mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: case neg
α : Type u
ι : Type u_1
s : ι → Set α
t : Set ι
ht : t.Finite
hs : ∀ i ∈ t, (s i).Finite
he : ∀ i ∉ t, s i = ∅
i : ι
x : α
hx : False
hi : i ∉ t
⊢ x ∈ ⋃ i ∈ t, s i

Tactic:
]

Goal state after:
case neg
α : Type u
ι : Type u_1
s : ι → Set α
t : Set ι
ht : t.Finite
hs : ∀ i ∈ t, (s i).Finite
he : ∀ i ∉ t, s i = ∅
i : ι
x : α
hx : False
hi : i ∉ t
⊢ x ∈ ⋃ i ∈ t, s i
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, sSup ∅ ≤ f x
h' : (image f s).Nonempty
⊢ sSup (Set.range fun i ↦ ⨆ (_ : i ∈ s), f i) = sSup (f '' ↑s)

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, sSup ∅ ≤ f x
h' : (image f s).Nonempty
⊢ sSup (Set.range fun i ↦ ⨆ (_ : i ∈ s), f i) = sSup (f '' ↑s)
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j ↦ Disjoint (s i) (s j)
h : (⋃ i, s i).Finite
⊢ {i | (s i).Nonempty}.Finite

Tactic:
let u (i : {i | (s i).Nonempty}) : ⋃ i, s i := ⟨i.2.choose, mem_iUnion.2 ⟨i.1, i.2.choose_spec⟩⟩

Goal state after:
α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j ↦ Disjoint (s i) (s j)
h : (⋃ i, s i).Finite
u : ↑{i | (s i).Nonempty} → ↑(⋃ i, s i) := fun i ↦ ⟨Exists.choose ⋯, ⋯⟩
⊢ {i | (s i).Nonempty}.Finite
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j ↦ Disjoint (s i) (s j)
h : (⋃ i, s i).Finite
u : ↑{i | (s i).Nonempty} → ↑(⋃ i, s i) := fun i ↦ ⟨Exists.choose ⋯, ⋯⟩
i : ι
hi : i ∈ {i | (s i).Nonempty}
j : ι
hj : j ∈ {i | (s i).Nonempty}
hij : u ⟨i, hi⟩ = u ⟨j, hj⟩
⊢ ↑(u ⟨j, hj⟩) ∈ s ↑⟨j, hj⟩

Tactic:
]

Goal state after:
α : Type u
ι : Type u_1
s : ι → Set α
hs : Pairwise fun i j ↦ Disjoint (s i) (s j)
h : (⋃ i, s i).Finite
u : ↑{i | (s i).Nonempty} → ↑(⋃ i, s i) := fun i ↦ ⟨Exists.choose ⋯, ⋯⟩
i : ι
hi : i ∈ {i | (s i).Nonempty}
j : ι
hj : j ∈ {i | (s i).Nonempty}
hij : u ⟨i, hi⟩ = u ⟨j, hj⟩
⊢ ↑(u ⟨j, hj⟩) ∈ s ↑⟨j, hj⟩
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
ι : Sort u_1
inst✝ : Subsingleton ι
s : ι → Set α
⊢ (∀ (i : PLift ι), (s i.down).Finite) ∧ {i | (s i.down).Nonempty}.Finite ↔ ∀ (i : ι), (s i).Finite

Tactic:
]

Goal state after:
α : Type u
ι : Sort u_1
inst✝ : Subsingleton ι
s : ι → Set α
⊢ (∀ (i : PLift ι), (s i.down).Finite) ∧ {i | (s i.down).Nonempty}.Finite ↔ ∀ (i : ι), (s i).Finite
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝² : Preorder α
inst✝¹ : IsEmpty ι
inst✝ : InfSet α
f : ι → WithTop α
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝² : Preorder α
inst✝¹ : Nonempty ι
inst✝ : InfSet α
f : ι → α
hf : BddBelow (range f)
⊢ sInf (range fun x ↦ ↑(f x)) = sInf (range fun i ↦ ↑(f i))

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
β : Type v
f : β → Set α
s : Set β
hs : s.PairwiseDisjoint f
⊢ (∀ (i : β), (⋃ (_ : i ∈ s), f i).Finite) ∧ {i | (⋃ (_ : i ∈ s), f i).Nonempty}.Finite ↔
    (∀ i ∈ s, (f i).Finite) ∧ {i | i ∈ s ∧ (f i).Nonempty}.Finite

