Semiquotients

A data type for semiquotients, which are classically equivalent to nonempty sets, but are useful for programming; the idea is that a semiquotient set S represents some (particular but unknown) element of S. This can be used to model nondeterministic functions, which return something in a range of values (represented by the predicate S) but are not completely determined.

structure Semiquot (α : Type u_1) : Type u_1

A member of Semiquot α is classically a nonempty Set α, and in the VM is represented by an element of α; the relation between these is that the VM element is required to be a member of the set s. The specific element of s that the VM computes is hidden by a quotient construction, allowing for the representation of nondeterministic functions.

• mk' :: (
• s : Set α

  Set containing some element of α

• val : Trunc ↑self.s

  Assertion of non-emptiness via Trunc

• )

instance Semiquot.instMembershipSemiquot {α : Type u_1} : Membership α (Semiquot α)

Equations

• Semiquot.instMembershipSemiquot = { mem := fun (a : α) (q : Semiquot α) => a ∈ q.s }

def Semiquot.mk {α : Type u_1} {a : α} {s : Set α} (h : a ∈ s) : Semiquot α

Construct a Semiquot α from h : a ∈ s where s : Set α.

Equations

• Semiquot.mk h = { s := s, val := Trunc.mk { val := a, property := h } }

theorem Semiquot.ext_s {α : Type u_1} {q₁ : Semiquot α} {q₂ : Semiquot α} : q₁ = q₂ ↔ q₁.s = q₂.s

theorem Semiquot.ext {α : Type u_1} {q₁ : Semiquot α} {q₂ : Semiquot α} : q₁ = q₂ ↔ ∀ (a : α), a ∈ q₁ ↔ a ∈ q₂

theorem Semiquot.exists_mem {α : Type u_1} (q : Semiquot α) : ∃ (a : α), a ∈ q

theorem Semiquot.eq_mk_of_mem {α : Type u_1} {q : Semiquot α} {a : α} (h : a ∈ q) : q = Semiquot.mk h

theorem Semiquot.nonempty {α : Type u_1} (q : Semiquot α) : Set.Nonempty q.s

def Semiquot.pure {α : Type u_1} (a : α) : Semiquot α

pure a is a reinterpreted as an unspecified element of {a}.

Equations

• Semiquot.pure a = Semiquot.mk (_ : a ∈ {a})

@[simp]

theorem Semiquot.mem_pure' {α : Type u_1} {a : α} {b : α} : a ∈ Semiquot.pure b ↔ a = b

def Semiquot.blur' {α : Type u_1} (q : Semiquot α) {s : Set α} (h : q.s ⊆ s) : Semiquot α

Replace s in a Semiquot with a superset.

Equations

• One or more equations did not get rendered due to their size.

def Semiquot.blur {α : Type u_1} (s : Set α) (q : Semiquot α) : Semiquot α

Replace s in a q : Semiquot α with a union s ∪ q.s

Equations

• Semiquot.blur s q = Semiquot.blur' q (_ : q.s ⊆ s ∪ q.s)

theorem Semiquot.blur_eq_blur' {α : Type u_1} (q : Semiquot α) (s : Set α) (h : q.s ⊆ s) : Semiquot.blur s q = Semiquot.blur' q h

@[simp]

theorem Semiquot.mem_blur' {α : Type u_1} (q : Semiquot α) {s : Set α} (h : q.s ⊆ s) {a : α} : a ∈ Semiquot.blur' q h ↔ a ∈ s

def Semiquot.ofTrunc {α : Type u_1} (q : Trunc α) : Semiquot α

Convert a Trunc α to a Semiquot α.

Equations

• Semiquot.ofTrunc q = { s := Set.univ, val := Trunc.map (fun (a : α) => { val := a, property := trivial }) q }

def Semiquot.toTrunc {α : Type u_1} (q : Semiquot α) : Trunc α

Convert a Semiquot α to a Trunc α.

Equations

• Semiquot.toTrunc q = Trunc.map Subtype.val q.val

def Semiquot.liftOn {α : Type u_1} {β : Type u_2} (q : Semiquot α) (f : α → β) (h : ∀ a ∈ q, ∀ b ∈ q, f a = f b) : β

If f is a constant on q.s, then q.liftOn f is the value of f at any point of q.

Equations

• Semiquot.liftOn q f h = Trunc.liftOn q.val (fun (x : ↑q.s) => f ↑x) (_ : ∀ (x y : ↑q.s), f ↑x = f ↑y)

theorem Semiquot.liftOn_ofMem {α : Type u_1} {β : Type u_2} (q : Semiquot α) (f : α → β) (h : ∀ a ∈ q, ∀ b ∈ q, f a = f b) (a : α) (aq : a ∈ q) : Semiquot.liftOn q f h = f a

def Semiquot.map {α : Type u_1} {β : Type u_2} (f : α → β) (q : Semiquot α) : Semiquot β

Apply a function to the unknown value stored in a Semiquot α.

