mathlib-initiative/mathlib-tactics huggingface dataset:
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
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫
        ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j =
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
r : ℝ
hr : HD (optimalGHDist X Y) < r
A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r
z : Y
I1 : ⨆ y, ⨅ x, (optimalGHDist X Y) (inl x, inr y) < r
⊢ BddAbove (range fun y ↦ ⨅ x, (optimalGHDist X Y) (inl x, inr y))

Tactic:
by

Goal state after:
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
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
        ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j =
      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
⊢ (ConcreteCategory.hom ((S.X₃.obj ⋙ CategoryTheory.forget AddCommGrpCat).map (homOfLE le).op)) ⟨op U, s⟩.snd =
    ((NatTrans.mapElements (Functor.whiskerRight S.g.hom (CategoryTheory.forget AddCommGrpCat))).obj
        ⟨op (iSup f), t₅⟩).snd

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f

Tactic:
by

Goal state after:
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
c d : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ d
⊢ (coneEquivInverse F U ⋙ coneEquivFunctor F U).map f ≫
      { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj d).pt, w := ⋯ },
          inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj d).pt, w := ⋯ },
          hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom =
    { hom := { hom := 𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt, w := ⋯ },
          inv := { hom := 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt, w := ⋯ },
          hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫
      (𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).map f

Tactic:
by

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

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
U U' : (Opens ↑X)ᵒᵖ
f : U ⟶ U'
⊢ (F.map f ≫ ↾fun f ↦ ⟨fun x ↦ (ConcreteCategory.hom (F.germ (unop U') ↑x ⋯)) f, ⋯⟩) =
    (↾fun f ↦ ⟨fun x ↦ (ConcreteCategory.hom (F.germ (unop U) ↑x ⋯)) f, ⋯⟩) ≫ F.sheafify.obj.map f

Tactic:
by

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

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ Function.Surjective ⇑(ConcreteCategory.hom (F.stalkToFiber x))

Tactic:
by

Goal state after:
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
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
⊢ ∀ (j : (CategoryTheory.Pairwise ι)ᵒᵖ),
    𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
        (Functor.mapCone F (cocone U).op).π.app j =
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app j

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
t₆ : Under S.g s :=
  StructuredArrow.mk
    (CategoryOfElements.homMk ⟨op U, s⟩
      ((NatTrans.mapElements (Functor.whiskerRight S.g.hom (CategoryTheory.forget AddCommGrpCat))).obj
        ⟨op (iSup f), t₅⟩)
      (homOfLE le).op ⋯)
⊢ StructuredArrow.hom t₆ ≫
      (NatTrans.mapElements (Functor.whiskerRight S.g.hom (CategoryTheory.forget AddCommGrpCat))).map
        (CategoryOfElements.homMk (StructuredArrow.right t₆) (StructuredArrow.right t) (homOfLE ⋯).op ⋯) =
    StructuredArrow.hom t

Tactic:
by

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

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this✝ : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
Aq : ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
this : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩)
⊢ ⌊ε⁻¹ * dist x y⌋ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋

Tactic:
by

Goal state after:
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
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
⊢ ∀ (j : (CategoryTheory.Pairwise ι)ᵒᵖ),
    𝟙 (Functor.mapCone F (cocone U).op).pt ≫
        ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app j =
      (Functor.mapCone F (cocone U).op).π.app j

Tactic:
by

Goal state after:
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
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
        (SheafConditionEqualizerProducts.fork F U).π.app j =
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app j

Tactic:
by

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

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this✝¹ : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
Aq : ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
this✝ : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩)
this : ⌊ε⁻¹ * dist x y⌋ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋
⊢ |ε⁻¹ * (dist x y - dist (Ψ x) (Ψ y))| = |ε⁻¹ * dist x y - ε⁻¹ * dist (Ψ x) (Ψ y)|

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
⊢ U ≤ unop (StructuredArrow.right t).fst

Tactic:
by

Goal state after:
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
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
        ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app j =
      (SheafConditionEqualizerProducts.fork F U).π.app j

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
b : β
y : ↑Y
x : ↑X
⊢ x ∈ sublevelLeft X f b ↑y ↔ f ↑x ↑y ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
a : E
b : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : b < max (f a ↑y) (f a ↑y')
⊢ Disjoint (sublevelLeft X f b ↑y) (sublevelLeft X f b ↑y')

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝ : TopologicalSpace E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
y : ↑Y
b : β
h : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
⊢ (sublevelLeft X f b ↑y).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝ : TopologicalSpace E
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
b : β
y : ↑Y
⊢ IsClosed (sublevelLeft X f b ↑y)

