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[Merged by Bors] - chore(algebra/order/monoid): move lemmas #16176

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[Merged by Bors] - chore(algebra/order/monoid): move lemmas #16176

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urkud Aug 20, 2022
0d251e5
Fix lint
urkud Aug 20, 2022
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8 changes: 8 additions & 0 deletions src/algebra/order/monoid.lean
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Expand Up @@ -1109,6 +1109,10 @@ instance [canonically_linear_ordered_add_monoid α] :
canonically_linear_ordered_add_monoid (with_top α) :=
{ ..with_top.canonically_ordered_add_monoid, ..with_top.linear_order }

@[simp, norm_cast] lemma coe_nat [add_monoid_with_one α] (n : ℕ) : ((n : α) : with_top α) = n := rfl
@[simp] lemma nat_ne_top [add_monoid_with_one α] (n : ℕ) : (n : with_top α) ≠ ⊤ := coe_ne_top
@[simp] lemma top_ne_nat [add_monoid_with_one α] (n : ℕ) : (⊤ : with_top α) ≠ n := top_ne_coe

/-- Coercion from `α` to `with_top α` as an `add_monoid_hom`. -/
def coe_add_hom [add_monoid α] : α →+ with_top α :=
⟨coe, rfl, λ _ _, rfl⟩
Expand Down Expand Up @@ -1175,6 +1179,10 @@ with_top.coe_eq_one
@[to_additive] protected lemma map_one {β} [has_one α] (f : α → β) :
(1 : with_bot α).map f = (f 1 : with_bot β) := rfl

@[norm_cast] lemma coe_nat [add_monoid_with_one α] (n : ℕ) : ((n : α) : with_bot α) = n := rfl
@[simp] lemma nat_ne_bot [add_monoid_with_one α] (n : ℕ) : (n : with_bot α) ≠ ⊥ := coe_ne_bot
@[simp] lemma bot_ne_nat [add_monoid_with_one α] (n : ℕ) : (⊥ : with_bot α) ≠ n := bot_ne_coe

section has_add
variables [has_add α] {a b c d : with_bot α} {x y : α}

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10 changes: 0 additions & 10 deletions src/data/nat/cast.lean
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Expand Up @@ -230,16 +230,6 @@ variables {α : Type*}

variables [add_monoid_with_one α]

@[simp, norm_cast] lemma coe_nat : ∀ (n : ℕ), ((n : α) : with_top α) = n
| 0 := rfl
| (n+1) := by { push_cast, rw [coe_nat n] }

@[simp] lemma nat_ne_top (n : nat) : (n : with_top α) ≠ ⊤ :=
by { rw [←coe_nat n], apply coe_ne_top }

@[simp] lemma top_ne_nat (n : nat) : (⊤ : with_top α) ≠ n :=
by { rw [←coe_nat n], apply top_ne_coe }

lemma add_one_le_of_lt {i n : with_top ℕ} (h : i < n) : i + 1 ≤ n :=
begin
cases n, { exact le_top },
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