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 fraortmoy dancorgar ferrenseg *) 
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
   marpoldia1*) 
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 anaprarod dancorgar ferrenseg *)
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 marpoldia1 fraortmoy *)
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 *) 

(* ivamenjim *)
fun longitud7 :: "'a list ⇒ nat" where
  "longitud7 [] = 0"  
| "longitud7 xs = longitud7 (drop 1 xs) + 1"
   
value "longitud7 [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)
  ------------------------------------------------------------------ *}

(* wilmorort marcarmor13 danrodcha crigomgom pablucoto rubgonmar
   manmorjim1 serrodcal fraortmoy anaprarod dancorgar *) 
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where
  "intercambia (x,y) = (y,x)"

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

(* jeamacpov ivamenjim marpoldia1*)
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 fraortmoy *)
(* @ :: "'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 anaprarod dancorgar *)
(* 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 ivamenjim *)
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 marpoldia1*)
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 ivamenjim marpoldia1*) 
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 fraortmoy dancorgar *)
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 anaprarod *)
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]"

(* ivamenjim *)
fun repite3 :: "nat ⇒ 'a ⇒ 'a list" where
  "repite3 n x = replicate n x"

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

(* Comentario: uso de replicate *) 

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 fraortmoy anaprarod *)
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]"

(* ivamenjim *)
fun conc4 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc4 xs ys = concat [xs,ys]"

value "conc4 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

(* Comentario: uso de concat *) 

(* dancorgar *)
fun conc5 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "conc5 [] [] = []"
| "conc5 [] (y#ys) = y#(conc5 [] ys)"
| "conc5 (x#xs) ys = x#(conc5 xs ys)"

value "conc5 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"

(* marpoldia1 *)
fun conc6 :: "'a list ⇒ 'a list ⇒ 'a list" where
 "conc6 [] [] = []"| 
 "conc6 [] ys = ys"|
 "conc6 xs ys = (hd xs)#(conc6 (tl xs) ys)"

value "conc6 [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)) "

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

(* danrodcha fraortmoy anaprarod*)
fun coge1 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge1 0 _            = []"
| "coge1 _ []           = []"
| "coge1 (Suc n) (x#xs) = x#(coge1 n xs)"

value "coge1 2 [a,c,d,b,e] = [a,c]"
value "coge1 2 [a,c,d,b,e] = [a,c]"

(* crimgomgom jeamacpov rubgonmar ivamenjim *)
fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge2 0 _      = []"|
  "coge2 _ []     = []" |
  "coge2 n (x#xs) = x#(coge2 (n-1) xs) "

value "coge2 2 [a,c,d,b,e] = [a,c]"
value "coge2 2 [a,c,d,b,e] = [a,c]"

(* 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))" 

value "coge3 2 [a,c,d,b,e] = [a,c]"
value "coge3 3 [a,c] = [a,c]"

(* serrodcal *)
fun coge4 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge4 0 xs = []"
| "coge4 n xs = [(hd xs)]@(coge4 (n-1) (tl xs))"

value "coge4 2 [a,c,d,b,e] = [a,c]"

(* ivamenjim *)
fun coge5 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge5 n xs = take n xs"

value "coge5 2 [a,c,d,b,e] = [a,c]"

(* Comentario: uso de take *)

(* dancorgar *)
fun coge6 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge6 0 xs = []"
| "coge6 (Suc m) [] = []"
| "coge6 (Suc m) (x#xs) = (x#(coge6 m xs))"

value "coge6 2 [a,c,d,b,e] = [a,c]"

(* marpoldia1 *)
fun coge7 :: "nat ⇒ 'a list ⇒ 'a list" where
  "coge7 0 xs = []"|
  "coge7 n [] = []"| 
  "coge7 n xs = (hd xs) # (coge7 (n-1) (tl xs))"

value "coge7 0 [a,b,c] = []"
value "coge7 2 [] = []"
value "coge7 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 marpoldia1*)
fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina 0 xs = xs"
| "elimina n xs = (elimina (n-1) (tl xs ))"

(* danrodcha crigomgom fraortmoy anaprarod *)
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 ivamenjim *)
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 "elimina4 2 [a,c,d,b,e] = [d,b,e]"

