mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.HolderNorm

Goal state before:
pp: case pos.refine_1
X : Type u_1
Y : Type u_2
inst✝⁴ : MetricSpace X
inst✝³ : NormedAddCommGroup Y
r : ℝ≥0
f : X → Y
α : Type u_3
inst✝² : NormedRing α
inst✝¹ : Module α Y
inst✝ : NormSMulClass α Y
c : α
hc : ¬‖c‖₊ = 0
hf : MemHolder r f
⊢ ↑(nnHolderNorm r f * ‖c‖₊) ≤ ↑‖c‖₊ * eHolderNorm r f

pp: case pos.refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : MetricSpace X
inst✝³ : NormedAddCommGroup Y
r : ℝ≥0
f : X → Y
α : Type u_3
inst✝² : NormedRing α
inst✝¹ : Module α Y
inst✝ : NormSMulClass α Y
c : α
hc : ¬‖c‖₊ = 0
hf : MemHolder r f
⊢ eHolderNorm r f ≤ eHolderNorm r (c • f) / ↑‖c‖₊

Tactic:
· rw [coe_mul, hf.coe_nnHolderNorm_eq_eHolderNorm, mul_comm]

Goal state after:
case pos.refine_2
X : Type u_1
Y : Type u_2
inst✝⁴ : MetricSpace X
inst✝³ : NormedAddCommGroup Y
r : ℝ≥0
f : X → Y
α : Type u_3
inst✝² : NormedRing α
inst✝¹ : Module α Y
inst✝ : NormSMulClass α Y
c : α
hc : ¬‖c‖₊ = 0
hf : MemHolder r f
⊢ eHolderNorm r f ≤ eHolderNorm r (c • f) / ↑‖c‖₊
Module: Mathlib.Topology.NatEmbedding

Goal state before:
pp: case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : Infinite X
U : ℕ → Set X
hne : ∀ (n : ℕ), (U n).Nonempty
ho : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hd : Pairwise (Disjoint on U)
n i j : ℕ
hij : U (Nat.pair n i) = U (Nat.pair n j)
⊢ ¬(Disjoint on U) (Nat.pair n i) (Nat.pair n j)

Tactic:
simpa [hij, onFun] using (hne _).ne_empty

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n ↦ ℝ) ∞)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n ↦ ℝ) ∞) :=
  fun x ↦
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p ↦ dist (f p.1) (f p.2)
x y : X ⊕ Y
A : ∀ (z : X ⊕ Y), f z ∈ range Φ ∪ range Ψ
⊢ F (x, y) ≤ diam (range Φ ∪ range Ψ)

Tactic:
refine dist_le_diam_of_mem ?_ (A _) (A _)

Goal state after:
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ↑p ↑q < diam univ + 1 + diam univ
Φ : X → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Φrange : range Φ = ↑p
Ψ : Y → ↥(lp (fun n ↦ ℝ) ∞)
Ψisom : Isometry Ψ
Ψrange : range Ψ = ↑q
I : diam (range Φ ∪ range Ψ) ≤ 2 * diam univ + 1 + 2 * diam univ
f : X ⊕ Y → ↥(lp (fun n ↦ ℝ) ∞) :=
  fun x ↦
    match x with
    | inl y => Φ y
    | inr z => Ψ z
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p ↦ dist (f p.1) (f p.2)
x y : X ⊕ Y
A : ∀ (z : X ⊕ Y), f z ∈ range Φ ∪ range Ψ
⊢ Bornology.IsBounded (range Φ ∪ range Ψ)
Module: Mathlib.Topology.Order.HullKernel

Goal state before:
pp: α : Type u_1
inst✝ : SemilatticeInf α
T : Set α
hT : ∀ p ∈ T, InfPrime p
a b : α
⊢ hull T (a ⊓ b) = hull T a ∪ hull T b

Tactic:
by
  grind [InfPrime.inf_le]

Goal state after:
Module: Mathlib.Topology.Order.NhdsSet

Goal state before:
pp: case inr
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a b : α
h : a ≤ b
hlt : a < b
⊢ 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b)

Tactic:
· rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc]

Goal state after:
Module: Mathlib.Topology.Order.PartialSups

Goal state before:
pp: L : Type u_1
inst✝² : SemilatticeSup L
inst✝¹ : TopologicalSpace L
inst✝ : ContinuousSup L
α : Type u_2
l : Filter α
f : ℕ → α → L
g : ℕ → L
n : ℕ
hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))
⊢ Tendsto (fun a ↦ (partialSups fun x ↦ f x a) n) l (𝓝 ((partialSups g) n))

