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
