File: Mathlib/Data/Nat/Cast/Order/Basic.lean

full_name: Nat.mono_cast
code: @[mono]
theorem mono_cast : Monotone (Nat.cast : ℕ → α) 
goal: Monotone Nat.cast

full_name: Nat.cast_le_cast
code: @[deprecated mono_cast (since := "2024-02-10")]
theorem cast_le_cast {a b : ℕ} (h : a ≤ b) : (a : α) ≤ b 
goal: ↑a ≤ ↑b

full_name: GCongr.natCast_le_natCast
code: @[gcongr]
theorem _root_.GCongr.natCast_le_natCast {a b : ℕ} (h : a ≤ b) : (a : α) ≤ b 
goal: ↑a ≤ ↑b

full_name: Nat.cast_nonneg'
code: /-- See also `Nat.cast_nonneg`, specialised for an `OrderedSemiring`. -/
@[simp low]
theorem cast_nonneg' (n : ℕ) : 0 ≤ (n : α) 
goal: 0 ≤ ↑n

full_name: Nat.ofNat_nonneg'
code: /-- See also `Nat.ofNat_nonneg`, specialised for an `OrderedSemiring`. -/
-- See note [no_index around OfNat.ofNat]
@[simp low]
theorem ofNat_nonneg' (n : ℕ) [n.AtLeastTwo] : 0 ≤ (no_index (OfNat.ofNat n : α)) 
goal: 0 ≤ OfNat.ofNat n

full_name: Nat.cast_add_one_pos
code: theorem cast_add_one_pos (n : ℕ) : 0 < (n : α) + 1 
goal: 0 < ↑n + 1

full_name: Nat.cast_pos'
code: /-- See also `Nat.cast_pos`, specialised for an `OrderedSemiring`. -/
@[simp low]
theorem cast_pos' {n : ℕ} : (0 : α) < n ↔ 0 < n 
goal: 0 < ↑n ↔ 0 < n

full_name: Nat.strictMono_cast
code: theorem strictMono_cast : StrictMono (Nat.cast : ℕ → α) 
goal: StrictMono Nat.cast

full_name: Nat.castOrderEmbedding
code: /-- `Nat.cast : ℕ → α` as an `OrderEmbedding` -/
@[simps! (config := .asFn)]
def castOrderEmbedding : ℕ ↪o α 
goal: ℕ ↪o α

full_name: Nat.castOrderEmbedding_apply
code: /-- `Nat.cast : ℕ → α` as an `OrderEmbedding` -/
@[simps! (config := .asFn)]
def castOrderEmbedding : ℕ ↪o α 
goal: ⇑Nat.castOrderEmbedding = Nat.cast

full_name: Nat.cast_le
code: @[simp, norm_cast]
theorem cast_le : (m : α) ≤ n ↔ m ≤ n 
goal: ↑m ≤ ↑n ↔ m ≤ n

full_name: Nat.cast_lt
code: @[simp, norm_cast, mono]
theorem cast_lt : (m : α) < n ↔ m < n 
goal: ↑m < ↑n ↔ m < n

full_name: Nat.one_lt_cast
code: @[simp, norm_cast]
theorem one_lt_cast : 1 < (n : α) ↔ 1 < n 
goal: 1 < ↑n ↔ 1 < n

full_name: Nat.one_le_cast
code: @[simp, norm_cast]
theorem one_le_cast : 1 ≤ (n : α) ↔ 1 ≤ n 
goal: 1 ≤ ↑n ↔ 1 ≤ n

full_name: Nat.cast_lt_one
code: @[simp, norm_cast]
theorem cast_lt_one : (n : α) < 1 ↔ n = 0 
goal: ↑n < 1 ↔ n = 0

full_name: Nat.cast_le_one
code: @[simp, norm_cast]
theorem cast_le_one : (n : α) ≤ 1 ↔ n ≤ 1 
goal: ↑n ≤ 1 ↔ n ≤ 1

full_name: Nat.ofNat_le
code: theorem ofNat_le :
    (no_index (OfNat.ofNat m : α)) ≤ (no_index (OfNat.ofNat n)) ↔
      (OfNat.ofNat m : ℕ) ≤ OfNat.ofNat n 
goal: OfNat.ofNat m ≤ OfNat.ofNat n ↔ OfNat.ofNat m ≤ OfNat.ofNat n

full_name: Nat.ofNat_lt
code: theorem ofNat_lt :
    (no_index (OfNat.ofNat m : α)) < (no_index (OfNat.ofNat n)) ↔
      (OfNat.ofNat m : ℕ) < OfNat.ofNat n 
goal: OfNat.ofNat m < OfNat.ofNat n ↔ OfNat.ofNat m < OfNat.ofNat n

full_name: Nat.ofNat_le_cast
code: @[simp]
theorem ofNat_le_cast : (no_index (OfNat.ofNat m : α)) ≤ n ↔ (OfNat.ofNat m : ℕ) ≤ n 
goal: OfNat.ofNat m ≤ ↑n ↔ OfNat.ofNat m ≤ n

full_name: Nat.ofNat_lt_cast
code: @[simp]
theorem ofNat_lt_cast : (no_index (OfNat.ofNat m : α)) < n ↔ (OfNat.ofNat m : ℕ) < n 
goal: OfNat.ofNat m < ↑n ↔ OfNat.ofNat m < n

full_name: Nat.cast_le_ofNat
code: @[simp]
theorem cast_le_ofNat : (m : α) ≤ (no_index (OfNat.ofNat n)) ↔ m ≤ OfNat.ofNat n 
goal: ↑m ≤ OfNat.ofNat n ↔ m ≤ OfNat.ofNat n

full_name: Nat.cast_lt_ofNat
code: @[simp]
theorem cast_lt_ofNat : (m : α) < (no_index (OfNat.ofNat n)) ↔ m < OfNat.ofNat n 
goal: ↑m < OfNat.ofNat n ↔ m < OfNat.ofNat n

full_name: Nat.one_lt_ofNat
code: @[simp]
theorem one_lt_ofNat : 1 < (no_index (OfNat.ofNat n : α)) 
goal: 1 < OfNat.ofNat n

full_name: Nat.one_le_ofNat
code: @[simp]
theorem one_le_ofNat : 1 ≤ (no_index (OfNat.ofNat n : α)) 
goal: 1 ≤ OfNat.ofNat n

full_name: Nat.not_ofNat_le_one
code: @[simp]
theorem not_ofNat_le_one : ¬(no_index (OfNat.ofNat n : α)) ≤ 1 
goal: ¬OfNat.ofNat n ≤ 1

full_name: Nat.not_ofNat_lt_one
code: @[simp]
theorem not_ofNat_lt_one : ¬(no_index (OfNat.ofNat n : α)) < 1 
goal: ¬OfNat.ofNat n < 1

full_name: instNontrivialOfCharZero
code: instance [AddMonoidWithOne α] [CharZero α] : Nontrivial α 
goal: Nontrivial α

full_name: NeZero.nat_of_injective
code: theorem NeZero.nat_of_injective {n : ℕ} [h : NeZero (n : R)] [RingHomClass F R S] {f : F}
    (hf : Function.Injective f) : NeZero (n : S) 
goal: NeZero ↑n