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feat(data/int/interval): Finite intervals in ℤ (#9526)
This proves that `ℤ` is a locally finite order.
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/- | ||
Copyright (c) 2021 Yaël Dillies. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yaël Dillies | ||
-/ | ||
import data.int.basic | ||
import data.nat.interval | ||
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/-! | ||
# Finite intervals of integers | ||
This file proves that `ℤ` is a `locally_finite_order` and calculates the cardinality of its | ||
intervals as finsets and fintypes. | ||
-/ | ||
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open finset int | ||
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instance : locally_finite_order ℤ := | ||
{ finset_Icc := λ a b, (finset.range (b + 1 - a).to_nat).map $ | ||
nat.cast_embedding.trans $ add_left_embedding a, | ||
finset_Ico := λ a b, (finset.range (b - a).to_nat).map $ | ||
nat.cast_embedding.trans $ add_left_embedding a, | ||
finset_Ioc := λ a b, (finset.range (b - a).to_nat).map $ | ||
nat.cast_embedding.trans $ add_left_embedding (a + 1), | ||
finset_Ioo := λ a b, (finset.range (b - a - 1).to_nat).map $ | ||
nat.cast_embedding.trans $ add_left_embedding (a + 1), | ||
finset_mem_Icc := λ a b x, begin | ||
simp_rw [mem_map, exists_prop, mem_range, int.lt_to_nat, function.embedding.trans_apply, | ||
nat.cast_embedding_apply, add_left_embedding_apply, nat_cast_eq_coe_nat], | ||
split, | ||
{ rintro ⟨a, h, rfl⟩, | ||
rw [lt_sub_iff_add_lt, int.lt_add_one_iff, add_comm] at h, | ||
exact ⟨int.le.intro rfl, h⟩ }, | ||
{ rintro ⟨ha, hb⟩, | ||
use (x - a).to_nat, | ||
rw ←lt_add_one_iff at hb, | ||
rw to_nat_sub_of_le ha, | ||
exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩ } | ||
end, | ||
finset_mem_Ico := λ a b x, begin | ||
simp_rw [mem_map, exists_prop, mem_range, int.lt_to_nat, function.embedding.trans_apply, | ||
nat.cast_embedding_apply, add_left_embedding_apply, nat_cast_eq_coe_nat], | ||
split, | ||
{ rintro ⟨a, h, rfl⟩, | ||
exact ⟨int.le.intro rfl, lt_sub_iff_add_lt'.mp h⟩ }, | ||
{ rintro ⟨ha, hb⟩, | ||
use (x - a).to_nat, | ||
rw to_nat_sub_of_le ha, | ||
exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩ } | ||
end, | ||
finset_mem_Ioc := λ a b x, begin | ||
simp_rw [mem_map, exists_prop, mem_range, int.lt_to_nat, function.embedding.trans_apply, | ||
nat.cast_embedding_apply, add_left_embedding_apply, nat_cast_eq_coe_nat], | ||
split, | ||
{ rintro ⟨a, h, rfl⟩, | ||
rw [←add_one_le_iff, le_sub_iff_add_le', add_comm _ (1 : ℤ), ←add_assoc] at h, | ||
exact ⟨int.le.intro rfl, h⟩ }, | ||
{ rintro ⟨ha, hb⟩, | ||
use (x - (a + 1)).to_nat, | ||
rw [to_nat_sub_of_le ha, ←add_one_le_iff, sub_add, add_sub_cancel], | ||
exact ⟨sub_le_sub_right hb _, add_sub_cancel'_right _ _⟩ } | ||
end, | ||
finset_mem_Ioo := λ a b x, begin | ||
simp_rw [mem_map, exists_prop, mem_range, int.lt_to_nat, function.embedding.trans_apply, | ||
nat.cast_embedding_apply, add_left_embedding_apply, nat_cast_eq_coe_nat], | ||
split, | ||
{ rintro ⟨a, h, rfl⟩, | ||
rw [sub_sub, lt_sub_iff_add_lt'] at h, | ||
exact ⟨int.le.intro rfl, h⟩ }, | ||
{ rintro ⟨ha, hb⟩, | ||
use (x - (a + 1)).to_nat, | ||
rw [to_nat_sub_of_le ha, sub_sub], | ||
exact ⟨sub_lt_sub_right hb _, add_sub_cancel'_right _ _⟩ } | ||
end } | ||
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namespace int | ||
variables (a b : ℤ) | ||
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lemma Icc_eq_finset_map : | ||
Icc a b = (finset.range (b + 1 - a).to_nat).map | ||
(nat.cast_embedding.trans $ add_left_embedding a) := rfl | ||
lemma Ioc_eq_finset_map : | ||
Ioc a b = (finset.range (b - a).to_nat).map | ||
(nat.cast_embedding.trans $ add_left_embedding (a + 1)) := rfl | ||
lemma Ioo_eq_finset_map : | ||
Ioo a b = (finset.range (b - a - 1).to_nat).map | ||
(nat.cast_embedding.trans $ add_left_embedding (a + 1)) := rfl | ||
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@[simp] lemma card_Icc : (Icc a b).card = (b + 1 - a).to_nat := | ||
by { change (finset.map _ _).card = _, rw [finset.card_map, finset.card_range] } | ||
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@[simp] lemma card_Ioc : (Ioc a b).card = (b - a).to_nat := | ||
by { change (finset.map _ _).card = _, rw [finset.card_map, finset.card_range] } | ||
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@[simp] lemma card_Ioo : (Ioo a b).card = (b - a - 1).to_nat := | ||
by { change (finset.map _ _).card = _, rw [finset.card_map, finset.card_range] } | ||
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lemma card_Icc_of_le (h : a ≤ b + 1) : ((Icc a b).card : ℤ) = b + 1 - a := | ||
by rw [card_Icc, to_nat_sub_of_le h] | ||
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lemma card_Ioc_of_le (h : a ≤ b) : ((Ioc a b).card : ℤ) = b - a := | ||
by rw [card_Ioc, to_nat_sub_of_le h] | ||
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lemma card_Ioo_of_lt (h : a < b) : ((Ioo a b).card : ℤ) = b - a - 1 := | ||
by rw [card_Ioo, sub_sub, to_nat_sub_of_le h] | ||
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@[simp] lemma card_fintype_Icc : fintype.card (set.Icc a b) = (b + 1 - a).to_nat := | ||
by rw [←card_Icc, fintype.card_of_finset] | ||
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@[simp] lemma card_fintype_Ioc : fintype.card (set.Ioc a b) = (b - a).to_nat := | ||
by rw [←card_Ioc, fintype.card_of_finset] | ||
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@[simp] lemma card_fintype_Ioo : fintype.card (set.Ioo a b) = (b - a - 1).to_nat := | ||
by rw [←card_Ioo, fintype.card_of_finset] | ||
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lemma card_fintype_Icc_of_le (h : a ≤ b + 1) : (fintype.card (set.Icc a b) : ℤ) = b + 1 - a := | ||
by rw [card_fintype_Icc, to_nat_sub_of_le h] | ||
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lemma card_fintype_Ioc_of_le (h : a ≤ b) : (fintype.card (set.Ioc a b) : ℤ) = b - a := | ||
by rw [card_fintype_Ioc, to_nat_sub_of_le h] | ||
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lemma card_fintype_Ioo_of_lt (h : a < b) : (fintype.card (set.Ioo a b) : ℤ) = b - a - 1 := | ||
by rw [card_fintype_Ioo, sub_sub, to_nat_sub_of_le h] | ||
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end int |