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : β → Set α
s : Set β
hs : s.PairwiseDisjoint f
⊢ (∀ (i : β), (⋃ (_ : i ∈ s), f i).Finite) ∧ {i | (⋃ (_ : i ∈ s), f i).Nonempty}.Finite ↔
    (∀ i ∈ s, (f i).Finite) ∧ {i | i ∈ s ∧ (f i).Nonempty}.Finite
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝¹ : Preorder α
inst✝ : SupSet α
f : ι → α
h : BddAbove (range f)
⊢ sSup (range fun x ↦ ↑(f x)) = sSup (range fun i ↦ ↑(f i))

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
β : Type v
f : α → β
s : Set β
h : s.Finite
hf : ∀ b ∈ s, (f ⁻¹' {b}).Finite
⊢ (⋃ y ∈ s, f ⁻¹' {y}).Finite

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : α → β
s : Set β
h : s.Finite
hf : ∀ b ∈ s, (f ⁻¹' {b}).Finite
⊢ (⋃ y ∈ s, f ⁻¹' {y}).Finite
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
β : Type v
f : α → Finset β
h : (range f).Finite
⊢ (⋃ x ∈ range f, ↑x).Finite

Tactic:
]

Goal state after:
α : Type u
β : Type v
f : α → Finset β
h : (range f).Finite
⊢ (⋃ x ∈ range f, ↑x).Finite
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
h' : (image f s).Nonempty
⊢ ⨆ i, OrderDual.toDual (⨅ (_ : i ∈ s), f i) = (image (⇑OrderDual.toDual) (image f s)).max' ⋯

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
h' : (image f s).Nonempty
⊢ ⨆ i, OrderDual.toDual (⨅ (_ : i ∈ s), f i) = (image (⇑OrderDual.toDual) (image f s)).max' ⋯
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
h' : (image f s).Nonempty
⊢ (image (fun i ↦ OrderDual.toDual (f i)) s).max' ⋯ = (image (⇑OrderDual.toDual) (image f s)).max' ⋯

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
h' : (image f s).Nonempty
⊢ (image (fun i ↦ OrderDual.toDual (f i)) s).max' ⋯ = (image (⇑OrderDual.toDual) (image f s)).max' ⋯
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLattice α
f : β → α
s : Set β
H : BddAbove (f '' s)
Hne : s.Nonempty
⊢ IsLUB (f '' s) (sSup (f '' s))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLattice α
f : β → α
s : Set β
H : BddAbove (f '' s)
Hne : s.Nonempty
⊢ IsLUB (f '' s) (sSup (f '' s))
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLattice α
inst✝ : Nonempty ι
f : ι → α
c : α
H : ∀ (x : ι), f x ≤ c
⊢ ∀ (i : ι), f i ≤ c

Tactic:
]

Goal state after:
α : Type u_1
ι : Sort u_4
inst✝¹ : ConditionallyCompleteLattice α
inst✝ : Nonempty ι
f : ι → α
c : α
H : ∀ (x : ι), f x ≤ c
⊢ ∀ (i : ι), f i ≤ c
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, sSup ∅ ≤ f x
⊢ (image f s).max' ⋯ ∈ image f s

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, sSup ∅ ≤ f x
⊢ (image f s).max' ⋯ ∈ image f s
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
s : Set α
hs : s.Finite
ι : Type u_1
t : ι → Set α
h : s ⊆ ⋃ i, t i
this : Finite ↑s
f : ↑s → ι
hf : ∀ (x : ↑s), ↑x ∈ t (f x)
x : α
hx : x ∈ s
⊢ ∃ i, x ∈ t (f i)

Tactic:
]

Goal state after:
α : Type u
s : Set α
hs : s.Finite
ι : Type u_1
t : ι → Set α
h : s ⊆ ⋃ i, t i
this : Finite ↑s
f : ↑s → ι
hf : ∀ (x : ↑s), ↑x ∈ t (f x)
x : α
hx : x ∈ s
⊢ ∃ i, x ∈ t (f i)
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
⊢ (image f s).min' ⋯ ∈ image f s

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
⊢ (image f s).min' ⋯ ∈ image f s
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
ι : Type u_1
s : ι → Set α
t : Set α
tfin : t.Finite
h : t ⊆ ⋃ i, s i
I : Set ι
Ifin : I.Finite
hI : t ⊆ ⋃ i ∈ I, s i
x : α
⊢ x ∈ t ↔ ∃ i, x ∈ s ↑i ∩ t

Tactic:
]

