<source lang="isar">
header {* 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
------------------------------------------------------------------- *}

-- "antmacrui maresccas4 diecabmen1"
fun factorial :: "nat ⇒ nat" where
"factorial 0 = 1"
| "factorial (Suc n) = Suc n * factorial n"

value "factorial 4" -- "24"

-- "irealetei domlloriv"
fun factorial2 :: "nat ⇒ nat" where
"factorial2 (0::nat)= (1::nat)"
| "factorial2 n = n * factorial2 (n - 1)"

value "factorial2 4" -- "24"

-- "julrobrel"
fun factorial3 :: "nat ⇒ nat" where
"factorial3 0 = 1"
| "factorial3 n = factorial3 (n - 1) * n"

value "factorial3 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
------------------------------------------------------------------- *}

-- "maresccas4 domlloriv"
fun longitud :: "'a list ⇒ nat" where
"longitud [] = 0"
| "longitud (x # xs) = Suc (longitud xs)"

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

-- "irealetei pabflomar julrobrel"
fun longitud2 :: "'a list ⇒ nat" where
"longitud2 [] = (0::nat)"
| "longitud2 xs = 1 + longitud2 (tl xs) "

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

-- "diecabmen1"
fun longitud3 :: "'a list ⇒ nat" where
"longitud3 [] = 0"
| "longitud3 (head#tail) = 1 + longitud3 tail"

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

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

-- "maresccas4 pabflomar"
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where
"intercambia p = (snd p, fst p)"

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

-- "irealetei diecabmen1 domlloriv julrobrel"
fun intercambia2 :: "'a × 'b ⇒ 'b × 'a" where
"intercambia2 (x,y) = (y,x)"

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

-- "maresccas4 diecabmen1 domlloriv julrobrel"
fun inversa :: "'a list ⇒ 'a list" where
"inversa [] = []"
| "inversa (x#xs) = inversa xs @ (x#[])"

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

-- "irealetei pabflomar"
fun inversa2 :: "'a list ⇒ 'a list" where
"inversa2 [] = [] "
| "inversa2 xs = inversa2 (tl xs) @ (hd xs#[])"

value "inversa2 [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]
------------------------------------------------------------------ *}

-- "maresccas4 julrobrel"
fun repite :: "nat ⇒ 'a ⇒ 'a list" where
"repite 0 x = []"
| "repite (Suc n) x = x # repite n x"

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

-- "irealetei"
fun repite2 :: "nat ⇒ 'a ⇒ 'a list" where
"repite2 (0::nat) x = []"
| "repite2 n x = x#(repite2 (n - 1) x) "

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

-- "diecabmen1"
fun repite3 :: "nat ⇒ 'a ⇒ 'a list" where
"repite3 0 x = []"
| "repite3 (Suc n) x = (x#[]) @ (repite3 n x)"

value "repite3 3 a" -- "= [a,a,a]"

-- "domlloriv pabflomar"
fun repite4 :: "nat ⇒ 'a ⇒ 'a list" where
"repite4 0 x = []"
| "repite4 n x = x# repite4 (n - 1) x"

value "repite4 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]
------------------------------------------------------------------ *}

-- "maresccas4 julrobrel"
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]"

-- irealetei
fun conc2 :: "'a list ⇒ 'a list ⇒ 'a list" where
"conc2 xs [] = xs"
| "conc2 xs ys = hd ys # conc2 xs (tl ys)"

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

-- "diecabmen1 domlloriv pabflomar"
fun conc3 :: "'a list ⇒ 'a list ⇒ 'a list" where
"conc3 xs ys = 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]
------------------------------------------------------------------ *}

-- "maresccas4 diecabmen1 domlloriv"
fun coge :: "nat ⇒ 'a list ⇒ 'a list" where
"coge 0 xs = []"
| "coge n [] = []"
| "coge (Suc n) (x#xs) = x # (coge n xs)"

