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instance {ι : Type u} {α : ι → Type v} [∀ i, LE (α i)] : LE (∀ i, α i) where | ||
le x y := ∀ i, x i ≤ y i | ||
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class Top (α : Type u) where | ||
top : α | ||
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class Bot (α : Type u) where | ||
bot : α | ||
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notation "⊤" => Top.top | ||
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notation "⊥" => Bot.bot | ||
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class Preorder (α : Type u) extends LE α, LT α where | ||
le_refl : ∀ a : α, a ≤ a | ||
le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c | ||
lt := λ a b => a ≤ b ∧ ¬ b ≤ a | ||
lt_iff_le_not_le : ∀ a b : α, a < b ↔ (a ≤ b ∧ ¬ b ≤ a) | ||
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class PartialOrder (α : Type u) extends Preorder α := | ||
(le_antisymm : ∀ a b : α, a ≤ b → b ≤ a → a = b) | ||
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def Set (α : Type u) := α → Prop | ||
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def setOf {α : Type u} (p : α → Prop) : Set α := | ||
p | ||
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namespace Set | ||
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protected def Mem (a : α) (s : Set α) : Prop := | ||
s a | ||
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instance : Membership α (Set α) := | ||
⟨Set.Mem⟩ | ||
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def range (f : ι → α) : Set α := | ||
setOf (λ x => ∃ y, f y = x) | ||
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end Set | ||
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class InfSet (α : Type _) where | ||
infₛ : Set α → α | ||
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class SupSet (α : Type _) where | ||
supₛ : Set α → α | ||
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export SupSet (supₛ) | ||
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export InfSet (infₛ) | ||
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open Set | ||
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def supᵢ {α : Type _} [SupSet α] {ι} (s : ι → α) : α := | ||
supₛ (range s) | ||
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def infᵢ {α : Type _} [InfSet α] {ι} (s : ι → α) : α := | ||
infₛ (range s) | ||
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class HasSup (α : Type u) where | ||
sup : α → α → α | ||
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class HasInf (α : Type u) where | ||
inf : α → α → α | ||
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@[inherit_doc] | ||
infixl:68 " ⊔ " => HasSup.sup | ||
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@[inherit_doc] | ||
infixl:69 " ⊓ " => HasInf.inf | ||
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class SemilatticeSup (α : Type u) extends HasSup α, PartialOrder α where | ||
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b | ||
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b | ||
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c | ||
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class SemilatticeInf (α : Type u) extends HasInf α, PartialOrder α where | ||
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a | ||
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b | ||
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c | ||
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class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α | ||
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class CompleteSemilatticeInf (α : Type _) extends PartialOrder α, InfSet α where | ||
infₛ_le : ∀ s, ∀ a, a ∈ s → infₛ s ≤ a | ||
le_infₛ : ∀ s a, (∀ b, b ∈ s → a ≤ b) → a ≤ infₛ s | ||
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class CompleteSemilatticeSup (α : Type _) extends PartialOrder α, SupSet α where | ||
le_supₛ : ∀ s, ∀ a, a ∈ s → a ≤ supₛ s | ||
supₛ_le : ∀ s a, (∀ b, b ∈ s → b ≤ a) → supₛ s ≤ a | ||
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class CompleteLattice (α : Type _) extends Lattice α, CompleteSemilatticeSup α, | ||
CompleteSemilatticeInf α, Top α, Bot α where | ||
protected le_top : ∀ x : α, x ≤ ⊤ | ||
protected bot_le : ∀ x : α, ⊥ ≤ x | ||
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class Frame (α : Type _) extends CompleteLattice α where | ||
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class Coframe (α : Type _) extends CompleteLattice α where | ||
infᵢ_sup_le_sup_infₛ (a : α) (s : Set α) : (infᵢ (λ b => a ⊔ b)) ≤ a ⊔ infₛ s | ||
-- should be ⨅ b ∈ s but I had problems with notation | ||
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class CompleteDistribLattice (α : Type _) extends Frame α where | ||
infᵢ_sup_le_sup_infₛ : ∀ a s, (infᵢ (λ b => a ⊔ b)) ≤ a ⊔ infₛ s | ||
-- similarly this is not quite right mathematically but this doesn't matter | ||
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-- See note [lower instance priority] | ||
instance (priority := 100) CompleteDistribLattice.toCoframe {α : Type _} [CompleteDistribLattice α] : | ||
Coframe α := | ||