Goal state after:
α : Type u
ι : Type u_1
s : ι → Set α
t : Set α
tfin : t.Finite
h : t ⊆ ⋃ i, s i
I : Set ι
Ifin : I.Finite
hI : t ⊆ ⋃ i ∈ I, s i
x : α
⊢ x ∈ t ↔ ∃ i, x ∈ s ↑i ∩ t
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inl
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
f g : ι → α
B : BddAbove (range g)
H : ∀ (x : ι), f x ≤ g x
h✝ : IsEmpty ι
⊢ sSup ∅ ≤ sSup ∅

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
ι : Type u_1
s : ι → Set α
a : Set ι
ha : a.Infinite
hs : InjOn s a
⊢ (⋃ x, s ↑x).Infinite

Tactic:
]

Goal state after:
α : Type u
ι : Type u_1
s : ι → Set α
a : Set ι
ha : a.Infinite
hs : InjOn s a
⊢ (⋃ x, s ↑x).Infinite
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
s : Set (Set α)
hs : s.Infinite
⊢ (⋃ i, ↑i).Infinite

Tactic:
]

Goal state after:
α : Type u
s : Set (Set α)
hs : s.Infinite
⊢ (⋃ i, ↑i).Infinite
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u
β : Type v
F : Type u_1
ι : Type u_2
inst✝⁴ : CompleteLattice α
inst✝³ : CompleteLattice β
inst✝² : FunLike F α β
inst✝¹ : SupBotHomClass F α β
inst✝ : Finite ι
f : F
g : ι → α
⊢ f (⨆ x ∈ univ, g x) = ⨆ x ∈ univ, f (g x)

Tactic:
]

Goal state after:
α : Type u
β : Type v
F : Type u_1
ι : Type u_2
inst✝⁴ : CompleteLattice α
inst✝³ : CompleteLattice β
inst✝² : FunLike F α β
inst✝¹ : SupBotHomClass F α β
inst✝ : Finite ι
f : F
g : ι → α
⊢ f (⨆ x ∈ univ, g x) = ⨆ x ∈ univ, f (g x)
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: case insert
ι : Type u_1
ι' : Type u_2
α : Type u_3
inst✝³ : Preorder ι'
inst✝² : Nonempty ι'
inst✝¹ : IsDirectedOrder ι'
inst✝ : Order.Frame α
s : Set ι
f : ι → ι' → α
a✝¹ : ι
s✝ : Set ι
a✝ : a✝¹ ∉ s✝
hs✝ : s✝.Finite
ihs : (∀ i ∈ s✝, Monotone (f i)) → ⨆ j, ⨅ i ∈ s✝, f i j = ⨅ i ∈ s✝, ⨆ j, f i j
hf : Monotone (f a✝¹) ∧ ∀ x ∈ s✝, Monotone (f x)
⊢ ⨆ j, ⨅ i ∈ insert a✝¹ s✝, f i j = ⨅ i ∈ insert a✝¹ s✝, ⨆ j, f i j

Tactic:
]

Goal state after:
case insert
ι : Type u_1
ι' : Type u_2
α : Type u_3
inst✝³ : Preorder ι'
inst✝² : Nonempty ι'
inst✝¹ : IsDirectedOrder ι'
inst✝ : Order.Frame α
s : Set ι
f : ι → ι' → α
a✝¹ : ι
s✝ : Set ι
a✝ : a✝¹ ∉ s✝
hs✝ : s✝.Finite
ihs : (∀ i ∈ s✝, Monotone (f i)) → ⨆ j, ⨅ i ∈ s✝, f i j = ⨅ i ∈ s✝, ⨆ j, f i j
hf : Monotone (f a✝¹) ∧ ∀ x ∈ s✝, Monotone (f x)
⊢ ⨆ j, ⨅ i ∈ insert a✝¹ s✝, f i j = ⨅ i ∈ insert a✝¹ s✝, ⨆ j, f i j
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : ∃ x, ∀ y ∈ range f, y ≤ x
h✝¹ : Nonempty β
h✝ : Nonempty γ
⊢ ∃ x, ∀ y ∈ range fun b ↦ ⨆ c, f (b, c), y ≤ x

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : ∃ x, ∀ y ∈ range f, y ≤ x
h✝¹ : Nonempty β
h✝ : Nonempty γ
⊢ ∃ x, ∀ y ∈ range fun b ↦ ⨆ c, f (b, c), y ≤ x
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : ∃ x, ∀ y ∈ range f, y ≤ x
h✝¹ : Nonempty β
h✝ : Nonempty γ
h₁ : BddAbove (range fun b ↦ ⨆ c, f (b, c))
b : β
⊢ ∃ x, ∀ y ∈ range fun c ↦ f (b, c), y ≤ x

