mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.Interval

Goal state before:
pp: α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg✝ : ∀ᶠ (i : ι) in p, (F i) ⊤ = (G i) ⊥
hfg' : f ⊤ = g ⊥
hf✝ :
  ∀ (K : Set ↑(Icc a b)),
    IsCompact K → ∀ (U : Set E), IsOpen[inst✝] U → MapsTo (⇑f) K U → ∀ᶠ (a_4 : ι) in p, MapsTo (⇑(F a_4)) K U
hg✝ :
  ∀ (K : Set ↑(Icc b c)),
    IsCompact K → ∀ (U : Set E), IsOpen[inst✝] U → MapsTo (⇑g) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(G a)) K U
K : Set ↑(Icc a c)
hK : IsCompact K
U : Set E
hU : IsOpen[inst✝] U
hfgU : MapsTo (⇑(f.concat g)) K U
h : b ∈ Icc a c
K₁ : Set ↑(Icc a b) := ⇑projIccCM '' Subtype.val '' (K ∩ Iic ⟨b, h⟩)
K₂ : Set ↑(Icc b c) := ⇑projIccCM '' Subtype.val '' (K ∩ Ici ⟨b, h⟩)
hK₁ : IsCompact K₁
hK₂ : IsCompact K₂
hfU : MapsTo (⇑f) K₁ U
hgU : MapsTo (⇑g) K₂ U
i : ι
hf : MapsTo (⇑(F i)) K₁ U
hg : MapsTo (⇑(G i)) K₂ U
hfg : (F i) ⊤ = (G i) ⊥
x : ↑(Icc a c)
⊢ x ∈ K → ((F i).concat (G i)) x ∈ U

Tactic:
hx

Goal state after:
α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg✝ : ∀ᶠ (i : ι) in p, (F i) ⊤ = (G i) ⊥
hfg' : f ⊤ = g ⊥
hf✝ :
  ∀ (K : Set ↑(Icc a b)),
    IsCompact K → ∀ (U : Set E), IsOpen[inst✝] U → MapsTo (⇑f) K U → ∀ᶠ (a_4 : ι) in p, MapsTo (⇑(F a_4)) K U
hg✝ :
  ∀ (K : Set ↑(Icc b c)),
    IsCompact K → ∀ (U : Set E), IsOpen[inst✝] U → MapsTo (⇑g) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(G a)) K U
K : Set ↑(Icc a c)
hK : IsCompact K
U : Set E
hU : IsOpen[inst✝] U
hfgU : MapsTo (⇑(f.concat g)) K U
h : b ∈ Icc a c
K₁ : Set ↑(Icc a b) := ⇑projIccCM '' Subtype.val '' (K ∩ Iic ⟨b, h⟩)
K₂ : Set ↑(Icc b c) := ⇑projIccCM '' Subtype.val '' (K ∩ Ici ⟨b, h⟩)
hK₁ : IsCompact K₁
hK₂ : IsCompact K₂
hfU : MapsTo (⇑f) K₁ U
hgU : MapsTo (⇑g) K₂ U
i : ι
hf : MapsTo (⇑(F i)) K₁ U
hg : MapsTo (⇑(G i)) K₂ U
hfg : (F i) ⊤ = (G i) ⊥
x : ↑(Icc a c)
hx : x ∈ K
⊢ ((F i).concat (G i)) x ∈ U
Module: Mathlib.Topology.ContinuousMap.SecondCountableSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen[inst✝] U
hfKU : MapsTo (⇑f) K U
t : Set (Set Y)
htT : t ⊆ T
htf : t.Finite
hTU : ∀ V ∈ t, V ⊆ U
hKT : K ⊆ ⇑f ⁻¹' ⋃₀ t
x : X
⊢ x ∈ K → ∃ L ∈ S, L ∈ 𝓝 x ∧ MapsTo (⇑f) L (⋃₀ t)

