Relación 1

chapter {* R1: Programación funcional en Isabelle *}

theory R1_Programacion_funcional_en_Isabelle
imports Main 
begin

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 [a,b,c] = 3
  ------------------------------------------------------------------- *}

(* edupalhid anddonram jescudero cesgongut luicedval rafcabgon diwu2 
   jospermon1 macmerflo*)
fun longitud :: "'a list ⇒ nat" where
  "longitud [] = 0" |
  "longitud (x#xs) = 1 + longitud xs "

(* rafferrod *)
fun longitud2 :: "'a list ⇒ nat" where
  "longitud2 [] = 0" |
  "longitud2 x = 1 + longitud2 (tl x)"

(* davperriv *)
fun longitud3 :: "'a list ⇒ nat" where
  "longitud3 [] = 0"
| "longitud3 xs = 1 + longitud3 (butlast xs)"

value "longitud [a,b,c] = 3"
value "longitud (x#(y#(z#[])))"

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)
  ------------------------------------------------------------------ *}

(* edupalhid anddonram cesgongut luicedval rafcabgon diwu2 jescudero
   rafferrod davperriv macmerflo jospermon1*) 
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where
  "intercambia (x,y) = (y,x)"

value "intercambia (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]
  ------------------------------------------------------------------ *}

(* edupalhid cesgongut rafcabgon diwu2 jescudero jospermon1*)
fun inversa :: "'a list ⇒ 'a list" where
  "inversa [] = []" |
  "inversa (x#xs) = inversa xs @[x] "

(* anddonram *)
fun conc1 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc1 [] b = b"
| "conc1 (x#xs) b = x # conc1 xs b"

value "conc1 [1::int,2] [3] = [1,2,3]"
value "conc1 [1::int,2] [] = [1,2]"

fun inversa2 :: "'a list ⇒ 'a list" where
  "inversa2 [] = []"
| "inversa2 (x#xs) = conc1 (inversa2 xs)  (x#[])"

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

(* luicedval *)
fun cuantos :: "'a list ⇒ nat" where
  "cuantos [] = 0" |
  "cuantos (x#xs) = 1 + cuantos xs"

fun invertir :: "nat  ⇒ 'a list ⇒ 'a list" where
  "invertir n [] = []" |
  "invertir 0 xs = xs" |
  "invertir n (x#xs) = invertir (n-1) xs@[x]"

fun inversa3 :: "'a list ⇒ 'a list" where
  "inversa3 [] = []" |
  "inversa3 xs = invertir (cuantos xs) xs"

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

(* rafferrod davperriv macmerflo *)
fun inversa4 :: "'a list ⇒ 'a list" where
  "inversa4 [] = []" |
  "inversa4 x = (last x) # (inversa4 (butlast x))"

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]
  ------------------------------------------------------------------ *}

(* edupalhid anddonram cesgongut luicedval rafcabgon diwu2 rafferrod 
   davperriv macmerflo jospermon1*)
fun repite :: "nat ⇒ 'a ⇒ 'a list" where
  "repite 0 x = []" |
  "repite n x = x # repite (n-1) x"

(* jescudero *)
fun repite2 :: "nat ⇒ 'a ⇒ 'a list" where
  "repite2 0 x = []" |
  "repite2 (Suc n) x = x # repite2 n x"

value "repite 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]
  ------------------------------------------------------------------ *}

(* edupalhid anddonram cesgongut luicedval rafferrod macmerflo jospermon1*)
fun conc :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc [] ys = ys" |
  "conc (x#xs) ys = x # conc xs ys"

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

(* rafcabgon diwu2 *) 
fun conc2 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc2 [] [] = []" |
  "conc2 xs ys = xs @ ys"

(* Comentario: El objetivo es mostrar la definición de @ *)

(* jescudero *)
fun conc3 :: "'a list ⇒ 'a list ⇒ 'a list" where
    "conc3 [] ys = ys" |
    "conc3 xs [] = xs" |
    "conc3 (x#xs) (y#ys) = x # (y #  (conc3 xs ys))"

(* Comentario: Se puede simplificar. *)

(* davperriv *)
fun conc4 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc4 [] ys = ys" |
  "conc4 xs ys = (hd xs) # conc4 (tl xs) ys"

