mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.Compacts

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ Nontrivial (NonemptyCompacts α) ↔ Nontrivial α

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
⊢ Nontrivial (NonemptyCompacts α) ↔ Nontrivial α
Module: Mathlib.Topology.Sober

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
x : α
S : Set α
h : IsGenericPoint x S
f : α → β
hf : Continuous f
⊢ closure {f x} = closure (f '' closure {x})

Tactic:
← image_singleton,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
x : α
S : Set α
h : IsGenericPoint x S
f : α → β
hf : Continuous f
⊢ closure (f '' {x}) = closure (f '' closure {x})
Module: Mathlib.Topology.Sober

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x : α
S : Set α
hS : IsClosed S
hxS : x ∈ S
this : closure {x} ⊆ S
⊢ IsGenericPoint x S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
x : α
S : Set α
hS : IsClosed S
hxS : x ∈ S
this : closure {x} ⊆ S
⊢ IsGenericPoint x S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z
Module: Mathlib.Topology.Sober

Goal state before:
pp: case a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ⊆ closure {↑H.genericPoint}

Tactic:
← image_singleton,

Goal state after:
case a
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
t : Set α
h : IsPreirreducible t
x : α
hx : x ∈ t
h' : IsClosed t
i : ι
hi : x ∈ U i
H : IsIrreducible (Subtype.val ⁻¹' t)
⊢ t ⊆ closure (Subtype.val '' {H.genericPoint})
Module: Mathlib.Topology.Constructible

Goal state before:
pp: case sdiff
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
P : (s : Set X) → IsConstructible s → Prop
inst✝¹ : QuasiSeparatedSpace X
ι : Type u_4
inst✝ : Nonempty ι
b : ι → Set X
basis : IsTopologicalBasis (range b)
isCompact_basis : ∀ (i : ι), IsCompact (b i)
sdiff : ∀ (i : ι) (s : Set ι) (hs : s.Finite), P (b i \ ⋃ j ∈ s, b j) ⋯
union : ∀ (s : Set X) (hs : IsConstructible s) (t : Set X) (ht : IsConstructible t), P s hs → P t ht → P (s ∪ t) ⋯
s✝ : Set X
this : ∀ (U : Set X), IsCompact U ∧ IsOpen[inst✝³] U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
s : Set ι
hs : s.Finite
hU : ⋃ i ∈ s, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
t : Set ι
ht : t.Finite
hV : ⋃ i ∈ t, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
⊢ P ((⋃ i ∈ s, b i) \ ⋃ i ∈ t, b i) ⋯

Tactic:
[

Goal state after:
case sdiff
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
P : (s : Set X) → IsConstructible s → Prop
inst✝¹ : QuasiSeparatedSpace X
ι : Type u_4
inst✝ : Nonempty ι
b : ι → Set X
basis : IsTopologicalBasis (range b)
isCompact_basis : ∀ (i : ι), IsCompact (b i)
sdiff : ∀ (i : ι) (s : Set ι) (hs : s.Finite), P (b i \ ⋃ j ∈ s, b j) ⋯
union : ∀ (s : Set X) (hs : IsConstructible s) (t : Set X) (ht : IsConstructible t), P s hs → P t ht → P (s ∪ t) ⋯
s✝ : Set X
this : ∀ (U : Set X), IsCompact U ∧ IsOpen[inst✝³] U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
s : Set ι
hs : s.Finite
hU : ⋃ i ∈ s, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
t : Set ι
ht : t.Finite
hV : ⋃ i ∈ t, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
⊢ P ((⋃ i ∈ s, b i) \ ⋃ i ∈ t, b i) ⋯
Module: Mathlib.Topology.Constructible

