Documentation

Mathlib.Data.Nat.PSub

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 #

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def Nat.ppred :
ℕ → Option ℕ

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
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@[simp]
theorem Nat.ppred_zero :
Nat.ppred 0 = none
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@[simp]
theorem Nat.ppred_succ {n : ℕ} :
n.succ.ppred = some n
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def Nat.psub (m : ℕ) :
ℕ → Option ℕ

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

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