Relación 1

chapter {* R1: Programación funcional en Isabelle *}

theory R1 imports Main begin

text {* ----------------------------------------------------------------

 Ejercicio 0. Definir, por recursión, la función
    factorial :: nat ⇒ nat
 tal que (factorial n) es el factorial de n. Por ejemplo,
    factorial 4 = 24
 ------------------------------------------------------------------- *}

(* danrodcha anaprarod ivamenjim serrodcal manmorjim1 *) fun factorial :: "nat ⇒ nat" where

 "factorial 0 = 1"

| "factorial (Suc n) = (Suc n) * factorial n"

value "factorial 4 = 24"

(* wilmorort pablucoto marcarmor13 crigomgom rubgonmar jeamacpov *) fun factorial1 :: "nat ⇒ nat" where

 "factorial1 0  = 1 "

| "factorial1 n = n * factorial1 (n-1)"

value "factorial1 4 = 24"

text {* ----------------------------------------------------------------

 Ejercicio 1. Definir, por recursión, la función
    longitud :: 'a list ⇒ nat
 tal que (longitud xs) es la longitud de la listas xs. Por ejemplo,
    longitud [4,2,5] = 3
 ------------------------------------------------------------------- *}

(* wilmorort *) (* Para usar las lista en forma de [a,b,c] *) translations

 "[x, xs]" == "x#[xs]"
 "[x]" == "x#[]"

(* wilmorort serrodcal crigomgom rubgonmar *) fun longitud :: "'a list ⇒ nat" where

 "longitud []       = 0" |
 "longitud (x # xs) = 1 + longitud xs "

value "longitud [4,2,5] = 3"

(* pablucoto *) fun longitud2 :: "'a list ⇒ nat " where

 "longitud2 []  = 0"

| "longitud2 xs = 1 + longitud2 (butlast xs) "

value "longitud2 [4,2,5] = 3"

(* pablucoto *) fun longitud3 :: "'a list ⇒ nat " where

 "longitud3 xs = (if xs = [] 
                  then 0 
                  else 1 + longitud3 (butlast xs))"  

value "longitud3 [4,2,5] = 3"

(* marcarmor13 manmorjim1 jeamacpov *) fun longitud4 :: "'a list ⇒ nat" where

 "longitud4 []  = 0 "

| "longitud4 xs = (1 + longitud4 (tl xs))"

value "longitud4 [4,2,5] = 3"

(* danrodcha *) fun longitud5 :: "'a list ⇒ nat" where

 "longitud5 []     = 0"

| "longitud5 (x#xs) = Suc (longitud5 xs)"

value "longitud5 [4,2,5] = 3"

(* serrodcal *) fun longitud6 :: "'a list ⇒ nat" where

 "longitud xs = length xs"

value "longitud6 [4,2,5] " -- "= 3"

(* Comentario: El objetivo es explicitar cómo está definida la función

  length *) 

text {* ---------------------------------------------------------------

 Ejercicio 2. Definir la función
    fun intercambia :: 'a × 'b ⇒ 'b × 'a
 tal que (intercambia p) es el par obtenido intercambiando las
 componentes del par p. Por ejemplo,
    intercambia (u,v) = (v,u)
 ------------------------------------------------------------------ *}

(* wilmorort marcarmor13 danrodcha crigomgom pablucoto rubgonmar

  manmorjim1 serrodcal *) 

fun intercambia :: "'a × 'b ⇒ 'b × 'a" where

 "intercambia (x,y) = (y,x)"

value "intercambia (u,v) = (v,u)"

(* jeamacpov *) fun intercambia2 :: "'a × 'b ⇒ 'b × 'a" where

 "intercambia2 (x,y) = (snd (x,y), fst (x,y))"

value "intercambia2 (u,v) = (v,u)"

text {* ---------------------------------------------------------------

 Ejercicio 3. Definir, por recursión, la función
    inversa :: 'a list ⇒ 'a list
 tal que (inversa xs) es la lista obtenida invirtiendo el orden de los
 elementos de xs. Por ejemplo,
    inversa [a,d,c] = [c,d,a]
 ------------------------------------------------------------------ *}

(* wilmorort *) (* @ :: "'a list => 'a list => 'a list", función agregación definida

  en Theory Main, concatena dos listas: [a,b] @ [c,d] = [a,b,c,d] *)

fun inversa :: "'a list ⇒ 'a list" where

 "inversa [] = []" |
 "inversa (x # xs) = (inversa xs) @ (x#[]) "

value "inversa [a,d,c] = [c,d,a]"

(*marcarmor13 manmorjim1*) fun inversa1 :: "'a list ⇒ 'a list" where

 "inversa1 [] = []"

| "inversa1 xs = inversa1 (tl xs)@ ((hd xs)#[])"

value "inversa1 [a,d,c] = [c,d,a]"

(* danrodcha pablucoto rubgonmar *) (* es igual que inversa sustituyendo x#[] por [x] *) fun inversa2 :: "'a list ⇒ 'a list" where

  "inversa2 []     = []"

| "inversa2 (x#xs) = (inversa2 xs) @ [x]"

value "inversa2 [a,d,c] = [c,d,a]"

(* danrodcha *) fun inversa3 :: "'a list ⇒ 'a list" where

  "inversa3 []     = []"

| "inversa3 (x#xs) = concat [(inversa3 xs),[x]] "

value "inversa3 [a,d,c] = [c,d,a]"

(* crigomgom serrodcal *) fun inversa4 :: "'a list ⇒ 'a list" where

 "inversa4 [] = []" |
 "inversa4 xs = (last xs) # (inversa4 (butlast xs)) "

value "inversa4 [a,d,c] = [c,d,a]"

(* jeamacpov *) fun inversa5 :: "'a list ⇒ 'a list" where

 "inversa5 [] = []"

| "inversa5 xs =

   (drop (length(xs)-1) xs) @ inversa5 (take (length(xs)-1) xs)"

value "inversa5 [a,d,c]" -- "= [c,d,a]"

text {* ---------------------------------------------------------------

 Ejercicio 4. Definir la función
    repite :: nat ⇒ 'a ⇒ 'a list
 tal que (repite n x) es la lista formada por n copias del elemento
 x. Por ejemplo, 
    repite 3 a = [a,a,a]
 ------------------------------------------------------------------ *}

(* wilmorort marcarmor13 crigomgom pablucoto rubgonmar manmorjim1

  jeamacpov *) 

fun repite :: "nat ⇒ 'a ⇒ 'a list" where

 "repite 0 x = [] " |
 "repite n x = x # (repite (n-1) x)"

value "repite 3 a = [a,a,a]"

(* danrodcha *) fun repite1 :: "nat ⇒ 'a ⇒ 'a list" where

 "repite1 0 x       = []"

| "repite1 (Suc n) x = x # (repite1 n x)"

value "repite1 3 a = [a,a,a]"

(*serrodcal*) fun repite2 :: "nat ⇒ 'a ⇒ 'a list" where

 "repite2 0 x       = []"

| "repite2 (Suc n) x = [x] @ (repite2 n x)"

value "repite2 3 a = [a,a,a]"

text {* ---------------------------------------------------------------

 Ejercicio 5. Definir la función
    conc :: 'a list ⇒ 'a list ⇒ 'a list
 tal que (conc xs ys) es la concatención de las listas xs e ys. Por
 ejemplo, 
    conc [a,d] [b,d,a,c] = [a,d,b,d,a,c]
 ------------------------------------------------------------------ *}

(* marcarmor13 serrodcal *) fun conc :: "'a list ⇒ 'a list ⇒ 'a list" where

 "conc xs ys = xs@ys"

value "conc [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

(* Comentario: El objetivo del ejercicio es explicitar la definición de

  @ *) 

(* danrodcha crigomgom rubgonmar*) fun conc1 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "conc1 [] ys = ys"

| "conc1 xs [] = xs" (*esta no hace falta*) | "conc1 (x#xs) ys = x# (conc1 xs ys)"

value "conc1 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

(* pablucoto jeamacpov *) fun conc2 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "conc2 [] ys     = ys" |
 "conc2 (x#xs) ys = x # (conc2 xs ys)"

value "conc2 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

(* manmorjim1 *) fun conc3 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "conc3 [] ys = ys"

| "conc3 xs ys = (hd xs)#[] @ (conc3 (tl xs) ys)"

value "conc3 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

text {* ---------------------------------------------------------------

 Ejercicio 6. Definir la función
    coge :: nat ⇒ 'a list ⇒ 'a list
 tal que (coge n xs) es la lista de los n primeros elementos de xs. Por 
 ejemplo, 
    coge 2 [a,c,d,b,e] = [a,c]
 ------------------------------------------------------------------ *}

(*marcarmor13, manmorjim1*) fun coge :: "nat ⇒ 'a list ⇒ 'a list" where

 "coge 0 xs = []"

| "coge n xs = (hd xs)#(coge (n-1) (tl xs)) "

(*danrodcha*) fun coge1 :: "nat ⇒ 'a list ⇒ 'a list" where

 "coge1 0 _ = []"

| "coge1 _ [] = []" | "coge1 (Suc n) (x#xs) = x#(coge1 n xs)"

(*crimgomgom, jeamacpov, rubgonmar*) fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where

 "coge2 0 _ = []"|
 "coge2 _ [] = []" |
 "coge2 n (x#xs) = x#(coge2 (n-1) xs) "

(*pablucoto*) fun coge3 :: "nat ⇒ 'a list ⇒ 'a list" where

 "coge3 0 xs = []" |
 "coge3 n (x#xs) = (if n>length(x#xs) then (x#xs) else x # (coge3 (n-1) xs))" 

(*serrodcal*) fun coge4 :: "nat ⇒ 'a list ⇒ 'a list" where

 "coge4 0 xs = []"

| "coge4 n xs = [(hd xs)]@(coge4 (n-1) (tl xs))"

value "coge 2 [a,c,d,b,e]" -- "= [a,c]"

text {* ---------------------------------------------------------------

 Ejercicio 7. Definir la función
    elimina :: nat ⇒ 'a list ⇒ 'a list
 tal que (elimina n xs) es la lista obtenida eliminando los n primeros
 elementos de xs. Por ejemplo, 
    elimina 2 [a,c,d,b,e] = [d,b,e]
 ------------------------------------------------------------------ *}

(*marcarmor13, manmorjim1, serrodcal*) fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where

 "elimina 0  xs = xs"

| "elimina n xs = (elimina (n-1) (tl xs ))"

(*danrodcha, crigomgom*) fun elimina1 :: "nat ⇒ 'a list ⇒ 'a list" where

 "elimina1 0 xs = xs"

| "elimina1 _ [] = []" | "elimina1 (Suc n) (x#xs) = elimina1 n xs"

(*pablucoto*) fun elimina2 :: "nat ⇒ 'a list ⇒ 'a list" where

  "elimina2 0 xs = xs" |
 "elimina2 n (x#xs) = (if n>length(x#xs) then [] else (elimina2 (n-1) xs))" 

(* jeamacpov *) fun elimina3 :: "nat ⇒ 'a list ⇒ 'a list" where

 "elimina3 0 xs = xs"

| "elimina3 n xs = (elimina3 (n-1) ((drop 1 xs))) "

(*rubgonmar*) fun elimina4 :: "nat ⇒ 'a list ⇒ 'a list" where

  "elimina4 0 xs = xs" |
  "elimina4 n xs = (if n>length(xs) then 
                               [] 
                     else 
                               (elimina4 (n-1) (tl xs)))"

value "elimina 2 [a,c,d,b,e]" -- "= [d,b,e]"

text {* ---------------------------------------------------------------

 Ejercicio 8. Definir la función
    esVacia :: 'a list ⇒ bool
 tal que (esVacia xs) se verifica si xs es la lista vacía. Por ejemplo,
    esVacia []  = True
    esVacia [1] = False
 ------------------------------------------------------------------ *}

(*marcarmor13, rubgonmar, danrodcha, crigomgom*) fun esVacia :: "'a list ⇒ bool" where

 "esVacia [] = True "

| "esVacia xs = False"

(*danrodcha*) fun esVacia1 :: "'a list ⇒ bool" where

 "esVacia1 xs = (xs = [])"

(*pablucoto, jeamacpov *) fun esVacia2 :: "'a list ⇒ bool" where "esVacia2 xs = (if xs=[] then True else False)"

(*manmorjim1, serrodcal*) fun esVacia3 :: "'a list ⇒ bool" where

 "esVacia3 xs = (length xs = 0)"

value "esVacia []" -- "= True" value "esVacia [1]" -- "= False"

text {* ---------------------------------------------------------------

 Ejercicio 9. Definir la función
    inversaAc :: 'a list ⇒ 'a list
 tal que (inversaAc xs) es a inversa de xs calculada usando
 acumuladores. Por ejemplo, 
    inversaAc [a,c,b,e] = [e,b,c,a]
 ------------------------------------------------------------------ *}

(*rubgonmar,marcarmor13, mamnorjim1*) fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where

 "inversaAcAux [] ys = ys"

| "inversaAcAux xs ys = inversaAcAux (tl xs) (hd xs#ys) "

(*danrodcha, crigomgom*) fun inversaAcAux1 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "inversaAcAux1 [] ys = ys"

| "inversaAcAux1 (x#xs) ys = inversaAcAux1 xs (x#ys)"

(*rubgonmar, danrodcha, crigomgom, manmorjim1*) fun inversaAc :: "'a list ⇒ 'a list" where

 "inversaAc xs = inversaAcAux xs []"

(*pablucoto*) fun inversaAcAux2 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "inversaAcAux2 [] [] = []"| 
 "inversaAcAux2 xs (y#ys) = (inversaAcAux2 [] ys) @ [y]" 

fun inversaAc2 :: "'a list ⇒ 'a list" where

 "inversaAc2 xs = inversaAcAux2 [] xs "

(* jeamacpov *) fun inversaAcAux3 :: "'a list ⇒ 'a list ⇒ 'a list" where

 "inversaAcAux3 ys [] = ys"

| "inversaAcAux3 ys (x#xs) = inversaAcAux3 (x#ys) xs" fun inversaAc3 :: "'a list ⇒ 'a list" where

 "inversaAc3 xs = inversaAcAux3 [] xs"

value "inversaAc [a,c,b,e]" -- "= [e,b,c,a]"

text {* ---------------------------------------------------------------

 Ejercicio 10. Definir la función
    sum :: nat list ⇒ nat
 tal que (sum xs) es la suma de los elementos de xs. Por ejemplo,
    sum [3,2,5] = 10
 ------------------------------------------------------------------ *}

(*rubgonmar,marcarmor13, manmorjim1*) fun sum :: "nat list ⇒ nat" where

 "sum [] = 0"
|"sum xs = hd xs + sum (tl xs)"

(*danrodcha, crigomgom, pablucoto*) fun sum1 :: "nat list ⇒ nat" where

 "sum1 [] = 0"

| "sum1 (x#xs) = x + sum1 xs"

(*danrodcha*) fun sum2 :: "nat list ⇒ nat" where

 "sum2 xs = fold (op +) xs 0"

(* jeamacpov *) fun sum3 :: "nat list ⇒ nat" where

 "sum3 [] = 0"

| "sum3 xs = (hd xs)+(sum3(drop 1 xs))"

value "sum [3,2,5]" -- "= 10"

text {* ---------------------------------------------------------------

 Ejercicio 11. Definir la función
    map :: ('a ⇒ 'b) ⇒ 'a list ⇒ 'b list
 tal que (map f xs) es la lista obtenida aplicando la función f a los
 elementos de xs. Por ejemplo,
    map (λx. 2*x) [3,2,5] = [6,4,10]
 ------------------------------------------------------------------ *}

(*rubgonmar,marcarmor13,manmorjim1*) fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where

 "map f [] = []" 

| "map f xs = f(hd xs)#map f (tl xs)"

(*wilmorort, danrodcha, crigomgom, pablucoto*) fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where

"map f [] = []" 

|"map f (x # xs) = f x # map f xs" (*yo pondría paréntesis, pero sin

                                                        ellos lo entiende*)

(* jeamacpov *) fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where

 "map f [] = []"

| "map f xs =f(hd xs)#(map f (drop 1 xs)) "

value "map (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"

end