Tactic:
hx

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen[inst✝] U
hfKU : MapsTo (⇑f) K U
t : Set (Set Y)
htT : t ⊆ T
htf : t.Finite
hTU : ∀ V ∈ t, V ⊆ U
hKT : K ⊆ ⇑f ⁻¹' ⋃₀ t
x : X
hx : x ∈ K
⊢ ∃ L ∈ S, L ∈ 𝓝 x ∧ MapsTo (⇑f) L (⋃₀ t)
Module: Mathlib.Topology.ContinuousMap.SecondCountableSpace

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : SecondCountableTopology Y
S : Set (Set X)
hScount : S.Countable
hScomp : ∀ K ∈ S, IsCompact K
hS : ∀ (f : C(X, Y)) (V : Set Y), IsOpen[inst✝¹] V → ∀ x ∈ ⇑f ⁻¹' V, ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
x : X
V : Set Y
hV : V ∈ countableBasis Y
⊢ f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V

Tactic:
hx

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : SecondCountableTopology Y
S : Set (Set X)
hScount : S.Countable
hScomp : ∀ K ∈ S, IsCompact K
hS : ∀ (f : C(X, Y)) (V : Set Y), IsOpen[inst✝¹] V → ∀ x ∈ ⇑f ⁻¹' V, ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
x : X
V : Set Y
hV : V ∈ countableBasis Y
hx : f x ∈ V
⊢ ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
Module: Mathlib.Topology.Germ

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f g : X → Y
A : Set X
P : (x : X) → (𝓝 x).Germ Y → Prop
hf : ∀ x ∈ A, ∀ᶠ (y : X) in 𝓝 x, P y ↑f
h : ∀ x ∈ A, ∀ᶠ (y : X) in 𝓝 x, g y = f y
x : X
⊢ x ∈ A → ∀ᶠ (y : X) in 𝓝 x, P y ↑g

Tactic:
hx

Goal state after:
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f g : X → Y
A : Set X
P : (x : X) → (𝓝 x).Germ Y → Prop
hf : ∀ x ∈ A, ∀ᶠ (y : X) in 𝓝 x, P y ↑f
h : ∀ x ∈ A, ∀ᶠ (y : X) in 𝓝 x, g y = f y
x : X
hx : x ∈ A
⊢ ∀ᶠ (y : X) in 𝓝 x, P y ↑g
Module: Mathlib.Topology.Germ

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), (↑f).IsConstant
s : Set Y
a : X
ha : a ∈ f ⁻¹' s
b : Y
hb : f =ᶠ[𝓝 a] fun x ↦ b
x : X
⊢ x ∈ {x | (fun x ↦ f x = (fun x ↦ b) x) x} → x ∈ f ⁻¹' s

Tactic:
hx

Goal state after:
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h : ∀ (x : X), (↑f).IsConstant
s : Set Y
a : X
ha : a ∈ f ⁻¹' s
b : Y
hb : f =ᶠ[𝓝 a] fun x ↦ b
x : X
hx : x ∈ {x | (fun x ↦ f x = (fun x ↦ b) x) x}
⊢ x ∈ f ⁻¹' s
Module: Mathlib.Topology.Homotopy.HSpaces

Goal state before:
pp: θ : ↑I
⊢ 1 * (1 + ↑θ) ≤ 2 * ↑1

Tactic:
coe_one,

Goal state after:
θ : ↑I
⊢ 1 * (1 + ↑θ) ≤ 2 * 1
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝ : DecidableEq T
ha : 1 < a
⊢ ↑(#(pairSet J a c)) ≤ a * ↑(#J)

Tactic:
by
  wlog hJ : J.Nonempty
  · simp [Finset.not_nonempty_iff_eq_empty.mp hJ]
  unfold pairSet
  grw [Finset.card_biUnion_le, Nat.cast_sum,
    Finset.sum_le_sum fun i _ ↦ card_pairSetSeq_le_logSizeRadius_mul hJ i ha,
    Finset.sum_le_sum fun _ _ ↦ mul_le_mul_right (pow_le_pow_right₀ ha.le le_tsub_add) _]
  conv_lhs => enter [2]; ext _; rw [pow_add, pow_one, ← mul_assoc, mul_comm]
  grw [← Finset.mul_sum]
  gcongr
  grw [Finset.sum_le_sum fun i _ ↦ logSizeRadius_le_card_smallBall hJ i ha, ← Nat.cast_sum,
    ← Finset.card_biUnion fun _ _ _ _ ↦ disjoint_smallBall_logSizeBallSeq hJ]
  gcongr
  unfold logSizeBallStruct.smallBall
  rw [Finset.biUnion_subset_iff_forall_subset]
  intro i _
  exact (Finset.filter_subset _ _).trans (finset_logSizeBallSeq_subset_logSizeBallSeq_init _ _)

