Cast of integers (additional theorems)

This file proves additional properties about the canonical homomorphism from the integers into an additive group with a one (Int.cast), particularly results involving algebraic homomorphisms or the order structure on ℤ which were not available in the import dependencies of Data.Int.Cast.Basic.

Main declarations

• castAddHom: cast bundled as an AddMonoidHom.
• castRingHom: cast bundled as a RingHom.

def Int.ofNatHom : ℕ →+* ℤ

Coercion ℕ → ℤ as a RingHom.

theorem Int.coe_nat_pos {n : ℕ} : 0 < ↑n ↔ 0 < n

theorem Int.coe_nat_succ_pos (n : ℕ) : 0 < ↑(Nat.succ n)

theorem Int.toNat_lt' {a : ℤ} {b : ℕ} (hb : b ≠ 0) : Int.toNat a < b ↔ a < ↑b

theorem Int.natMod_lt {a : ℤ} {b : ℕ} (hb : b ≠ 0) : Int.natMod a ↑b < b

@[simp]

theorem Int.cast_ite {α : Type u_3} [AddGroupWithOne α] (P : Prop) [Decidable P] (m : ℤ) (n : ℤ) : ↑(if P then m else n) = if P then ↑m else ↑n

def Int.castAddHom (α : Type u_5) [AddGroupWithOne α] : ℤ →+ α

coe : ℤ → α as an AddMonoidHom.

@[simp]

theorem Int.coe_castAddHom {α : Type u_3} [AddGroupWithOne α] : ↑(Int.castAddHom α) = fun x => ↑x

def Int.castRingHom (α : Type u_5) [NonAssocRing α] : ℤ →+* α

coe : ℤ → α as a RingHom.

@[simp]

theorem Int.coe_castRingHom {α : Type u_3} [NonAssocRing α] : ↑(Int.castRingHom α) = fun x => ↑x

theorem Int.cast_commute {α : Type u_3} [NonAssocRing α] (m : ℤ) (x : α) : Commute (↑m) x

theorem Int.cast_comm {α : Type u_3} [NonAssocRing α] (m : ℤ) (x : α) : ↑m * x = x * ↑m

theorem Int.commute_cast {α : Type u_3} [NonAssocRing α] (x : α) (m : ℤ) : Commute x ↑m

theorem Int.cast_mono {α : Type u_3} [OrderedRing α] : Monotone fun x => ↑x

@[simp]

theorem Int.cast_nonneg {α : Type u_3} [OrderedRing α] [Nontrivial α] {n : ℤ} : 0 ≤ ↑n ↔ 0 ≤ n

@[simp]

theorem Int.cast_le {α : Type u_3} [OrderedRing α] [Nontrivial α] {m : ℤ} {n : ℤ} : ↑m ≤ ↑n ↔ m ≤ n

theorem Int.cast_strictMono {α : Type u_3} [OrderedRing α] [Nontrivial α] : StrictMono fun x => ↑x

@[simp]

theorem Int.cast_lt {α : Type u_3} [OrderedRing α] [Nontrivial α] {m : ℤ} {n : ℤ} : ↑m < ↑n ↔ m < n

@[simp]

theorem Int.cast_nonpos {α : Type u_3} [OrderedRing α] [Nontrivial α] {n : ℤ} : ↑n ≤ 0 ↔ n ≤ 0

@[simp]

theorem Int.cast_pos {α : Type u_3} [OrderedRing α] [Nontrivial α] {n : ℤ} : 0 < ↑n ↔ 0 < n

@[simp]

theorem Int.cast_lt_zero {α : Type u_3} [OrderedRing α] [Nontrivial α] {n : ℤ} : ↑n < 0 ↔ n < 0

@[simp]

theorem Int.cast_min {α : Type u_3} [LinearOrderedRing α] {a : ℤ} {b : ℤ} : ↑(min a b) = min ↑a ↑b

@[simp]

theorem Int.cast_max {α : Type u_3} [LinearOrderedRing α] {a : ℤ} {b : ℤ} : ↑(max a b) = max ↑a ↑b

@[simp]

theorem Int.cast_abs {α : Type u_3} [LinearOrderedRing α] {a : ℤ} : ↑|a| = |↑a|

theorem Int.cast_one_le_of_pos {α : Type u_3} [LinearOrderedRing α] {a : ℤ} (h : 0 < a) : 1 ≤ ↑a

theorem Int.cast_le_neg_one_of_neg {α : Type u_3} [LinearOrderedRing α] {a : ℤ} (h : a < 0) : ↑a ≤ -1

theorem Int.cast_le_neg_one_or_one_le_cast_of_ne_zero (α : Type u_3) [LinearOrderedRing α] {n : ℤ} (hn : n ≠ 0) : ↑n ≤ -1 ∨ 1 ≤ ↑n

theorem Int.nneg_mul_add_sq_of_abs_le_one {α : Type u_3} [LinearOrderedRing α] (n : ℤ) {x : α} (hx : |x| ≤ 1) : 0 ≤ ↑n * x + ↑n * ↑n

theorem Int.cast_natAbs {α : Type u_3} [LinearOrderedRing α] (n : ℤ) : ↑(Int.natAbs n) = ↑|n|

theorem Int.coe_int_dvd {α : Type u_3} [CommRing α] (m : ℤ) (n : ℤ) (h : m ∣ n) : ↑m ∣ ↑n

theorem AddMonoidHom.ext_int {A : Type u_5} [AddMonoid A] {f : ℤ →+ A} {g : ℤ →+ A} (h1 : ↑f 1 = ↑g 1) : f = g

