The positive natural numbers

This file contains the definitions, and basic results. Most algebraic facts are deferred to Data.PNat.Basic, as they need more imports.

instance instPNatLinearOrder : LinearOrder ℕ+

Equations

• instPNatLinearOrder = Subtype.instLinearOrder fun (n : ℕ) => 0 < n

instance instOnePNat : One ℕ+

Equations

• instOnePNat = { one := ⟨1, Nat.zero_lt_one⟩ }

instance instOfNatPNatOfNeZeroNat (n : ℕ) [NeZero n] : OfNat ℕ+ n

Equations

• instOfNatPNatOfNeZeroNat n = { ofNat := ⟨n, ⋯⟩ }

@[simp]

theorem PNat.mk_coe (n : ℕ) (h : 0 < n) : ↑⟨n, h⟩ = n

def PNat.natPred (i : ℕ+) : ℕ

Predecessor of a ℕ+, as a ℕ.

Equations

• i.natPred = ↑i - 1

@[simp]

theorem PNat.natPred_eq_pred {n : ℕ} (h : 0 < n) : PNat.natPred ⟨n, h⟩ = n.pred

def Nat.toPNat (n : ℕ) (h : 0 < n := by decide) : ℕ+

Convert a natural number to a positive natural number. The positivity assumption is inferred by dec_trivial.

Equations

• n.toPNat h = ⟨n, h⟩

def Nat.succPNat (n : ℕ) : ℕ+

Write a successor as an element of ℕ+.

Equations

• n.succPNat = ⟨n.succ, ⋯⟩

@[simp]

theorem Nat.succPNat_coe (n : ℕ) : ↑n.succPNat = n.succ

@[simp]

theorem Nat.natPred_succPNat (n : ℕ) : n.succPNat.natPred = n

@[simp]

theorem PNat.succPNat_natPred (n : ℕ+) : n.natPred.succPNat = n

def Nat.toPNat' (n : ℕ) : ℕ+

Convert a natural number to a PNat. n+1 is mapped to itself, and 0 becomes 1.

Equations

• n.toPNat' = n.pred.succPNat

@[simp]

theorem Nat.toPNat'_zero : Nat.toPNat' 0 = 1

@[simp]

theorem Nat.toPNat'_coe (n : ℕ) : ↑n.toPNat' = if 0 < n then n else 1

theorem PNat.mk_le_mk (n k : ℕ) (hn : 0 < n) (hk : 0 < k) : ⟨n, hn⟩ ≤ ⟨k, hk⟩ ↔ n ≤ k

We now define a long list of structures on ℕ+ induced by similar structures on ℕ. Most of these behave in a completely obvious way, but there are a few things to be said about subtraction, division and powers.

theorem PNat.mk_lt_mk (n k : ℕ) (hn : 0 < n) (hk : 0 < k) : ⟨n, hn⟩ < ⟨k, hk⟩ ↔ n < k

@[simp]

theorem PNat.coe_le_coe (n k : ℕ+) : ↑n ≤ ↑k ↔ n ≤ k

@[simp]

theorem PNat.coe_lt_coe (n k : ℕ+) : ↑n < ↑k ↔ n < k

@[simp]

theorem PNat.pos (n : ℕ+) : 0 < ↑n

theorem PNat.eq {m n : ℕ+} : ↑m = ↑n → m = n

theorem PNat.coe_injective : Function.Injective PNat.val

@[simp]

theorem PNat.ne_zero (n : ℕ+) : ↑n ≠ 0

theorem NeZero.pnat {a : ℕ+} : NeZero ↑a

theorem PNat.toPNat'_coe {n : ℕ} : 0 < n → ↑n.toPNat' = n

@[simp]

theorem PNat.coe_toPNat' (n : ℕ+) : (↑n).toPNat' = n

@[simp]

theorem PNat.one_le (n : ℕ+) : 1 ≤ n

@[simp]

theorem PNat.not_lt_one (n : ℕ+) : ¬n < 1

instance PNat.instInhabited : Inhabited ℕ+

Equations

• PNat.instInhabited = { default := 1 }

@[simp]

theorem PNat.mk_one {h : 0 < 1} : ⟨1, h⟩ = 1

theorem PNat.one_coe : ↑1 = 1

@[simp]

theorem PNat.coe_eq_one_iff {m : ℕ+} : ↑m = 1 ↔ m = 1

instance PNat.instWellFoundedRelation : WellFoundedRelation ℕ+

Equations

• PNat.instWellFoundedRelation = measure fun (a : ℕ+) => ↑a

@[irreducible]

def PNat.strongInductionOn {p : ℕ+ → Sort u_1} (n : ℕ+) : ((k : ℕ+) → ((m : ℕ+) → m < k → p m) → p k) → p n

Strong induction on ℕ+.

Equations

• n.strongInductionOn x = x n fun (a : ℕ+) (x_1 : a < n) => a.strongInductionOn x

def PNat.modDivAux : ℕ+ → ℕ → ℕ → ℕ+ × ℕ

We define m % k and m / k in the same way as for ℕ except that when m = n * k we take m % k = k and m / k = n - 1. This ensures that m % k is always positive and m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

Equations

• x✝.modDivAux 0 x = (x✝, x.pred)
• x✝.modDivAux r.succ x = (⟨r + 1, ⋯⟩, x)

def PNat.modDiv (m k : ℕ+) : ℕ+ × ℕ

mod_div m k = (m % k, m / k). We define m % k and m / k in the same way as for ℕ except that when m = n * k we take m % k = k and m / k = n - 1. This ensures that m % k is always positive and m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

Equations

• m.modDiv k = k.modDivAux (↑m % ↑k) (↑m / ↑k)

def PNat.mod (m k : ℕ+) : ℕ+

We define m % k in the same way as for ℕ except that when m = n * k we take m % k = k This ensures that m % k is always positive.

Equations

• m.mod k = (m.modDiv k).fst

def PNat.div (m k : ℕ+) : ℕ

We define m / k in the same way as for ℕ except that when m = n * k we take m / k = n - 1. This ensures that m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

Equations

• m.div k = (m.modDiv k).snd

theorem PNat.mod_coe (m k : ℕ+) : ↑(m.mod k) = if ↑m % ↑k = 0 then ↑k else ↑m % ↑k

theorem PNat.div_coe (m k : ℕ+) : m.div k = if ↑m % ↑k = 0 then (↑m / ↑k).pred else ↑m / ↑k

def PNat.divExact (m k : ℕ+) : ℕ+

If h : k | m, then k * (div_exact m k) = m. Note that this is not equal to m / k.

Equations

• m.divExact k = ⟨(m.div k).succ, ⋯⟩

theorem Nat.canLiftPNat : CanLift ℕ ℕ+ PNat.val fun (n : ℕ) => 0 < n

theorem Int.canLiftPNat : CanLift ℤ ℕ+ (fun (x : ℕ+) => ↑↑x) fun (x : ℤ) => 0 < x