{ ‹CompleteDistribLattice α› with } | ||
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class OrderTop (α : Type u) [LE α] extends Top α where | ||
le_top : ∀ a : α, a ≤ ⊤ | ||
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class OrderBot (α : Type u) [LE α] extends Bot α where | ||
bot_le : ∀ a : α, ⊥ ≤ a | ||
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class BoundedOrder (α : Type u) [LE α] extends OrderTop α, OrderBot α | ||
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instance(priority := 100) CompleteLattice.toBoundedOrder {α : Type _} [h : CompleteLattice α] : | ||
BoundedOrder α := | ||
{ h with } | ||
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namespace Pi | ||
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variable {ι : Type _} {α' : ι → Type _} | ||
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instance [∀ i, Bot (α' i)] : Bot (∀ i, α' i) := | ||
⟨fun _ => ⊥⟩ | ||
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instance [∀ i, Top (α' i)] : Top (∀ i, α' i) := | ||
⟨fun _ => ⊤⟩ | ||
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protected instance LE {ι : Type u} {α : ι → Type v} [∀ i, LE (α i)] : LE (∀ i, α i) where | ||
le x y := ∀ i, x i ≤ y i | ||
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instance Preorder {ι : Type u} {α : ι → Type v} [∀ i, Preorder (α i)] : Preorder (∀ i, α i) := | ||
{ Pi.LE with | ||
le_refl := sorry | ||
le_trans := sorry | ||
lt_iff_le_not_le := sorry } | ||
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instance PartialOrder {ι : Type u} {α : ι → Type v} [∀ i, PartialOrder (α i)] : | ||
PartialOrder (∀ i, α i) := | ||
{ Pi.Preorder with | ||
le_antisymm := sorry } | ||
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instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where | ||
le_sup_left _ _ _ := SemilatticeSup.le_sup_left _ _ | ||
le_sup_right _ _ _ := SemilatticeSup.le_sup_right _ _ | ||
sup_le _ _ _ ac bc i := SemilatticeSup.sup_le _ _ _ (ac i) (bc i) | ||
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instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where | ||
inf_le_left _ _ _ := SemilatticeInf.inf_le_left _ _ | ||
inf_le_right _ _ _ := SemilatticeInf.inf_le_right _ _ | ||
le_inf _ _ _ ac bc i := SemilatticeInf.le_inf _ _ _ (ac i) (bc i) | ||
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instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) := | ||
{ Pi.semilatticeSup, Pi.semilatticeInf with } | ||
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instance orderTop [∀ i, LE (α' i)] [∀ i, OrderTop (α' i)] : OrderTop (∀ i, α' i) := | ||
{ inferInstanceAs (Top (∀ i, α' i)) with le_top := sorry } | ||
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instance orderBot [∀ i, LE (α' i)] [∀ i, OrderBot (α' i)] : OrderBot (∀ i, α' i) := | ||
{ inferInstanceAs (Bot (∀ i, α' i)) with bot_le := sorry } | ||
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instance boundedOrder [∀ i, LE (α' i)] [∀ i, BoundedOrder (α' i)] : BoundedOrder (∀ i, α' i) := | ||
{ Pi.orderTop, Pi.orderBot with } | ||
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instance SupSet {α : Type _} {β : α → Type _} [∀ i, SupSet (β i)] : SupSet (∀ i, β i) := | ||
⟨fun s i => supᵢ (λ (f : {f : ∀ i, β i // f ∈ s}) => f.1 i)⟩ | ||
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instance InfSet {α : Type _} {β : α → Type _} [∀ i, InfSet (β i)] : InfSet (∀ i, β i) := | ||
⟨fun s i => infᵢ (λ (f : {f : ∀ i, β i // f ∈ s}) => f.1 i)⟩ | ||
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instance completeLattice {α : Type _} {β : α → Type _} [∀ i, CompleteLattice (β i)] : | ||
CompleteLattice (∀ i, β i) := | ||
{ Pi.boundedOrder, Pi.lattice with | ||
le_supₛ := sorry | ||
infₛ_le := sorry | ||
supₛ_le := sorry | ||
le_infₛ := sorry | ||
} | ||
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instance frame {ι : Type _} {π : ι → Type _} [∀ i, Frame (π i)] : Frame (∀ i, π i) := | ||
{ Pi.completeLattice with } | ||
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instance coframe {ι : Type _} {π : ι → Type _} [∀ i, Coframe (π i)] : Coframe (∀ i, π i) := | ||
{ Pi.completeLattice with infᵢ_sup_le_sup_infₛ := sorry } | ||
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end Pi | ||
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-- very quick (instantaneous) in Lean 4 | ||
instance Pi.completeDistribLattice' {ι : Type _} {π : ι → Type _} | ||
[∀ i, CompleteDistribLattice (π i)] : CompleteDistribLattice (∀ i, π i) := | ||
CompleteDistribLattice.mk (Pi.coframe.infᵢ_sup_le_sup_infₛ) | ||
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-- takes around 2 seconds wall clock time on my PC (but very quick in Lean 3) | ||
set_option maxHeartbeats 400 -- make sure it stays fast | ||
set_option synthInstance.maxHeartbeats 400 | ||
instance Pi.completeDistribLattice'' {ι : Type _} {π : ι → Type _} | ||
[∀ i, CompleteDistribLattice (π i)] : CompleteDistribLattice (∀ i, π i) := | ||
{ Pi.frame, Pi.coframe with } | ||
-- quick Lean 3 version: | ||
-- https://github.com/leanprover-community/mathlib/blob/b26e15a46f1a713ce7410e016d50575bb0bc3aa4/src/order/complete_boolean_algebra.lean#L210 |