(* dancorgar *)
fun elimina5 :: "nat ⇒ 'a list ⇒ 'a list" where
  "elimina5 0 xs = xs"
| "elimina5 (Suc m) [] = []"
| "elimina5 (Suc m) (x#xs) = elimina5 m xs"

value "elimina5 2 [a,c,d,b,e]= [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 fraortmoy dancorgar *)
fun esVacia :: "'a list ⇒ bool" where
  "esVacia [] = True "
| "esVacia xs = False"

(* danrodcha anaprarod*)
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 "esVacia3 []  = True"
value "esVacia3 [1] = False"

(* ivamenjim marpoldia1*)
fun esVacia4 :: "'a list ⇒ bool" where
  "esVacia4 xs = (if length (xs) > 0 
                  then False 
                  else True)"

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

(* rubgonmar marcarmor13 mamnorjim1 serrodcal ivamenjim marpoldia1*)
fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux [] ys = ys"
| "inversaAcAux xs ys = inversaAcAux (tl xs) (hd xs#ys) "

(* rubgonmar danrodcha crigomgom manmorjim1 serrodcal ivamenjim anaprarod marpoldia1*)
fun inversaAc :: "'a list ⇒ 'a list" where
  "inversaAc xs = inversaAcAux xs []"

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

(* danrodcha crigomgom anaprarod *)
fun inversaAcAux1 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux1 [] ys     = ys"
| "inversaAcAux1 (x#xs) ys = inversaAcAux1 xs (x#ys)"

(* 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 "

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

(* 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 "inversaAc3 [a,c,b,e]= [e,b,c,a]"


(* fraortmoy *)

fun inversaAcAux4 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux4 xs [] = xs"
| "inversaAcAux4 xs (x#ys) = inversaAcAux4 (x#xs) ys"

fun inversaAc4 :: "'a list ⇒ 'a list" where
  "inversaAc4 [] = []"
| "inversaAc4 (x#xs)= (inversaAcAux4 [x] xs)"

value "inversaAc4 [a,c,b,e]" -- "= [e,b,c,a]"

(* dancorgar *)
fun inversaAcAux5 :: "'a list ⇒ 'a list ⇒ 'a list" where
  "inversaAcAux5 [] [] = []"
| "inversaAcAux5 [] (y#ys) = y#(inversaAcAux5 [] ys)"
| "inversaAcAux5 (x#xs) ys = x#(inversaAcAux5 xs ys)"

(* dancorgar *)
fun inversaAc5 :: "'a list ⇒ 'a list" where
  "inversaAc5 [] = []"
| "inversaAc5 (x#xs) = inversaAcAux5 (inversaAc5 xs) [x]"

value "inversaAc5 [a,c,b,e] = [e,b,c,a]" -- "= [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 serrodcal ivamenjim *)
fun sum :: "nat list ⇒ nat" where
  "sum [] = 0"
| "sum xs = hd xs + sum (tl xs)"

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

(* danrodcha crigomgom pablucoto fraortmoy anaprarod*)
fun sum1 :: "nat list ⇒ nat" where
  "sum1 []     = 0"
| "sum1 (x#xs) = x + sum1 xs"

value "sum1 [3,2,5] = 10"

(* danrodcha *)
fun sum2 :: "nat list ⇒ nat" where
  "sum2 xs = fold (op +) xs 0"

value "sum2 [3,2,5] = 10"

(* jeamacpov *)
fun sum3 :: "nat list ⇒ nat" where
  "sum3 [] = 0"
| "sum3 xs = (hd xs) + (sum3(drop 1 xs))"

value "sum3 [3,2,5] = 10"

(* ivamenjim *)
fun sum4 :: "nat list ⇒ nat" where
  "sum4 xs = listsum xs"

value "sum4 [3,2,5] = 10"

(* Comentario: uso de listsum *)

(* dancorgar *)
fun sum5 :: "nat list ⇒ nat" where
  "sum5 [] = 0"
| "sum5 (x#[]) = x"
| "sum5 (x#(y#xs)) = sum5 ((x + y)#xs)"

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

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

(* wilmorort danrodcha crigomgom pablucoto fraortmoy anaprarod dancorgar *)
fun map2 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map2 f [] = []" 
| "map2 f (x # xs) = f x # map2 f xs" (* yo pondría paréntesis, pero sin
                                         ellos lo entiende*)

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

(* jeamacpov *)
fun map3 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
  "map3 f [] = []"
| "map3 f xs = f (hd xs) # (map3 f (drop 1 xs)) "

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

end