Documentation

Mathlib.Data.Nat.Cast.Order.Basic

Cast of natural numbers: lemmas about order #

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theorem Nat.mono_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] :
Monotone Nat.cast
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@[deprecated Nat.mono_cast]
theorem Nat.cast_le_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] {a : ℕ} {b : ℕ} (h : a ≤ b) :
↑a ≤ ↑b
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theorem GCongr.natCast_le_natCast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] {a : ℕ} {b : ℕ} (h : a ≤ b) :
↑a ≤ ↑b
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@[simp]
theorem Nat.cast_nonneg' {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] (n : ℕ) :
0 ≤ ↑n

See also Nat.cast_nonneg, specialised for an OrderedSemiring.

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@[simp]
theorem Nat.ofNat_nonneg' {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] (n : ℕ) [n.AtLeastTwo] :
0 ≤ OfNat.ofNat n

See also Nat.ofNat_nonneg, specialised for an OrderedSemiring.

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theorem Nat.cast_add_one_pos {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [NeZero 1] (n : ℕ) :
0 < ↑n + 1
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@[simp]
theorem Nat.cast_pos' {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [NeZero 1] {n : ℕ} :
0 < ↑n ↔ 0 < n

See also Nat.cast_pos, specialised for an OrderedSemiring.

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theorem Nat.strictMono_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] :
StrictMono Nat.cast
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theorem GCongr.natCast_lt_natCast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {a : ℕ} {b : ℕ} (h : a < b) :
↑a < ↑b
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@[simp]
theorem Nat.castOrderEmbedding_apply {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] :
⇑Nat.castOrderEmbedding = Nat.cast
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def Nat.castOrderEmbedding {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] :
ℕ ↪o α

Nat.cast : ℕ → α as an OrderEmbedding

Equations
Instances For
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    @[simp]
    theorem Nat.cast_le {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} :
    ↑m ≤ ↑n ↔ m ≤ n
    source
    @[simp]
    theorem Nat.cast_lt {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} :
    ↑m < ↑n ↔ m < n
    source
    @[simp]
    theorem Nat.one_lt_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} :
    1 < ↑n ↔ 1 < n
    source
    @[simp]
    theorem Nat.one_le_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} :
    1 ≤ ↑n ↔ 1 ≤ n
    source
    @[simp]
    theorem Nat.cast_lt_one {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} :
    ↑n < 1 ↔ n = 0
    source
    @[simp]
    theorem Nat.cast_le_one {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} :
    ↑n ≤ 1 ↔ n ≤ 1
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    @[simp]
    theorem Nat.ofNat_le_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [m.AtLeastTwo] :
    OfNat.ofNat m ≤ ↑n ↔ OfNat.ofNat m ≤ n
    source
    @[simp]
    theorem Nat.ofNat_lt_cast {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [m.AtLeastTwo] :
    OfNat.ofNat m < ↑n ↔ OfNat.ofNat m < n
    source
    @[simp]
    theorem Nat.cast_le_ofNat {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [n.AtLeastTwo] :
    ↑m ≤ OfNat.ofNat n ↔ m ≤ OfNat.ofNat n
    source
    @[simp]
    theorem Nat.cast_lt_ofNat {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [n.AtLeastTwo] :
    ↑m < OfNat.ofNat n ↔ m < OfNat.ofNat n
    source
    @[simp]
    theorem Nat.one_lt_ofNat {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} [n.AtLeastTwo] :
    1 < OfNat.ofNat n
    source
    @[simp]
    theorem Nat.one_le_ofNat {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} [n.AtLeastTwo] :
    1 ≤ OfNat.ofNat n
    source
    @[simp]
    theorem Nat.not_ofNat_le_one {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} [n.AtLeastTwo] :
    ¬OfNat.ofNat n ≤ 1
    source
    @[simp]
    theorem Nat.not_ofNat_lt_one {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {n : ℕ} [n.AtLeastTwo] :
    ¬OfNat.ofNat n < 1
    source
    theorem Nat.ofNat_le {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [n.AtLeastTwo] [m.AtLeastTwo] :
    OfNat.ofNat m ≤ OfNat.ofNat n ↔ OfNat.ofNat m ≤ OfNat.ofNat n
    source
    theorem Nat.ofNat_lt {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 ≤ x2] [ZeroLEOneClass α] [CharZero α] {m : ℕ} {n : ℕ} [n.AtLeastTwo] [m.AtLeastTwo] :
    OfNat.ofNat m < OfNat.ofNat n ↔ OfNat.ofNat m < OfNat.ofNat n
    source
    instance instNontrivialOfCharZero {α : Type u_1} [AddMonoidWithOne α] [CharZero α] :
    Nontrivial α
    Equations
    • ⋯ = ⋯
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    theorem NeZero.nat_of_injective {R : Type u_2} {S : Type u_3} {F : Type u_4} [NonAssocSemiring R] [NonAssocSemiring S] [FunLike F R S] {n : ℕ} [NeZero ↑n] [RingHomClass F R S] {f : F} (hf : Function.Injective ⇑f) :
    NeZero ↑n