Two-pointings

This file defines TwoPointing α, the type of two pointings of α. A two-pointing is the data of two distinct terms.

This is morally a Type-valued Nontrivial. Another type which is quite close in essence is Sym2. Categorically speaking, prod is a cospan in the category of types. This forms the category of bipointed types. Two-pointed types form a full subcategory of those.

References

• [nLab, Coalgebra of the real interval] (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)

theorem TwoPointing.ext_iff {α : Type u_3} (x : TwoPointing α) (y : TwoPointing α) : x = y ↔ x.toProd.fst = y.toProd.fst ∧ x.toProd.snd = y.toProd.snd

theorem TwoPointing.ext {α : Type u_3} (x : TwoPointing α) (y : TwoPointing α) (fst : x.toProd.fst = y.toProd.fst) (snd : x.toProd.snd = y.toProd.snd) : x = y

structure TwoPointing (α : Type u_3) extends Prod : Type u_3

• fst : α
• snd : α
• fst_ne_snd : s.toProd.fst ≠ s.toProd.snd

  fst and snd are distinct terms

Two-pointing of a type. This is a Type-valued termed Nontrivial.

instance instDecidableEqTwoPointing : {α : Type u_3} → [inst : DecidableEq α] → DecidableEq (TwoPointing α)

theorem TwoPointing.snd_ne_fst {α : Type u_1} (p : TwoPointing α) : p.toProd.snd ≠ p.toProd.fst

@[simp]

theorem TwoPointing.swap_toProd {α : Type u_1} (p : TwoPointing α) : (TwoPointing.swap p).toProd = (p.toProd.snd, p.toProd.fst)

def TwoPointing.swap {α : Type u_1} (p : TwoPointing α) : TwoPointing α

Swaps the two pointed elements.

theorem TwoPointing.swap_fst {α : Type u_1} (p : TwoPointing α) : (TwoPointing.swap p).toProd.fst = p.toProd.snd

theorem TwoPointing.swap_snd {α : Type u_1} (p : TwoPointing α) : (TwoPointing.swap p).toProd.snd = p.toProd.fst

@[simp]

theorem TwoPointing.swap_swap {α : Type u_1} (p : TwoPointing α) : TwoPointing.swap (TwoPointing.swap p) = p

@[reducible]

theorem TwoPointing.to_nontrivial {α : Type u_1} (p : TwoPointing α) : Nontrivial α

instance TwoPointing.instNonemptyTwoPointing {α : Type u_1} [Nontrivial α] : Nonempty (TwoPointing α)

@[simp]

theorem TwoPointing.nonempty_two_pointing_iff {α : Type u_1} : Nonempty (TwoPointing α) ↔ Nontrivial α

def TwoPointing.pi (α : Type u_1) {β : Type u_2} (q : TwoPointing β) [Nonempty α] : TwoPointing (α → β)

The two-pointing of constant functions.

@[simp]

theorem TwoPointing.pi_fst (α : Type u_1) {β : Type u_2} (q : TwoPointing β) [Nonempty α] : (TwoPointing.pi α q).toProd.fst = Function.const α q.toProd.fst

@[simp]

theorem TwoPointing.pi_snd (α : Type u_1) {β : Type u_2} (q : TwoPointing β) [Nonempty α] : (TwoPointing.pi α q).toProd.snd = Function.const α q.toProd.snd

def TwoPointing.prod {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : TwoPointing (α × β)

The product of two two-pointings.

@[simp]

theorem TwoPointing.prod_fst {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : (TwoPointing.prod p q).toProd.fst = (p.toProd.fst, q.toProd.fst)

@[simp]

theorem TwoPointing.prod_snd {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : (TwoPointing.prod p q).toProd.snd = (p.toProd.snd, q.toProd.snd)

def TwoPointing.sum {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : TwoPointing (α ⊕ β)

The sum of two pointings. Keeps the first point from the left and the second point from the right.

@[simp]

theorem TwoPointing.sum_fst {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : (TwoPointing.sum p q).toProd.fst = Sum.inl p.toProd.fst

@[simp]

theorem TwoPointing.sum_snd {α : Type u_1} {β : Type u_2} (p : TwoPointing α) (q : TwoPointing β) : (TwoPointing.sum p q).toProd.snd = Sum.inr q.toProd.snd

def TwoPointing.bool : TwoPointing Bool

The false, true two-pointing of Bool.

@[simp]

theorem TwoPointing.bool_fst : TwoPointing.bool.toProd.fst = false

@[simp]

theorem TwoPointing.bool_snd : TwoPointing.bool.toProd.snd = true

instance TwoPointing.instInhabitedTwoPointingBool : Inhabited (TwoPointing Bool)

def TwoPointing.prop : TwoPointing Prop

The False, True two-pointing of Prop.

@[simp]

theorem TwoPointing.prop_fst : TwoPointing.prop.toProd.fst = False

@[simp]

theorem TwoPointing.prop_snd : TwoPointing.prop.toProd.snd = True