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

Goal state before:
pp: G : GrpCat
P : ProfiniteGrp.{u}
f g : completion G ⟶ P
h : eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map f = eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map g
⊢ f = g

Tactic:
by

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

Goal state before:
pp: G : GrpCat
⊢ ProfiniteCompletion.lift (𝟙 G ≫ ProfiniteCompletion.eta G) = 𝟙 (ProfiniteCompletion.completion G)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x y : G
⊢ (MapClusterPt x⁻¹ atTop fun x ↦ y ^ x) ↔ MapClusterPt x atTop fun x ↦ y ^ x

Tactic:
by

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

Goal state before:
pp: X✝ Y✝ Z✝ : GrpCat
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ProfiniteCompletion.lift ((f ≫ g) ≫ ProfiniteCompletion.eta Z✝) =
    ProfiniteCompletion.lift (f ≫ ProfiniteCompletion.eta Y✝) ≫
      ProfiniteCompletion.lift (g ≫ ProfiniteCompletion.eta Z✝)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x : G
⊢ closure[inst✝²] (range fun x_1 ↦ x ^ x_1) = closure[inst✝²] (range fun x_1 ↦ x ^ x_1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x : G
⊢ (DenseRange fun x_1 ↦ x ^ x_1) ↔ DenseRange fun x_1 ↦ x ^ x_1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
⊢ IsOpenMap fun g ↦ g • x

Tactic:
by

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

Goal state before:
pp: G : GrpCat
P : ProfiniteGrp.{u}
f : completion G ⟶ P
⊢ (fun f ↦ lift f) ((fun f ↦ eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map f) f) = f

Tactic:
by

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

Goal state before:
pp: G : GrpCat
P : ProfiniteGrp.{u}
f : G ⟶ GrpCat.of ↑P.toProfinite.toTop
⊢ (fun f ↦ eta G ≫ (forget₂ ProfiniteGrp.{u} GrpCat).map f) ((fun f ↦ lift f) f) = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : SigmaCompactSpace G
H : Type u_3
inst✝⁴ : Group H
inst✝³ : TopologicalSpace H
inst✝² : BaireSpace H
inst✝¹ : T2Space H
inst✝ : ContinuousMul H
f : G →* H
hf : Surjective ⇑f
h'f : Continuous[inst✝⁸, inst✝³] ⇑f
A : MulAction G H := MulAction.compHom H f
this✝ : ContinuousSMul G H
this : IsPretransitive G H
⊢ ⇑f = fun g ↦ g • 1

Tactic:
by

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

Goal state before:
pp: X'✝ X✝ : GrpCat
Y✝ : ProfiniteGrp.{?u.22}
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget₂ ProfiniteGrp.{?u.22} GrpCat).obj Y✝
⊢ (homEquiv X'✝ Y✝).symm (f ≫ g) = profiniteCompletion.map f ≫ (homEquiv X✝ Y✝).symm g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.OpenMapping

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : SigmaCompactSpace G
H : Type u_3
inst✝⁴ : Group H
inst✝³ : TopologicalSpace H
inst✝² : BaireSpace H
inst✝¹ : T2Space H
inst✝ : ContinuousMul H
f : G →* H
hf : Surjective ⇑f
h'f : Continuous[inst✝⁸, inst✝³] ⇑f
⊢ IsOpenMap ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
s : Set G
⊢ (Subgroup.closure s).topologicalClosure = (Submonoid.closure s).topologicalClosure

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.SubmonoidClosure

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
s : Set G
⊢ Dense ↑(Submonoid.closure s) ↔ Dense ↑(Subgroup.closure s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ascoli

