Diferencia entre revisiones de «Relación 1»
De Razonamiento automático (2018-19)
Línea 2: | Línea 2: | ||
chapter {* R1: Programación funcional en Isabelle *} | chapter {* R1: Programación funcional en Isabelle *} | ||
− | theory | + | theory R1_Programacion_funcional_en_Isabelle |
imports Main | imports Main | ||
begin | begin |
Revisión del 14:03 28 feb 2019
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 raffergon2 aribatval juacanrod josgomrom4
marfruman1 gleherlop benber alfmarcua enrparlav manperjim chrgencar
antramhur pabbergue alikan*)
fun longitud :: "'a list ⇒ nat" where
"longitud [] = 0 "
| "longitud (x#xs) = 1 + longitud xs "
value "longitud [a,b,c] = 3"
(* hugrubsan giafus1 *)
fun longitud2 :: "'a list ⇒ nat" where
"longitud2 [] = 0 "
| "longitud2 xs = 1 + longitud2 (tl xs) "
value "longitud2 [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)
------------------------------------------------------------------ *}
(* manperjim pabalagon cammonagu raffergon2 aribatval juacanrod
marfruman1 gleherlop benber hugrubsan alfmarcua enrparalv giafus1
chrgencar antramhur alikan pabbergue *)
fun intercambia :: "'a × 'b ⇒ 'b × 'a" where
"intercambia (x,y) = (y, x)"
value "intercambia (u,v) = (v,u)"
(* josgomrom4 *)
fun intercambia2 :: "'a × 'b ⇒ 'b × 'a" where
"intercambia2 xs = (snd xs, fst xs) "
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]
------------------------------------------------------------------ *}
(* 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 []"
value "inversa [a,d,c] = [c,d,a]"
(* manperjim pabalagon raffergon2 cammonagu josgomrom4 marfruman1
gleherlop alfmarcua enrparalv chrgencar antramhur alikan pabbergue *)
fun inversa2 :: "'a list ⇒ 'a list" where
"inversa2 [] = []"
| "inversa2 (x#xs) = inversa2 xs @ [x]"
value "inversa2 [a,d,c] = [c,d,a]"
(* juacanrod hugrubsan giafus1 *)
fun inversa3 :: "'a list ⇒ 'a list" where
"inversa3 [] = []"
| "inversa3 (xs) = inversa3(tl xs) @ [ hd (xs)]"
value "inversa3 [a,d,c] = [c,d,a]"
(* aribatval *)
fun inversa4 :: "'a list ⇒ 'a list" where
"inversa4 [] = []"
| "inversa4 xs = last xs # (inversa4 (butlast xs))"
value "inversa4 [a,d,c] = [c,d,a]"
(* benber *)
fun inversa5aux :: "'a list ⇒ 'a ⇒ 'a list" where
"inversa5aux [] y = [y]"
| "inversa5aux (x#xs) y = x#(inversa5aux xs y)"
fun inversa5 :: "'a list ⇒ 'a list" where
"inversa5 [] = []"
| "inversa5 (x#xs) = inversa5aux (inversa5 xs) x"
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]
------------------------------------------------------------------ *}
(* pabalag aribatval antramhur *)
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]"
(* manperjim raffergon2 cammonagu josgomrom4 marfruman1 benber alfmarcua
gleherlop giafus1 chrgencar pabbergue alikan *)
fun repite2 :: "nat ⇒ 'a ⇒ 'a list" where
"repite2 0 x = [] "
| "repite2 n x = x # repite2 (n-1) x "
value "repite2 3 a = [a,a,a]"
(* juacanrod hugrubsan *)
fun repite3 :: "nat ⇒ 'a ⇒ 'a list" where
"repite3 0 a = []"
| "repite3 n a = [a] @ repite3 (n-1) a"
value "repite3 3 a = [a,a,a]"
(* enrparalv *)
fun repite4 :: "nat ⇒ 'a ⇒ 'a list" where
"repite4 n x = (if n = 0 then [] else repite4 (n-1) x @ [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]
------------------------------------------------------------------ *}
(* manperjim pabalagon raffergon2 josgomrom4 aribatval gleherlop
chrgencar benber antramhur*)
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]"
(* cammonagu marfruman1 pabbergue *)
fun conc2 :: "'a list ⇒ 'a list ⇒ 'a list" where
"conc2 ys [] = ys" |
"conc2 xs (y#ys) = xs @y # ys"
value "conc2 [a,d] [b,d,a,c] = [a,d,b,d,a,c]"
(* juacanrod *)
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]"
(* hugrubsan enrparalv giafus1 alikan *)
fun conc4 :: "'a list ⇒ 'a list ⇒ 'a list" where
"conc4 xs ys = xs @ ys"
(* alfmarcua *)
fun conc5 :: "'a list ⇒ 'a list ⇒ 'a list" where
"conc5 [] ys = ys"
| "conc5 xs ys = conc5 (butlast xs) ((last xs)#ys)"
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 raffergon2 antramhur *)
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]"
(* manperjim cammonagu josgomrom4 marfruman1 benber alfmarcua chrgencar
gleherlop giafus1 pabbergue *)
fun coge2 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge2 0 xs = []" |
"coge2 n [] = []" |
"coge2 n (x#xs) = x # coge2 (n-1) xs"
value "coge2 2 [a,c,d,b,e] = [a,c]"
(* juacanrod *)