Goal state before:
pp: case sdiff.insert
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
P : (s : Set X) → IsConstructible s → Prop
inst✝¹ : QuasiSeparatedSpace X
ι : Type u_4
inst✝ : Nonempty ι
b : ι → Set X
basis : IsTopologicalBasis (range b)
isCompact_basis : ∀ (i : ι), IsCompact (b i)
sdiff : ∀ (i : ι) (s : Set ι) (hs : s.Finite), P (b i \ ⋃ j ∈ s, b j) ⋯
union : ∀ (s : Set X) (hs : IsConstructible s) (t : Set X) (ht : IsConstructible t), P s hs → P t ht → P (s ∪ t) ⋯
s✝¹ : Set X
this : ∀ (U : Set X), IsCompact U ∧ IsOpen[inst✝³] U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
s✝ t : Set ι
ht : t.Finite
hV : ⋃ i ∈ t, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
i : ι
s : Set ι
hi : i ∉ s
hs : s.Finite
ih : ∀ (hU : ⋃ i ∈ s, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}), P (⋃ i ∈ s, b i \ ⋃ i ∈ t, b i) ⋯
hU : ⋃ i_1 ∈ insert i s, b i_1 ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
⊢ P (⋃ i_1 ∈ insert i s, b i_1 \ ⋃ i ∈ t, b i) ⋯

Tactic:
[

Goal state after:
case sdiff.insert
X : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : CompactSpace X
P : (s : Set X) → IsConstructible s → Prop
inst✝¹ : QuasiSeparatedSpace X
ι : Type u_4
inst✝ : Nonempty ι
b : ι → Set X
basis : IsTopologicalBasis (range b)
isCompact_basis : ∀ (i : ι), IsCompact (b i)
sdiff : ∀ (i : ι) (s : Set ι) (hs : s.Finite), P (b i \ ⋃ j ∈ s, b j) ⋯
union : ∀ (s : Set X) (hs : IsConstructible s) (t : Set X) (ht : IsConstructible t), P s hs → P t ht → P (s ∪ t) ⋯
s✝¹ : Set X
this : ∀ (U : Set X), IsCompact U ∧ IsOpen[inst✝³] U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
s✝ t : Set ι
ht : t.Finite
hV : ⋃ i ∈ t, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
i : ι
s : Set ι
hi : i ∉ s
hs : s.Finite
ih : ∀ (hU : ⋃ i ∈ s, b i ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}), P (⋃ i ∈ s, b i \ ⋃ i ∈ t, b i) ⋯
hU : ⋃ i_1 ∈ insert i s, b i_1 ∈ {U | IsOpen[inst✝³] U ∧ IsRetrocompact U}
⊢ P (⋃ i_1 ∈ insert i s, b i_1 \ ⋃ i ∈ t, b i) ⋯
Module: Mathlib.Topology.Constructible

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
hs : IsLocallyConstructible s
ht : IsLocallyConstructible t
x : X
U : Set X
hxU : U ∈ 𝓝 x
hU : IsOpen[inst✝] U
hsU : IsConstructible (U ↓∩ s)
V : Set X
hxV : V ∈ 𝓝 x
hV : IsOpen[inst✝] V
htV : IsConstructible (V ↓∩ t)
this : (U ∩ V) ↓∩ (s ∪ t) = inclusion ⋯ ⁻¹' U ↓∩ s ∪ inclusion ⋯ ⁻¹' V ↓∩ t
⊢ IsConstructible ((U ∩ V) ↓∩ (s ∪ t))

Tactic:
this

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
s t : Set X
hs : IsLocallyConstructible s
ht : IsLocallyConstructible t
x : X
U : Set X
hxU : U ∈ 𝓝 x
hU : IsOpen[inst✝] U
hsU : IsConstructible (U ↓∩ s)
V : Set X
hxV : V ∈ 𝓝 x
hV : IsOpen[inst✝] V
htV : IsConstructible (V ↓∩ t)
this : (U ∩ V) ↓∩ (s ∪ t) = inclusion ⋯ ⁻¹' U ↓∩ s ∪ inclusion ⋯ ⁻¹' V ↓∩ t
⊢ IsConstructible (inclusion ⋯ ⁻¹' U ↓∩ s ∪ inclusion ⋯ ⁻¹' V ↓∩ t)
Module: Mathlib.RingTheory.RingHom.Flat

