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

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
x✝ x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (x✝, i) ≤ f (x✝, x) + f (x, i)}

Tactic:
apply I6 _ _

Goal state after:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
x✝ x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (x✝, i) ≤ f (x✝, x) + f (x, i)}
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
⊢ ∃ β x F, ∀ (x y : ↑t), F x = F y → dist ↑x ↑y < δ

Tactic:
have u_le_ε : u n ≤ ε := by
    have := hn n le_rfl
    simp only [Real.dist_eq, add_zero, sub_eq_add_neg, neg_zero] at this
    exact le_of_lt (lt_of_le_of_lt (le_abs_self _) this)

Goal state after:
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
⊢ ∃ β x F, ∀ (x y : ↑t), F x = F y → dist ↑x ↑y < δ
Module: Mathlib.Topology.Sheaves.MayerVietoris

Goal state before:
pp: T : Type u
inst✝ : TopologicalSpace T
sq : Square (Opens T)
h₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃
h₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃
x : T
hx : x ∈ sq.X₄
⊢ ∃ U f, (Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄).arrows f ∧ x ∈ U

Tactic:
by
      rw [h₄] at hx
      obtain (hx | hx) := hx
      · exact ⟨_, _, ⟨Sieve.ofArrows_mk _ _ WalkingPair.left, hx⟩⟩
      · exact ⟨_, _, ⟨Sieve.ofArrows_mk _ _ WalkingPair.right, hx⟩⟩

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (i, i) = 0}

Tactic:
apply I6 _ _

Goal state after:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (i, i) = 0}
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ y, {f | f (i, y) ≤ ↑(maxVar X Y)})

Tactic:
apply I6 _ _

Goal state after:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
⊢ ∀ (i : X ⊕ Y), IsClosed (⋂ y, {f | f (i, y) ≤ ↑(maxVar X Y)})
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (x, i) ≤ ↑(maxVar X Y)}

Tactic:
apply I6 _ _

Goal state after:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
x : X ⊕ Y
⊢ ∀ (i : X ⊕ Y), IsClosed {f | f (x, i) ≤ ↑(maxVar X Y)}
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
I1 : ∀ (x y : X), IsClosed {f | f (inl x, inl y) = dist x y}
I2 : ∀ (x y : Y), IsClosed {f | f (inr x, inr y) = dist x y}
I3 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) = f (y, x)}
I4 : ∀ (x y z : X ⊕ Y), IsClosed {f | f (x, z) ≤ f (x, y) + f (y, z)}
I5 : ∀ (x : X ⊕ Y), IsClosed {f | f (x, x) = 0}
I6 : ∀ (x y : X ⊕ Y), IsClosed {f | f (x, y) ≤ ↑(maxVar X Y)}
this :
  candidatesB X Y =
    (((((⋂ x, ⋂ y, {f | f (inl x, inl y) = dist x y}) ∩ ⋂ x, ⋂ y, {f | f (inr x, inr y) = dist x y}) ∩
            ⋂ x, ⋂ y, {f | f (x, y) = f (y, x)}) ∩
          ⋂ x, ⋂ y, ⋂ z, {f | f (x, z) ≤ f (x, y) + f (y, z)}) ∩
        ⋂ x, {f | f (x, x) = 0}) ∩
      ⋂ x, ⋂ y, {f | f (x, y) ≤ ↑(maxVar X Y)}
x✝ x : X ⊕ Y
⊢ IsClosed {f | f (x✝, x) ≤ ↑(maxVar X Y)}

Tactic:
apply I6 _ _

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
b : β
h : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
y : F
hy : y ∈ Y
x : E
hx : x ∈ X
hx_le : IsMinOn (fun x ↦ f x y) X x
x' : E
hx' : x' ∈ X
hx'b : f x' y ≤ b
⊢ (sublevelLeft X f b ↑⟨y, hy⟩).Nonempty

Tactic:
exact ⟨⟨x, hx⟩, le_trans (hx_le hx') hx'b⟩

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

Goal state before:
pp: case neg
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
h1 : p₀ ∈ unop U
h2 : p₀ ∉ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
· simp
        grind

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

Goal state before:
pp: case refine_2
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
t : Set X
ht : t ∈ s
⊢ IsCompact t

Tactic:
exact isCompact_of_mem_constructibleTopologySubbasis (hs ht)

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
n : ℕ
γ : Path a b
t : Fin (n + 1) → ↑I
⊢ (concat (⇑γ ∘ t) fun k ↦ γ.subpath (t k.castSucc) (t k.succ)).Homotopy (γ.subpath (t 0) (t (last n)))

Tactic:
by
  induction n with
  | zero =>
    simp only [concat_zero, reduceLast, subpath_self]
    exact refl _
  | succ n ih =>
    rw [concat_succ]
    exact trans ((ih (t ∘ castSucc)).hcomp (refl _)) (subpathTransSubpath γ _ _ _)

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case a
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
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
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
x : E
hx : x ∈ X
y : F
hy : y ∈ Y
⊢ inf_x y ≤ sup_y x

Tactic:
trans f x y

Goal state after:
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
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
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
x : E
hx : x ∈ X
y : F
hy : y ∈ Y
⊢ inf_x y ≤ f x y

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
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
sup_y : E → β
hsup_y : ∀ x ∈ X, IsLUB {x_1 | ∃ y ∈ Y, f x y = x_1} (sup_y x)
inf_sup : β
hinf_sup : IsGLB {x | ∃ x_1 ∈ X, sup_y x_1 = x} inf_sup
inf_x : F → β
hinf_x : ∀ y ∈ Y, IsGLB {x | ∃ x_1 ∈ X, f x_1 y = x} (inf_x y)
sup_inf : β
hsup_inf : IsLUB {x | ∃ y ∈ Y, inf_x y = x} sup_inf
x : E
hx : x ∈ X
y : F
hy : y ∈ Y
⊢ f x y ≤ sup_y x
Module: Mathlib.Topology.Sion

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

Tactic:
minimax' ne_X cX kX hfy hfy' cY hfx hfx'

Goal state after:
case right
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : CompleteLinearOrder β
inst✝¹⁰ : DenselyOrdered β
X : Set E
Y : Set F
f : E → F → β
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul ℝ E
ne_X : X.Nonempty
cX : Convex ℝ X
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✝⁴ : TopologicalSpace F
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cY : Convex ℝ Y
ne_Y : Y.Nonempty
kY : IsCompact Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
hlsc : LowerSemicontinuousOn (fun x ↦ ⨆ y ∈ Y, f x y) X
husc : UpperSemicontinuousOn (fun y ↦ ⨅ x ∈ X, f x y) Y
a : E
ha : a ∈ X
ha' : IsMinOn (fun x ↦ ⨆ y ∈ Y, f x y) X a
b : F
hb : b ∈ Y
hb' : IsMaxOn (fun y ↦ ⨅ x ∈ X, f x y) Y b
⊢ ⨆ y ∈ Y, ⨅ x ∈ X, f x y ≤ ⨆ i ∈ Y, (fun y ↦ ⨅ x ∈ X, f x y) i