Two additive monoid homomorphisms f, g from ℤ to an additive monoid are equal if f 1 = g 1.

theorem AddMonoidHom.eq_int_castAddHom {A : Type u_5} [AddGroupWithOne A] (f : ℤ →+ A) (h1 : ↑f 1 = 1) : f = Int.castAddHom A

theorem eq_intCast' {F : Type u_1} {α : Type u_3} [AddGroupWithOne α] [AddMonoidHomClass F ℤ α] (f : F) (h₁ : ↑f 1 = 1) (n : ℤ) : ↑f n = ↑n

@[simp]

theorem Int.castAddHom_int : Int.castAddHom ℤ = AddMonoidHom.id ℤ

theorem MonoidHom.ext_mint {M : Type u_5} [Monoid M] {f : Multiplicative ℤ →* M} {g : Multiplicative ℤ →* M} (h1 : ↑f (↑Multiplicative.ofAdd 1) = ↑g (↑Multiplicative.ofAdd 1)) : f = g

theorem MonoidHom.ext_int {M : Type u_5} [Monoid M] {f : ℤ →* M} {g : ℤ →* M} (h_neg_one : ↑f (-1) = ↑g (-1)) (h_nat : MonoidHom.comp f ↑Int.ofNatHom = MonoidHom.comp g ↑Int.ofNatHom) : f = g

If two MonoidHoms agree on -1 and the naturals then they are equal.

theorem MonoidWithZeroHom.ext_int {M : Type u_5} [MonoidWithZero M] {f : ℤ →*₀ M} {g : ℤ →*₀ M} (h_neg_one : ↑f (-1) = ↑g (-1)) (h_nat : MonoidWithZeroHom.comp f (RingHom.toMonoidWithZeroHom Int.ofNatHom) = MonoidWithZeroHom.comp g (RingHom.toMonoidWithZeroHom Int.ofNatHom)) : f = g

If two MonoidWithZeroHoms agree on -1 and the naturals then they are equal.

theorem ext_int' {F : Type u_1} {α : Type u_3} [MonoidWithZero α] [MonoidWithZeroHomClass F ℤ α] {f : F} {g : F} (h_neg_one : ↑f (-1) = ↑g (-1)) (h_pos : ∀ (n : ℕ), 0 < n → ↑f ↑n = ↑g ↑n) : f = g

If two MonoidWithZeroHoms agree on -1 and the positive naturals then they are equal.

@[simp]

theorem eq_intCast {F : Type u_1} {α : Type u_3} [NonAssocRing α] [RingHomClass F ℤ α] (f : F) (n : ℤ) : ↑f n = ↑n

@[simp]

theorem map_intCast {F : Type u_1} {α : Type u_3} {β : Type u_4} [NonAssocRing α] [NonAssocRing β] [RingHomClass F α β] (f : F) (n : ℤ) : ↑f ↑n = ↑n

theorem RingHom.eq_intCast' {α : Type u_3} [NonAssocRing α] (f : ℤ →+* α) : f = Int.castRingHom α

theorem RingHom.ext_int {R : Type u_5} [NonAssocSemiring R] (f : ℤ →+* R) (g : ℤ →+* R) : f = g

instance RingHom.Int.subsingleton_ringHom {R : Type u_5} [NonAssocSemiring R] : Subsingleton (ℤ →+* R)

@[simp]

theorem Int.castRingHom_int : Int.castRingHom ℤ = RingHom.id ℤ

instance Pi.intCast {ι : Type u_2} {π : ι → Type u_5} [(i : ι) → IntCast (π i)] : IntCast ((i : ι) → π i)

theorem Pi.int_apply {ι : Type u_2} {π : ι → Type u_5} [(i : ι) → IntCast (π i)] (n : ℤ) (i : ι) : ↑n i = ↑n

@[simp]

theorem Pi.coe_int {ι : Type u_2} {π : ι → Type u_5} [(i : ι) → IntCast (π i)] (n : ℤ) : ↑n = fun x => ↑n

theorem Sum.elim_intCast_intCast {α : Type u_5} {β : Type u_6} {γ : Type u_7} [IntCast γ] (n : ℤ) : Sum.elim ↑n ↑n = ↑n

Order dual

instance instIntCastOrderDual {α : Type u_3} [h : IntCast α] : IntCast αᵒᵈ

instance instAddGroupWithOneOrderDual {α : Type u_3} [h : AddGroupWithOne α] : AddGroupWithOne αᵒᵈ

instance instAddCommGroupWithOneOrderDual {α : Type u_3} [h : AddCommGroupWithOne α] : AddCommGroupWithOne αᵒᵈ

@[simp]

theorem toDual_intCast {α : Type u_3} [IntCast α] (n : ℤ) : ↑OrderDual.toDual ↑n = ↑n

@[simp]

theorem ofDual_intCast {α : Type u_3} [IntCast α] (n : ℤ) : ↑(↑OrderDual.ofDual n) = ↑n

Lexicographic order

instance instIntCastLex {α : Type u_3} [h : IntCast α] : IntCast (Lex α)

instance instAddGroupWithOneLex {α : Type u_3} [h : AddGroupWithOne α] : AddGroupWithOne (Lex α)

instance instAddCommGroupWithOneLex {α : Type u_3} [h : AddCommGroupWithOne α] : AddCommGroupWithOne (Lex α)

@[simp]

theorem toLex_intCast {α : Type u_3} [IntCast α] (n : ℤ) : ↑toLex ↑n = ↑n

@[simp]

theorem ofLex_intCast {α : Type u_3} [IntCast α] (n : ℤ) : ↑(↑ofLex n) = ↑n