Goal state after:
Module: Mathlib.Topology.Instances.CantorSet

Goal state before:
pp: x : ℝ
hx : x ∈ cantorSet
⊢ ∑' (n : ℕ), ofDigitsTerm (cantorToTernary x).get n = x

Tactic:
rw [HasSum.tsum_eq]

Goal state after:
x : ℝ
hx : x ∈ cantorSet
⊢ HasSum (ofDigitsTerm (cantorToTernary x).get) x
Module: Mathlib.Topology.List

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
l : List α
s : Set (List α)
u✝ : List (Set α)
hu✝¹ : List.Forall₂ (fun b s ↦ s ∈ 𝓝 b) l u✝
hus : sequence u✝ ⊆ s
v : List (Set α)
hv : List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) l v
hvs : sequence v ⊆ s
this✝ : ∀ᶠ (y : List α) in traverse 𝓝 l, y ∈ sequence v
u : List α
hu✝ : u ∈ sequence v
hu : List.Forall₂ (fun b a ↦ b ∈ id a) u v
this : List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) u v
⊢ s ∈ traverse 𝓝 u

Tactic:
grw [← hvs]

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
l : List α
s : Set (List α)
u✝ : List (Set α)
hu✝¹ : List.Forall₂ (fun b s ↦ s ∈ 𝓝 b) l u✝
hus : sequence u✝ ⊆ s
v : List (Set α)
hv : List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) l v
hvs : sequence v ⊆ s
this✝ : ∀ᶠ (y : List α) in traverse 𝓝 l, y ∈ sequence v
u : List α
hu✝ : u ∈ sequence v
hu : List.Forall₂ (fun b a ↦ b ∈ id a) u v
this : List.Forall₂ (fun a s ↦ IsOpen[inst✝] s ∧ a ∈ s) u v
⊢ sequence v ∈ traverse 𝓝 u
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
hJ : J.Nonempty
P : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J) (hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c),
    let P := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    s ∈ (logSizeBallSeq J hJ a c l).finset →
      t ∈ (logSizeBallSeq J hJ a c l).finset →
        s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset →
          edist (f ↑⟨s, hs⟩) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
⊢ edist t s ≤ c

Tactic:
edist_comm

Goal state after:
T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
hJ : J.Nonempty
P : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J) (hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c),
    let P := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    s ∈ (logSizeBallSeq J hJ a c l).finset →
      t ∈ (logSizeBallSeq J hJ a c l).finset →
        s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset →
          edist (f ↑⟨s, hs⟩) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
⊢ edist s t ≤ c
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: case inr
T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
hJ : J.Nonempty
P : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J) (hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c),
    let P := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    s ∈ (logSizeBallSeq J hJ a c l).finset →
      t ∈ (logSizeBallSeq J hJ a c l).finset →
        s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset →
          edist (f ↑⟨s, hs⟩) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hts : edist t s ≤ c
⊢ edist (f ↑⟨s, hs⟩) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)

Tactic:
edist_comm

Goal state after:
case inr
T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
hJ : J.Nonempty
P : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J) (hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c),
    let P := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    s ∈ (logSizeBallSeq J hJ a c l).finset →
      t ∈ (logSizeBallSeq J hJ a c l).finset →
        s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset →
          edist (f ↑⟨s, hs⟩) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hts : edist t s ≤ c
⊢ edist (f ↑↑⟨⟨t, ht⟩, hst⟩) (f ↑⟨s, hs⟩) ≤ 2 * ⨆ p, edist (f (↑p).1) (f (↑p).2)
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
x✝ : α × α
| edist x✝.1 x✝.2 < δ

Tactic:
edist_comm

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h : ⨆ x ∈ s, infEDist x t < δ ∧ ⨆ y ∈ t, infEDist y s < δ
x✝ : α × α
| edist x✝.2 x✝.1 < δ
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
s✝ t✝ : Set α
δ : ℝ≥0∞
h✝ : ⨆ x ∈ s✝, infEDist x t✝ < δ ∧ ⨆ y ∈ t✝, infEDist y s✝ < δ
s t : Set α
h : ⨆ x ∈ s, infEDist x t < δ
x : α
⊢ x ∈ s → x ∈ SetRel.preimage {p | edist p.1 p.2 < δ} t

Tactic:
hx

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
s✝ t✝ : Set α
δ : ℝ≥0∞
h✝ : ⨆ x ∈ s✝, infEDist x t✝ < δ ∧ ⨆ y ∈ t✝, infEDist y s✝ < δ
s t : Set α
h : ⨆ x ∈ s, infEDist x t < δ
x : α
hx : x ∈ s
⊢ x ∈ SetRel.preimage {p | edist p.1 p.2 < δ} t
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
hJ : J.Nonempty
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
P : ℕ → Prop := ⋯
l : ℕ := ⋯
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
h : edist (logSizeBallSeq J hJ a c l).point s ≤ (↑(logSizeBallSeq J hJ a c l).radius - 1) * c
hsB : s ∈ (logSizeBallSeq J hJ a c l).ball c
htB : t ∈ (logSizeBallSeq J hJ a c l).ball c
hsP : ((logSizeBallSeq J hJ a c l).point, s) ∈ pairSetSeq J a c l
htP : ((logSizeBallSeq J hJ a c l).point, t) ∈ pairSetSeq J a c l
sup_bound : ∀ {x y : T}, (x, y) ∈ pairSetSeq J a c l → edist (f x) (f y) ≤ ⨆ p, edist (f (↑p).1) (f (↑p).2)
⊢ edist (f ↑⟨s, hs⟩) (f (logSizeBallSeq J hJ a c l).point) +
      edist (f (logSizeBallSeq J hJ a c l).point) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤
    (⨆ p, edist (f (↑p).1) (f (↑p).2)) + ⨆ p, edist (f (↑p).1) (f (↑p).2)

Tactic:
edist_comm

Goal state after:
T : Type u_1
inst✝² : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝¹ : DecidableEq T
E : Type u_2
inst✝ : PseudoEMetricSpace E
ha : 1 < a
f : T → E
hJ : J.Nonempty
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c
P : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
h : edist (logSizeBallSeq J hJ a c l).point s ≤ (↑(logSizeBallSeq J hJ a c l).radius - 1) * c
hsB : s ∈ (logSizeBallSeq J hJ a c l).ball c
htB : t ∈ (logSizeBallSeq J hJ a c l).ball c
hsP : ((logSizeBallSeq J hJ a c l).point, s) ∈ pairSetSeq J a c l
htP : ((logSizeBallSeq J hJ a c l).point, t) ∈ pairSetSeq J a c l
sup_bound : ∀ {x y : T}, (x, y) ∈ pairSetSeq J a c l → edist (f x) (f y) ≤ ⨆ p, edist (f (↑p).1) (f (↑p).2)
⊢ edist (f (logSizeBallSeq J hJ a c l).point) (f ↑⟨s, hs⟩) +
      edist (f (logSizeBallSeq J hJ a c l).point) (f ↑↑⟨⟨t, ht⟩, hst⟩) ≤
    (⨆ p, edist (f (↑p).1) (f (↑p).2)) + ⨆ p, edist (f (↑p).1) (f (↑p).2)
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h :
  (s, t).1 ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} (s, t).2 ∧ (s, t).2 ⊆ SetRel.image {p | edist p.1 p.2 ≤ δ} (s, t).1
x✝ : α × α
| edist x✝.1 x✝.2 ≤ δ