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : ∃ x, ∀ y ∈ range f, y ≤ x
h✝¹ : Nonempty β
h✝ : Nonempty γ
h₁ : BddAbove (range fun b ↦ ⨆ c, f (b, c))
b : β
⊢ ∃ x, ∀ y ∈ range fun c ↦ f (b, c), y ≤ x
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inr.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : BddAbove (range f)
h✝¹ : Nonempty β
h✝ : Nonempty γ
h₁ : BddAbove (range fun b ↦ ⨆ c, f (b, c))
h₂ : ∀ (b : β), BddAbove (range fun c ↦ f (b, c))
c : α
⊢ (∀ (i : β × γ), f i ≤ c) ↔ ∀ (i : β), ⨆ c, f (i, c) ≤ c

Tactic:
]

Goal state after:
case inr.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : ConditionallyCompleteLattice α
f : β × γ → α
hf : BddAbove (range f)
h✝¹ : Nonempty β
h✝ : Nonempty γ
h₁ : BddAbove (range fun b ↦ ⨆ c, f (b, c))
h₂ : ∀ (b : β), BddAbove (range fun c ↦ f (b, c))
c : α
⊢ (∀ (i : β × γ), f i ≤ c) ↔ ∀ (i : β), ⨆ c, f (i, c) ≤ c
Module: Mathlib.Order.Hom.WithTopBot

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PartialOrder α
inst✝¹ : OrderTop α
inst✝ : DecidablePred fun x ↦ x = ⊤
a : α
⊢ (fun a ↦ untopD ⊤ (map Subtype.val a)) ((fun a ↦ if h : a = ⊤ then ⊤ else ↑⟨a, h⟩) a) = a

Tactic:
by dsimp only; split_ifs <;> simp [*]

Goal state after:
Module: Mathlib.Order.Hom.WithTopBot

Goal state before:
pp: α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : OrderTop α
inst✝ : DecidablePred fun x ↦ x = ⊤
a : α
h : a ≠ ⊤
⊢ a = subtypeOrderIso ↑⟨a, h⟩

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : OrderTop α
inst✝ : DecidablePred fun x ↦ x = ⊤
a : α
h : a ≠ ⊤
⊢ a = subtypeOrderIso ↑⟨a, h⟩
Module: Mathlib.Order.Hom.WithTopBot

Goal state before:
pp: α : Type u_1
inst✝ : Lattice α
⊢ ⇑(LatticeHom.id α).withTopWithBot = ⇑(BoundedLatticeHom.id (WithTop (WithBot α)))

Tactic:
by simp [WithTop.map_id, WithBot.map_id]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: case mpr
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
a : α
s : Set ι
f : ι → α
hs : s.Finite
h : ∃ x ∈ s, f x ≤ sInf ∅
H : ∀ x ∈ s, a < f x
this : ∃ x ∈ s, f x = ⨅ i ∈ s, f i
⊢ a < ⨅ i ∈ s, f i

Tactic:
obtain ⟨_, hmem, hx⟩ := this

Goal state after:
case mpr
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
a : α
s : Set ι
f : ι → α
hs : s.Finite
h : ∃ x ∈ s, f x ≤ sInf ∅
H : ∀ x ∈ s, a < f x
w✝ : ι
hmem : w✝ ∈ s
hx : f w✝ = ⨅ i ∈ s, f i
⊢ a < ⨅ i ∈ s, f i
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inl
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
p : ι → Prop
f : Subtype p → α
hf : BddAbove (range f)
hf' : sSup ∅ ≤ iSup f
h✝ : IsEmpty (Subtype p)
⊢ sSup ∅ = sSup ∅

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ univ = univ

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
⊢ Ioo ↑a ⊤ = Ioo ↑a ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
⊢ Ico ↑a ⊤ = Ico ↑a ⊤

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
⊢ Iio ↑a = Iio ↑a

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: case mpr
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
a : α
s : Set ι
f : ι → α
hs : s.Finite
h : ∃ x ∈ s, f x ≤ sInf ∅
H : ∀ x ∈ s, a < f x
w✝ : ι
hmem : w✝ ∈ s
hx : f w✝ = ⨅ i ∈ s, f i
⊢ a < f w✝

Tactic:
]

Goal state after:
case mpr
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
a : α
s : Set ι
f : ι → α
hs : s.Finite
h : ∃ x ∈ s, f x ≤ sInf ∅
H : ∀ x ∈ s, a < f x
w✝ : ι
hmem : w✝ ∈ s
hx : f w✝ = ⨅ i ∈ s, f i
⊢ a < f w✝
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a : α
⊢ Iic ↑a = Iic ↑a

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ Icc ↑a ↑b = Icc ↑a ↑b

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inr.refine_1
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
p : ι → Prop
f : Subtype p → α
hf : BddAbove (range f)
hf' : sSup ∅ ≤ iSup f
h✝ : Nonempty (Subtype p)
this✝ : Nonempty ι
i : ι
h : p i
this : f ⟨i, h⟩ = (fun i ↦ ⨆ (h : p i), f ⟨i, h⟩) i
⊢ (fun i ↦ ⨆ (h : p i), f ⟨i, h⟩) i ≤ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩

Tactic:
]

Goal state after:
case inr.refine_1
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
p : ι → Prop
f : Subtype p → α
hf : BddAbove (range f)
hf' : sSup ∅ ≤ iSup f
h✝ : Nonempty (Subtype p)
this✝ : Nonempty ι
i : ι
h : p i
this : f ⟨i, h⟩ = (fun i ↦ ⨆ (h : p i), f ⟨i, h⟩) i
⊢ (fun i ↦ ⨆ (h : p i), f ⟨i, h⟩) i ≤ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ Ico ↑a ↑b = Ico ↑a ↑b

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ Ioc ↑a ↑b = Ioc ↑a ↑b

Tactic:
]

Goal state after:
Module: Mathlib.Order.Interval.Set.WithBotTop

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ Ioo ↑a ↑b = Ioo ↑a ↑b

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLattice α
s : Set β
f : β → α
hf : BddAbove (range fun i ↦ f ↑i)
hf' : sSup ∅ ≤ ⨆ i, f ↑i
⊢ sSup (range fun x ↦ f ↑x) = sSup (range fun i ↦ f ↑i)

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddAbove s
h : sSup ∅ ≤ sSup s
⊢ sSup s = sSup s

pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddAbove s
h : sSup ∅ ≤ sSup s
⊢ sSup ∅ ≤ ⨆ i, ↑i

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddAbove s
h : sSup ∅ ≤ sSup s
⊢ sSup ∅ ≤ ⨆ i, ↑i
Module: Mathlib.Order.ConditionallyCompleteLattice.Finset

Goal state before:
pp: α : Type u_2
inst✝ : ConditionallyCompleteLattice α
s : Finset α
hs : s.Nonempty
⊢ sSup ↑s = sSup ↑s

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddAbove s
h : sSup ∅ ≤ sSup s
⊢ sSup s = sSup s

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddBelow s
h : sInf s ≤ sInf ∅
⊢ sInf s = sInf s

pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddBelow s
h : sInf s ≤ sInf ∅
⊢ ⨅ i, ↑i ≤ sInf ∅

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddBelow s
h : sInf s ≤ sInf ∅
⊢ ⨅ i, ↑i ≤ sInf ∅
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
s : Set α
hs : BddBelow s
h : sInf s ≤ sInf ∅
⊢ sInf s = sInf s

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → Set α
c : Set ι
hn : c.Nonempty
hc : DirectedOn (fun i j ↦ f i ⊆ f j) c
s : Finset α
hs : ↑s ⊆ ⋃ x, f ↑x
⊢ ∃ i ∈ c, ↑s ⊆ f i

Tactic:
]

Goal state after:
α : Type u_1
ι : Type u_2
f : ι → Set α
c : Set ι
hn : c.Nonempty
hc : DirectedOn (fun i j ↦ f i ⊆ f j) c
s : Finset α
hs : ↑s ⊆ ⋃ x, f ↑x
⊢ ∃ i ∈ c, ↑s ⊆ f i
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inl
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
ι : Type u_5
ι' : Type u_6
f : ι → ι'
g : ι' → α
hf : BddAbove (range fun i ↦ g (f ↑i))
hg' : sSup ∅ ≤ ⨆ i, g (f ↑i)
⊢ sSup ∅ = sSup ∅

Tactic:
]

Goal state after:
Module: Mathlib.Data.Set.Finite.Lattice

Goal state before:
pp: α : Type u_1
c : Set (Set α)
hn : c.Nonempty
hc : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) c
s : Set α
hs : s.Finite
hsc : ↑hs.toFinset ⊆ ⋃ i ∈ c, i
⊢ ∃ t ∈ c, s ⊆ t

