mathlib documentation

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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@[simp]
def nat.ppred (a : ℕ) :
option ℕ

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

Equations
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@[simp]
def nat.psub (m a : ℕ) :
option ℕ

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

Equations
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theorem nat.pred_eq_ppred (n : ℕ) :
n.pred = n.ppred.get_or_else 0

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theorem nat.sub_eq_psub (m n : ℕ) :
m - n = (m.psub n).get_or_else 0

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@[simp]
theorem nat.ppred_eq_some {m 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 k : ℕ} :
m.psub n = some k ↔ k + n = m

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theorem nat.psub_eq_none (m 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} (h : n ≤ m) :
m.psub n = some (m - n)

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theorem nat.psub_add (m n k : ℕ) :
m.psub (n + k) = m.psub n >>= λ (x : ℕ), x.psub k