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
A : Type u_4
B : Type u_5
C : Type u_6
D : Type u_7
inst✝¹¹ : CommRing A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Algebra S A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R B
inst✝⁵ : CommRing C
inst✝⁴ : Algebra R C
inst✝³ : Algebra S C
inst✝² : IsScalarTower R S C
inst✝¹ : CommRing D
inst✝ : Algebra R D
f : A →ₐ[S] C
g : B →ₐ[R] D
hf : f.Flat
hg : g.Flat
heq :
  Algebra.TensorProduct.map f g =
    (Algebra.TensorProduct.map f (AlgHom.id R D)).comp (Algebra.TensorProduct.map (AlgHom.id S A) g)
this :
  AlgHom.restrictScalars R (Algebra.TensorProduct.map f (AlgHom.id R D)) =
    (↑(Algebra.TensorProduct.comm R D C)).comp
      ((Algebra.TensorProduct.lTensor D (AlgHom.restrictScalars R f)).comp ↑(Algebra.TensorProduct.comm R A D))
⊢ (AlgHom.restrictScalars R (Algebra.TensorProduct.map f (AlgHom.id R D))).Flat

Tactic:
this

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
A : Type u_4
B : Type u_5
C : Type u_6
D : Type u_7
inst✝¹¹ : CommRing A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Algebra S A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R B
inst✝⁵ : CommRing C
inst✝⁴ : Algebra R C
inst✝³ : Algebra S C
inst✝² : IsScalarTower R S C
inst✝¹ : CommRing D
inst✝ : Algebra R D
f : A →ₐ[S] C
g : B →ₐ[R] D
hf : f.Flat
hg : g.Flat
heq :
  Algebra.TensorProduct.map f g =
    (Algebra.TensorProduct.map f (AlgHom.id R D)).comp (Algebra.TensorProduct.map (AlgHom.id S A) g)
this :
  AlgHom.restrictScalars R (Algebra.TensorProduct.map f (AlgHom.id R D)) =
    (↑(Algebra.TensorProduct.comm R D C)).comp
      ((Algebra.TensorProduct.lTensor D (AlgHom.restrictScalars R f)).comp ↑(Algebra.TensorProduct.comm R A D))
⊢ ((↑(Algebra.TensorProduct.comm R D C)).comp
      ((Algebra.TensorProduct.lTensor D (AlgHom.restrictScalars R f)).comp ↑(Algebra.TensorProduct.comm R A D))).Flat
Module: Mathlib.RingTheory.RingHom.Flat

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
g : S →+* T
hg : Function.Bijective ⇑g
hf : (g.comp f).Flat
e : S ≃+* T := RingEquiv.ofBijective g hg
this : f = e.symm.toRingHom.comp (e.toRingHom.comp f)
⊢ f.Flat

Tactic:
this

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
g : S →+* T
hg : Function.Bijective ⇑g
hf : (g.comp f).Flat
e : S ≃+* T := RingEquiv.ofBijective g hg
this : f = e.symm.toRingHom.comp (e.toRingHom.comp f)
⊢ (e.symm.toRingHom.comp (e.toRingHom.comp f)).Flat
Module: Mathlib.RingTheory.RingHom.Flat

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
g : T →+* R
hg : Function.Bijective ⇑g
hf : (f.comp g).Flat
e : T ≃+* R := RingEquiv.ofBijective g hg
this : f = (f.comp e.toRingHom).comp e.symm.toRingHom
⊢ f.Flat

Tactic:
this

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
g : T →+* R
hg : Function.Bijective ⇑g
hf : (f.comp g).Flat
e : T ≃+* R := RingEquiv.ofBijective g hg
this : f = (f.comp e.toRingHom).comp e.symm.toRingHom
⊢ ((f.comp e.toRingHom).comp e.symm.toRingHom).Flat
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mp
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
h : zeroLocus ↑I = zeroLocus ↑J
⊢ I.radical = J.radical

Tactic:
[

Goal state after:
case mp
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
h : zeroLocus ↑I = zeroLocus ↑J
⊢ I.radical = J.radical
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case neg
R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
h : IsField R
x : PrimeSpectrum R
hx : ¬x.asIdeal = ⊥
⊢ x.asIdeal.IsMaximal

Tactic:
exact absurd h (Ring.not_isField_iff_exists_prime.2 ⟨x.asIdeal, ⟨hx, x.2⟩⟩)

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ IsPreirreducible (zeroLocus ↑I) ↔
    ∀ (x y : Ideal R),
      zeroLocus ↑I ⊆ zeroLocus ↑x ∪ zeroLocus ↑y → zeroLocus ↑I ⊆ zeroLocus ↑x ∨ zeroLocus ↑I ⊆ zeroLocus ↑y