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]
  ------------------------------------------------------------------ *}

(* edupalhid anddonram cesgongut luicedval rafcabgon diwu2 
   rafferrod macmerflo *)
fun coge :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge n [] = []" |
  "coge 0 xs = []" |
  "coge n (x#xs) = x # coge (n-1) xs "

(* jescudero *)
fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge2 0 xs = []"|
  "coge2 n [] = []"|
  "coge2 (Suc n) (x#xs) = x # coge2 n xs"

(* davperriv *)
fun coge3 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge3 0 xs = []" |
  "coge3 n xs = (hd xs) # coge3 (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]
  ------------------------------------------------------------------ *}

(* edupalhid anddonram cesgongut luicedval rafcabgon diwu2 rafferrod
   jescudero macmerflo jospermon1*) 
fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina 0 xs = xs" |
  "elimina n [] = []" |
  "elimina n (x#xs) = elimina (n-1) xs"

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

(* davperriv *)
fun elimina2 :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina2 0 xs = xs" |
  "elimina2 n xs = elimina2 (n-1) (tl xs)"

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 [a] = False
  ------------------------------------------------------------------ *}

(* edupalhid jescudero*)
fun esVacia :: "'a list ⇒ bool" where
  "esVacia xs = (if xs = [] then True else False)"

(* anddonram diwu2 *)
fun esVacia2 :: "'a list ⇒ bool" where
  "esVacia2 x = (x=[])"

(* cesgongut luicedval rafcabgon rafferrod davperriv macmerflo*)
fun esVacia3 :: "'a list ⇒ bool" where
  "esVacia3 [] = True" |
  "esVacia3 xs = False"

value "esVacia [a] = 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]
  ------------------------------------------------------------------ *}

(* edupalhid anddonram rafcabgon diwu2 rafferrod davperriv *)
fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux [] ys = ys" |
  "inversaAcAux (x#xs) ys = inversaAcAux xs (x#ys)"

fun inversaAc :: "'a list ⇒ 'a list" where
  "inversaAc xs = inversaAcAux xs []"

(* cesgongut *)
fun inversaAcAux2 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux2 [] ys = []" |
  "inversaAcAux2 xs [] = xs" |
  "inversaAcAux2 (x # xs) ys = inversaAcAux2 xs (x # ys)"

(* Comentario: Se puede simplificar. *)

fun inversaAc2 :: "'a list ⇒ 'a list" where
  "inversaAc2 xs = inversaAcAux2 xs []"

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

(* luicedval *)
fun elementos :: "'a list ⇒ nat" where
  "elementos [] = 0" |
  "elementos (x#xs) = 1 + elementos xs"

fun inversaAcAux3 :: "nat  ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux3 n [] = []" |
  "inversaAcAux3 0 xs = xs" |
  "inversaAcAux3 n (x#xs) = inversaAcAux3 (n-1) xs@[x]"
 
fun inversaAc3 :: "'a list ⇒ 'a list" where
  "inversaAc3 xs = inversaAcAux3 (elementos xs) xs"

value "inversaAc3 [a,c,b,e] = [e,b,c,a]"
value "inversaAc3 [] = []"

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
  ------------------------------------------------------------------ *}

(* anddonram edupalhid cesgongut luicedval rafcabgon diwu2 rafferrod
   jescudero macmerflo jospermon1*) 
fun sum :: "nat list ⇒ nat" where
  "sum [] = 0"
 |"sum (x#xs) = x+sum xs"

(* davperriv *)
fun sum2 :: "nat list ⇒ nat" where
  "sum2 [] = 0" |
  "sum2 xs = (hd xs) + sum2 (tl 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]
  ------------------------------------------------------------------ *}

(* anddonram edupalhid cesgongut luicedval rafcabgon diwu2 rafferrod
   jescudero macmerflo jospermon1*) 
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map f [] = []"
 |"map f (x#xs) = f x # (map f xs)"

(* davperriv *)
fun map2 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map2 f [] = []" |
  "map2 f xs = f (hd xs) # map2 f (tl xs)"

value "map (λx. x+1) [3::nat,2,4]=[4,3,5]"

end