Partial predecessor and partial subtraction on the natural numbers

The usual definition of natural number subtraction (Nat.sub) returns 0 as a "garbage value" for a - b when a < b. Similarly, Nat.pred 0 is defined to be 0. The functions in this file wrap the result in an Option type instead:

Main definitions

• Nat.ppred: a partial predecessor operation
• Nat.psub: a partial subtraction operation

def Nat.ppred : ℕ → Option ℕ

Partial predecessor operation. Returns ppred n = some m if n = m + 1, otherwise none.

Equations

• Nat.ppred 0 = none
• n.succ.ppred = some n

@[simp]

theorem Nat.ppred_zero : ppred 0 = none

@[simp]

theorem Nat.ppred_succ {n : ℕ} : n.succ.ppred = some n

def Nat.psub (m : ℕ) : ℕ → Option ℕ

Partial subtraction operation. Returns psub m n = some k if m = n + k, otherwise none.

Equations

• m.psub 0 = some m
• m.psub n.succ = m.psub n >>= Nat.ppred

@[simp]

theorem Nat.psub_zero {m : ℕ} : m.psub 0 = some m

@[simp]

theorem Nat.psub_succ {m n : ℕ} : m.psub n.succ = m.psub n >>= ppred

theorem Nat.pred_eq_ppred (n : ℕ) : n.pred = n.ppred.getD 0

theorem Nat.sub_eq_psub (m n : ℕ) : m - n = (m.psub n).getD 0

@[simp]

theorem Nat.ppred_eq_some {m n : ℕ} : n.ppred = some m ↔ m.succ = n

@[simp]

theorem Nat.ppred_eq_none {n : ℕ} : n.ppred = none ↔ n = 0

theorem Nat.psub_eq_some {m n k : ℕ} : m.psub n = some k ↔ k + n = m

theorem Nat.psub_eq_none {m n : ℕ} : m.psub n = none ↔ m < n

theorem Nat.ppred_eq_pred {n : ℕ} (h : 0 < n) : n.ppred = some n.pred

theorem Nat.psub_eq_sub {m n : ℕ} (h : n ≤ m) : m.psub n = some (m - n)

theorem Nat.psub_add (m n k : ℕ) : m.psub (n + k) = do let __do_lift ← m.psub n __do_lift.psub k

@[inline]

def Nat.psub' (m n : ℕ) : Option ℕ

Same as psub, but with a more efficient implementation.

Equations

• m.psub' n = if n ≤ m then some (m - n) else none

theorem Nat.psub'_eq_psub (m n : ℕ) : m.psub' n = m.psub n