Tactic:
[

Goal state after:
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ IsPreirreducible (zeroLocus ↑I) ↔
    ∀ (x y : Ideal R),
      zeroLocus ↑I ⊆ zeroLocus ↑x ∪ zeroLocus ↑y → zeroLocus ↑I ⊆ zeroLocus ↑x ∨ zeroLocus ↑I ⊆ zeroLocus ↑y
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ (x y : Ideal R),
      zeroLocus ↑I ⊆ zeroLocus ↑x ∪ zeroLocus ↑y → zeroLocus ↑I ⊆ zeroLocus ↑x ∨ zeroLocus ↑I ⊆ zeroLocus ↑y) ↔
    ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I

Tactic:
[

Goal state after:
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ (x y : Ideal R),
      zeroLocus ↑I ⊆ zeroLocus ↑x ∪ zeroLocus ↑y → zeroLocus ↑I ⊆ zeroLocus ↑x ∨ zeroLocus ↑I ⊆ zeroLocus ↑y) ↔
    ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mp
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I) → ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I

Tactic:
[

Goal state after:
case mp
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I) → ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mpr
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I) → ∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I

Tactic:
[

Goal state after:
case mpr
R : Type u
inst✝ : CommSemiring R
I : Ideal R
hI : I.IsRadical
⊢ (∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I) → ∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S✝ : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S✝
ι✝ : Type u
S : ι✝ → Set (PrimeSpectrum R)
S_closed : ∀ (i : ι✝), IsClosed (S i)
S_empty : ⋂ i, S i = ∅
I : ι✝ → Ideal R
hI : ∀ (i : ι✝), S i = zeroLocus ↑(I i)
⊢ ∃ u, ⋂ i ∈ u, S i = ∅

Tactic:
[

Goal state after:
R : Type u
S✝ : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S✝
ι✝ : Type u
S : ι✝ → Set (PrimeSpectrum R)
S_closed : ∀ (i : ι✝), IsClosed (S i)
S_empty : ⋂ i, S i = ∅
I : ι✝ → Ideal R
hI : ∀ (i : ι✝), S i = zeroLocus ↑(I i)
⊢ ∃ u, ⋂ i ∈ u, S i = ∅
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
Z : Set (PrimeSpectrum S)
⊢ IsClosed Z ↔ IsClosed[TopologicalSpace.induced (comap (algebraMap R S)) zariskiTopology] Z

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
Z : Set (PrimeSpectrum S)
⊢ IsClosed Z ↔ IsClosed[TopologicalSpace.induced (comap (algebraMap R S)) zariskiTopology] Z
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosed (Set.range (comap (RingHom.fst R S)))

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosed (Set.range (comap (RingHom.fst R S)))
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosed (Set.range (comap (RingHom.snd R S)))

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosed (Set.range (comap (RingHom.snd R S)))
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_2
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosedMap ⇑(primeSpectrumProd R S).symm

Tactic:
[

Goal state after:
case refine_2
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ IsClosedMap ⇑(primeSpectrumProd R S).symm
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
f g : R
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away f S
⊢ basicOpen f ≤ basicOpen g ↔ IsUnit ((algebraMap R S) g)

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
f g : R
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away f S
⊢ basicOpen f ≤ basicOpen g ↔ IsUnit ((algebraMap R S) g)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
f : R
p : PrimeSpectrum R
h : (basicOpen f).carrier = {p}
r : R
hr : ∀ m ∈ Submonoid.powers f, ¬r ∣ m
⊢ r ∉ p.asIdeal.primeCompl

Tactic:
[

Goal state after:
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
f : R
p : PrimeSpectrum R
h : (basicOpen f).carrier = {p}
r : R
hr : ∀ m ∈ Submonoid.powers f, ¬r ∣ m
⊢ r ∉ p.asIdeal.primeCompl
Module: Mathlib.Data.ZMod.QuotientGroup

Goal state before:
pp: n : ℕ
A : Type u_1
R : Type u_2
inst✝³ : AddGroup A
inst✝² : Ring R
α : Type u_3
β : Type u_4
inst✝¹ : AddGroup α
a : α
inst✝ : AddAction α β
b : β
⊢ minimalPeriod (fun x ↦ a +ᵥ x) b • ⟨a, ⋯⟩ +ᵥ b = b