Tactic:
]

Goal state after:
α : Type u_1
c : Set (Set α)
hn : c.Nonempty
hc : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) c
s : Set α
hs : s.Finite
hsc : ↑hs.toFinset ⊆ ⋃ i ∈ c, i
⊢ ∃ t ∈ c, s ⊆ t
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
ι : Type u_5
ι' : Type u_6
s : Set ι
f : ι → ι'
g : ι' → α
hf : BddAbove (range fun i ↦ g (f ↑i))
hg' : sSup ∅ ≤ ⨆ i, g (f ↑i)
hs : s.Nonempty
hg : BddAbove (range fun i ↦ g ↑i)
this : Nonempty ↑s
i : ι
h : i ∈ s
t : ↑(f '' s)
ht : g ↑t = g (f ↑⟨i, h⟩)
⊢ g ↑t ≤ ⨆ i, g ↑i

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
ι : Type u_5
ι' : Type u_6
s : Set ι
f : ι → ι'
g : ι' → α
hf : BddAbove (range fun i ↦ g (f ↑i))
hg' : sSup ∅ ≤ ⨆ i, g (f ↑i)
hs : s.Nonempty
hg : BddAbove (range fun i ↦ g ↑i)
this : Nonempty ↑s
i : ι
h : i ∈ s
t : ↑(f '' s)
ht : g ↑t = g (f ↑⟨i, h⟩)
⊢ g ↑t ≤ ⨆ i, g ↑i
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inr
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
ι : Type u_5
ι' : Type u_6
s : Set ι
f : ι → ι'
g : ι' → α
hf : BddAbove (range fun i ↦ g (f ↑i))
hg' : sSup ∅ ≤ ⨆ i, g (f ↑i)
hs : s.Nonempty
hg : BddAbove (range fun i ↦ g ↑i)
hf' : sSup ∅ ≤ ⨆ i, g ↑i
⊢ sSup ((fun x ↦ g (f x)) '' s) = sSup ((fun x ↦ g (f x)) '' s)

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
p : ι → Prop
f : Exists p → α
h : Exists p
⊢ ⨆ (ih : Exists p), f ih ≤ ⨆ i, ⨆ (h : p i), f ⋯

Tactic:
have : Nonempty <| Exists p := ⟨h⟩

Goal state after:
case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLattice α
p : ι → Prop
f : Exists p → α
h : Exists p
this : Nonempty (Exists p)
⊢ ⨆ (ih : Exists p), f ih ≤ ⨆ i, ⨆ (h : p i), f ⋯
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
f g : ι → α
hf : ¬BddAbove (range f)
⊢ sSup ∅ ≤ max (sSup ∅) (⨆ x, g x)

Tactic:
]

Goal state after:
case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
f g : ι → α
hf : ¬BddAbove (range f)
⊢ sSup ∅ ≤ max (sSup ∅) (⨆ x, g x)
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
f g : ι → α
hf : BddAbove (range f)
hg : ¬BddAbove (range g)
⊢ sSup ∅ ≤ max (⨆ x, f x) (sSup ∅)

Tactic:
]

Goal state after:
case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
f g : ι → α
hf : BddAbove (range f)
hg : ¬BddAbove (range g)
⊢ sSup ∅ ≤ max (⨆ x, f x) (sSup ∅)
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case inl
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
le : sSup ∅ ≤ iSup f
⊢ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = iSup f

Tactic:
]

Goal state after:
case inl
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
le : sSup ∅ ≤ iSup f
⊢ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = iSup f
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case pos
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
le : sSup ∅ ≤ iSup f
bdd : BddAbove (range f)
⊢ iSup f = iSup f

Tactic:
]

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Indexed

Goal state before:
pp: case neg
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrder α
p : ι → Prop
f : Subtype p → α
hp : ¬∀ (i : ι), p i
le : sSup ∅ ≤ iSup f
bdd : ¬BddAbove (range f)
⊢ sSup ∅ = sSup ∅

Tactic:
]

Goal state after:
Module: Mathlib.Order.CompleteLattice.Finset

Goal state before:
pp: α : Type u_2
ι' : Sort u_7
inst✝ : CompleteLattice α
s : ι' → α
⊢ ⨆ x, s (Equiv.plift x) = ⨆ i, s i.down

Tactic:
]

Goal state after:
α : Type u_2
ι' : Sort u_7
inst✝ : CompleteLattice α
s : ι' → α
⊢ ⨆ x, s (Equiv.plift x) = ⨆ i, s i.down