Goal state before:
pp: X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
S : Set C(X, α)
hS1 : IsCompact (ContinuousMap.toFun '' S)
hS2 : Equicontinuous fun x ↦ ⇑↑x
⊢ IsCompact S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
x : M
ab : M × M
⊢ ab ∈ mulFiber x ↔ ab.1 * ab.2 = x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
inst✝ : Monoid M
⊢ (1, 1) ∈ mulFiber 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝⁸ : Monoid M
inst✝⁷ : CommSemiring S
inst✝⁶ : AddCommMonoid E
inst✝⁵ : AddCommMonoid E'
inst✝⁴ : AddCommMonoid F
inst✝³ : Module S E
inst✝² : Module S E'
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : E →ₗ[S] E' →ₗ[S] F
f : M → E'
⊢ 0 ⋆[L] f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝⁸ : Monoid M
inst✝⁷ : CommSemiring S
inst✝⁶ : AddCommMonoid E
inst✝⁵ : AddCommMonoid E'
inst✝⁴ : AddCommMonoid F
inst✝³ : Module S E
inst✝² : Module S E'
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : E →ₗ[S] E' →ₗ[S] F
f : M → E
⊢ f ⋆[L] 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : E →ₗ[S] F →ₗ[S] F
e : E
f : M → F
hL : ∀ (y : F), (L e) y = y
x : M
⊢ (1, x) ∈ mulFiber x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : E →ₗ[S] F →ₗ[S] F
e : E
f : M → F
hL : ∀ (y : F), (L e) y = y
x : M
⊢ ∀ (b' : { x_1 // x_1 ∈ mulFiber x }), b' ≠ ⟨(1, x), ⋯⟩ → (L (if (↑b').1 ∈ {1} then e else 0)) (f (↑b').2) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : E →ₗ[S] F →ₗ[S] F
e : E
f : M → F
hL : ∀ (y : F), (L e) y = y
⊢ ({1}.indicator fun x ↦ e) ⋆[L] f = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : F →ₗ[S] E →ₗ[S] F
f : M → F
e : E
hL : ∀ (y : F), (L y) e = y
x : M
⊢ (x, 1) ∈ mulFiber x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : F →ₗ[S] E →ₗ[S] F
f : M → F
e : E
hL : ∀ (y : F), (L y) e = y
x : M
⊢ ∀ (b' : { x_1 // x_1 ∈ mulFiber x }), b' ≠ ⟨(x, 1), ⋯⟩ → (L (f (↑b').1)) (if (↑b').2 ∈ {1} then e else 0) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
F : Type u_6
inst✝⁶ : Monoid M
inst✝⁵ : CommSemiring S
inst✝⁴ : AddCommMonoid E
inst✝³ : AddCommMonoid F
inst✝² : Module S E
inst✝¹ : Module S F
inst✝ : TopologicalSpace F
L : F →ₗ[S] E →ₗ[S] F
f : M → F
e : E
hL : ∀ (y : F), (L y) e = y
⊢ (f ⋆[L] {1}.indicator fun x ↦ e) = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
E : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜₁
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜₁ E
t₁ t₂ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : ContinuousSMul 𝕜₁ E
V : Set E
𝓕₁ : Filter E := 𝓝 0
V_mem : V ∈ 𝓕₁
𝓕₂ : Filter E := 𝓝 0
H : 𝓕₁ ⊓ 𝓟 V = 𝓕₂ ⊓ 𝓟 V
⊢ 𝓕₁ = 𝓕₂ ⊓ 𝓟 V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
inst : TopologicalSpace E
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
hB : B.IsWeak
α : Type u_7
inst✝ : TopologicalSpace α
f : α → E
x✝ : Continuous[inst✝, inst] f
⊢ ∀ (y : F), Continuous[inst✝, inst✝⁵] fun x ↦ (B (f x)) y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : AddCommMonoid F
inst✝⁵ : Module S E
inst✝⁴ : Module S E'
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : T2Space F
inst✝ : ContinuousAdd F
f : M → E
g g' : M → E'
x : M
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExistsAt L f g x
hfg' : ConvolutionExistsAt L f g' x
⊢ (f ⋆[L] (g + g')) x = (f ⋆[L] g) x + (f ⋆[L] g') x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : AddCommMonoid E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommMonoid F
inst✝ : Module 𝕜 F
inst : TopologicalSpace E
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
hB : B.IsWeak
hB_inj : Function.Injective ⇑B
⊢ IsEmbedding fun x1 x2 ↦ (B x1) x2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : AddCommMonoid F