fun coge3 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge3 0 xs = []" |
"coge3 n xs = [hd (xs)] @ coge3 (n-1) (tl (xs))"
value "coge3 2 [a,c,d,b,e] = [a,c]"
(* aribatval hugrubsan *)
fun coge4 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge4 0 xs = []" |
"coge4 n [] = []" |
"coge4 n xs = (hd xs) # coge4 (n-1) (tl xs)"
value "coge4 2 [a,c,d,b,e] = [a,c]"
(* enrparalv *)
fun coge5 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge5 n [] = []"
| "coge5 n (x#xs) = (if (n=0) then [] else [x] @ coge5 (n-1) xs )"
value "coge5 2 [a,c,d,b,e] = [a,c]"
(* alikan *)
fun coge6 :: "nat ⇒ 'a list ⇒ 'a list" where
"coge6 n [] = []"|
"coge6 n (x#xs) = (case n of 0 ⇒ [] | Suc n ⇒ x # coge6 n 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]
------------------------------------------------------------------ *}
(* pabalagon antramhur *)
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]"
(* manperjim raffergon2 cammonagu josgomrom4 marfruman1 benber alfmarcua
enrparalv aribatval gleherlop chrgencar giafus1 pabbergue*)
fun elimina2 :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina2 0 xs = xs" |
"elimina2 n [] = []" |
"elimina2 n (x#xs) = elimina2 (n-1) xs"
value "elimina2 2 [a,c,d,b,e] = [d,b,e]"
(* juacanrod hugrubsan *)
fun elimina3 :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina3 0 xs = xs" |
"elimina3 n xs = elimina3 (n-1) (tl xs)"
value "elimina3 2 [a,c,d,b,e] = [d,b,e]"
(* alikan *)
fun elimina4 :: "nat ⇒ 'a list ⇒ 'a list" where
"elimina4 n [] = []"|
"elimina4 n (x#xs) = (case n of 0 ⇒ x#xs | Suc n ⇒ 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 [a] = False
------------------------------------------------------------------ *}
(* manperjim pabalagon raffergon2 josgomrom4 marfruman1 benber hugrubsan
alfmarcua enrparalv aribatval chrgencar giafus1 alikan pabbergue
antramhur *)
fun esVacia :: "'a list ⇒ bool" where
"esVacia [] = True" |
"esVacia xs = False"
value "esVacia [a] = False"
(* cammonagu juacanrod *)
fun esVacia2 :: "'a list ⇒ bool" where
"esVacia2 xs = (longitud xs = 0)"
value "esVacia2 [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]
------------------------------------------------------------------ *}
(* manperjim pabalagon cammonagu josgomrom4 marfruman1 benber alfmarcua
enrparalv aribatval gleherlop chrgencar giafus1 alikan pabbergue
antramhur *)
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]"
(* juacanrod *)
fun inversaAcAux2 :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAcAux2 [] ys = ys" |
"inversaAcAux2 xs ys = inversaAcAux2 (tl xs) ([hd xs]) @ ys"
value "inversaAcAux2 [a,b,c] []"
fun inversaAc2 :: "'a list ⇒ 'a list" where
"inversaAc2 xs = inversaAcAux2 xs []"
value "inversaAc2 [a,c,b,e] = [e,b,c,a]"
(* hugrubsan *)
fun inversaAcAux3 :: "'a list ⇒ 'a list ⇒ 'a list" where
"inversaAcAux3 [] ys = ys"
|"inversaAcAux3 xs ys = inversaAcAux3 (tl xs) [(hd xs)] @ ys"
fun inversaAc3 :: "'a list ⇒ 'a list" where
"inversaAc3 xs = inversaAcAux3 (tl xs) [(hd xs)]"
value "inversaAc3 [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
------------------------------------------------------------------ *}
(* manperjim pabalagon raffergon2 josgomrom4 marfruman1 benber alfmarcua
enrparalv aribatval gleherlop chrgencar giafus1 pabbergue antramhur
alikan *)
fun sum :: "nat list ⇒ nat" where
"sum [] = 0" |
"sum (x#xs) = x + sum xs"
value "sum [3,2,5] = 10"
(* cammonagu *)
fun sum2:: "nat list ⇒ nat" where
"sum2 [] = 0" |
"sum2 [x] = x" |
"sum2 (x#xs) = x + sum2 xs"
value "sum2 [3,2,5] = 10"
(* juacanrod hugrubsan *)
fun sum3 :: "nat list ⇒ nat" where
"sum3 [] = 0" |
"sum3 xs = (hd xs) + sum3 (tl xs)"
value "sum3 [3,2,5,1]"
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]
------------------------------------------------------------------ *}
(* manperjim pabalagon raffergon2 cammonagu josgomrom4 marfruman1 benber
alfmarcua aribatval gleherlop chrgencar giafus1 pabbergue antramhur *)
fun map :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map f [] = []" |
"map f (x#xs) = f x # map f xs"
value "map (λn. Suc n) [2,3,4,5]"
(* juacanrod *)
fun map2 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map2 f [] = []" |
"map2 f (x#xs) = [(f x)] @ map2 f xs"
value "map2 (λn. Suc n) [2,3,4,5]"
(* hugrubsan *)
fun map3 :: "('a ⇒ 'b) ⇒ 'a list ⇒ 'b list" where
"map3 f [] = []" |
"map3 f xs = f (hd xs) # map3 f (tl xs)"
value "map3 (λn. Suc n) [2,3,4,5]"
end