mathlib3 documentation

order.order_iso_nat

Relation embeddings from the naturals #

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

This file allows translation from monotone functions ℕ → α to order embeddings ℕ ↪ α and defines the limit value of an eventually-constant sequence.

Main declarations #

def rel_embedding.nat_lt {α : Type u_1} {r : α α Prop} [is_strict_order α r] (f : α) (H : (n : ), r (f n) (f (n + 1))) :

If f is a strictly r-increasing sequence, then this returns f as an order embedding.

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@[simp]
theorem rel_embedding.coe_nat_lt {α : Type u_1} {r : α α Prop} [is_strict_order α r] {f : α} {H : (n : ), r (f n) (f (n + 1))} :
def rel_embedding.nat_gt {α : Type u_1} {r : α α Prop} [is_strict_order α r] (f : α) (H : (n : ), r (f (n + 1)) (f n)) :

If f is a strictly r-decreasing sequence, then this returns f as an order embedding.

Equations
@[simp]
theorem rel_embedding.coe_nat_gt {α : Type u_1} {r : α α Prop} [is_strict_order α r] {f : α} {H : (n : ), r (f (n + 1)) (f n)} :
theorem rel_embedding.exists_not_acc_lt_of_not_acc {α : Type u_1} {a : α} {r : α α Prop} (h : ¬acc r a) :
(b : α), ¬acc r b r b a
theorem rel_embedding.acc_iff_no_decreasing_seq {α : Type u_1} {r : α α Prop} [is_strict_order α r] {x : α} :

A value is accessible iff it isn't contained in any infinite decreasing sequence.

theorem rel_embedding.not_acc_of_decreasing_seq {α : Type u_1} {r : α α Prop} [is_strict_order α r] (f : gt ↪r r) (k : ) :
¬acc r (f k)

A relation is well-founded iff it doesn't have any infinite decreasing sequence.

theorem rel_embedding.not_well_founded_of_decreasing_seq {α : Type u_1} {r : α α Prop} [is_strict_order α r] (f : gt ↪r r) :
noncomputable def nat.subtype.order_iso_of_nat (s : set ) [infinite s] :

nat.subtype.of_nat as an order isomorphism between and an infinite subset. See also nat.nth for a version where the subset may be finite.

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theorem nat.exists_subseq_of_forall_mem_union {α : Type u_1} {s t : set α} (e : α) (he : (n : ), e n s t) :
(g : ↪o ), ( (n : ), e (g n) s) (n : ), e (g n) t
theorem exists_increasing_or_nonincreasing_subseq' {α : Type u_1} (r : α α Prop) (f : α) :
(g : ↪o ), ( (n : ), r (f (g n)) (f (g (n + 1)))) (m n : ), m < n ¬r (f (g m)) (f (g n))
theorem exists_increasing_or_nonincreasing_subseq {α : Type u_1} (r : α α Prop) [is_trans α r] (f : α) :
(g : ↪o ), ( (m n : ), m < n r (f (g m)) (f (g n))) (m n : ), m < n ¬r (f (g m)) (f (g n))

This is the infinitary Erdős–Szekeres theorem, and an important lemma in the usual proof of Bolzano-Weierstrass for .

theorem well_founded.monotone_chain_condition' {α : Type u_1} [preorder α] :
well_founded gt (a : →o α), (n : ), (m : ), n m ¬a n < a m
theorem well_founded.monotone_chain_condition {α : Type u_1} [partial_order α] :
well_founded gt (a : →o α), (n : ), (m : ), n m a n = a m

The "monotone chain condition" below is sometimes a convenient form of well foundedness.

noncomputable def monotonic_sequence_limit_index {α : Type u_1} [preorder α] (a : →o α) :

Given an eventually-constant monotone sequence a₀ ≤ a₁ ≤ a₂ ≤ ... in a partially-ordered type, monotonic_sequence_limit_index a is the least natural number n for which aₙ reaches the constant value. For sequences that are not eventually constant, monotonic_sequence_limit_index a is defined, but is a junk value.

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noncomputable def monotonic_sequence_limit {α : Type u_1} [preorder α] (a : →o α) :
α

The constant value of an eventually-constant monotone sequence a₀ ≤ a₁ ≤ a₂ ≤ ... in a partially-ordered type.

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