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
this : U ≤ unop (StructuredArrow.right t).fst
⊢ (ConcreteCategory.hom (S.g.hom.app (op U))) (((StructuredArrow.right t).snd |_ U) this) = s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝² : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
b : β
y y' : ↑Y
z : ↑(segment ℝ ↑y ↑y')
x : ↑X
hx : x ∈ sublevelLeft X f b ↑z
⊢ x ∈ sublevelLeft X f b ↑y ∪ sublevelLeft X f b ↑y'

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
⊢ Epi (S.g.hom.app (op U))

Tactic:
by

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

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this✝¹ : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
Aq : ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
this✝ : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩)
this : ⌊ε⁻¹ * dist x y⌋ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋
I : |ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)| ≤ 1
⊢ |dist x y - dist (Ψ x) (Ψ y)| = ε * ε⁻¹ * |dist x y - dist (Ψ x) (Ψ y)|

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝⁷ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝⁶ : TopologicalSpace E
inst✝⁵ : AddCommGroup E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : Module ℝ E
inst✝² : ContinuousSMul ℝ E
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
cY : Convex ℝ Y
a : E
b : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : b < max (f a ↑y) (f a ↑y')
z : ↑(segment ℝ ↑y ↑y')
⊢ sublevelLeft X f b ↑z ⊆ sublevelLeft X f b ↑y ∨ sublevelLeft X f b ↑z ⊆ sublevelLeft X f b ↑y'

Tactic:
by

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

Goal state before:
pp: X : TopCat
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝¹ : S.X₁.IsFlasque
inst✝ : S.X₂.IsFlasque
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
⊢ Epi (S.g.hom.app U ≫ S.X₃.obj.map i)

Tactic:
by

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

Goal state before:
pp: X : TopCat
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝¹ : S.X₁.IsFlasque
inst✝ : S.X₂.IsFlasque
U V : (Opens ↑X)ᵒᵖ
i : U ⟶ V
⊢ Epi (S.X₃.obj.map i)

Tactic:
by

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

Goal state before:
pp: X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
by

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

Goal state before:
pp: X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ Function.Injective ⇑(ConcreteCategory.hom (F.stalkToFiber x))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
h : HasSmallInductiveDimensionLT X 0
⊢ IsEmpty X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: m : ℕ
IH :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] {m_1 : ℕ},
    m_1 ≤ m → HasSmallInductiveDimensionLT X m_1 → HasSmallInductiveDimensionLT X m
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsEmpty X
hmn : 0 ≤ m + 1
⊢ IsTopologicalBasis ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: m : ℕ
IH :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] {m_1 : ℕ},
    m_1 ≤ m → HasSmallInductiveDimensionLT X m_1 → HasSmallInductiveDimensionLT X m
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsEmpty X
hmn : 0 ≤ m + 1
⊢ ∀ U ∈ ∅, HasSmallInductiveDimensionLT (↑(frontier U)) m

Tactic:
by

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

Goal state before:
pp: X : TopCat
p₀ : ↑X
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Limits.HasColimits C
inst✝¹ : Limits.HasTerminal C
𝓕 : Presheaf C X
inst✝ : HasWeakSheafify (Opens.grothendieckTopology ↑X) C
⊢ IsIso ((stalkFunctor C p₀).map (CategoryTheory.toSheafify (Opens.grothendieckTopology ↑X) 𝓕))

Tactic:
by

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

Goal state before:
pp: t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this✝¹ : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
Aq : ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
this✝ : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩)
this : ⌊ε⁻¹ * dist x y⌋ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋
I : |ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)| ≤ 1
⊢ ε * ε⁻¹ * |dist x y - dist (Ψ x) (Ψ y)| = ε * (|ε⁻¹| * |dist x y - dist (Ψ x) (Ψ y)|)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
m n : ℕ
hmn : m ≤ n
H : HasSmallInductiveDimensionLT X m
⊢ HasSmallInductiveDimensionLT X n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
m n : ℕ
hmn : m ≤ n
H : HasSmallInductiveDimensionLE X m
⊢ HasSmallInductiveDimensionLE X n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
ho : IsOpen[inst✝] s
⊢ IsOpen[constructibleTopology X] s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact sᶜ
ho : IsClosed[inst✝] s
⊢ IsOpen[constructibleTopology X] s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ sᶜ ∈ constructibleTopologySubbasis X ↔ s ∈ constructibleTopologySubbasis X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ constructibleTopologySubbasis X
⊢ IsCompact s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
⊢ IsCompact (⋂₀ s)

Tactic:
by

Goal state after: