Relación 1
De Razonamiento automático (2016-17)
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 *) 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 *) 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 *) 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 *) 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 *)
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 *)
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where
"intercambia (x,y) = (y,x)"
value "intercambia (u,v) = (v,u)"
(* jeamacpov *) 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 *) (* @ :: "'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 *) (* 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 *) 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 *) 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 *)
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 *) 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*) 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]"
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 *) 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]"
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)) "
(*danrodcha*) fun coge1 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge1 0 _ = []"
| "coge1 _ [] = []" | "coge1 (Suc n) (x#xs) = x#(coge1 n xs)"
(*crimgomgom, jeamacpov, rubgonmar*) fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge2 0 _ = []"| "coge2 _ [] = []" | "coge2 n (x#xs) = x#(coge2 (n-1) xs) "
(*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))"
(*serrodcal*) fun coge4 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge4 0 xs = []"
| "coge4 n xs = [(hd xs)]@(coge4 (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]
------------------------------------------------------------------ *}
(*marcarmor13, manmorjim1, serrodcal*) fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina 0 xs = xs"
| "elimina n xs = (elimina (n-1) (tl xs ))"
(*danrodcha, crigomgom*) 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 *) 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 "elimina 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
------------------------------------------------------------------ *}
(*marcarmor13, rubgonmar, danrodcha, crigomgom*) fun esVacia :: "'a list ⇒ bool" where
"esVacia [] = True "
| "esVacia xs = False"
(*danrodcha*) 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 "esVacia []" -- "= True" value "esVacia [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*) fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAcAux [] ys = ys"
| "inversaAcAux xs ys = inversaAcAux (tl xs) (hd xs#ys) "
(*danrodcha, crigomgom*) fun inversaAcAux1 :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAcAux1 [] ys = ys"
| "inversaAcAux1 (x#xs) ys = inversaAcAux1 xs (x#ys)"
(*rubgonmar, danrodcha, crigomgom, manmorjim1*) fun inversaAc :: "'a list ⇒ 'a list" where
"inversaAc xs = inversaAcAux xs []"
(*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 "
(* 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 "inversaAc [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
------------------------------------------------------------------ *}
(*rubgonmar,marcarmor13, manmorjim1*) fun sum :: "nat list ⇒ nat" where
"sum [] = 0" |"sum xs = hd xs + sum (tl xs)"
(*danrodcha, crigomgom, pablucoto*) fun sum1 :: "nat list ⇒ nat" where
"sum1 [] = 0"
| "sum1 (x#xs) = x + sum1 xs"
(*danrodcha*) fun sum2 :: "nat list ⇒ nat" where
"sum2 xs = fold (op +) xs 0"
(* jeamacpov *) fun sum3 :: "nat list ⇒ nat" where
"sum3 [] = 0"
| "sum3 xs = (hd xs)+(sum3(drop 1 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]
------------------------------------------------------------------ *}
(*rubgonmar,marcarmor13,manmorjim1*) fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []"
| "map f xs = f(hd xs)#map f (tl xs)"
(*wilmorort, danrodcha, crigomgom, pablucoto*)
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []"
|"map f (x # xs) = f x # map f xs" (*yo pondría paréntesis, pero sin
ellos lo entiende*)
(* jeamacpov *) fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []"
| "map f xs =f(hd xs)#(map f (drop 1 xs)) "
value "map (λx. 2*x) [3::nat,2,5]" -- "= [6,4,10]"
end