Successors and predecessors of naturals

In this file, we show that ℕ is both an archimedean succOrder and an archimedean predOrder.

@[reducible, inline]

instance Nat.instSuccOrder : SuccOrder ℕ

Equations

• Nat.instSuccOrder = SuccOrder.ofSuccLeIff Nat.succ ⋯

instance Nat.instSuccAddOrder : SuccAddOrder ℕ

Equations

• Nat.instSuccAddOrder = SuccAddOrder.mk Nat.instSuccAddOrder.proof_1

@[reducible, inline]

instance Nat.instPredOrder : PredOrder ℕ

Equations

• Nat.instPredOrder = { pred := Nat.pred, pred_le := Nat.pred_le, min_of_le_pred := @Nat.instPredOrder.proof_1, le_pred_of_lt := @Nat.instPredOrder.proof_2 }

instance Nat.instPredSubOrder : PredSubOrder ℕ

Equations

• Nat.instPredSubOrder = PredSubOrder.mk Nat.instPredSubOrder.proof_1

@[simp]

theorem Nat.succ_eq_succ : Order.succ = Nat.succ

@[simp]

theorem Nat.pred_eq_pred : Order.pred = Nat.pred

theorem Nat.succ_iterate (a n : ℕ) : Nat.succ^[n] a = a + n

theorem Nat.pred_iterate (a n : ℕ) : Nat.pred^[n] a = a - n

theorem Nat.le_succ_iff_eq_or_le {m n : ℕ} : m ≤ n.succ ↔ m = n.succ ∨ m ≤ n

instance Nat.instIsSuccArchimedean : IsSuccArchimedean ℕ

Equations

• Nat.instIsSuccArchimedean = ⋯

instance Nat.instIsPredArchimedean : IsPredArchimedean ℕ

Equations

• Nat.instIsPredArchimedean = ⋯

theorem Nat.forall_ne_zero_iff (P : ℕ → Prop) : (∀ (i : ℕ), i ≠ 0 → P i) ↔ ∀ (i : ℕ), P (i + 1)

Covering relation

@[deprecated Order.covBy_iff_add_one_eq]

theorem Nat.covBy_iff_succ_eq {m n : ℕ} : m ⋖ n ↔ m + 1 = n

@[simp]

theorem Fin.coe_covBy_iff {n : ℕ} {a b : Fin n} : ↑a ⋖ ↑b ↔ a ⋖ b

theorem CovBy.coe_fin {n : ℕ} {a b : Fin n} : a ⋖ b → ↑a ⋖ ↑b

Alias of the reverse direction of Fin.coe_covBy_iff.