Archimedean groups and fields. #

This file defines the archimedean property for ordered groups and proves several results connected to this notion. Being archimedean means that for all elements x and y>0 there exists a natural number n such that x ≤ n • y.

Main definitions #

• Archimedean is a typeclass for an ordered additive commutative monoid to have the archimedean property.
• Archimedean.floorRing defines a floor function on an archimedean linearly ordered ring making it into a floorRing.

Main statements #

• ℕ, ℤ, and ℚ are archimedean.
class Archimedean (α : Type u_2) :

An ordered additive commutative monoid is called Archimedean if for any two elements x, y such that 0 < y, there exists a natural number n such that x ≤ n • y.

• arch : ∀ (x : α) {y : α}, 0 < y∃ (n : ), x n y

For any two elements x, y such that 0 < y, there exists a natural number n such that x ≤ n • y.

Instances
theorem Archimedean.arch {α : Type u_2} [self : ] (x : α) {y : α} :
0 < y∃ (n : ), x n y

For any two elements x, y such that 0 < y, there exists a natural number n such that x ≤ n • y.

instance OrderDual.archimedean {α : Type u_1} [] :
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theorem exists_lt_nsmul {M : Type u_2} [] [CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x < x_1] {a : M} (ha : 0 < a) (b : M) :
∃ (n : ), b < n a
theorem existsUnique_zsmul_near_of_pos {α : Type u_1} [] {a : α} (ha : 0 < a) (g : α) :
∃! k : , k a g g < (k + 1) a

An archimedean decidable linearly ordered AddCommGroup has a version of the floor: for a > 0, any g in the group lies between some two consecutive multiples of a.

theorem existsUnique_zsmul_near_of_pos' {α : Type u_1} [] {a : α} (ha : 0 < a) (g : α) :
∃! k : , 0 g - k a g - k a < a
theorem existsUnique_sub_zsmul_mem_Ico {α : Type u_1} [] {a : α} (ha : 0 < a) (b : α) (c : α) :
∃! m : , b - m a Set.Ico c (c + a)
theorem existsUnique_add_zsmul_mem_Ico {α : Type u_1} [] {a : α} (ha : 0 < a) (b : α) (c : α) :
∃! m : , b + m a Set.Ico c (c + a)
theorem existsUnique_add_zsmul_mem_Ioc {α : Type u_1} [] {a : α} (ha : 0 < a) (b : α) (c : α) :
∃! m : , b + m a Set.Ioc c (c + a)
theorem existsUnique_sub_zsmul_mem_Ioc {α : Type u_1} [] {a : α} (ha : 0 < a) (b : α) (c : α) :
∃! m : , b - m a Set.Ioc c (c + a)
theorem exists_nat_ge {α : Type u_1} [] [] (x : α) :
∃ (n : ), x n
@[instance 100]
instance instIsDirectedLeOfArchimedean {α : Type u_1} [] [] :
IsDirected α fun (x x_1 : α) => x x_1
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theorem exists_nat_gt {α : Type u_1} [] (x : α) :
∃ (n : ), x < n
theorem add_one_pow_unbounded_of_pos {α : Type u_1} [] {y : α} (x : α) (hy : 0 < y) :
∃ (n : ), x < (y + 1) ^ n
theorem pow_unbounded_of_one_lt {α : Type u_1} [] {y : α} [] (x : α) (hy1 : 1 < y) :
∃ (n : ), x < y ^ n
theorem exists_int_gt {α : Type u_1} [] (x : α) :
∃ (n : ), x < n
theorem exists_int_lt {α : Type u_1} [] (x : α) :
∃ (n : ), n < x
theorem exists_floor {α : Type u_1} [] (x : α) :
∃ (fl : ), ∀ (z : ), z fl z x
theorem exists_nat_pow_near {α : Type u_1} [] [] {x : α} {y : α} (hx : 1 x) (hy : 1 < y) :
∃ (n : ), y ^ n x x < y ^ (n + 1)

Every x greater than or equal to 1 is between two successive natural-number powers of every y greater than one.

theorem exists_mem_Ico_zpow {α : Type u_1} [] [] {x : α} {y : α} (hx : 0 < x) (hy : 1 < y) :
∃ (n : ), x Set.Ico (y ^ n) (y ^ (n + 1))

