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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instance Nat.instSuccOrderNatToPreorderToPartialOrderStrictOrderedSemiring : SuccOrder ℕ

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• Nat.instSuccOrderNatToPreorderToPartialOrderStrictOrderedSemiring = SuccOrder.ofSuccLeIff Nat.succ @Nat.instSuccOrderNatToPreorderToPartialOrderStrictOrderedSemiring.proof_1

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instance Nat.instPredOrderNatToPreorderToPartialOrderStrictOrderedSemiring : PredOrder ℕ

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• One or more equations did not get rendered due to their size.

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@[simp]

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

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@[simp]

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

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

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

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instance Nat.instIsSuccArchimedeanNatToPreorderToPartialOrderStrictOrderedSemiringInstSuccOrderNatToPreorderToPartialOrderStrictOrderedSemiring : IsSuccArchimedean ℕ

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• One or more equations did not get rendered due to their size.

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instance Nat.instIsPredArchimedeanNatToPreorderToPartialOrderStrictOrderedSemiringInstPredOrderNatToPreorderToPartialOrderStrictOrderedSemiring : IsPredArchimedean ℕ

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• One or more equations did not get rendered due to their size.

Covering relation

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

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@[simp]

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

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

Alias of the reverse direction of Fin.coe_covby_iff.