mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Span

Goal state before:
pp: case cons
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
s : Finset I
inst✝ : WellFoundedLT I
x : ↑(π C fun x ↦ x ∈ s)
l : List I := s.sort fun x1 x2 ↦ x1 ≥ x2
a : I
as : List I
ha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)
c : Products I →₀ ℤ
hc : ↑c.support ⊆ {m | ↑m ≤ as}
hc' :
  (c.sum fun i a ↦ a • Products.eval (π C fun x ↦ x ∈ s) i) =
    (List.map (fun i ↦ if ↑x i = true then e (π C fun x ↦ x ∈ s) i else 1 - e (π C fun x ↦ x ∈ s) i) as).prod
⊢ ((if ↑x a = true then e (π C fun x ↦ x ∈ s) a else 1 - e (π C fun x ↦ x ∈ s) a) *
      c.sum fun i a ↦ a • Products.eval (π C fun x ↦ x ∈ s) i) ∈
    Submodule.span ℤ (Products.eval (π C fun x ↦ x ∈ s) '' {m | ↑m ≤ a :: as})

Tactic:
clear hc'

Goal state after:
case cons
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
s : Finset I
inst✝ : WellFoundedLT I
x : ↑(π C fun x ↦ x ∈ s)
l : List I := s.sort fun x1 x2 ↦ x1 ≥ x2
a : I
as : List I
ha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)
c : Products I →₀ ℤ
hc : ↑c.support ⊆ {m | ↑m ≤ as}
⊢ ((if ↑x a = true then e (π C fun x ↦ x ∈ s) a else 1 - e (π C fun x ↦ x ∈ s) a) *
      c.sum fun i a ↦ a • Products.eval (π C fun x ↦ x ∈ s) i) ∈
    Submodule.span ℤ (Products.eval (π C fun x ↦ x ∈ s) '' {m | ↑m ≤ a :: as})
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: case inr
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
a : Products I
ha : Products.isGood C a
b : Products I
hb : Products.isGood C b
h : Products.eval C a = Products.eval C b
hne : ⟨a, ha⟩ ≠ ⟨b, hb⟩
h' : ⟨b, hb⟩ < ⟨a, ha⟩
⊢ False

Tactic:
apply ha

Goal state after:
case inr
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
a : Products I
ha : Products.isGood C a
b : Products I
hb : Products.isGood C b
h : Products.eval C a = Products.eval C b
hne : ⟨a, ha⟩ ≠ ⟨b, hb⟩
h' : ⟨b, hb⟩ < ⟨a, ha⟩
⊢ Products.eval C a ∈ Submodule.span ℤ (Products.eval C '' {m | m < a})
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: e : forget CompHaus ≅ compactumToCompHaus.inv ⋙ Compactum.forget :=
  (((forget CompHaus).leftUnitor.symm ≪≫
        Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) ≪≫
      compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) ≪≫
    compactumToCompHaus.inv.isoWhiskerLeft compactumToCompHausCompForget
⊢ CreatesLimits (forget CompHaus)

Tactic:
exact createsLimitsOfNatIso e.symm

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
⊢ ord I i < o

Tactic:
by_contra h'
  apply h
  suffices eval C l = 0 by simp [this]
  ext x
  simp only [eval_eq, LocallyConstant.coe_zero, Pi.zero_apply, ite_eq_right_iff, one_ne_zero]
  contrapose! h'
  exact hsC x.val x.prop i (h'.1 i hi)

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: case refine_3.inl
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
f : LocallyConstant ↑C ℤ
hf :
  ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) =
    ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x ↦ ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x ↦ ord I x < o) → ↑C := fun x ↦ ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
x : I → Bool
hx : x ∈ C
hx₀ : x ∈ C0 C ho
⊢ ((πs C o)
        (LocallyConstant.piecewise' ⋯ ⋯ ⋯ (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f)
          (LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⋯))
      ⟨x, hx⟩ =
    f ⟨x, hx⟩

pp: case refine_3.inr
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
f : LocallyConstant ↑C ℤ
hf :
  ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) =
    ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x ↦ ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x ↦ ord I x < o) → ↑C := fun x ↦ ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
x : I → Bool
hx : x ∈ C
hx₁ : x ∈ C1 C ho
⊢ ((πs C o)
        (LocallyConstant.piecewise' ⋯ ⋯ ⋯ (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f)
          (LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⋯))
      ⟨x, hx⟩ =
    f ⟨x, hx⟩

