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If x is the limit of u and y is an upper bound of u, then x ≤ y

Limits

In Lean4, we can define that \(a\) is the limit of the sequence \(u\) by:

   def is_limit (u : ℕ → ℝ) (a : ℝ) :=
     ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| < ε

and that \(a\) is an upper bound of \(u\) by:

   def is_upper_bound (u : ℕ → ℝ) (a : ℝ) :=
     ∀ n, u n ≤ a

Prove that if \(x\) is the limit of the sequence \(u\) and \(y\) is an upper bound of \(u\), then \(x ≤ y\).

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic
import Mathlib.Tactic

variable  (u : ℕ → ℝ)
variable (x y : ℝ)

def is_limit (u : ℕ → ℝ) (a : ℝ) :=
  ∀ ε > 0, ∃ k, ∀ n ≥ k, |u n - a| < ε

def is_upper_bound (u : ℕ → ℝ) (a : ℝ) :=
  ∀ n, u n ≤ a

example
  (hx : is_limit u x)
  (hy : is_upper_bound u y)
  : x ≤ y :=
by sorry

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If (∀ ε > 0, y ≤ x + ε), then y ≤ x

Real numbers

Let \(x, y ∈ ℝ\). Prove that \[ (∀ ε > 0, y ≤ x + ε) → y ≤ x \]

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic
import Mathlib.Tactic

variable {x y : ℝ}

example :
  (∀ ε > 0, y ≤ x + ε) → y ≤ x :=
by sorry

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If x is the supremum of set A, then ((∀ y)[y < x → (∃ a ∈ A)[y < a]]

Real numbers

In Lean4, one can define that \(x\) is an upper bound of \(A\) by

   def is_upper_bound (A : Set ℝ) (x : ℝ) :=
     ∀ a ∈ A, a ≤ x

and \(x\) is supremum of \(A\) by

   def is_supremum (A : Set ℝ) (x : ℝ) :=
     is_upper_bound A x ∧ ∀ y, is_upper_bound A y → x ≤ y

Prove that if \(x\) is the supremum of \(A\), then \[ (∀ y)[y < x → (∃ a ∈ A)[y < a]] \]

To do this, complete the following Lean4 theory:

import Mathlib.Data.Real.Basic
variable {A : Set ℝ}
variable {x : ℝ}

def is_upper_bound (A : Set ℝ) (x : ℝ) :=
  ∀ a ∈ A, a ≤ x

def is_supremum (A : Set ℝ) (x : ℝ) :=
  is_upper_bound A x ∧ ∀ y, is_upper_bound A y → x ≤ y

example
  (hx : is_supremum A x)
  : ∀ y, y < x → ∃ a ∈ A, y < a :=
by sorry

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Equivalence of definitions of the Fibonacci function

Induction

In Lean4, the Fibonacci function can be defined as

   def fibonacci : Nat → Nat
     | 0     => 0
     | 1     => 1
     | n + 2 => fibonacci n + fibonacci (n+1)

Another more efficient definition is

   def fib (n : Nat) : Nat :=
     (loop n).1
   where
     loop : Nat → Nat × Nat
       | 0   => (0, 1)
       | n + 1 =>
         let p := loop n
         (p.2, p.1 + p.2)

Prove that both definitions are equivalent; that is,

   fibonacci = fib

To do this, complete the following Lean4 theory:

open Nat

set_option pp.fieldNotation false

def fibonacci : Nat → Nat
  | 0     => 0
  | 1     => 1
  | n + 2 => fibonacci n + fibonacci (n+1)

def fib (n : Nat) : Nat :=
  (loop n).1
where
  loop : Nat → Nat × Nat
    | 0   => (0, 1)
    | n + 1 =>
      let p := loop n
      (p.2, p.1 + p.2)

example : fibonacci = fib :=
by sorry

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Proofs of "flatten (mirror a) = reverse (flatten a)"

