Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)
this✝¹ : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P).IsMaximal
this✝ : P'.IsMaximal
this : P'.height = 1
⊢ (Ideal.map e (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P)).height = 1