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)

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

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

• fst : α
• snd : α
• fst_ne_snd : self.fst ≠ self.snd

  fst and snd are distinct terms

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

theorem TwoPointing.fst_ne_snd {α : Type u_3} (self : TwoPointing α) : self.fst ≠ self.snd

fst and snd are distinct terms

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

Equations

• instDecidableEqTwoPointing = decEqTwoPointing✝

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

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

Swaps the two pointed elements.

Equations

• p.swap = { toProd := (p.snd, p.fst), fst_ne_snd := ⋯ }

@[simp]

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

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

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

@[simp]

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

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

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

Equations

• ⋯ = ⋯

@[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.

Equations

• TwoPointing.pi α q = { toProd := (fun (x : α) => q.fst, fun (x : α) => q.snd), fst_ne_snd := ⋯ }

@[simp]

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

@[simp]

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

Equations

• p.prod q = { toProd := ((p.fst, q.fst), p.snd, q.snd), fst_ne_snd := ⋯ }

@[simp]

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

@[simp]

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

Equations

• p.sum q = { toProd := (Sum.inl p.fst, Sum.inr q.snd), fst_ne_snd := ⋯ }

@[simp]

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

@[simp]

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

def TwoPointing.bool : TwoPointing Bool

The false, true two-pointing of Bool.

Equations

• TwoPointing.bool = { toProd := (false, true), fst_ne_snd := Bool.false_ne_true }

@[simp]

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

@[simp]

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

instance TwoPointing.instInhabitedBool : Inhabited (TwoPointing Bool)

Equations

• TwoPointing.instInhabitedBool = { default := TwoPointing.bool }

def TwoPointing.prop : TwoPointing Prop

The False, True two-pointing of Prop.

Equations

• TwoPointing.prop = { toProd := (False, True), fst_ne_snd := false_ne_true }

@[simp]

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

@[simp]

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