Tactic:
[

Goal state after:
n : ℕ
A : Type u_1
R : Type u_2
inst✝³ : AddGroup A
inst✝² : Ring R
α : Type u_3
β : Type u_4
inst✝¹ : AddGroup α
a : α
inst✝ : AddAction α β
b : β
⊢ minimalPeriod (fun x ↦ a +ᵥ x) b • ⟨a, ⋯⟩ +ᵥ b = b
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case pos
G : Type u
inst✝ : Monoid G
g : G
h : ExponentExists G
⊢ g ^ exponent G = 1

Tactic:
[

Goal state after:
case pos
G : Type u
inst✝ : Monoid G
g : G
h : ExponentExists G
⊢ g ^ exponent G = 1
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case neg
G : Type u
inst✝ : Monoid G
g : G
h : ¬ExponentExists G
⊢ g ^ exponent G = 1

Tactic:
[

Goal state after:
case neg
G : Type u
inst✝ : Monoid G
g : G
h : ¬ExponentExists G
⊢ g ^ exponent G = 1
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: α : Type u_1
inst✝ : Group α
H : Subgroup α
⊢ (∃ g, zpowers g = ⊤) ↔ ∃ g, zpowers g = H

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : Group α
H : Subgroup α
⊢ (∃ g, zpowers g = ⊤) ↔ ∃ g, zpowers g = H
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : Group α
inst✝ : Finite α
⊢ IsCyclic α ↔ ∃ g, orderOf g = Nat.card α

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : Group α
inst✝ : Finite α
⊢ IsCyclic α ↔ ∃ g, orderOf g = Nat.card α
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : DecidableEq ι
i : ι
f : R i
p : PrimeSpectrum ((i : ι) → R i)
⊢ p ∈ ↑(basicOpen (Pi.single i f)) → p ∈ comap (Pi.evalRingHom R i) '' ↑(basicOpen f)

Tactic:
intro hp

Goal state after:
case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : DecidableEq ι
i : ι
f : R i
p : PrimeSpectrum ((i : ι) → R i)
hp : p ∈ ↑(basicOpen (Pi.single i f))
⊢ p ∈ comap (Pi.evalRingHom R i) '' ↑(basicOpen f)
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : Monoid G
⊢ exponent G = 1 ↔ Subsingleton G

Tactic:
refine ⟨fun eq_one => ⟨fun a b => ?a_eq_b⟩, fun h => le_antisymm ?le ?ge⟩
  · rw [← pow_one a, ← pow_one b, ← eq_one, Monoid.pow_exponent_eq_one, Monoid.pow_exponent_eq_one]
  · apply exponent_min' _ Nat.one_pos
    simp [eq_iff_true_of_subsingleton]
  · apply Nat.succ_le_of_lt
    apply exponent_pos_of_exists 1 Nat.one_pos
    simp [eq_iff_true_of_subsingleton]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: G : Type u_2
inst✝ : Group G
p : ℕ
hp : Fact (Nat.Prime p)
h : Nat.card G = p
g g' : G
hg : g ≠ 1
⊢ g' ∈ zpowers g

Tactic:
[

Goal state after:
G : Type u_2
inst✝ : Group G
p : ℕ
hp : Fact (Nat.Prime p)
h : Nat.card G = p
g g' : G
hg : g ≠ 1
⊢ g' ∈ zpowers g
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : Monoid G
n : ℕ
hpos : n > 0
hG : ∀ (g : G), g ^ n = 1
h : 0 < n % exponent G
h₂ : n % exponent G < exponent G
⊢ ∀ (g : G), g ^ (n % exponent G) = 1

Tactic:
[

Goal state after:
G : Type u
inst✝ : Monoid G
n : ℕ
hpos : n > 0
hG : ∀ (g : G), g ^ n = 1
h : 0 < n % exponent G
h₂ : n % exponent G < exponent G
⊢ ∀ (g : G), g ^ (n % exponent G) = 1
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (f (e.symm a)) (Localization.Away ↑a)

Tactic:
[

Goal state after:
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (f (e.symm a)) (Localization.Away ↑a)
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : Monoid G
n : ℕ
⊢ exponent G ∣ n ↔ ∀ (g : G), orderOf g ∣ n

