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

• mk' :: (
• s : Set α

  Set containing some element of α

• val : Trunc ↑s.s

  Assertion of non-emptiness via Trunc

• )

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.

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

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.

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, 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}.

@[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.

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

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.

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

Convert a Semiquot α to a Trunc α.

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

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

@[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, 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.

@[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, a ∈ q ∧ b ∈ f a

instance Semiquot.instMonadSemiquot : Monad Semiquot

@[simp]

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

@[simp]

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

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

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

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

@[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.

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

Extract the value from an IsPure semiquotient.

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, 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.

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

@[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)