Binary trees

The tree corresponding to

    3
   / \
  2   4
 / \
1   5

can be represented by the term

Node 3 (Node 2 (Leaf 1) (Leaf 5)) (Leaf 4)

using the datatype defined by

   inductive Tree' (α : Type) : Type
     | Leaf : α → Tree' α
     | Node : α → Tree' α → Tree' α → Tree' α

The mirror image of the previous tree is

  3
 / \
4   2
   / \
  5   1

and the list obtained by flattening it (traversing it in infix order) is

[4, 3, 5, 2, 1]

The definition of the function that calculates the mirror image is

   def mirror : Tree' α → Tree' α
     | Leaf x     => Leaf x
     | Node x l r => Node x (mirror r) (mirror l)

and the one that flattens the tree is

   def flatten : Tree' α → List α
     | Leaf x     => [x]
     | Node x l r => (flatten l) ++ [x] ++ (flatten r)

Prove that

   flatten (mirror a) = reverse (flatten a)

To do this, complete the following Lean4 theory:

import Mathlib.Tactic

open List

variable {α : Type}

set_option pp.fieldNotation false

inductive Tree' (α : Type) : Type
  | Leaf : α → Tree' α
  | Node : α → Tree' α → Tree' α → Tree' α

namespace Tree'

variable (a l r : Tree' α)
variable (x : α)

def mirror : Tree' α → Tree' α
  | Leaf x     => Leaf x
  | Node x l r => Node x (mirror r) (mirror l)

def flatten : Tree' α → List α
  | Leaf x     => [x]
  | Node x l r => (flatten l) ++ [x] ++ (flatten r)

example :
  flatten (mirror a) = reverse (flatten a) :=
by sorry

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Proofs that the mirror function of binary trees is involutive

Binary trees

The tree corresponding to

    3
   / \
  2   4
 / \
1   5

can be represented by the term

Node 3 (Node 2 (Leaf 1) (Leaf 5)) (Leaf 4)

using the datatype defined by

   inductive Tree' (α : Type) : Type
     | Leaf : α → Tree' α
     | Node : α → Tree' α → Tree' α → Tree' α

The mirror image of the previous tree is

  3
 / \
4   2
   / \
  5   1

whose term is

Node 3 (Leaf 4) (Node 2 (Leaf 5) (Leaf 1))

The definition of the function that calculates the mirror image is

   def mirror : Tree' α → Tree' α
     | (Leaf x)     => Leaf x
     | (Node x l r) => Node x (mirror r) (mirror l)

Prove that the mirror function is involutive; that is,

   mirror (mirror a) = a

To do this, complete the following Lean4 theory:

import Mathlib.Tactic

variable {α : Type}

set_option pp.fieldNotation false

inductive Tree' (α : Type) : Type
  | Leaf : α → Tree' α
  | Node : α → Tree' α → Tree' α → Tree' α

namespace Tree'

variable (a l r : Tree' α)
variable (x : α)

def mirror : Tree' α → Tree' α
  | (Leaf x)     => Leaf x
  | (Node x l r) => Node x (mirror r) (mirror l)

example :
  mirror (mirror a) = a :=
by sorry

end Tree'

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Equivalence of reverse definitions

Lists

In Lean4, the function

    reverse : List α → List α

is defined such that (reverse xs) is the list obtained by reversing the order of elements in xs, such that the first element becomes the last, the second element becomes the second to last, and so on, resulting in a new list with the same elements but in the opposite sequence. For example,

reverse  [3,2,5,1] = [1,5,2,3]

Its definition is

    def reverseAux : List α → List α → List α
      | [],    ys => ys
      | x::xs, ys => reverseAux xs (x::ys)

    def reverse (xs : List α) : List α :=
      reverseAux xs []

The following lemmas characterize its behavior:

    reverseAux_nil : reverseAux [] ys = ys
    reverseAux_cons : reverseAux (x::xs) ys = reverseAux xs (x::ys)

An alternative definition is

    def reverse2 : List α → List α
      | []        => []
      | (x :: xs) => reverse2 xs ++ [x]

