# 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 :

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

Equations
• x.ppred = match x with | 0 => none | n.succ => some n
Instances For
@[simp]
theorem Nat.ppred_zero :
= none
@[simp]
theorem Nat.ppred_succ {n : } :
n.succ.ppred = some n
def Nat.psub (m : ) :

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

Equations
Instances For
@[simp]
theorem Nat.psub_zero {m : } :
m.psub 0 = some m
@[simp]
theorem Nat.psub_succ {m : } {n : } :
m.psub n.succ = m.psub n >>= Nat.ppred
theorem Nat.pred_eq_ppred (n : ) :
n.pred = n.ppred.getD 0
theorem Nat.sub_eq_psub (m : ) (n : ) :
m - n = (m.psub n).getD 0
@[simp]
theorem Nat.ppred_eq_some {m : } {n : } :
n.ppred = some m m.succ = n
@[simp]
theorem Nat.ppred_eq_none {n : } :
n.ppred = none n = 0
theorem Nat.psub_eq_some {m : } {n : } {k : } :
m.psub n = some k k + n = m
theorem Nat.psub_eq_none {m : } {n : } :
m.psub n = none m < n
theorem Nat.ppred_eq_pred {n : } (h : 0 < n) :
n.ppred = some n.pred
theorem Nat.psub_eq_sub {m : } {n : } (h : n m) :
m.psub n = some (m - n)
theorem Nat.psub_add (m : ) (n : ) (k : ) :
m.psub (n + k) = do let __do_liftm.psub n __do_lift.psub k
@[inline]
def Nat.psub' (m : ) (n : ) :

Same as psub, but with a more efficient implementation.

Equations
• m.psub' n = if n m then some (m - n) else none
Instances For
theorem Nat.psub'_eq_psub (m : ) (n : ) :
m.psub' n = m.psub n