mathlib documentation

data.rat.cast

Casts for Rational Numbers #

Summary #

We define the canonical injection from ℚ into an arbitrary division ring and prove various casting lemmas showing the well-behavedness of this injection.

Notations #

Tags #

rat, rationals, field, ℚ, numerator, denominator, num, denom, cast, coercion, casting

@[instance]
def rat.cast_coe {α : Type u_1} [division_ring α] :

Construct the canonical injection from into an arbitrary division ring. If the field has positive characteristic p, we define 1 / p = 1 / 0 = 0 for consistency with our division by zero convention.

Equations
theorem rat.cast_def {α : Type u_1} [division_ring α] (r : ) :
r = (r.num) / (r.denom)
@[simp]
theorem rat.cast_of_int {α : Type u_1} [division_ring α] (n : ) :
@[simp]
theorem rat.cast_coe_int {α : Type u_1} [division_ring α] (n : ) :
@[simp]
theorem rat.cast_coe_nat {α : Type u_1} [division_ring α] (n : ) :
@[simp]
theorem rat.cast_zero {α : Type u_1} [division_ring α] :
0 = 0
@[simp]
theorem rat.cast_one {α : Type u_1} [division_ring α] :
1 = 1
theorem rat.cast_commute {α : Type u_1} [division_ring α] (r : ) (a : α) :
theorem rat.cast_comm {α : Type u_1} [division_ring α] (r : ) (a : α) :
(r) * a = a * r
theorem rat.commute_cast {α : Type u_1} [division_ring α] (a : α) (r : ) :
theorem rat.cast_mk_of_ne_zero {α : Type u_1} [division_ring α] (a b : ) (b0 : b 0) :
(a /. b) = a / b
theorem rat.cast_add_of_ne_zero {α : Type u_1} [division_ring α] {m n : } :
(m.denom) 0(n.denom) 0(m + n) = m + n
@[simp]
theorem rat.cast_neg {α : Type u_1} [division_ring α] (n : ) :
theorem rat.cast_sub_of_ne_zero {α : Type u_1} [division_ring α] {m n : } (m0 : (m.denom) 0) (n0 : (n.denom) 0) :
(m - n) = m - n
theorem rat.cast_mul_of_ne_zero {α : Type u_1} [division_ring α] {m n : } :
(m.denom) 0(n.denom) 0m * n = (m) * n
@[simp]
theorem rat.cast_inv_nat {α : Type u_1} [division_ring α] (n : ) :
@[simp]
theorem rat.cast_inv_int {α : Type u_1} [division_ring α] (n : ) :
theorem rat.cast_inv_of_ne_zero {α : Type u_1} [division_ring α] {n : } :
(n.num) 0(n.denom) 0n⁻¹ = (n)⁻¹
theorem rat.cast_div_of_ne_zero {α : Type u_1} [division_ring α] {m n : } (md : (m.denom) 0) (nn : (n.num) 0) (nd : (n.denom) 0) :
(m / n) = m / n
@[simp]
theorem rat.cast_inj {α : Type u_1} [division_ring α] [char_zero α] {m n : } :
m = n m = n
@[simp]
theorem rat.cast_eq_zero {α : Type u_1} [division_ring α] [char_zero α] {n : } :
n = 0 n = 0
theorem rat.cast_ne_zero {α : Type u_1} [division_ring α] [char_zero α] {n : } :
n 0 n 0
@[simp]
theorem rat.cast_add {α : Type u_1} [division_ring α] [char_zero α] (m n : ) :
(m + n) = m + n
@[simp]
theorem rat.cast_sub {α : Type u_1} [division_ring α] [char_zero α] (m n : ) :
(m - n) = m - n
@[simp]
theorem rat.cast_mul {α : Type u_1} [division_ring α] [char_zero α] (m n : ) :
m * n = (m) * n
@[simp]
theorem rat.cast_bit0 {α : Type u_1} [division_ring α] [char_zero α] (n : ) :
@[simp]
theorem rat.cast_bit1 {α : Type u_1} [division_ring α] [char_zero α] (n : ) :
def rat.cast_hom (α : Type u_1) [division_ring α] [char_zero α] :

Coercion ℚ → α as a ring_hom.

Equations
@[simp]
theorem rat.coe_cast_hom {α : Type u_1} [division_ring α] [char_zero α] :
@[simp]
theorem rat.cast_inv {α : Type u_1} [division_ring α] [char_zero α] (n : ) :
@[simp]
theorem rat.cast_div {α : Type u_1} [division_ring α] [char_zero α] (m n : ) :
(m / n) = m / n
theorem rat.cast_mk {α : Type u_1} [division_ring α] [char_zero α] (a b : ) :
(a /. b) = a / b
@[simp]
theorem rat.cast_pow {α : Type u_1} [division_ring α] [char_zero α] (q : ) (k : ) :
(q ^ k) = q ^ k
@[simp]
theorem rat.cast_nonneg {α : Type u_1} [linear_ordered_field α] {n : } :
0 n 0 n
@[simp]
theorem rat.cast_le {α : Type u_1} [linear_ordered_field α] {m n : } :
m n m n
@[simp]
theorem rat.cast_lt {α : Type u_1} [linear_ordered_field α] {m n : } :
m < n m < n
@[simp]
theorem rat.cast_nonpos {α : Type u_1} [linear_ordered_field α] {n : } :
n 0 n 0
@[simp]
theorem rat.cast_pos {α : Type u_1} [linear_ordered_field α] {n : } :
0 < n 0 < n
@[simp]
theorem rat.cast_lt_zero {α : Type u_1} [linear_ordered_field α] {n : } :
n < 0 n < 0
@[simp]
theorem rat.cast_id (n : ) :
n = n
@[simp]
theorem rat.cast_min {α : Type u_1} [linear_ordered_field α] {a b : } :
(min a b) = min a b
@[simp]
theorem rat.cast_max {α : Type u_1} [linear_ordered_field α] {a b : } :
(max a b) = max a b
@[simp]
theorem rat.cast_abs {α : Type u_1} [linear_ordered_field α] {q : } :
theorem ring_hom.eq_rat_cast {k : Type u_1} [division_ring k] (f : →+* k) (r : ) :
f r = r
theorem ring_hom.map_rat_cast {k : Type u_1} {k' : Type u_2} [division_ring k] [char_zero k] [division_ring k'] (f : k →+* k') (r : ) :
theorem ring_hom.ext_rat {R : Type u_1} [semiring R] (f g : →+* R) :
f = g
@[instance]
def rat.subsingleton_ring_hom {R : Type u_1} [semiring R] :
@[ext]

If f and g agree on the integers then they are equal φ.

See note [partially-applied ext lemmas] for why comp is used here.

theorem monoid_with_zero_hom.ext_rat_on_pnat {M : Type u_1} [group_with_zero M] {f g : monoid_with_zero_hom M} (same_on_neg_one : f (-1) = g (-1)) (same_on_pnat : ∀ (n : ), 0 < nf n = g n) :
f = g

Positive integer values of a morphism φ and its value on -1 completely determine φ.