Tactic:
· have hx₀' : ProjRestrict C (ord I · < o) ⟨x, hx⟩ = x := by
        simpa only [ProjRestrict, Set.MapsTo.val_restrict_apply] using! C0_projOrd C hsC ho hx₀
      simp only [C₀C, πs_apply_apply, hx₀', hx₀, LocallyConstant.piecewise'_apply_left,
        LocallyConstant.coe_comap, ContinuousMap.coe_mk, Function.comp_apply]

Goal state after:
case refine_3.inr
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
f : LocallyConstant ↑C ℤ
hf :
  ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) =
    ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x ↦ ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x ↦ ord I x < o) → ↑C := fun x ↦ ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
x : I → Bool
hx : x ∈ C
hx₁ : x ∈ C1 C ho
⊢ ((πs C o)
        (LocallyConstant.piecewise' ⋯ ⋯ ⋯ (LocallyConstant.comap { toFun := C₀C, continuous_toFun := h₀ } f)
          (LocallyConstant.comap { toFun := C₁C, continuous_toFun := h₁ } f) ⋯))
      ⟨x, hx⟩ =
    f ⟨x, hx⟩
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
n : ℕ
⊢ ↑(skeletonLT C ↑n) ∪ ⋃ j, closedCell n j = ↑(skeletonLT C (↑n + 1))

Tactic:
rw [coe_skeletonLT, coe_skeletonLT, union_assoc]

Goal state after:
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
n : ℕ
⊢ D ∪ ((⋃ m, ⋃ (_ : ↑m < ↑n), ⋃ j, closedCell m j) ∪ ⋃ j, closedCell n j) =
    D ∪ ⋃ m, ⋃ (_ : ↑m < ↑n + 1), ⋃ j, closedCell m j
Module: Mathlib.Topology.ContinuousMap.BoundedCompactlySupported

Goal state before:
pp: case pos
α : Type u_1
γ : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : NonUnitalNormedRing γ
c : Nonempty α
f : α →ᵇ γ
h : f ∈ C_cb(α, γ)
hs : (tsupport ⇑f).Nonempty
x : α
left✝ : x ∈ tsupport ⇑f
hmax : IsMaxOn (fun x ↦ ‖f x‖) (tsupport ⇑f) x
⊢ ∃ x, ‖f x‖ = ‖f‖

Tactic:
refine ⟨x, le_antisymm (norm_coe_le_norm f x) (norm_le (norm_nonneg _) |>.mpr fun y ↦ ?_)⟩

Goal state after:
case pos
α : Type u_1
γ : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : NonUnitalNormedRing γ
c : Nonempty α
f : α →ᵇ γ
h : f ∈ C_cb(α, γ)
hs : (tsupport ⇑f).Nonempty
x : α
left✝ : x ∈ tsupport ⇑f
hmax : IsMaxOn (fun x ↦ ‖f x‖) (tsupport ⇑f) x
y : α
⊢ ‖f y‖ ≤ ‖f x‖
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))
m n : ↑(MaxProducts C ho)
h : MaxToGood C hC hsC ho h₁ m = MaxToGood C hC hsC ho h₁ n
⊢ ↑↑m = ↑↑n

Tactic:
rw [Subtype.ext_iff] at h

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))
m n : ↑(MaxProducts C ho)
h : ↑(MaxToGood C hC hsC ho h₁ m) = ↑(MaxToGood C hC hsC ho h₁ n)
⊢ ↑↑m = ↑↑n
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))
m n : ↑(MaxProducts C ho)
h : (↑m).Tail = (↑n).Tail
⊢ term I ho :: ↑(↑m).Tail = ↑↑n

Tactic:
max_eq_o_cons_tail C hsC ho n,

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))
m n : ↑(MaxProducts C ho)
h : (↑m).Tail = (↑n).Tail
⊢ term I ho :: ↑(↑m).Tail = term I ho :: ↑(↑n).Tail
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: N : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
x✝ : X
f : ↑(Ω^ N X x✝)
g : (N → ↑I) → X
h : g = ⇑f
x : N → ↑I
⊢ (copy f g h) x = f x

