# 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.

def PNat :

ℕ+ is the type of positive natural numbers. It is defined as a subtype, and the VM representation of ℕ+ is the same as ℕ because the proof is not stored.

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Instances For

ℕ+ is the type of positive natural numbers. It is defined as a subtype, and the VM representation of ℕ+ is the same as ℕ because the proof is not stored.

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instance instOnePNat :
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def PNat.val :
ℕ+

The underlying natural number

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instance coePNatNat :
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instance instReprPNat :
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instance instOfNatPNatOfNeZeroNat (n : ) [] :
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• = { ofNat := n, }
@[simp]
theorem PNat.mk_coe (n : ) (h : 0 < n) :
n, h = n
def PNat.natPred (i : ℕ+) :

Predecessor of a ℕ+, as a ℕ.

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• i.natPred = i - 1
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@[simp]
theorem PNat.natPred_eq_pred {n : } (h : 0 < n) :
PNat.natPred n, h = n.pred
def Nat.toPNat (n : ) (h : autoParam (0 < n) _auto✝) :

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

Equations
• n.toPNat h = n, h
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def Nat.succPNat (n : ) :

Write a successor as an element of ℕ+.

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• n.succPNat = n.succ,
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@[simp]
theorem Nat.succPNat_coe (n : ) :
n.succPNat = n.succ
@[simp]
theorem Nat.natPred_succPNat (n : ) :
n.succPNat.natPred = n
@[simp]
theorem PNat.succPNat_natPred (n : ℕ+) :
n.natPred.succPNat = n
def Nat.toPNat' (n : ) :

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

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• n.toPNat' = n.pred.succPNat
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@[simp]
theorem Nat.toPNat'_zero :
= 1
@[simp]
theorem Nat.toPNat'_coe (n : ) :
n.toPNat' = if 0 < n then n else 1
theorem PNat.mk_le_mk (n : ) (k : ) (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 : ) (hn : 0 < n) (hk : 0 < k) :
n, hn < k, hk n < k
@[simp]
theorem PNat.coe_le_coe (n : ℕ+) (k : ℕ+) :
n k n k
@[simp]
theorem PNat.coe_lt_coe (n : ℕ+) (k : ℕ+) :
n < k n < k
@[simp]
theorem PNat.pos (n : ℕ+) :
0 < n
theorem PNat.eq {m : ℕ+} {n : ℕ+} :
m = nm = n
@[simp]
theorem PNat.ne_zero (n : ℕ+) :
n 0
instance NeZero.pnat {a : ℕ+} :
NeZero a
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• =
theorem PNat.toPNat'_coe {n : } :
0 < nn.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
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@[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
@[irreducible]
def PNat.strongInductionOn {p : ℕ+Sort u_1} (n : ℕ+) :
((k : ℕ+) → ((m : ℕ+) → m < kp m)p k)p n

Strong induction on ℕ+.

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

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• x✝¹.modDivAux x✝ x = match x✝¹, x✝, x with | k, 0, q => (k, q.pred) | x, r.succ, q => (r + 1, , q)
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def PNat.modDiv (m : ℕ+) (k : 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)
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def PNat.mod (m : ℕ+) (k : 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.

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• m.mod k = (m.modDiv k).1
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def PNat.div (m : ℕ+) (k : 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.

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• m.div k = (m.modDiv k).2
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theorem PNat.mod_coe (m : ℕ+) (k : ℕ+) :
(m.mod k) = if m % k = 0 then k else m % k
theorem PNat.div_coe (m : ℕ+) (k : ℕ+) :
m.div k = if m % k = 0 then (m / k).pred else m / k
def PNat.divExact (m : ℕ+) (k : ℕ+) :

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

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• m.divExact k = (m.div k).succ,
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instance Nat.canLiftPNat :
CanLift ℕ+ PNat.val fun (n : ) => 0 < n
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instance Int.canLiftPNat :
CanLift ℕ+ (fun (x : ℕ+) => x) fun (x : ) => 0 < x
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