Tactic:
simpa only [← Pi.partialSups_apply] using Tendsto.partialSups hf

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: x y z : GHSpace
⊢ dist x z ≤ dist x y + dist y z

Tactic:
let X := x.Rep

Goal state after:
x y z : GHSpace
X : Type := x.Rep
⊢ dist x z ≤ dist x y + dist y z
Module: Mathlib.Topology.Sets.BaseChangeNhds

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
K : Compacts α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
⊢ ⋯.functor.Initial

Tactic:
Monotone.initial_functor_iff

Goal state after:
α : Type u_1
inst✝² : TopologicalSpace α
K : Compacts α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
⊢ ∀ (j₁ : ↑K.openNhds), ∃ j₂, K.openRcNhdsToOpenNhds j₂ ≤ j₁
Module: Mathlib.Topology.Sets.BaseChangeNhds

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
K : Compacts α
inst✝ : T2Space α
⊢ ⋯.functor.Initial

Tactic:
Monotone.initial_functor_iff

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
K : Compacts α
inst✝ : T2Space α
⊢ ∀ (j₁ : ↑K.compactNhds), ∃ j₂, K.openRcNhdsToCompactNhds j₂ ≤ j₁
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
i : ι
⊢ res F U ≫ limit.π (Discrete.functor fun i ↦ F.obj (op (U i))) { as := i } = F.map (leSupr U i).op

Tactic:
by
  rw [res, limit.lift_π, Fan.mk_π_app]

Goal state after:
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (U V : OpenNhds x) (fU : (y : ↥↑U) → F.stalk ↑y),
    (Sheafify.isLocallyGerm F).pred fU →
      ∀ (fV : (y : ↥↑V) → F.stalk ↑y),
        (Sheafify.isLocallyGerm F).pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥↑W), fU (iU w) = fV (iV w)

Tactic:
intro U V fU hU fV hV e

Goal state after:
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
⊢ ∃ W iU iV, ∀ (w : ↥↑W), fU (iU w) = fV (iV w)
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ HasSmallInductiveDimensionLT X 1 ↔ IsTopologicalBasis {s | IsClopen s}

Tactic:
by
  constructor
  · intro (.succ _ s hs h)
    refine hs.of_isOpen_of_subset (fun _ hU ↦ hU.isOpen) (fun U hU ↦ ⟨?_, hs.isOpen hU⟩)
    rw [← closure_subset_iff_isClosed]
    cases h U hU
    rwa [isEmpty_coe_sort, (hs.isOpen hU).frontier_eq, sdiff_eq_empty] at ‹_›
  · exact fun h ↦ .succ 0 _ h fun _ hU ↦ hU.frontier_eq ▸ .zero

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ smallInductiveDimension X < ↑0 ↔ smallInductiveDimension X < 0

Tactic:
WithBot.coe_zero

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Tactic:
let Φ := toInductiveLimit I

Goal state after:
X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
⊢ ∃ x, Tendsto u atTop (𝓝 x)
Module: Mathlib.Topology.UniformSpace.CompareReals

Goal state before:
pp: ⊢ IsUniformInducing Rat.cast

Tactic:
by
      rw [Rat.uniformSpace_eq]
      exact Rat.isUniformEmbedding_coe_real.isUniformInducing

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : UniformSpace X
x : X
V : SetRel X X
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ IsOpen[inst✝¹.toTopologicalSpace] (ball x V)

Tactic:
rw [isOpen_iff_ball_subset]

Goal state after:
X : Type u_1
inst✝¹ : UniformSpace X
x : X
V : SetRel X X
inst✝ : V.IsTrans
h' : V ∈ 𝓤 X
⊢ ∀ x_1 ∈ ball x V, ∃ V_1 ∈ 𝓤 X, ball x_1 V_1 ⊆ ball x V
Module: Mathlib.Topology.UniformSpace.Ultra.Constructions

Goal state before:
pp: X : Type u_1
ι : Type u_3
U : ι → UniformSpace X
hU : ∀ (i : ι), IsUltraUniformity X
this : UniformSpace X := ⨅ i, U i
⊢ IsUltraUniformity X

Tactic:
refine .mk_of_hasBasis (iInf_uniformity ▸ Filter.HasBasis.iInf fun i ↦ (hU i).hasBasis) ?_ ?_ <;>
  · simp only [forall_and, Subtype.forall, id_eq, Set.iInter_coe_set, and_imp]
    rintro _ _ _ _ _
    infer_instance

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case h
X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this✝¹ : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun n ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this✝ : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
⊢ Tendsto NonemptyCompacts.toGHSpace (𝓝 L) (𝓝 L.toGHSpace)

Tactic:
apply toGHSpace_continuous.tendsto

Goal state after: