mathlib3 documentation

data.rat.order

Order for Rational Numbers #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

Summary #

We define the order on ℚ, prove that ℚ is a discrete, linearly ordered field, and define functions such as abs and sqrt that depend on this order.

Notations #

Tags #

rat, rationals, field, ℚ, numerator, denominator, num, denom, order, ordering, sqrt, abs

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@[protected]
def rat.nonneg (r : ℚ) :
Prop

A rational number is called nonnegative if its numerator is nonnegative.

Equations
Instances for rat.nonneg
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@[simp]
theorem rat.mk_nonneg (a : ℤ) {b : ℤ} (h : 0 < b) :
(rat.mk a b).nonneg ↔ 0 ≤ a
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@[protected]
theorem rat.nonneg_add {a b : ℚ} :
a.nonneg → b.nonneg → (a + b).nonneg
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@[protected]
theorem rat.nonneg_mul {a b : ℚ} :
a.nonneg → b.nonneg → (a * b).nonneg
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@[protected]
theorem rat.nonneg_antisymm {a : ℚ} :
a.nonneg → (-a).nonneg → a = 0
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@[protected]
theorem rat.nonneg_total (a : ℚ) :
a.nonneg ∨ (-a).nonneg
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@[protected, instance]
def rat.decidable_nonneg (a : ℚ) :
decidable a.nonneg
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@[protected, irreducible]
def rat.le (a b : ℚ) :
Prop

Relation a ≤ b on ℚ defined as a ≤ b ↔ rat.nonneg (b - a). Use a ≤ b instead of rat.le a b.

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@[protected, instance]
def rat.has_le  :
has_le ℚ
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@[protected, instance]
def rat.decidable_le  :
decidable_rel has_le.le
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@[protected]
theorem rat.le_def {a b c d : ℤ} (b0 : 0 < b) (d0 : 0 < d) :
rat.mk a b ≤ rat.mk c d ↔ a * d ≤ c * b
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@[protected]
theorem rat.le_refl (a : ℚ) :
a ≤ a
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@[protected]
theorem rat.le_total (a b : ℚ) :
a ≤ b ∨ b ≤ a
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@[protected]
theorem rat.le_antisymm {a b : ℚ} (hab : a ≤ b) (hba : b ≤ a) :
a = b
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@[protected]
theorem rat.le_trans {a b c : ℚ} (hab : a ≤ b) (hbc : b ≤ c) :
a ≤ c
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@[protected, instance]
def rat.linear_order  :
linear_order ℚ
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@[protected, instance]
def rat.has_lt  :
has_lt ℚ
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@[protected, instance]
def rat.distrib_lattice  :
distrib_lattice ℚ
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@[protected, instance]
def rat.lattice  :
lattice ℚ
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@[protected, instance]
def rat.semilattice_inf  :
semilattice_inf ℚ
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@[protected, instance]
def rat.semilattice_sup  :
semilattice_sup ℚ
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@[protected, instance]
def rat.has_inf  :
has_inf ℚ
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@[protected, instance]
def rat.has_sup  :
has_sup ℚ
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@[protected, instance]
def rat.partial_order  :
partial_order ℚ
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@[protected, instance]
def rat.preorder  :
preorder ℚ
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@[protected]
theorem rat.le_def' {p q : ℚ} :
p ≤ q ↔ p.num * ↑(q.denom) ≤ q.num * ↑(p.denom)
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@[protected]
theorem rat.lt_def {p q : ℚ} :
p < q ↔ p.num * ↑(q.denom) < q.num * ↑(p.denom)
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theorem rat.nonneg_iff_zero_le {a : ℚ} :
a.nonneg ↔ 0 ≤ a
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theorem rat.num_nonneg_iff_zero_le {a : ℚ} :
0 ≤ a.num ↔ 0 ≤ a
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@[protected]
theorem rat.add_le_add_left {a b c : ℚ} :
c + a ≤ c + b ↔ a ≤ b
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@[protected]
theorem rat.mul_nonneg {a b : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
0 ≤ a * b
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@[protected, instance]
def rat.linear_ordered_field  :
linear_ordered_field ℚ
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@[protected, instance]
def rat.linear_ordered_comm_ring  :
linear_ordered_comm_ring ℚ
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@[protected, instance]
def rat.linear_ordered_ring  :
linear_ordered_ring ℚ
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@[protected, instance]
def rat.ordered_ring  :
ordered_ring ℚ
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@[protected, instance]
def rat.linear_ordered_semiring  :
linear_ordered_semiring ℚ
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@[protected, instance]
def rat.ordered_semiring  :
ordered_semiring ℚ
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@[protected, instance]
def rat.linear_ordered_add_comm_group  :
linear_ordered_add_comm_group ℚ
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@[protected, instance]
def rat.ordered_add_comm_group  :
ordered_add_comm_group ℚ
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@[protected, instance]
def rat.ordered_cancel_add_comm_monoid  :
ordered_cancel_add_comm_monoid ℚ
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@[protected, instance]
def rat.ordered_add_comm_monoid  :
ordered_add_comm_monoid ℚ
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theorem rat.num_pos_iff_pos {a : ℚ} :
0 < a.num ↔ 0 < a
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theorem rat.div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
↑a / ↑b < ↑c / ↑d ↔ a * d < c * b
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theorem rat.lt_one_iff_num_lt_denom {q : ℚ} :
q < 1 ↔ q.num < ↑(q.denom)
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theorem rat.abs_def (q : ℚ) :
|q| = rat.mk ↑(q.num.nat_abs) ↑(q.denom)