Tactic:
exact congr_fun h x

Goal state after:
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst✝ : DecidableEq T
ha : 1 < a
hJ_card : ↑(#J) ≤ a ^ n
s t : T
h : (s, t) ∈ pairSet J a c
i : ℕ
hiJ : i < #J
h' : (s, t) ∈ pairSetSeq J a c i
hJ : J.Nonempty
⊢ edist s t ≤ ↑n * c

Tactic:
wlog! hn : 1 ≤ n

Goal state after:
case inr
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst✝ : DecidableEq T
ha : 1 < a
hJ_card : ↑(#J) ≤ a ^ n
s t : T
h : (s, t) ∈ pairSet J a c
i : ℕ
hiJ : i < #J
h' : (s, t) ∈ pairSetSeq J a c i
hJ : J.Nonempty
this :
  ∀ {T : Type u_1} [inst : PseudoEMetricSpace T] {a c : ℝ≥0∞} {n : ℕ} {J : Finset T} [inst_1 : DecidableEq T],
    1 < a →
      ↑(#J) ≤ a ^ n →
        ∀ {s t : T},
          (s, t) ∈ pairSet J a c → ∀ i < #J, (s, t) ∈ pairSetSeq J a c i → J.Nonempty → 1 ≤ n → edist s t ≤ ↑n * c
hn : n < 1
⊢ edist s t ≤ ↑n * c

T✝ : Type u_1
inst✝³ : PseudoEMetricSpace T✝
a✝ c✝ : ℝ≥0∞
n✝ : ℕ
J✝ : Finset T✝
inst✝² : DecidableEq T✝
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
n : ℕ
J : Finset T
inst✝ : DecidableEq T
ha : 1 < a
hJ_card : ↑(#J) ≤ a ^ n
s t : T
h : (s, t) ∈ pairSet J a c
i : ℕ
hiJ : i < #J
h' : (s, t) ∈ pairSetSeq J a c i
hJ : J.Nonempty
hn : 1 ≤ n
⊢ edist s t ≤ ↑n * c
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: N : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
x✝ : X
M : Type ?u.12
x : X
p : ↑(Ω^ (M ⊕ N) X x)
⊢ (fun p ↦ ⟨(GenLoop.uncurry x p).comp { toFun := sumArrowHomeomorphProdArrow.toFun, continuous_toFun := ⋯ }, ⋯⟩)
      ((fun q ↦ ⟨currySum x q, ⋯⟩) p) =
    p

Tactic:
by ext; simp

Goal state after:
Module: Mathlib.Topology.IsClosedRestrict

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
s : Set ((i : ι) → α i)
S : Set ι
p : ↑(Sᶜ.domRestrict '' s) × ((i : ↑S) → α ↑i)
⊢ Sᶜ.domRestrict (reorderRestrictProd S s p) = ↑p.1

Tactic:
by ext; simp

Goal state after:
Module: Mathlib.Topology.IsClosedRestrict

Goal state before:
pp: ι : Type u_1
α✝ : ι → Type u_2
s✝ : Set ((i : ι) → α✝ i)
i : ι
S✝ : Set ι
α : ι → Type u_3
inst✝ : (i : ι) → TopologicalSpace (α i)
S : Set ι
s : Set ((j : ι) → α j)
x : ↑(Sᶜ.domRestrict ⁻¹' Sᶜ.domRestrict '' s)
⊢ (fun p ↦ ⟨reorderRestrictProd S s p, ⋯⟩) ((fun x ↦ (⟨Sᶜ.domRestrict ↑x, ⋯⟩, fun i ↦ ↑x ↑i)) x) = x

Tactic:
by ext; simp

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

Goal state before:
pp: a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h✝ : ofDigits a = ofDigits b
h : ∃ a_1, a a_1 ≠ b a_1
n0 : ℕ := Nat.find h
⊢ False

Tactic:
have h1 (n) (hn : n < n0) : a n = b n := by simpa using Nat.find_min h hn

Goal state after:
a b : ℕ → Fin 3
ha : ∀ (n : ℕ), a n ≠ 1
hb : ∀ (n : ℕ), b n ≠ 1
h✝ : ofDigits a = ofDigits b
h : ∃ a_1, a a_1 ≠ b a_1
n0 : ℕ := Nat.find h
h1 : ∀ n < n0, a n = b n
⊢ False
Module: Mathlib.Topology.MetricSpace.CoveringNumbers