inst✝⁵ : Module S E
inst✝⁴ : Module S E'
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : T2Space F
inst✝ : ContinuousAdd F
f : M → E
g g' : M → E'
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExists L f g
hfg' : ConvolutionExists L f g'
⊢ f ⋆[L] (g + g') = f ⋆[L] g + f ⋆[L] g'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : AddCommMonoid F
inst✝⁵ : Module S E
inst✝⁴ : Module S E'
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : T2Space F
inst✝ : ContinuousAdd F
f f' : M → E
g : M → E'
x : M
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExistsAt L f g x
hfg' : ConvolutionExistsAt L f' g x
⊢ ((f + f') ⋆[L] g) x = (f ⋆[L] g) x + (f' ⋆[L] g) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
F : Type u_6
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : AddCommMonoid F
inst✝⁵ : Module S E
inst✝⁴ : Module S E'
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : T2Space F
inst✝ : ContinuousAdd F
f f' : M → E
g : M → E'
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExists L f g
hfg' : ConvolutionExists L f' g
⊢ (f + f') ⋆[L] g = f ⋆[L] g + f' ⋆[L] g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : AddCommMonoid E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommMonoid F
inst✝ : Module 𝕜 F
inst : TopologicalSpace E
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
hB : B.IsWeak
α : Type u_7
l : Filter α
f : α → E
x : E
hB_inj : Function.Injective ⇑B
⊢ Tendsto f l (𝓝 x) ↔ ∀ (y : F), Tendsto (fun i ↦ (B (f i)) y) l (𝓝 ((B x) y))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module S E
inst✝⁵ : Module S E'
F : Type u_10
inst✝⁴ : AddCommMonoid F
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousConstSMul S F
inst✝ : T2Space F
c : S
f : M → E
g : M → E'
x : M
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExistsAt L f g x
⊢ (c • f ⋆[L] g) x = c • (f ⋆[L] g) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
E : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜₁
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜₁ E
t₁ t₂ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : ContinuousSMul 𝕜₁ E
V : Set E
𝓕₁ : Filter E := 𝓝 0
V_mem : V ∈ 𝓕₁
𝓕₂ : Filter E := 𝓝 0
H : 𝓕₁ = 𝓕₂ ⊓ 𝓟 V
this : 𝓕₁ = 𝓕₂
⊢ t₁ = t₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
inst✝¹⁰ : Monoid M
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module S E
inst✝⁵ : Module S E'
F : Type u_10
inst✝⁴ : AddCommMonoid F
inst✝³ : Module S F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousConstSMul S F
inst✝ : T2Space F
c : S
f : M → E
g : M → E'
x : M
L : E →ₗ[S] E' →ₗ[S] F
hfg : ConvolutionExistsAt L f g x
⊢ (f ⋆[L] c • g) x = c • (f ⋆[L] g) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
E' : Type u_4
E'' : Type u_5
F : Type u_6
F' : Type u_7
G : Type u_8
R : Type u_9
inst✝⁴ : CommMonoid M
inst✝³ : CommSemiring S
inst✝² : AddCommMonoid E
inst✝¹ : Module S E
inst✝ : TopologicalSpace E
x : M
x✝ : ↑(mulFiber x)
p : M × M
h : p ∈ mulFiber x
⊢ p.swap ∈ mulFiber x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
E : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜₁
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜₁ E
t₁ t₂ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : ContinuousSMul 𝕜₁ E
V : Set E
𝓕₁ : Filter E := 𝓝 0
V_mem : V ∈ 𝓕₁
𝓕₂ : Filter E := 𝓝 0
H : 𝓕₁ = 𝓕₂ ⊓ 𝓟 V
this : V ∈ 𝓕₂
⊢ 𝓕₁ = 𝓕₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: M : Type u_1
inst✝³ : GroupWithZero M
inst✝² : TopologicalSpace M
inst✝¹ : ContinuousInv₀ M
inst✝ : T1Space M
⊢ IsOpenEmbedding Units.val

