Relación 1

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 juaruipav marescpla"
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"

-- "jaisalmen zoiruicha"
fun factorial4 :: "nat ⇒ nat" where
  "factorial4 0 = 1"
| "factorial4 (n) =  n * factorial4 (n- 1)" 
 
value "factorial4 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 juaruipav"
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"

-- "marescpla"
fun longitud4 :: "'a list ⇒ nat" where
  "longitud4 [] = 0"|
  "longitud4 xs = 1 + longitud4 (tl xs) "

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

-- ""jaisalmen  zoiruicha"
fun longitud5 :: "'a list ⇒ nat" where
  "longitud5 [] = 0"
| "longitud5 (x#a) = 1 + longitud5 a"
   
value "longitud5 [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 juaruipav marescpla jaisalmen"
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 juaruipav"
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]"

-- "jaisalmen zoiruicha"
fun inversa3 :: "'a list ⇒ 'a list" where
  "inversa3 [] = []"
| "inversa3 (x#a) = inversa3 a @ [x]"

value "inversa3 [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 juaruipav marescpla"
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 jaisalmen zoiruicha"
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 juaruipav"
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]"

-- "marescpla"
fun conc4 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc4 [] [] = []"|
  "conc4 [] ys = (hd ys) # conc4 [] (tl ys)"|
  "conc4 xs ys = (hd xs) # conc4 (tl xs) ys"
 
value "conc4 [a,d] [b,d,a,c]" -- "= [a,d,b,d,a,c]"

-- "jaisalmen zoiruicha"
fun conc5 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc5 [] ys = ys"
| "conc5 xs [] = xs"
| "conc5 (x#xs) ys = x # conc5 xs ys"

value "conc5 [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,juaruipav"
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]"

-- "marescpla"
fun coge5 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge5 0 xs = []"|
  "coge5 (Suc n) xs = hd xs # coge5 n (tl xs)"
 
value "coge5 2 [a,c,d,b,e]" -- "= [a,c]"

-- "jaisalmen zoiruicha"
fun coge6 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge6 0 xs = []"
| "coge6 n [] = []"
| "coge6 n (x#xs) = x # (coge6 (n-(1::nat)) xs)"

value "coge6 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, juaruipav"
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]"

-- "marescpla"
fun elimina5 :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina5 0 xs = xs"|
  "elimina5 (Suc n) xs = elimina5 n (tl xs)"
 
value "elimina5 2 [a,c,d,b,e]" 

-- "jaisalmen zoiruicha"
fun elimina6 :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina6 0 xs = xs"
| "elimina6 n [] = []"
| "elimina6 n (x#xs) = elimina6 (n-(1::nat)) xs"

value "elimina6 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 juaruipav marescpla"
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"

-- "jaisalmen zoiruicha"
fun esVacia4 :: "'a list ⇒ bool" where
  "esVacia4 x = (if longitud(x)=0 then True else False)"

value "esVacia4 []"  -- "= True"
value "esVacia4 [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,juaruipav"
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]"


-- "marescpla"
fun inversaAcAux5 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux5 [] ys = ys"|
  "inversaAcAux5 xs ys = inversaAcAux5 (elimina 1 xs) (conc (coge 1 xs) ys)"
 
fun inversaAc5 :: "'a list ⇒ 'a list" where
  "inversaAc5 xs = inversaAcAux5 xs []"
 
value "inversaAc5 [a,c,b,e]" -- "= [e,b,c,a]"


-- "jaisalmen"
fun inverseAcAux6 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inverseAcAux6 xs ys = xs@ys"

fun inversaAc6 :: "'a list ⇒ 'a list" where
  "inversaAc6 [] = []"
| "inversaAc6 (x#xs) = inverseAcAux6 (inversaAc6 xs) [x]"

value "inversaAc6 [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 juaruipav jaisalmen zoiruicha"
fun sum :: "nat list ⇒ nat" where
  "sum []      = 0"
| "sum (x#xs) = x + sum xs"

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

-- "irealetei pabflomar marescpla"
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 juaruipav"
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]"

-- "marescpla"
fun map4 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map4 f [] = []"|
  "map4 f xs = value f (coge 1 xs) # map4 f (elimina 1 xs)"
 
value "map4 (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"

-- "jaisalmen zoiruicha"
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map5 f [] = []"
| "map5 f (x#xs) = f(x) # map5 f xs "

value "map5 (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"

end