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
⊢ ∃ C' ⊆ A, C = f '' C'

Tactic:
have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
A : Set X
ε : ℝ≥0
f : X → Y
hf : Isometry f
hf_inj : InjOn f A
C : Set Y
hC_subset : C ⊆ f '' A
hC_cover : IsCover ε (f '' A) C
this : ∀ (x : ↑C), ∃ y ∈ A, f y = ↑x
⊢ ∃ C' ⊆ A, C = f '' C'
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝ : EDist α
x y : α
⊢ ∀ (x_1 : α),
    x_1 = x ∨ x_1 = y →
      ∀ (y_1 : α), y_1 = x ∨ y_1 = y → x_1 ≠ y_1 → edist x y ≤ edist x_1 y_1 ∨ edist y x ≤ edist x_1 y_1

Tactic:
rintro a (rfl | rfl) b (rfl | rfl) hab <;> (try simp only [le_refl, true_or, or_true]) <;>
    contradiction

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

Goal state before:
pp: X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
⊢ ⟦p⟧ = toGHSpace X ↔ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑p

Tactic:
simp only [toGHSpace, Quotient.eq]

Goal state after:
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
⊢ IsometryRel.setoid p (NonemptyCompacts.kuratowskiEmbedding X) ↔ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑p
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
S : Set (Set α)
hS1 : S ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}
hS2 : U = ⋃₀ S
hUS' : univ ∈ S
⊢ False

Tactic:
apply hU

Goal state after:
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
S : Set (Set α)
hS1 : S ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}
hS2 : U = ⋃₀ S
hUS' : univ ∈ S
⊢ U = univ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
a : α
inst✝ : IsScott α univ
⊢ ↑(lowerClosure {a}) ⊆ closure[inst✝¹] {a}

Tactic:
apply lowerClosure_subset_closure

Goal state after:
Module: Mathlib.Topology.Order.Category.FrameAdjunction

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
L : Locale
S : Set ↑(Opposite.unop L)
⊢ { carrier := (openOfElementHom ↑(Opposite.unop L)) (sSup S), is_open' := ⋯ } =
    sSup ((fun u ↦ { carrier := (openOfElementHom ↑(Opposite.unop L)) u, is_open' := ⋯ }) '' S)

Tactic:
by ext; simp

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

Goal state before:
pp: α : Type u_1
inst✝¹ : SemilatticeInf α
T : Set α
inst✝ : OrderTop α
hT : ∀ p ∈ T, InfPrime p
F : Finset α
⊢ hull T (F.inf id) = T ↓∩ ↑(upperClosure ↑F)

Tactic:
coe_upperClosure

Goal state after:
α : Type u_1
inst✝¹ : SemilatticeInf α
T : Set α
inst✝ : OrderTop α
hT : ∀ p ∈ T, InfPrime p
F : Finset α
⊢ hull T (F.inf id) = T ↓∩ ⋃ a ∈ ↑F, Set.Ici a
Module: Mathlib.Topology.Order.HullKernel

Goal state before:
pp: case mpr
α : Type u_1
inst✝² : CompleteLattice α
T : Set α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
hT : ∀ p ∈ T, InfPrime p
S : Set ↑T
a : α
ha : S = (hull T a)ᶜ
⊢ IsOpen[instTopologicalSpaceSubtype] S

Tactic:
exact ⟨(Ici a)ᶜ, isClosed_Ici.isOpen_compl, ha.symm⟩

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

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a : α
⊢ 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a)

Tactic:
rw [← Ioi_insert, nhdsSet_insert, nhdsSet_Ioi]

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

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝¹ : Category.{u, v} C
A : C
inst✝ : HasTerminal C
⊢ IsTerminal ((skyscraperPresheaf PUnit.unit A).obj (op ⊥))

Tactic:
dsimp [skyscraperPresheaf]

Goal state after:
X : TopCat
p₀ : ↑X
inst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝¹ : Category.{u, v} C
A : C
inst✝ : HasTerminal C
⊢ IsTerminal (if PUnit.unit ∈ ⊥ then A else ⊤_ C)
Module: Mathlib.Topology.Sheaves.Skyscraper