Prove that the two definitions are equivalent; that is,

    reverse xs = reverse2 xs

To do this, complete the following Lean4 theory:

import Mathlib.Data.List.Basic
open List

variable {α : Type}
variable (xs : List α)

example : reverse xs = reverse2 xs :=
by sorry

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Proofs of take n xs ++ drop n xs = xs

Lists

In Lean4, the functions

    take : Nat → List α → Nat
    drop : Nat → List α → Nat
    (++) : List α → List α → List α

are defined such that

  • (take n xs) is the list consisting of the first n elements of xs. For example,

    take 2 [3,5,1,9,7] = [3,5]

  • (drop n xs) is the list containing all elements of xs except the first n elements. For example,

    drop 2 [3,5,1,9,7] = [1,9,7]

  • (xs ++ ys) is the list obtained by concatenating xs and ys. For example,

    [3,5] ++ [1,9,7] = [3,5,1,9,7]

These functions are characterized by the following lemmas:

take_zero      : take 0 xs = []
take_nil       : take n [] = []
take_cons      : take (succ n) (x :: xs) = x :: take n xs
drop_zero      : drop 0 xs = xs
drop_nil       : drop n [] = []
drop_succ_cons : drop (n + 1) (x :: xs) = drop n xs
nil_append     : [] ++ ys = ys
cons_append    : (x :: xs) ++ y = x :: (xs ++ ys)

Prove that

take n xs ++ drop n xs = xs

To do this, complete the following Lean4 theory:

import Mathlib.Data.List.Basic
import Mathlib.Tactic
open List Nat

variable {α : Type}
variable (n : ℕ)
variable (x : α)
variable (xs : List α)

example : take n xs ++ drop n xs = xs :=
by sorry

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Proofs of the equality length(xs ++ ys) = length(xs) + length(ys)

Lists

In Lean, the functions

    length : List α → Nat
    (++)   : List α → List α → List α

are defined such that

  • (length xs) is the length of xs. For example,

    length [2,3,5,3] = 4

  • (xs ++ ys) is the list obtained by concatenating xs and ys. For example.

    [1,2] ++ [2,3,5,3] = [1,2,2,3,5,3]

These functions are characterized by the following lemmas:

length_nil  : length [] = 0
length_cons : length (x :: xs) = length xs + 1
nil_append  : [] ++ ys = ys
cons_append : (x :: xs) ++ y = x :: (xs ++ ys)

To prove that

length (xs ++ ys) = length xs + length ys

To that end, complete the following Lean4 theory

import Mathlib.Tactic
open List

variable {α : Type}
variable (xs ys : List α)

example :
  length (xs ++ ys) = length xs + length ys :=
by sorry

Read more…

Asociatividad de la concatenación de listas

Listas

En Lean4 la operación de concatenación de listas se representa por (++) y está caracterizada por los siguientes lemas

nil_append  : [] ++ ys = ys
cons_append : (x :: xs) ++ y = x :: (xs ++ ys)

Demostrar que la concatenación es asociativa; es decir,

xs ++ (ys ++ zs) = (xs ++ ys) ++ zs

Para ello, completar la siguiente teoría de Lean4:

import Mathlib.Data.List.Basic
open List

variable {α : Type}
variable (xs ys zs : List α)

-- 1ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as HI
  . calc [] ++ (ys ++ zs)
         = ys ++ zs                := nil_append (ys ++ zs)
       _ = ([] ++ ys) ++ zs        := congrArg (. ++ zs) (nil_append ys).symm
  . calc (a :: as) ++ (ys ++ zs)
         = a :: (as ++ (ys ++ zs)) := cons_append a as (ys ++ zs)
       _ = a :: ((as ++ ys) ++ zs) := congrArg (a :: .) HI
       _ = (a :: (as ++ ys)) ++ zs := (cons_append a (as ++ ys) zs).symm