Tactic:
edist_comm

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
s t : Set α
δ : ℝ≥0∞
h :
  (s, t).1 ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} (s, t).2 ∧ (s, t).2 ⊆ SetRel.image {p | edist p.1 p.2 ≤ δ} (s, t).1
x✝ : α × α
| edist x✝.2 x✝.1 ≤ δ
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
s✝ t✝ : Set α
δ : ℝ≥0∞
h✝ :
  (s✝, t✝).1 ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} (s✝, t✝).2 ∧
    (s✝, t✝).2 ⊆ SetRel.image {x | edist x.2 x.1 ≤ δ} (s✝, t✝).1
s t : Set α
h : s ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} t
x : α
⊢ x ∈ s → infEDist x t ≤ δ

Tactic:
hx

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
s✝ t✝ : Set α
δ : ℝ≥0∞
h✝ :
  (s✝, t✝).1 ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} (s✝, t✝).2 ∧
    (s✝, t✝).2 ⊆ SetRel.image {x | edist x.2 x.1 ≤ δ} (s✝, t✝).1
s t : Set α
h : s ⊆ SetRel.preimage {p | edist p.1 p.2 ≤ δ} t
x : α
hx : x ∈ s
⊢ infEDist x t ≤ δ
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
⊢ x ∈ s n → ∃ y ∈ t0, edist x y ≤ 2 * B n

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s✝ : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
B_pos : ∀ (n : ℕ), 0 < B n
B_ne_top : ∀ (n : ℕ), B n ≠ ∞
s : ℕ → Closeds α
hs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N
t0 : Set α := ⋂ n, closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ m, ⋃ (_ : m ≥ n), ↑(s m))
t : Closeds α := { carrier := t0, isClosed' := ⋯ }
n : ℕ
x : α
hx : x ∈ s n
⊢ ∃ y ∈ t0, edist x y ≤ 2 * B n
Module: Mathlib.Topology.MetricSpace.Kuratowski

Goal state before:
pp: case inr.refine_2
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
ι : Type u_1
f : α → ↥(lp (fun x ↦ ℝ) ∞)
K : ℝ≥0
hfl : ∀ (i : ι), LipschitzOnWith K (fun a ↦ ↑(f a) i) s
g : ι → α → ℝ
hgl : ∀ (i : ι), LipschitzWith K (g i)
hgeq : ∀ (i : ι), EqOn (fun x ↦ ↑(f x) i) (g i) s
a₀ : α
ha₀_in_s : a₀ ∈ s
hf_extb : ∀ (a : α), Memℓp (swap g a) ∞
f_ext' : α → ↥(lp (fun x ↦ ℝ) ∞) := fun i ↦ ⟨swap g i, ⋯⟩
⊢ EqOn f f_ext' s

Tactic:
· intro a hyp
      ext i
      exact (hgeq i) hyp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
⊢ eHolderNorm r f = ∞ ↔ ¬MemHolder r f

Tactic:
by
  rw [← eHolderNorm_ne_top, not_not]

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

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
h : eHolderNorm r f = 0
x₁ x₂ : X
hx : x₁ = x₂
⊢ f x₁ = f x₂

Tactic:
hx

Goal state after:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
h : eHolderNorm r f = 0
x₁ x₂ : X
hx : x₁ = x₂
⊢ f x₂ = f x₂
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case e'_4
α : Type u_1
inst✝ : PseudoEMetricSpace α
x y : α
hxy : x ≠ y
⊢ edist x y = edist y x

Tactic:
edist_comm

Goal state after:
case e'_4
α : Type u_1
inst✝ : PseudoEMetricSpace α
x y : α
hxy : x ≠ y
⊢ edist y x = edist y x
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝ : PseudoEMetricSpace α
s : Set α
y : α
hy : y ∈ s
z : α
hyz : y ≠ z
⊢ ⨅ y ∈ s, ⨅ (_ : z ≠ y), edist z y ≤ edist y z ∨ s.einfsep ≤ edist y z

