Apostol. Begin working through floor/ceiling exercises.

Also discovered the basic interval module so replaced custom
interval syntax with it.
finite-set-exercises
Joshua Potter 2023-05-08 16:44:52 -06:00
parent fe6cb7e074
commit ed89078e76
7 changed files with 27 additions and 68 deletions

View File

@ -1,7 +1,5 @@
import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic
import Common.Real.Int
/-! # Apostol.Chapter_1_11 -/ /-! # Apostol.Chapter_1_11 -/
namespace Apostol.Chapter_1_11 namespace Apostol.Chapter_1_11
@ -15,18 +13,35 @@ Prove that the greatest-integer function has the properties indicated.
`⌊x + n⌋ = ⌊x⌋ + n` for every integer `n`. `⌊x + n⌋ = ⌊x⌋ + n` for every integer `n`.
-/ -/
theorem exercise_4a (x : ) (n : ) : ⌊x + n⌋ = ⌊x⌋ + n := by theorem exercise_4a (x : ) (n : ) : ⌊x + n⌋ = ⌊x⌋ + n :=
sorry Int.floor_add_int x n
/-- ### Exercise 4b /-- ### Exercise 4b.1
`⌊-x⌋ = -⌊x⌋` if `x` is an integer. `⌊-x⌋ = -⌊x⌋` if `x` is an integer.
-/
theorem exercise_4b_1 (x : ) : ⌊-x⌋ = -⌊x⌋ := by
simp only [Int.floor_int, id_eq]
/-- ### Exercise 4b.2
`⌊-x⌋ = -⌊x⌋ - 1` otherwise. `⌊-x⌋ = -⌊x⌋ - 1` otherwise.
-/ -/
theorem exercise_4b (x : ) theorem exercise_4b_2 (x : ) (h : ∃ n : , x ∈ Set.Ioo ↑n (↑n + (1 : )))
: (Real.isInt x → ⌊-x⌋ = -⌊x⌋) : ⌊-x⌋ = -⌊x⌋ - 1 := by
(¬Real.isInt x → ⌊-x⌋ = -⌊x⌋ - 1) := by rw [Int.floor_neg]
sorry suffices ⌈x⌉ = ⌊x⌋ + 1 by
have := congrArg (HMul.hMul (-1)) this
simp only [neg_mul, one_mul, neg_add_rev, add_comm] at this
exact this
have ⟨n, hn⟩ := h
have hn' : x ∈ Set.Ico ↑n (↑n + (1 : )) :=
Set.mem_of_subset_of_mem Set.Ioo_subset_Ico_self hn
rw [Int.ceil_eq_iff, Int.floor_eq_on_Ico n x hn']
simp only [Int.cast_add, Int.cast_one, add_sub_cancel]
apply And.intro
· exact (Set.mem_Ioo.mp hn).left
· exact le_of_lt (Set.mem_Ico.mp hn').right
/-- ### Exercise 4c /-- ### Exercise 4c

View File

@ -1,7 +1,6 @@
import Common.Real.Basic import Common.Real.Basic
import Common.Real.Function import Common.Real.Function
import Common.Real.Geometry import Common.Real.Geometry
import Common.Real.Int
import Common.Real.Rational import Common.Real.Rational
import Common.Real.Sequence import Common.Real.Sequence
import Common.Real.Set import Common.Real.Set

View File

@ -14,7 +14,7 @@ open Partition
Any member of a subinterval of a partition `P` must also be a member of `P`. Any member of a subinterval of a partition `P` must also be a member of `P`.
-/ -/
lemma mem_open_subinterval_imp_mem_partition {p : Partition} lemma mem_open_subinterval_imp_mem_partition {p : Partition}
(hI : I ∈ p.xs.pairwise (fun x₁ x₂ => i(x₁, x₂))) (hI : I ∈ p.xs.pairwise (fun x₁ x₂ => Set.Ioo x₁ x₂))
(hy : y ∈ I) : y ∈ p := by (hy : y ∈ I) : y ∈ p := by
cases h : p.xs with cases h : p.xs with
| nil => | nil =>
@ -50,7 +50,7 @@ structure Step where
p : Partition p : Partition
f : ∀ x ∈ p, f : ∀ x ∈ p,
const_open_subintervals : const_open_subintervals :
∀ (hI : I ∈ p.xs.pairwise (fun x₁ x₂ => i(x₁, x₂))), ∀ (hI : I ∈ p.xs.pairwise (fun x₁ x₂ => Set.Ioo x₁ x₂)),
∃ c : , ∀ (hy : y ∈ I), ∃ c : , ∀ (hy : y ∈ I),
f y (mem_open_subinterval_imp_mem_partition hI hy) = c f y (mem_open_subinterval_imp_mem_partition hI hy) = c

View File

@ -1,15 +0,0 @@
import Mathlib.Data.Real.Basic
/-! # Common.Real.Int
Additional theorems and definitions useful in the context of integers.
-/
namespace Real
/--
Check whether a real number is an integer.
-/
def isInt (x : ) := x = Int.floor x
end Real

View File

@ -1,3 +1,2 @@
import Common.Real.Set.Basic import Common.Real.Set.Basic
import Common.Real.Set.Interval
import Common.Real.Set.Partition import Common.Real.Set.Partition

View File

@ -1,39 +0,0 @@
import Mathlib.Data.Real.Basic
/-! # Common.Real.Set.Interval
A syntactic description of the various types of continuous intervals permitted
on the real number line.
-/
/--
Representation of a closed interval.
-/
syntax (priority := high) "i[" term "," term "]" : term
macro_rules
| `(i[$a, $b]) => `({ z | $a ≤ z ∧ z ≤ $b })
/--
Representation of an open interval.
-/
syntax (priority := high) "i(" term "," term ")" : term
macro_rules
| `(i($a, $b)) => `({ z | $a < z ∧ z < $b })
/--
Representation of a left half-open interval.
-/
syntax (priority := high) "i(" term "," term "]" : term
macro_rules
| `(i($a, $b]) => `({ z | $a < z ∧ z ≤ $b })
/--
Representation of a right half-open interval.
-/
syntax (priority := high) "i[" term "," term ")" : term
macro_rules
| `(i[$a, $b)) => `({ z | $a ≤ z ∧ z < $b })

View File

@ -1,7 +1,7 @@
import Mathlib.Data.Real.Basic
import Mathlib.Data.List.Sort import Mathlib.Data.List.Sort
import Common.List.Basic import Common.List.Basic
import Common.Real.Set.Interval
/-! # Common.Real.Set.Partition /-! # Common.Real.Set.Partition