Tactic:
[

Goal state after:
G : Type u
inst✝ : Monoid G
n : ℕ
⊢ exponent G ∣ n ↔ ∀ (g : G), orderOf g ∣ n
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
loc : ∀ (a : ↑(Set.range f)), IsLocalization.AtPrime (Localization.Away ↑a) (e.symm a).asIdeal
algE : (a : ↑(Set.range f)) → Localization.AtPrime (e.symm a).asIdeal ≃ₐ[R] Localization.Away ↑a :=
  fun a ↦
    IsLocalization.algEquiv (e.symm a).asIdeal.primeCompl (Localization.AtPrime (e.symm a).asIdeal)
      (Localization.Away ↑a)
span_eq : Ideal.span (Set.range f) = ⊤
a : ↑(Set.range f)
⊢ ↑(basicOpen ↑a) = {e.symm a}

Tactic:
[

Goal state after:
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
loc : ∀ (a : ↑(Set.range f)), IsLocalization.AtPrime (Localization.Away ↑a) (e.symm a).asIdeal
algE : (a : ↑(Set.range f)) → Localization.AtPrime (e.symm a).asIdeal ≃ₐ[R] Localization.Away ↑a :=
  fun a ↦
    IsLocalization.algEquiv (e.symm a).asIdeal.primeCompl (Localization.AtPrime (e.symm a).asIdeal)
      (Localization.Away ↑a)
span_eq : Ideal.span (Set.range f) = ⊤
a : ↑(Set.range f)
⊢ ↑(basicOpen ↑a) = {e.symm a}
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf✝ : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
loc : ∀ (a : ↑(Set.range f)), IsLocalization.AtPrime (Localization.Away ↑a) (e.symm a).asIdeal
algE : (a : ↑(Set.range f)) → Localization.AtPrime (e.symm a).asIdeal ≃ₐ[R] Localization.Away ↑a :=
  fun a ↦
    IsLocalization.algEquiv (e.symm a).asIdeal.primeCompl (Localization.AtPrime (e.symm a).asIdeal)
      (Localization.Away ↑a)
span_eq : Ideal.span (Set.range f) = ⊤
hf : ∀ (a : ↑(Set.range f)), ↑(basicOpen ↑a) = {e.symm a}
a b : ↑(Set.range f)
ne : a ≠ b
⊢ basicOpen (↑a * ↑b) = ⊥

Tactic:
[

Goal state after:
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf✝ : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
loc : ∀ (a : ↑(Set.range f)), IsLocalization.AtPrime (Localization.Away ↑a) (e.symm a).asIdeal
algE : (a : ↑(Set.range f)) → Localization.AtPrime (e.symm a).asIdeal ≃ₐ[R] Localization.Away ↑a :=
  fun a ↦
    IsLocalization.algEquiv (e.symm a).asIdeal.primeCompl (Localization.AtPrime (e.symm a).asIdeal)
      (Localization.Away ↑a)
span_eq : Ideal.span (Set.range f) = ⊤
hf : ∀ (a : ↑(Set.range f)), ↑(basicOpen ↑a) = {e.symm a}
a b : ↑(Set.range f)
ne : a ≠ b
⊢ basicOpen (↑a * ↑b) = ⊥
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal

Tactic:
[

Goal state after:
R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderGeneralization {x} ↔ x.asIdeal ∈ minimalPrimes R

Tactic:
[

Goal state after:
R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderGeneralization {x} ↔ x.asIdeal ∈ minimalPrimes R
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : Finite α
inst✝ : IsCyclic α
⊢ ∃ x, ∀ (y : α), y ∈ Submonoid.powers x

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Group α
inst✝¹ : Finite α
inst✝ : IsCyclic α
⊢ ∃ x, ∀ (y : α), y ∈ Submonoid.powers x
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
⊢ IsPGroup p G ↔ ∀ (g : G), ∃ k, orderOf g ∣ p ^ k

Tactic:
[

Goal state after:
p : ℕ
G : Type u_1
inst✝ : Group G
⊢ IsPGroup p G ↔ ∀ (g : G), ∃ k, orderOf g ∣ p ^ k