Diferencia entre revisiones de «Relación 1»
De Razonamiento automático (2018-19)
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Línea 28: | Línea 28: | ||
intercambia (u,v) = (v,u) | intercambia (u,v) = (v,u) | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where | fun intercambia :: "'a × 'b ⇒ 'b × 'a" where | ||
− | "intercambia (x,y) = | + | "intercambia (x,y) = (y, x)" |
value "intercambia (u,v) = (v,u)" | value "intercambia (u,v) = (v,u)" | ||
Línea 41: | Línea 43: | ||
inversa [a,d,c] = [c,d,a] | inversa [a,d,c] = [c,d,a] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
+ | |||
+ | fun aux :: "'a list ⇒ 'a list ⇒ 'a list" where | ||
+ | "aux [] a = a" | | ||
+ | "aux (x#xs) a = aux xs (x#a)" | ||
fun inversa :: "'a list ⇒ 'a list" where | fun inversa :: "'a list ⇒ 'a list" where | ||
− | "inversa xs = | + | "inversa xs = aux xs []" |
+ | |||
+ | (* pabalagon *) | ||
+ | |||
+ | fun inversa2 :: "'a list ⇒ 'a list" where | ||
+ | "inversa2 [] = []" | | ||
+ | "inversa2 (x#xs) = inversa2 xs @ [x]" | ||
value "inversa [a,d,c] = [c,d,a]" | value "inversa [a,d,c] = [c,d,a]" | ||
Línea 54: | Línea 68: | ||
repite 3 a = [a,a,a] | repite 3 a = [a,a,a] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun repite :: "nat ⇒ 'a ⇒ 'a list" where | fun repite :: "nat ⇒ 'a ⇒ 'a list" where | ||
− | "repite n x = | + | "repite 0 x = []" |
+ | | "repite (Suc n) x = x # repite n x" | ||
value "repite 3 a = [a,a,a]" | value "repite 3 a = [a,a,a]" | ||
Línea 67: | Línea 84: | ||
conc [a,d] [b,d,a,c] = [a,d,b,d,a,c] | conc [a,d] [b,d,a,c] = [a,d,b,d,a,c] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun conc :: "'a list ⇒ 'a list ⇒ 'a list" where | fun conc :: "'a list ⇒ 'a list ⇒ 'a list" where | ||
− | "conc xs ys = | + | "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]" | value "conc [a,d] [b,d,a,c] = [a,d,b,d,a,c]" | ||
Línea 80: | Línea 100: | ||
coge 2 [a,c,d,b,e] = [a,c] | coge 2 [a,c,d,b,e] = [a,c] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun coge :: "nat ⇒ 'a list ⇒ 'a list" where | fun coge :: "nat ⇒ 'a list ⇒ 'a list" where | ||
− | "coge n xs = | + | "coge 0 xs = []" | |
+ | "coge n [] = []" | | ||
+ | "coge (Suc n) (x#xs) = x # coge n xs" | ||
value "coge 2 [a,c,d,b,e] = [a,c]" | value "coge 2 [a,c,d,b,e] = [a,c]" | ||
Línea 93: | Línea 117: | ||
elimina 2 [a,c,d,b,e] = [d,b,e] | elimina 2 [a,c,d,b,e] = [d,b,e] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where | fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where | ||
− | "elimina n xs = | + | "elimina 0 xs = []" | |
+ | "elimina n [] = []" | | ||
+ | "elimina (Suc n) (x#xs) = elimina n xs" | ||
value "elimina 2 [a,c,d,b,e] = [d,b,e]" | value "elimina 2 [a,c,d,b,e] = [d,b,e]" | ||
Línea 105: | Línea 133: | ||
esVacia [a] = False | esVacia [a] = False | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun esVacia :: "'a list ⇒ bool" where | fun esVacia :: "'a list ⇒ bool" where | ||
− | "esVacia xs = | + | "esVacia [] = True" | |
+ | "esVacia xs = False" | ||
value "esVacia [a] = False" | value "esVacia [a] = False" | ||
Línea 118: | Línea 149: | ||
inversaAc [a,c,b,e] = [e,b,c,a] | inversaAc [a,c,b,e] = [e,b,c,a] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where | fun inversaAcAux :: "'a list ⇒ 'a list ⇒ 'a list" where | ||
− | "inversaAcAux xs ys = | + | "inversaAcAux [] ys = ys" | |
+ | "inversaAcAux (x#xs) ys = inversaAcAux xs (x#ys)" | ||
fun inversaAc :: "'a list ⇒ 'a list" where | fun inversaAc :: "'a list ⇒ 'a list" where | ||
− | "inversaAc xs = | + | "inversaAc xs = inversaAcAux xs []" |
value "inversaAc [a,c,b,e] = [e,b,c,a]" | value "inversaAc [a,c,b,e] = [e,b,c,a]" | ||
