File: Mathlib/Data/Fintype/Order.lean

full_name: Fintype.toOrderBot
code: /-- Constructs the `⊥` of a finite nonempty `SemilatticeInf`. -/
abbrev toOrderBot [SemilatticeInf α] : OrderBot α 
goal: OrderBot α

full_name: Fintype.toOrderTop
code: /-- Constructs the `⊤` of a finite nonempty `SemilatticeSup` -/
abbrev toOrderTop [SemilatticeSup α] : OrderTop α 
goal: OrderTop α

full_name: Fintype.toBoundedOrder
code: /-- Constructs the `⊤` and `⊥` of a finite nonempty `Lattice`. -/
abbrev toBoundedOrder [Lattice α] : BoundedOrder α 
goal: BoundedOrder α

full_name: Fintype.toCompleteLattice
code: /-- A finite bounded lattice is complete. -/
noncomputable abbrev toCompleteLattice [Lattice α] [BoundedOrder α] : CompleteLattice α 
goal: CompleteLattice α

full_name: Fintype.toCompleteDistribLatticeMinimalAxioms
code: /-- A finite bounded distributive lattice is completely distributive. -/
noncomputable abbrev toCompleteDistribLatticeMinimalAxioms [DistribLattice α] [BoundedOrder α] :
    CompleteDistribLattice.MinimalAxioms α 
goal: CompleteDistribLattice.MinimalAxioms α

full_name: Fintype.toCompleteDistribLattice
code: /-- A finite bounded distributive lattice is completely distributive. -/
noncomputable abbrev toCompleteDistribLattice [DistribLattice α] [BoundedOrder α] :
    CompleteDistribLattice α 
goal: CompleteDistribLattice α

full_name: Fintype.toCompleteLinearOrder
code: /-- A finite bounded linear order is complete. -/
noncomputable abbrev toCompleteLinearOrder
    [LinearOrder α] [BoundedOrder α] : CompleteLinearOrder α 
goal: CompleteLinearOrder α

full_name: Fintype.toCompleteBooleanAlgebra
code: /-- A finite boolean algebra is complete. -/
noncomputable abbrev toCompleteBooleanAlgebra [BooleanAlgebra α] : CompleteBooleanAlgebra α 
goal: CompleteBooleanAlgebra α

full_name: Fintype.toCompleteAtomicBooleanAlgebra
code: /-- A finite boolean algebra is complete and atomic. -/
noncomputable abbrev toCompleteAtomicBooleanAlgebra [BooleanAlgebra α] :
    CompleteAtomicBooleanAlgebra α 
goal: CompleteAtomicBooleanAlgebra α

full_name: Fintype.toCompleteLatticeOfNonempty
code: /-- A nonempty finite lattice is complete. If the lattice is already a `BoundedOrder`, then use
`Fintype.toCompleteLattice` instead, as this gives definitional equality for `⊥` and `⊤`. -/
noncomputable abbrev toCompleteLatticeOfNonempty [Lattice α] : CompleteLattice α 
goal: CompleteLattice α

full_name: Fintype.toCompleteLinearOrderOfNonempty
code: /-- A nonempty finite linear order is complete. If the linear order is already a `BoundedOrder`,
then use `Fintype.toCompleteLinearOrder` instead, as this gives definitional equality for `⊥` and
`⊤`. -/
noncomputable abbrev toCompleteLinearOrderOfNonempty [LinearOrder α] : CompleteLinearOrder α 
goal: CompleteLinearOrder α

full_name: Fin.completeLinearOrder
code: noncomputable instance Fin.completeLinearOrder {n : ℕ} [NeZero n] : CompleteLinearOrder (Fin n) 
goal: CompleteLinearOrder (Fin n)

full_name: Bool.completeLinearOrder
code: noncomputable instance Bool.completeLinearOrder : CompleteLinearOrder Bool 
goal: CompleteLinearOrder Bool

full_name: Bool.completeBooleanAlgebra
code: noncomputable instance Bool.completeBooleanAlgebra : CompleteBooleanAlgebra Bool 
goal: CompleteBooleanAlgebra Bool

full_name: Bool.completeAtomicBooleanAlgebra
code: noncomputable instance Bool.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra Bool 
goal: CompleteAtomicBooleanAlgebra Bool

full_name: Directed.finite_set_le
code: theorem Directed.finite_set_le (D : Directed r f) {s : Set γ} (hs : s.Finite) :
    ∃ z, ∀ i ∈ s, r (f i) (f z) 
goal: ∃ z, ∀ i ∈ s, r (f i) (f z)

full_name: Directed.finite_le
code: theorem Directed.finite_le (D : Directed r f) (g : β → γ) : ∃ z, ∀ i, r (f (g i)) (f z) 
goal: ∃ z, ∀ (i : β), r (f (g i)) (f z)

full_name: Finite.exists_le
code: theorem Finite.exists_le [IsDirected α (· ≤ ·)] (f : β → α) : ∃ M, ∀ i, f i ≤ M 
goal: ∃ M, ∀ (i : β), f i ≤ M

full_name: Finite.exists_ge
code: theorem Finite.exists_ge [IsDirected α (· ≥ ·)] (f : β → α) : ∃ M, ∀ i, M ≤ f i 
goal: ∃ M, ∀ (i : β), M ≤ f i

full_name: Set.Finite.exists_le
code: theorem Set.Finite.exists_le [IsDirected α (· ≤ ·)] {s : Set α} (hs : s.Finite) :
    ∃ M, ∀ i ∈ s, i ≤ M 
goal: ∃ M, ∀ i ∈ s, i ≤ M

full_name: Set.Finite.exists_ge
code: theorem Set.Finite.exists_ge [IsDirected α (· ≥ ·)] {s : Set α} (hs : s.Finite) :
    ∃ M, ∀ i ∈ s, M ≤ i 
goal: ∃ M, ∀ i ∈ s, M ≤ i

full_name: Finite.bddAbove_range
code: theorem Finite.bddAbove_range [IsDirected α (· ≤ ·)] (f : β → α) : BddAbove (Set.range f) 
goal: BddAbove (Set.range f)

full_name: Finite.bddBelow_range
code: theorem Finite.bddBelow_range [IsDirected α (· ≥ ·)] (f : β → α) : BddBelow (Set.range f) 
goal: BddBelow (Set.range f)