Documentation

Mathlib.Data.Fintype.Order

Order structures on finite types #

This file provides order instances on fintypes.

Computable instances #

On a Fintype, we can construct

Those are marked as def to avoid defeqness issues.

Completion instances #

Those instances are noncomputable because the definitions of sSup and sInf use Set.toFinset and set membership is undecidable in general.

On a Fintype, we can promote:

Those are marked as def to avoid typeclass loops.

Concrete instances #

We provide a few instances for concrete types:

@[reducible, inline]
abbrev Fintype.toOrderBot (α : Type u_2) [Fintype α] [Nonempty α] [SemilatticeInf α] :

Constructs the of a finite nonempty SemilatticeInf.

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@[reducible, inline]
abbrev Fintype.toOrderTop (α : Type u_2) [Fintype α] [Nonempty α] [SemilatticeSup α] :

Constructs the of a finite nonempty SemilatticeSup

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@[reducible, inline]
abbrev Fintype.toBoundedOrder (α : Type u_2) [Fintype α] [Nonempty α] [Lattice α] :

Constructs the and of a finite nonempty Lattice.

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@[reducible, inline]
noncomputable abbrev Fintype.toCompleteLattice (α : Type u_2) [Fintype α] [Lattice α] [BoundedOrder α] :

A finite bounded lattice is complete.

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@[reducible, inline]

A finite bounded distributive lattice is completely distributive.

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@[reducible, inline]

A finite bounded distributive lattice is completely distributive.

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@[reducible, inline]
noncomputable abbrev Fintype.toCompleteLinearOrder (α : Type u_2) [Fintype α] [LinearOrder α] [BoundedOrder α] :

A finite bounded linear order is complete.

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@[reducible, inline]

A finite boolean algebra is complete.

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@[reducible, inline]

A finite boolean algebra is complete and atomic.

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@[reducible, inline]
noncomputable abbrev Fintype.toCompleteLatticeOfNonempty (α : Type u_2) [Fintype α] [Nonempty α] [Lattice α] :

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 .

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@[reducible, inline]

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 .

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Properties for PartialOrders #

theorem Finite.exists_minimal_le {α : Type u_1} [PartialOrder α] {a : α} {p : αProp} [Finite α] (h : p a) :
ba, Minimal p b
@[deprecated Finite.exists_minimal_le (since := "2024-09-23")]
theorem Finite.exists_ge_minimal {α : Type u_1} [PartialOrder α] {a : α} {p : αProp} [Finite α] (h : p a) :
ba, Minimal p b

Alias of Finite.exists_minimal_le.

theorem Finite.exists_le_maximal {α : Type u_1} [PartialOrder α] {a : α} {p : αProp} [Finite α] (h : p a) :
∃ (b : α), a b Maximal p b
theorem Finset.exists_minimal_le {α : Type u_1} [PartialOrder α] {a : α} (s : Finset α) (h : a s) :
ba, Minimal (fun (x : α) => x s) b
theorem Finset.exists_le_maximal {α : Type u_1} [PartialOrder α] {a : α} (s : Finset α) (h : a s) :
∃ (b : α), a b Maximal (fun (x : α) => x s) b
theorem Set.Finite.exists_minimal_le {α : Type u_1} [PartialOrder α] {a : α} {s : Set α} (hs : s.Finite) (h : a s) :
ba, Minimal (fun (x : α) => x s) b
theorem Set.Finite.exists_le_maximal {α : Type u_1} [PartialOrder α] {a : α} {s : Set α} (hs : s.Finite) (h : a s) :
∃ (b : α), a b Maximal (fun (x : α) => x s) b

Concrete instances #

Directed Orders #

theorem Directed.finite_set_le {α : Type u_1} {r : ααProp} [IsTrans α r] {γ : Type u_3} [Nonempty γ] {f : γα} (D : Directed r f) {s : Set γ} (hs : s.Finite) :
∃ (z : γ), is, r (f i) (f z)
theorem Directed.finite_le {α : Type u_1} {r : ααProp} [IsTrans α r] {β : Type u_2} {γ : Type u_3} [Nonempty γ] {f : γα} [Finite β] (D : Directed r f) (g : βγ) :
∃ (z : γ), ∀ (i : β), r (f (g i)) (f z)
theorem Finite.exists_le {α : Type u_1} {β : Type u_2} [Finite β] [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] (f : βα) :
∃ (M : α), ∀ (i : β), f i M
theorem Finite.exists_ge {α : Type u_1} {β : Type u_2} [Finite β] [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] (f : βα) :
∃ (M : α), ∀ (i : β), M f i
theorem Set.Finite.exists_le {α : Type u_1} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] {s : Set α} (hs : s.Finite) :
∃ (M : α), is, i M
theorem Set.Finite.exists_ge {α : Type u_1} [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] {s : Set α} (hs : s.Finite) :
∃ (M : α), is, M i
@[simp]
theorem Finite.bddAbove_range {α : Type u_1} {β : Type u_2} [Finite β] [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] (f : βα) :
@[simp]
theorem Finite.bddBelow_range {α : Type u_1} {β : Type u_2} [Finite β] [Nonempty α] [Preorder α] [IsDirected α fun (x1 x2 : α) => x1 x2] (f : βα) :