Línea 133: | Línea 167: | ||
sum [3,2,5] = 10 | sum [3,2,5] = 10 | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun sum :: "nat list ⇒ nat" where | fun sum :: "nat list ⇒ nat" where | ||
− | "sum xs = | + | "sum [] = 0" | |
+ | "sum (x#xs) = x + sum xs" | ||
text {* --------------------------------------------------------------- | text {* --------------------------------------------------------------- | ||
Línea 144: | Línea 181: | ||
map (λx. 2*x) [3,2,5] = [6,4,10] | map (λx. 2*x) [3,2,5] = [6,4,10] | ||
------------------------------------------------------------------ *} | ------------------------------------------------------------------ *} | ||
+ | |||
+ | (* pabalagon *) | ||
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where | fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where | ||
− | "map f xs = | + | "map f [] = []" | |
+ | "map f (x#xs) = f x # map f xs" | ||
end | end | ||
</source> | </source> |
Revisión del 22:52 8 nov 2018
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
------------------------------------------------------------------- *}
(* cammonagu pabalagon*)
fun longitud :: "'a list ⇒ nat" where
"longitud [] = 0 "
| "longitud (x#xs) = 1 + longitud xs "
value "longitud [a,b,c] = 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)
------------------------------------------------------------------ *}
(* pabalagon *)
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]
------------------------------------------------------------------ *}
(* pabalagon *)
fun aux :: "'a list ⇒ 'a list ⇒ 'a list" where
"aux [] a = a" |
"aux (x#xs) a = aux xs (x#a)"
fun inversa :: "'a list ⇒ 'a list" where
"inversa xs = aux xs []"
(* pabalagon *)
fun inversa2 :: "'a list ⇒ 'a list" where
"inversa2 [] = []" |
"inversa2 (x#xs) = inversa2 xs @ [x]"
value "inversa [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]
------------------------------------------------------------------ *}
(* pabalagon *)
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]"
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]
------------------------------------------------------------------ *}
(* pabalagon *)
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]"
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]
------------------------------------------------------------------ *}
(* pabalagon *)
fun coge :: "nat ⇒ 'a list ⇒ 'a list" where
"coge 0 xs = []" |
"coge n [] = []" |
"coge (Suc n) (x#xs) = x # coge n 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]
------------------------------------------------------------------ *}
(* pabalagon *)
fun elimina :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina 0 xs = []" |
"elimina n [] = []" |
"elimina (Suc n) (x#xs) = elimina n 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 [a] = False
------------------------------------------------------------------ *}
(* pabalagon *)
fun esVacia :: "'a list ⇒ bool" where
"esVacia [] = True" |
"esVacia 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]
------------------------------------------------------------------ *}
(* pabalagon *)
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 "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
------------------------------------------------------------------ *}
(* pabalagon *)
fun sum :: "nat list ⇒ nat" where
"sum [] = 0" |
"sum (x#xs) = x + sum 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]
------------------------------------------------------------------ *}
(* pabalagon *)
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []" |
"map f (x#xs) = f x # map f xs"
end