Documentation

Mathlib.Data.Nat.SuccPred

Successors and predecessors of naturals #

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

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@[implicit_reducible]
instance Nat.instSuccOrder :
SuccOrder ℕ
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@[implicit_reducible]
instance Nat.instSuccAddOrder :
SuccAddOrder ℕ
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@[implicit_reducible]
instance Nat.instPredOrder :
PredOrder ℕ
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@[implicit_reducible]
instance Nat.instPredSubOrder :
PredSubOrder ℕ
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@[simp]
theorem Nat.succ_eq_succ :
Order.succ = succ
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@[simp]
theorem Nat.pred_eq_pred :
Order.pred = pred
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theorem Nat.succ_iterate (a n : ℕ) :
succ^[n] a = a + n
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theorem Nat.pred_iterate (a n : ℕ) :
pred^[n] a = a - n
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@[simp]
theorem Nat.covBy_iff_add_one_eq {m n : ℕ} :
m ⋖ n ↔ m + 1 = n

A special case of Order.covBy_iff_add_one_eq for use by simp.

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theorem Nat.le_succ_iff_eq_or_le {m n : ℕ} :
m ≤ n.succ ↔ m = n.succ ∨ m ≤ n
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instance Nat.instIsSuccArchimedean :
IsSuccArchimedean ℕ
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instance Nat.instIsPredArchimedean :
IsPredArchimedean ℕ
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theorem Nat.forall_ne_zero_iff (P : ℕ → Prop) :
(∀ (i : ℕ), i ≠ 0 → P i) ↔ ∀ (i : ℕ), P (i + 1)
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@[simp]
theorem Fin.covBy_iff {n : ℕ} {a b : Fin n} :
a ⋖ b ↔ ↑a ⋖ ↑b
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@[deprecated Fin.covBy_iff "use Fin.covBy_iff.symm instead" (since := "2026-02-13")]
theorem Fin.coe_covBy_iff {n : ℕ} {a b : Fin n} :
↑a ⋖ ↑b ↔ a ⋖ b
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theorem CovBy.coe_fin {n : ℕ} {a b : Fin n} :
a ⋖ b → ↑a ⋖ ↑b

Alias of the forward direction of Fin.covBy_iff.

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@[simp]
theorem withBotSucc_zero :
WithBot.succ 0 = 1
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@[simp]
theorem withBotSucc_one :
WithBot.succ 1 = 2