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
E : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜₁
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜₁ E
t₁ t₂ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : ContinuousSMul 𝕜₁ E
V : Set E
𝓕₁ : Filter E := 𝓝 0
V_mem : V ∈ 𝓕₁
𝓕₂ : Filter E := 𝓝 0
H : 𝓕₁ = 𝓕₂ ⊓ 𝓟 V
c : 𝕜₁
hc₀ : 0 < ‖c‖
hc₁ : ‖c‖ < 1
c_ne : c ≠ 0
⊢ c • V ∈ 𝓕₂ ⊓ 𝓟 V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution

Goal state before:
pp: M : Type u_1
S : Type u_2
E : Type u_3
inst✝⁴ : CommMonoid M
inst✝³ : CommSemiring S
inst✝² : AddCommMonoid E
inst✝¹ : Module S E
inst✝ : TopologicalSpace E
L : E →ₗ[S] E →ₗ[S] E
f g : M → E
hL : ∀ (x y : E), (L x) y = (L y) x
⊢ f ⋆[L] g = g ⋆[L] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
⊢ ∃ n₁ n₂, ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H✝ : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
n₁ n₂ : ℕ
H : ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
⊢ max (max n₁ n₂) 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.EmbeddingOfLocal

Goal state before:
pp: 𝕜₁ : Type u_1
E : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜₁
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜₁ E
t₁ t₂ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : ContinuousSMul 𝕜₁ E
V : Set E
𝓕₁ : Filter E := 𝓝 0
V_mem : V ∈ 𝓕₁
𝓕₂ : Filter E := 𝓝 0
H : 𝓕₁ = 𝓕₂ ⊓ 𝓟 V
c : 𝕜₁
hc₀ : 0 < ‖c‖
hc₁ : ‖c‖ < 1
c_ne : c ≠ 0
cV_mem : c • V ∈ 𝓕₂ ⊓ 𝓟 V
basis_𝓕₂ : 𝓕₂.HasBasis (fun s ↦ s ∈ 𝓕₂ ∧ Balanced 𝕜₁ s) id
W : Set E
hW : id W ∩ V ⊆ c • V
W_mem_𝓕₂ : W ∈ 𝓕₂
W_bal : Balanced 𝕜₁ W
w : E
w_in_W : w ∈ W
n : ℕ
hn : c ^ n • w ∈ V
this : c ^ 0 • w ∈ V
⊢ w ∈ V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
F : Type u_4
E' : Type u_5
F' : Type u_6
inst✝¹⁰ : CommSemiring 𝕜
inst✝⁹ : TopologicalSpace 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : AddCommMonoid F
inst✝⁵ : Module 𝕜 F
inst : TopologicalSpace E
inst✝⁴ : AddCommMonoid E'
inst✝³ : Module 𝕜 E'
inst✝² : AddCommMonoid F'
inst✝¹ : Module 𝕜 F'
inst✝ : TopologicalSpace E'
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
B' : E' →ₗ[𝕜] F' →ₗ[𝕜] 𝕜
e : E ≃L[𝕜] E'
f : F ≃ₗ[𝕜] F'
hBB' : ((↑e).arrowCongr (f.arrowCongr (LinearEquiv.refl 𝕜 𝕜))) B = B'
hB : B.IsWeak
⊢ inst✝ = TopologicalSpace.induced (fun x1 x2 ↦ (B' x1) x2) Pi.topologicalSpace

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H✝ : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
n₁ n₂ : ℕ
H : ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
u : Rˣ
h₁ : -1 + ↑u ∈ I ^ max (max n₁ n₂) 1
h₂ : -1 + ↑u⁻¹ ∈ I ^ max (max n₁ n₂) 1
⊢ n₁ ≤ max (max n₁ n₂) 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
E' : Type u_5
F' : Type u_6
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommMonoid F
inst✝² : Module 𝕜 F
inst : TopologicalSpace E
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
hB : B.IsWeak
inst✝¹ : ContinuousAdd 𝕜
inst✝ : ContinuousConstSMul 𝕜 𝕜
f : F
⊢ Continuous[inst, inst✝⁶] (B.flip f).toFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H✝ : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
n₁ n₂ : ℕ
H : ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
u : Rˣ
h₁ : -1 + ↑u ∈ I ^ max (max n₁ n₂) 1
h₂ : -1 + ↑u⁻¹ ∈ I ^ max (max n₁ n₂) 1
⊢ n₂ ≤ max (max n₁ n₂) 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Module.IsWeak

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
E' : Type u_5
F' : Type u_6
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommMonoid F
inst✝² : Module 𝕜 F
inst : TopologicalSpace E
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
hB : B.IsWeak
inst✝¹ : ContinuousAdd 𝕜
inst✝ : ContinuousConstSMul 𝕜 𝕜
x✝¹ x✝ : F
⊢ { toLinearMap := B.flip (x✝¹ + x✝), cont := ⋯ } =
    { toLinearMap := B.flip x✝¹, cont := ⋯ } + { toLinearMap := B.flip x✝, cont := ⋯ }

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this✝ :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
this : ∃ n₁ n₂, ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
⊢ ∃ n, n ≠ 0 ∧ ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n → -1 + ↑u⁻¹ ∈ I ^ n → ↑u ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.IsOpenUnits

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this✝ :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
this : ∃ n₁ n₂, ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
n : ℕ
hn : n ≠ 0
hn' : ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n → -1 + ↑u⁻¹ ∈ I ^ n → ↑u ∈ s
⊢ s ∈ 𝓝 1

Tactic:
H.1

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
I : Ideal R
hR : IsAdic I
hI : I ≤ ⊥.jacobson
s : Set R
hs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))
H : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(I ^ n)
this✝ :
  (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ True) fun i ↦
    ((fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.1)) ×ˢ (⇑MulOpposite.opHomeomorph.symm ⁻¹' (fun x ↦ -1 + x) ⁻¹' ↑(I ^ i.2))
this : ∃ n₁ n₂, ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n₁ → -1 + ↑u⁻¹ ∈ I ^ n₂ → ↑u ∈ s
n : ℕ
hn : n ≠ 0
hn' : ∀ (u : Rˣ), -1 + ↑u ∈ I ^ n → -1 + ↑u⁻¹ ∈ I ^ n → ↑u ∈ s
⊢ ∃ i, True ∧ (fun y ↦ 1 + y) '' ↑(I ^ i) ⊆ s