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[Merged by Bors] - feat(data/nat/basic): lt_one_iff #8224

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[Merged by Bors] - feat(data/nat/basic): lt_one_iff #8224

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b86e9b5
feat(data/nat/basic): lt_one_iff
pechersky Jul 8, 2021
172a07a
Use an existing proof
eric-wieser Jul 8, 2021
310bdf1
Use existing lemma name
Vierkantor Jul 8, 2021
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5 changes: 4 additions & 1 deletion src/data/nat/basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -314,7 +314,7 @@ lemma not_succ_lt_self {n : ℕ} : ¬succ n < n :=
not_lt_of_ge (nat.le_succ _)

theorem lt_succ_iff {m n : ℕ} : m < succ n ↔ m ≤ n :=
succ_le_succ_iff
⟨le_of_lt_succ, lt_succ_of_le⟩

lemma succ_le_iff {m n : ℕ} : succ m ≤ n ↔ m < n :=
⟨lt_of_succ_le, succ_le_of_lt⟩
Expand Down Expand Up @@ -344,6 +344,9 @@ H.lt_or_eq_dec.imp le_of_lt_succ id
lemma succ_lt_succ_iff {m n : ℕ} : succ m < succ n ↔ m < n :=
⟨lt_of_succ_lt_succ, succ_lt_succ⟩

@[simp] lemma lt_one_iff {n : ℕ} : n < 1 ↔ n = 0 :=
lt_succ_iff.trans le_zero_iff

lemma div_le_iff_le_mul_add_pred {m n k : ℕ} (n0 : 0 < n) : m / n ≤ k ↔ m ≤ n * k + (n - 1) :=
begin
rw [← lt_succ_iff, div_lt_iff_lt_mul _ _ n0, succ_mul, mul_comm],
Expand Down