Tactic:
edist_comm

Goal state after:
case inr.inl
α : Type u_1
inst✝ : PseudoEMetricSpace α
s : Set α
y : α
hy : y ∈ s
z : α
hyz : y ≠ z
⊢ ⨅ y ∈ s, ⨅ (_ : z ≠ y), edist z y ≤ edist z y ∨ s.einfsep ≤ edist z y
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoMetricSpace α
s : Set α
d : ℝ
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
hd : d ≤ 0
hs : s.Subsingleton
⊢ d ≤ dist x y

Tactic:
exact le_trans hd dist_nonneg

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

Goal state before:
pp: α : Type u
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
s : Set α
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
this : ∀ x ∈ s, (𝓝 x).HasBasis (fun r ↦ 0 < r ∧ r < R x) fun r ↦ ball x r
⊢ ∃ ι c r r',
    (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
      (LocallyFinite fun i ↦ ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i)

Tactic:
rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with
    ⟨ι, c, r', hr', hsub', hfin⟩

Goal state after:
α : Type u
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
s : Set α
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
this : ∀ x ∈ s, (𝓝 x).HasBasis (fun r ↦ 0 < r ∧ r < R x) fun r ↦ ball x r
ι : Type u
c : ι → α
r' : ι → ℝ
hr' : ∀ (a : ι), c a ∈ s ∧ 0 < r' a ∧ r' a < R (c a)
hsub' : s ⊆ ⋃ a, ball (c a) (r' a)
hfin : LocallyFinite fun a ↦ ball (c a) (r' a)
⊢ ∃ ι c r r',
    (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
      (LocallyFinite fun i ↦ ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i)
Module: Mathlib.Topology.MetricSpace.Snowflaking

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
x y : Snowflaking X α hα₀ hα₁
⊢ edist (ofSnowflaking x) (ofSnowflaking y) ^ α = edist (ofSnowflaking y) (ofSnowflaking x) ^ α

Tactic:
edist_comm

Goal state after:
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
x y : Snowflaking X α hα₀ hα₁
⊢ edist (ofSnowflaking y) (ofSnowflaking x) ^ α = edist (ofSnowflaking y) (ofSnowflaking x) ^ α
Module: Mathlib.Topology.MetricSpace.Snowflaking

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
⊢ (𝓤 X).HasBasis (fun x ↦ 0 < x) fun x ↦ {p | edist p.1 p.2 < x ^ α⁻¹}

Tactic:
refine EMetric.mk_uniformity_basis (fun _ _ ↦ by positivity) fun ε hε ↦
        ⟨ε ^ α, by positivity, ?_⟩

Goal state after:
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
ε : ℝ≥0∞
hε : 0 < ε
⊢ (ε ^ α) ^ α⁻¹ ≤ ε
Module: Mathlib.Topology.MetricSpace.Snowflaking

Goal state before:
pp: case h
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
aux : PseudoMetricSpace (Snowflaking X α hα₀ hα₁) := PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist ⋯ ⋯
s : Set (Snowflaking X α hα₀ hα₁)
C : ℝ
hC : ∀ ⦃x : Snowflaking X α hα₀ hα₁⦄, x ∈ s → ∀ ⦃y : Snowflaking X α hα₀ hα₁⦄, y ∈ s → dist x y ≤ C
x : X
⊢ x ∈ ⇑toSnowflaking ⁻¹' s → ∀ ⦃y : X⦄, y ∈ ⇑toSnowflaking ⁻¹' s → dist x y ≤ C ^ α⁻¹

Tactic:
hx

Goal state after:
case h
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
aux : PseudoMetricSpace (Snowflaking X α hα₀ hα₁) := PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist ⋯ ⋯
s : Set (Snowflaking X α hα₀ hα₁)
C : ℝ
hC : ∀ ⦃x : Snowflaking X α hα₀ hα₁⦄, x ∈ s → ∀ ⦃y : Snowflaking X α hα₀ hα₁⦄, y ∈ s → dist x y ≤ C
x : X
hx : x ∈ ⇑toSnowflaking ⁻¹' s
⊢ ∀ ⦃y : X⦄, y ∈ ⇑toSnowflaking ⁻¹' s → dist x y ≤ C ^ α⁻¹