Razonamiento automático (2013-14) Razonamiento automático (2013-14)

header {* R6: Sustitución, inversión y eliminación *}

theory R6
imports Main 
begin

section {* Sustitución, inversión y eliminación *}

text {* 
  --------------------------------------------------------------------- 
  Ejercicio 1. Definir la función 
     sust :: "'a ⇒ 'a ⇒ 'a list ⇒ 'a list"
  tal que (sust x y zs) es la lista obtenida sustituyendo cada
  occurrencia de x por y en la lista zs. Por ejemplo,
     sust (1::nat) 2 [1,2,3,4,1,2,3,4] = [2,2,3,4,2,2,3,4]
  --------------------------------------------------------------------- 
*}

fun sust :: "'a ⇒ 'a ⇒ 'a list ⇒ 'a list" where
  "sust x y zs = undefined"

text {*
  --------------------------------------------------------------------- 
  Ejercicio 2.1. Demostrar o refutar automáticamente 
     sust x y (xs@ys) = (sust x y xs)@(sust x y ys)"
  ---------------------------------------------------------------------  
*}

lemma 
  "sust x y (xs@ys) = (sust x y xs)@(sust x y ys)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 2.2. Demostrar o refutar detalladamente
     sust x y (xs@ys) = (sust x y xs)@(sust x y ys)"
  ---------------------------------------------------------------------  
*}

lemma sust_append: 
  "sust x y (xs@ys) = (sust x y xs)@(sust x y ys)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 3.1. Demostrar o refutar automáticamente
     rev (sust x y zs) = sust x y (rev zs)
  --------------------------------------------------------------------- 
*}

lemma 
  "rev(sust x y zs) = sust x y (rev zs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 3.2. Demostrar o refutar detalladamente
     rev (sust x y zs) = sust x y (rev zs)
  --------------------------------------------------------------------- 
*}

lemma rev_sust: 
  "rev(sust x y zs) = sust x y (rev zs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 4. Demostrar o refutar:
     sust x y (sust u v zs) = sust u v (sust x y zs)
  --------------------------------------------------------------------- 
*}

lemma 
  "sust x y (sust u v zs) = sust u v (sust x y zs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 5. Demostrar o refutar:
     sust y z (sust x y zs) = sust x z zs
  --------------------------------------------------------------------- 
*}

lemma 
  "sust y z (sust x y zs) = sust x z zs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 6. Definir la función
     borra :: "'a ⇒ 'a list ⇒ 'a list"
  tal que (borra x ys) es la lista obtenida borrando la primera
  ocurrencia del elemento x en la lista ys. Por ejemplo,
     borra (2::nat) [1,2,3,2] = [1,3,2]

  Nota: La función borra es equivalente a la predefinida remove1. 
  --------------------------------------------------------------------- 
*}

fun borra :: "'a ⇒ 'a list ⇒ 'a list" where
  "borra x ys = undefined"

text {* 
  --------------------------------------------------------------------- 
  Ejercicio 7. Definir la función
     borraTodas :: "'a ⇒ 'a list ⇒ 'a list"
  tal que (borraTodas x ys) es la lista obtenida borrando todas las
  ocurrencias del elemento x en la lista ys. Por ejemplo,
     borraTodas (2::nat) [1,2,3,2] = [1,3]
  --------------------------------------------------------------------- 
*}

fun borraTodas :: "'a ⇒ 'a list ⇒ 'a list" where
  "borraTodas x ys = undefined"

text {*
  --------------------------------------------------------------------- 
  Ejercicio 8.1. Demostrar o refutar automáticamente
     borra x (borraTodas x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borra x (borraTodas x xs) = borraTodas x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 8.2. Demostrar o refutar detalladamente 
     borra x (borraTodas x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borra x (borraTodas x xs) = borraTodas x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 9.1. Demostrar o refutar automáticamente
     borraTodas x (borraTodas x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borraTodas x xs) = borraTodas x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 9.2. Demostrar o refutar detalladamente
     borraTodas x (borraTodas x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borraTodas x xs) = borraTodas x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 10.1. Demostrar o refutar automáticamente
     borraTodas x (borra x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borra x xs) = borraTodas x xs"
oops
    
text {*
  --------------------------------------------------------------------- 
  Ejercicio 10.2. Demostrar o refutar detalladamente
     borraTodas x (borra x xs) = borraTodas x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borra x xs) = borraTodas x xs"
oops
    
text {*
  --------------------------------------------------------------------- 
  Ejercicio 11.1. