mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case h.left
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
a : α
ha : (Ici a)ᶜ = U
⊢ {U} ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}

pp: case h.right
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
a : α
ha : (Ici a)ᶜ = U
⊢ U = ⋃₀ {U}

Tactic:
·

Goal state after:
case h.left
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
a : α
ha : (Ici a)ᶜ = U
⊢ {U} ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case h.right
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
a : α
ha : (Ici a)ᶜ = U
⊢ U = ⋃₀ {U}

Tactic:
·

Goal state after:
case h.right
α : Type u_1
inst✝¹ : CompleteLinearOrder α
t : TopologicalSpace α
inst✝ : IsLower α
U : Set α
hU : ¬U = univ
hO : IsOpen[t] U
a : α
ha : (Ici a)ᶜ = U
⊢ U = ⋃₀ {U}
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case mpr
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 ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}, U = ⋃₀ S) → ∃ a, (Ici a)ᶜ = U

Tactic:
·

Goal state after:
case mpr
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 ⊆ insert univ {s | ∃ a, (Ici a)ᶜ = s}, U = ⋃₀ S) → ∃ a, (Ici a)ᶜ = U
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case h.a
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
⊢ Ici (sSup {a | (Ici a)ᶜ ∈ S}) ⊆ ⋂₀ (compl '' S)

pp: case a
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
⊢ ⋂₀ (compl '' S) ⊆ Ici (sSup {a | (Ici a)ᶜ ∈ S})

Tactic:
·

Goal state after:
case h.a
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
⊢ Ici (sSup {a | (Ici a)ᶜ ∈ S}) ⊆ ⋂₀ (compl '' S)
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: case mp
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsUpperSet αᵒᵈ → Topology.IsLowerSet α

pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsLowerSet α → Topology.IsUpperSet αᵒᵈ

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsUpperSet αᵒᵈ → Topology.IsLowerSet α
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsLowerSet α → Topology.IsUpperSet αᵒᵈ

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsLowerSet α → Topology.IsUpperSet αᵒᵈ
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case a
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
⊢ ⋂₀ (compl '' S) ⊆ Ici (sSup {a | (Ici a)ᶜ ∈ S})

Tactic:
·

Goal state after:
case a
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
⊢ ⋂₀ (compl '' S) ⊆ Ici (sSup {a | (Ici a)ᶜ ∈ S})
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case e'_3
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : IsScott α univ
⊢ ↑(lowerClosure s) = closure[upperSet α] s

pp: case e'_4.e'_2
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : IsScott α univ
⊢ inst✝¹ = scott α univ

Tactic:
·

Goal state after:
case e'_3
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : IsScott α univ
⊢ ↑(lowerClosure s) = closure[upperSet α] s
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case e'_4.e'_2
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : IsScott α univ
⊢ inst✝¹ = scott α univ

Tactic:
·

Goal state after:
case e'_4.e'_2
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : IsScott α univ
⊢ inst✝¹ = scott α univ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
h : ScottContinuousOn D f
u : Set β
hu : IsOpen[inst✝²] u
⊢ IsOpen[inst✝⁴] (f ⁻¹' u)

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
⊢ Continuous[inst✝⁴, inst✝²] f → ScottContinuousOn D f

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
h : ScottContinuousOn D f
u : Set β
hu : IsOpen[inst✝²] u
⊢ IsOpen[inst✝⁴] (f ⁻¹' u)
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
⊢ Continuous[inst✝⁴, inst✝²] f → ScottContinuousOn D f

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
⊢ Continuous[inst✝⁴, inst✝²] f → ScottContinuousOn D f
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
⊢ ∀ s ∈ {s | ∃ a, (Ici a)ᶜ = s}, IsOpen[instTopologicalSpaceProd] s

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
⊢ lower (α × β) ≤ instTopologicalSpaceProd

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
⊢ ∀ s ∈ {s | ∃ a, (Ici a)ᶜ = s}, IsOpen[instTopologicalSpaceProd] s
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mp
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
⊢ IsOpen[inst✝¹] U → U = univ ∨ ∃ a, (Iic a)ᶜ = U

pp: case mpr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
⊢ (U = univ ∨ ∃ a, (Iic a)ᶜ = U) → IsOpen[inst✝¹] U

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
⊢ IsOpen[inst✝¹] U → U = univ ∨ ∃ a, (Iic a)ᶜ = U
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mp.inl
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
hU : IsOpen[inst✝¹] U
eUc : Uᶜ = ∅
⊢ U = univ ∨ ∃ a, (Iic a)ᶜ = U

pp: case mp.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
hU : IsOpen[inst✝¹] U
neUc : Uᶜ.Nonempty
⊢ U = univ ∨ ∃ a, (Iic a)ᶜ = U

Tactic:
·

Goal state after:
case mp.inl
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
hU : IsOpen[inst✝¹] U
eUc : Uᶜ = ∅
⊢ U = univ ∨ ∃ a, (Iic a)ᶜ = U
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case refine_2.refine_1
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
x✝¹ : TopologicalSpace (α × β) := lower (α × β)
a : α
x✝ : a ∈ s
⊢ IsOpen[lower (α × β)] (Prod.fst ⁻¹' Ici a)ᶜ

pp: case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
x✝¹ : TopologicalSpace (α × β) := lower (α × β)
b : β
x✝ : b ∈ t
⊢ IsOpen[lower (α × β)] (Prod.snd ⁻¹' Ici b)ᶜ

Tactic:
·

Goal state after:
case refine_2.refine_1
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
x✝¹ : TopologicalSpace (α × β) := lower (α × β)
a : α
x✝ : a ∈ s
⊢ IsOpen[lower (α × β)] (Prod.fst ⁻¹' Ici a)ᶜ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mp.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
hU : IsOpen[inst✝¹] U
neUc : Uᶜ.Nonempty
⊢ U = univ ∨ ∃ a, (Iic a)ᶜ = U

Tactic:
·

Goal state after:
case mp.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
hU : IsOpen[inst✝¹] U
neUc : Uᶜ.Nonempty
⊢ U = univ ∨ ∃ a, (Iic a)ᶜ = U
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
x✝¹ : TopologicalSpace (α × β) := lower (α × β)
b : β
x✝ : b ∈ t
⊢ IsOpen[lower (α × β)] (Prod.snd ⁻¹' Ici b)ᶜ

Tactic:
·

Goal state after:
case refine_2.refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
x✝¹ : TopologicalSpace (α × β) := lower (α × β)
b : β
x✝ : b ∈ t
⊢ IsOpen[lower (α × β)] (Prod.snd ⁻¹' Ici b)ᶜ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mpr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
⊢ (U = univ ∨ ∃ a, (Iic a)ᶜ = U) → IsOpen[inst✝¹] U

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
U : Set α
⊢ (U = univ ∨ ∃ a, (Iic a)ᶜ = U) → IsOpen[inst✝¹] U
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mpr.inl
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
⊢ IsOpen[inst✝¹] univ

pp: case mpr.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
a : α
⊢ IsOpen[inst✝¹] (Iic a)ᶜ

Tactic:
·

Goal state after:
case mpr.inl
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
⊢ IsOpen[inst✝¹] univ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case mpr.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
a : α
⊢ IsOpen[inst✝¹] (Iic a)ᶜ

Tactic:
·

Goal state after:
case mpr.inr
α : Type u_1
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α univ
a : α
⊢ IsOpen[inst✝¹] (Iic a)ᶜ
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLinearOrder α
this✝ : TopologicalSpace α := upper α
U : Set α
this : TopologicalSpace α := scott α univ
⊢ IsOpen[scott α univ] U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U

Tactic:
@isOpen_iff_Iic_compl_or_univ _ _ (scott α univ) ({ topology_eq_scott := rfl }) U

Goal state after:
α : Type u_1
inst✝ : CompleteLinearOrder α
this✝ : TopologicalSpace α := upper α
U : Set α
this : TopologicalSpace α := scott α univ
⊢ (U = univ ∨ ∃ a, (Iic a)ᶜ = U) ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case mp
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsUpper αᵒᵈ → IsLower α

pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsLower α → IsUpper αᵒᵈ

Tactic:
·

Goal state after:
case mp
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsUpper αᵒᵈ → IsLower α
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsLower α → IsUpper αᵒᵈ

Tactic:
·

Goal state after:
case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsLower α → IsUpper αᵒᵈ
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case inl
⊢ ∃ a, Ioi a = Ioi True

pp: case inr
⊢ ∃ a, Ioi a = Ioi False

Tactic:
·

Goal state after:
case inl
⊢ ∃ a, Ioi a = Ioi True
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case inr
⊢ ∃ a, Ioi a = Ioi False

Tactic:
·

Goal state after:
case inr
⊢ ∃ a, Ioi a = Ioi False
Module: Mathlib.Topology.OmegaCompletePartialOrder

Goal state before:
pp: case left
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (i : ℕ), c i ≤ ωSup c

pp: case right
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → ωSup c ≤ y

Tactic:
·

Goal state after:
case left
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (i : ℕ), c i ≤ ωSup c
Module: Mathlib.Topology.OmegaCompletePartialOrder

Goal state before:
pp: case right
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → ωSup c ≤ y

Tactic:
·

Goal state after:
case right
α : Type u_1
inst✝ : OmegaCompletePartialOrder α
c : Chain α
⊢ ∀ (y : α), (∀ (i : ℕ), c i ≤ y) → ωSup c ≤ y
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case left.left.left.right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) = dist (f y) (f x)

pp: case left.left.right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y z : X ⊕ Y), dist (f x) (f z) ≤ dist (f x) (f y) + dist (f y) (f z)

pp: case right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) ≤ ↑(GromovHausdorff.maxVar✝ X Y)

Tactic:
·

Goal state after:
case left.left.left.right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) = dist (f y) (f x)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case left.left.right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y z : X ⊕ Y), dist (f x) (f z) ≤ dist (f x) (f y) + dist (f y) (f z)

pp: case right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) ≤ ↑(GromovHausdorff.maxVar✝ X Y)

Tactic:
·

Goal state after:
case left.left.right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y z : X ⊕ Y), dist (f x) (f z) ≤ dist (f x) (f y) + dist (f y) (f z)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) ≤ ↑(GromovHausdorff.maxVar✝ X Y)

Tactic:
·

Goal state after:
case right
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) ∞) := ⋯
F : (X ⊕ Y) × (X ⊕ Y) → ℝ := ⋯
⊢ ∀ (x y : X ⊕ Y), dist (f x) (f y) ≤ ↑(GromovHausdorff.maxVar✝ X Y)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case inl
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
val✝ : X
⊢ f (inl val✝) ∈ range Φ ∪ range Ψ

pp: case inr
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
val✝ : Y
⊢ f (inr val✝) ∈ range Φ ∪ range Ψ

Tactic:
·

Goal state after:
case inl
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
val✝ : X
⊢ f (inl val✝) ∈ range Φ ∪ range Ψ