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
𝓕 : Presheaf C X
c : C
f : 𝓕.stalk p₀ ⟶ c
⊢ fromStalk p₀ (toSkyscraperPresheaf p₀ f) = f

Tactic:
refine 𝓕.stalk_hom_ext fun U hxU ↦ ?_

Goal state after:
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
𝓕 : Presheaf C X
c : C
f : 𝓕.stalk p₀ ⟶ c
U : Opens ↑X
hxU : p₀ ∈ U
⊢ 𝓕.germ U p₀ hxU ≫ fromStalk p₀ (toSkyscraperPresheaf p₀ f) = 𝓕.germ U p₀ hxU ≫ f
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
x : X
⊢ Monotone (id + fun x ↦ dist f g)

Tactic:
intro x y hx
      simpa

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

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
E1 : ∀ (x : X), (⨅ y, g (inl x, inr y)) + dist f g = ⨅ y, g (inl x, inr y) + dist f g
⊢ Monotone (id + fun x ↦ dist f g)

Tactic:
intro x y hx
      simpa

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

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
y : Y
⊢ Monotone (id + fun x ↦ dist f g)

Tactic:
intro x y hx
      simpa

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

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
E1 : ∀ (y : Y), (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g
⊢ Monotone (id + fun x ↦ dist f g)

Tactic:
intro x y hx
      simpa

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
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Tactic:
let Y := auxGluing X

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
⊢ ∃ x, Tendsto u atTop (𝓝 x)
Module: Mathlib.Topology.UniformSpace.Dini

Goal state before:
pp: ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h_tendsto : ∀ (x : α), Tendsto (fun x_1 ↦ F x_1 x) atTop (𝓝 (f x))
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
⊢ ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in atTop, ∀ y ∈ t, ‖F n y - f y‖ < ε

Tactic:
simp_rw +singlePass [tendsto_iff_norm_sub_tendsto_zero] at h_tendsto

Goal state after:
ι : Type u_1
α : Type u_2
G : Type u_3
inst✝⁵ : Preorder ι
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup G
inst✝² : Lattice G
inst✝¹ : HasSolidNorm G
inst✝ : IsOrderedAddMonoid G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F i)
hF_mono : Monotone F
hf : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
x✝ : atTop.NeBot
F_le_f : ∀ (x : α) (n : ι), F n x ≤ f x
ε : ℝ
ε_pos : ε > 0
x : α
h_tendsto : ∀ (x : α), Tendsto (fun e ↦ ‖F e x - f x‖) atTop (𝓝 0)
⊢ ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in atTop, ∀ y ∈ t, ‖F n y - f y‖ < ε
Module: Mathlib.Topology.Sion

Goal state before:
pp: case right
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
kY : IsCompact Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
ne_Y : Y.Nonempty
η : E → F
η_mem : ∀ x ∈ X, η x ∈ Y
η_max : ∀ x ∈ X, IsMaxOn (f x) Y (η x)
hlsc : LowerSemicontinuousOn (fun x ↦ f x (η x)) X
a : E
ha : a ∈ X
ha' : IsMinOn (fun x ↦ f x (η x)) X a
hmin_x : ∀ y ∈ Y, ∃ x ∈ X, IsMinOn (fun x ↦ f x y) X x
⊢ ∃ b ∈ Y, IsSaddlePointOn X Y f a b

Tactic:
choose! ξ ξ_mem ξ_min using hmin_x

Goal state after:
case right
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
cY : Convex ℝ Y
kY : IsCompact Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
inst✝² : DenselyOrdered β
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cX : Convex ℝ X
ne_Y : Y.Nonempty
η : E → F
η_mem : ∀ x ∈ X, η x ∈ Y
η_max : ∀ x ∈ X, IsMaxOn (f x) Y (η x)
hlsc : LowerSemicontinuousOn (fun x ↦ f x (η x)) X
a : E
ha : a ∈ X
ha' : IsMinOn (fun x ↦ f x (η x)) X a
ξ : F → E
ξ_mem : ∀ y ∈ Y, ξ y ∈ X
ξ_min : ∀ y ∈ Y, IsMinOn (fun x ↦ f x y) X (ξ y)
⊢ ∃ b ∈ Y, IsSaddlePointOn X Y f a b