Calculemus (Posts about Discrete Structures)

Pigeonhole principle

José A. Alonso — Mon, 07 Oct 2024 04:00:00 GMT

Prove the pigeonhole principle; that is, if \(S\) is a finite set and \(T\) and \(U\) are subsets of \(S\) such that the number of elements of \(S\) is less than the sum of the number of elements of \(T\) and \(U\), then the intersection of \(T\) and \(U\) is non-empty.

To do this, complete the following Lean4 theory:

import Mathlib.Data.Finset.Card

open Finset

variable [DecidableEq α]
variable {s t u : Finset α}

set_option pp.fieldNotation false

example
  (hts : t ⊆ s)
  (hus : u ⊆ s)
  (hstu : card s < card t + card u)
  : Finset.Nonempty (t ∩ u) :=
by sorry

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

José A. Alonso — Wed, 28 Aug 2024 04:00:00 GMT

The tree corresponding to

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

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

Proofs that the mirror function of binary trees is involutive

José A. Alonso — Mon, 26 Aug 2024 04:00:00 GMT

The tree corresponding to

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

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'

Equivalence of reverse definitions

José A. Alonso — Mon, 19 Aug 2024 04:00:00 GMT

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

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

José A. Alonso — Wed, 14 Aug 2024 04:00:00 GMT

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

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

José A. Alonso — Wed, 07 Aug 2024 04:00:00 GMT

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