Equations

• Semiquot.map f q = { s := f '' q.s, val := Trunc.map (fun (x : ↑q.s) => { val := f ↑x, property := (_ : f ↑x ∈ f '' q.s) }) q.val }

@[simp]

theorem Semiquot.mem_map {α : Type u_1} {β : Type u_2} (f : α → β) (q : Semiquot α) (b : β) : b ∈ Semiquot.map f q ↔ ∃ a ∈ q, f a = b

def Semiquot.bind {α : Type u_1} {β : Type u_2} (q : Semiquot α) (f : α → Semiquot β) : Semiquot β

Apply a function returning a Semiquot to a Semiquot.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Semiquot.mem_bind {α : Type u_1} {β : Type u_2} (q : Semiquot α) (f : α → Semiquot β) (b : β) : b ∈ Semiquot.bind q f ↔ ∃ a ∈ q, b ∈ f a

instance Semiquot.instMonadSemiquot : Monad Semiquot

Equations

• Semiquot.instMonadSemiquot = Monad.mk

@[simp]

theorem Semiquot.map_def {α : Type u_1} {β : Type u_1} : (fun (x : α → β) (x_1 : Semiquot α) => x <$> x_1) = Semiquot.map

@[simp]

theorem Semiquot.bind_def {α : Type u_1} {β : Type u_1} : (fun (x : Semiquot α) (x_1 : α → Semiquot β) => x >>= x_1) = Semiquot.bind

@[simp]

theorem Semiquot.mem_pure {α : Type u_1} {a : α} {b : α} : a ∈ pure b ↔ a = b

theorem Semiquot.mem_pure_self {α : Type u_1} (a : α) : a ∈ pure a

@[simp]

theorem Semiquot.pure_inj {α : Type u_1} {a : α} {b : α} : pure a = pure b ↔ a = b

instance Semiquot.instLawfulMonadSemiquotInstMonadSemiquot : LawfulMonad Semiquot

Equations

• Semiquot.instLawfulMonadSemiquotInstMonadSemiquot = (_ : LawfulMonad Semiquot)

instance Semiquot.instLESemiquot {α : Type u_1} : LE (Semiquot α)

Equations

• Semiquot.instLESemiquot = { le := fun (s t : Semiquot α) => s.s ⊆ t.s }

instance Semiquot.partialOrder {α : Type u_1} : PartialOrder (Semiquot α)

Equations

• Semiquot.partialOrder = PartialOrder.mk (_ : ∀ (s t : Semiquot α), s ≤ t → t ≤ s → s = t)

instance Semiquot.instSemilatticeSupSemiquot {α : Type u_1} : SemilatticeSup (Semiquot α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Semiquot.pure_le {α : Type u_1} {a : α} {s : Semiquot α} : pure a ≤ s ↔ a ∈ s

def Semiquot.IsPure {α : Type u_1} (q : Semiquot α) : Prop

Assert that a Semiquot contains only one possible value.

Equations

• Semiquot.IsPure q = ∀ a ∈ q, ∀ b ∈ q, a = b

def Semiquot.get {α : Type u_1} (q : Semiquot α) (h : Semiquot.IsPure q) : α

Extract the value from an IsPure semiquotient.

Equations

• Semiquot.get q h = Semiquot.liftOn q id h

theorem Semiquot.get_mem {α : Type u_1} {q : Semiquot α} (p : Semiquot.IsPure q) : Semiquot.get q p ∈ q

theorem Semiquot.eq_pure {α : Type u_1} {q : Semiquot α} (p : Semiquot.IsPure q) : q = pure (Semiquot.get q p)

@[simp]

theorem Semiquot.pure_isPure {α : Type u_1} (a : α) : Semiquot.IsPure (pure a)

theorem Semiquot.isPure_iff {α : Type u_1} {s : Semiquot α} : Semiquot.IsPure s ↔ ∃ (a : α), s = pure a

theorem Semiquot.IsPure.mono {α : Type u_1} {s : Semiquot α} {t : Semiquot α} (st : s ≤ t) (h : Semiquot.IsPure t) : Semiquot.IsPure s

theorem Semiquot.IsPure.min {α : Type u_1} {s : Semiquot α} {t : Semiquot α} (h : Semiquot.IsPure t) : s ≤ t ↔ s = t

theorem Semiquot.isPure_of_subsingleton {α : Type u_1} [Subsingleton α] (q : Semiquot α) : Semiquot.IsPure q

def Semiquot.univ {α : Type u_1} [Inhabited α] : Semiquot α

univ : Semiquot α represents an unspecified element of univ : Set α.

Equations

• Semiquot.univ = Semiquot.mk (_ : default ∈ Set.univ)

instance Semiquot.instInhabitedSemiquot {α : Type u_1} [Inhabited α] : Inhabited (Semiquot α)

Equations

• Semiquot.instInhabitedSemiquot = { default := Semiquot.univ }

@[simp]

theorem Semiquot.mem_univ {α : Type u_1} [Inhabited α] (a : α) : a ∈ Semiquot.univ

theorem Semiquot.univ_unique {α : Type u_1} (I : Inhabited α) (J : Inhabited α) : Semiquot.univ = Semiquot.univ

@[simp]

theorem Semiquot.isPure_univ {α : Type u_1} [Inhabited α] : Semiquot.IsPure Semiquot.univ ↔ Subsingleton α

instance Semiquot.instOrderTopSemiquotInstLESemiquot {α : Type u_1} [Inhabited α] : OrderTop (Semiquot α)

Equations

• Semiquot.instOrderTopSemiquotInstLESemiquot = OrderTop.mk (_ : ∀ (x : Semiquot α), x.s ⊆ Set.univ)