-- 2ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as HI
  . calc [] ++ (ys ++ zs)
         = ys ++ zs                := by rw [nil_append]
       _ = ([] ++ ys) ++ zs        := by rw [nil_append]
  . calc (a :: as) ++ (ys ++ zs)
         = a :: (as ++ (ys ++ zs)) := by rw [cons_append]
       _ = a :: ((as ++ ys) ++ zs) := by rw [HI]
       _ = (a :: (as ++ ys)) ++ zs := by rw [cons_append]
       _ = ((a :: as) ++ ys) ++ zs := by rw [← cons_append]

-- 3ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as HI
  . calc [] ++ (ys ++ zs)
         = ys ++ zs                := by exact rfl
       _ = ([] ++ ys) ++ zs        := by exact rfl
  . calc (a :: as) ++ (ys ++ zs)
         = a :: (as ++ (ys ++ zs)) := rfl
       _ = a :: ((as ++ ys) ++ zs) := by rw [HI]
       _ = (a :: (as ++ ys)) ++ zs := rfl
       _ = ((a :: as) ++ ys) ++ zs := rfl

-- 4ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as HI
  . calc [] ++ (ys ++ zs)
         = ys ++ zs                := by simp
       _ = ([] ++ ys) ++ zs        := by simp
  . calc (a :: as) ++ (ys ++ zs)
         = a :: (as ++ (ys ++ zs)) := by simp
       _ = a :: ((as ++ ys) ++ zs) := congrArg (a :: .) HI
       _ = (a :: (as ++ ys)) ++ zs := by simp
       _ = ((a :: as) ++ ys) ++ zs := by simp

-- 5ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as
  . simp
  . exact (append_assoc (a :: as) ys zs).symm

-- 6ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction' xs with a as HI
  . -- ⊢ [] ++ (ys ++ zs) = ([] ++ ys) ++ zs
    rw [nil_append]
    -- ⊢ ys ++ zs = ([] ++ ys) ++ zs
    rw [nil_append]
  . -- a : α
    -- as : List α
    -- HI : as ++ (ys ++ zs) = as ++ ys ++ zs
    -- ⊢ (a :: as) ++ (ys ++ zs) = ((a :: as) ++ ys) ++ zs
    rw [cons_append]
    -- ⊢ a :: (as ++ (ys ++ zs)) = ((a :: as) ++ ys) ++ zs
    rw [HI]
    -- ⊢ a :: ((as ++ ys) ++ zs) = ((a :: as) ++ ys) ++ zs
    rw [cons_append]
    -- ⊢ a :: ((as ++ ys) ++ zs) = (a :: (as ++ ys)) ++ zs
    rw [cons_append]

-- 7ª demostración
-- ===============

lemma conc_asoc_1 (xs ys zs : List α) :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by
  induction xs with
  | nil => calc
    [] ++ (ys ++ zs)
        = ys ++ zs         := by rw [nil_append]
      _ = ([] ++ ys) ++ zs := by rw [nil_append]
  | cons a as HI => calc
    (a :: as) ++ (ys ++ zs)
        = a :: (as ++ (ys ++ zs)) := by rw [cons_append]
      _ = a :: ((as ++ ys) ++ zs) := by rw [HI]
      _ = (a :: (as ++ ys)) ++ zs := by rw [cons_append]
      _ = ((a :: as) ++ ys) ++ zs := congrArg (. ++ zs) (cons_append a as ys)

-- 8ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs  :=
(append_assoc xs ys zs).symm

-- 9ª demostración
-- ===============

example :
  xs ++ (ys ++ zs) = (xs ++ ys) ++ zs :=
by simp

-- Lemas usados
-- ============

-- variable (x : α)
-- #check (append_assoc xs ys zs : (xs ++ ys) ++ zs = xs ++ (ys ++ zs))
-- #check (cons_append x xs ys : (x :: xs) ++ ys = x :: (xs ++ ys))
-- #check (cons_inj x : x :: xs = x :: ys ↔ xs = ys)
-- #check (nil_append xs : [] ++ xs = xs)

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