-- "irealetei"
fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge2 0 xs = []"
| "coge2 n xs = (hd xs) # (coge2 (n - 1) (tl xs))"

-- "pabflomar"
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]"

-- "julrobrel"
fun coge4 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge4 0 xs = []"
|"coge4 (Suc n) [] = []"
|"coge4 (Suc n) (x#xs) = x # (coge4 n xs)"

value "coge4 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]
------------------------------------------------------------------ *}

-- "maresccas4 domlloriv"
fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina 0 xs = xs"
| "elimina n [] = []"
| "elimina (Suc n) (x#xs) = elimina n xs"

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

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

value "elimina2 2 [a,c,d,b,e]" -- "= [d,b,e]"

-- "diecabmen1"
fun elimina3 :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina3 0 xs = xs"
| "elimina3 n [] = []"
| "elimina3 n (x#xs) = elimina3 (n - 1) xs"

value "elimina3 2 [a,c,d,b,e]" -- "= [d,b,e]"

-- "julrobrel"
fun elimina4 :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina4 0 xs = xs"
|"elimina4 (Suc n) [] = []"
|"elimina4 (Suc n) (x#xs) = (elimina4 n xs)"

value "elimina4 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
------------------------------------------------------------------ *}

-- "maresccas4"
fun esVacia :: "'a list ⇒ bool" where
"esVacia [] = True"
| "esVacia (x#xs) = False"

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

-- "irealetei domlloriv pabflomar julrobrel"
fun esVacia2 :: "'a list ⇒ bool" where
"esVacia2 [] = True"
| "esVacia2 xs = False"

value "esVacia2 []" -- "= True"
value "esVacia2 [1]" -- "= False"

-- "diecabmen1"
fun esVacia3 :: "'a list ⇒ bool" where
"esVacia3 xs = (if xs = [] then True else False)"

value "esVacia3 []" -- "= True"
value "esVacia3 [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]
------------------------------------------------------------------ *}

-- "maresccas4 domlloriv"
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 []"

value "inversaAc2 [a,c,b,e]" -- "= [e,b,c,a]"

-- "irealetei"
fun inversaAc2Aux :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAc2Aux xs [] = xs"
| "inversaAc2Aux xs ys = inversaAc2Aux ((hd ys) # xs) (tl ys)"

fun inversaAc2 :: "'a list ⇒ 'a list" where
"inversaAc2 xs = inversaAc2Aux [] xs"

value "inversaAc2 [a,c,b,e]" -- "= [e,b,c,a]"

-- "diecabmen1 julrobrel"
fun inversaAc3Aux :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAc3Aux xs ys = xs@ys"

fun inversaAc3 :: "'a list ⇒ 'a list" where
"inversaAc3 [] = []"
| "inversaAc3 (x#xs) = inversaAc3Aux (inversaAc3 xs) (x#[])"

value "inversaAc3 [a,c,b,e]" -- "= [e,b,c,a]"
-- "pabflomar"
fun inversaAcAux4 :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAcAux4 [] ys = ys"
| "inversaAcAux4 xs ys = inversaAcAux4 (tl xs) ((hd xs) # ys)"

fun inversaAc4 :: "'a list ⇒ 'a list" where
"inversaAc4 xs = inversaAcAux4 xs []"

value "inversaAc4 [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
------------------------------------------------------------------ *}

-- "maresccas4 diecabmen1 domlloriv julrobrel"
fun sum :: "nat list ⇒ nat" where
"sum [] = 0"
| "sum (x#xs) = x + sum xs"

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

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

value "sum2 [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]
------------------------------------------------------------------ *}

-- "maresccas4 domlloriv julrobrel"
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []"
| "map f (x#xs) = (f x) # map f xs"

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

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

value "map2 (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"

-- "diecabmen1"
fun map3 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map3 f [] = []"
| "map3 f (x#xs) = ((f x)#[]) @ (map3 f xs)"

value "map3 (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"

end
</source>