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[Merged by Bors] - feat: a few lemmas on WithTop ℕ∞ #17164

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[Merged by Bors] - feat: a few lemmas on WithTop ℕ∞ #17164

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sgouezel Sep 26, 2024
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22 changes: 22 additions & 0 deletions Mathlib/Data/ENat/Basic.lean
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Expand Up @@ -259,4 +259,26 @@ theorem nat_induction {P : ℕ∞ → Prop} (a : ℕ∞) (h0 : P 0) (hsuc : ∀
· exact htop A
· exact A _

lemma add_one_nat_le_withTop_of_lt {m : ℕ} {n : WithTop ℕ∞} (h : m < n) : (m + 1 : ℕ) ≤ n := by
match n with
| ⊤ => exact le_top
| (⊤ : ℕ∞) => exact WithTop.coe_le_coe.2 (OrderTop.le_top _)
| (n : ℕ) => simpa only [Nat.cast_le, ge_iff_le, Nat.cast_lt] using h

@[simp] lemma coe_top_add_one : ((⊤ : ℕ∞) : WithTop ℕ∞) + 1 = (⊤ : ℕ∞) := rfl

@[simp] lemma add_one_eq_coe_top_iff (n : WithTop ℕ∞) :
n + 1 = (⊤ : ℕ∞) ↔ n = (⊤ : ℕ∞) := by
match n with
| ⊤ => exact Iff.rfl
| (⊤ : ℕ∞) => exact Iff.rfl
| (n : ℕ) => norm_cast; simp only [coe_ne_top, iff_false, ne_eq]

@[simp] lemma nat_ne_coe_top (n : ℕ) : (n : WithTop ℕ∞) ≠ (⊤ : ℕ∞) := ne_of_beq_false rfl

lemma one_le_iff_ne_zero_withTop {n : WithTop ℕ∞} :
1 ≤ n ↔ n ≠ 0 :=
⟨fun h ↦ (zero_lt_one.trans_le h).ne',
fun h ↦ add_one_nat_le_withTop_of_lt (pos_iff_ne_zero.mpr h)⟩

end ENat
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