Every positive x is between two successive integer powers of another y greater than one. This is the same as exists_mem_Ioc_zpow, but with ≤ and < the other way around.

theorem exists_mem_Ioc_zpow {α : Type u_1} [] [] {x : α} {y : α} (hx : 0 < x) (hy : 1 < y) :
∃ (n : ), x Set.Ioc (y ^ n) (y ^ (n + 1))

Every positive x is between two successive integer powers of another y greater than one. This is the same as exists_mem_Ico_zpow, but with ≤ and < the other way around.

theorem exists_pow_lt_of_lt_one {α : Type u_1} [] [] {x : α} {y : α} (hx : 0 < x) (hy : y < 1) :
∃ (n : ), y ^ n < x

For any y < 1 and any positive x, there exists n : ℕ with y ^ n < x.

theorem exists_nat_pow_near_of_lt_one {α : Type u_1} [] [] {x : α} {y : α} (xpos : 0 < x) (hx : x 1) (ypos : 0 < y) (hy : y < 1) :
∃ (n : ), y ^ (n + 1) < x x y ^ n

Given x and y between 0 and 1, x is between two successive powers of y. This is the same as exists_nat_pow_near, but for elements between 0 and 1

theorem exists_nat_one_div_lt {α : Type u_1} [] {ε : α} (hε : 0 < ε) :
∃ (n : ), 1 / (n + 1) < ε
theorem exists_rat_gt {α : Type u_1} [] (x : α) :
∃ (q : ), x < q
theorem exists_rat_lt {α : Type u_1} [] (x : α) :
∃ (q : ), q < x
theorem exists_rat_btwn {α : Type u_1} [] {x : α} {y : α} (h : x < y) :
∃ (q : ), x < q q < y
theorem le_of_forall_rat_lt_imp_le {α : Type u_1} [] {x : α} {y : α} (h : ∀ (q : ), q < xq y) :
x y
theorem le_of_forall_lt_rat_imp_le {α : Type u_1} [] {x : α} {y : α} (h : ∀ (q : ), y < qx q) :
x y
theorem le_iff_forall_rat_lt_imp_le {α : Type u_1} [] {x : α} {y : α} :
x y ∀ (q : ), q < xq y
theorem le_iff_forall_lt_rat_imp_le {α : Type u_1} [] {x : α} {y : α} :
x y ∀ (q : ), y < qx q
theorem eq_of_forall_rat_lt_iff_lt {α : Type u_1} [] {x : α} {y : α} (h : ∀ (q : ), q < x q < y) :
x = y
theorem eq_of_forall_lt_rat_iff_lt {α : Type u_1} [] {x : α} {y : α} (h : ∀ (q : ), x < q y < q) :
x = y
theorem exists_pos_rat_lt {α : Type u_1} [] {x : α} (x0 : 0 < x) :
∃ (q : ), 0 < q q < x
theorem exists_rat_near {α : Type u_1} [] {ε : α} (x : α) (ε0 : 0 < ε) :
∃ (q : ), |x - q| < ε
theorem archimedean_iff_nat_lt {α : Type u_1} :
∀ (x : α), ∃ (n : ), x < n
theorem archimedean_iff_nat_le {α : Type u_1} :
∀ (x : α), ∃ (n : ), x n
theorem archimedean_iff_int_lt {α : Type u_1} :
∀ (x : α), ∃ (n : ), x < n
theorem archimedean_iff_int_le {α : Type u_1} :
∀ (x : α), ∃ (n : ), x n
theorem archimedean_iff_rat_lt {α : Type u_1} :
∀ (x : α), ∃ (q : ), x < q
theorem archimedean_iff_rat_le {α : Type u_1} :
∀ (x : α), ∃ (q : ), x q
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instance Nonneg.archimedean {α : Type u_1} [] :
Archimedean { x : α // 0 x }
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noncomputable def Archimedean.floorRing (α : Type u_3) [] :

A linear ordered archimedean ring is a floor ring. This is not an instance because in some cases we have a computable floor function.

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Instances For
@[instance 100]
instance FloorRing.archimedean (α : Type u_3) [] :

A linear ordered field that is a floor ring is archimedean.

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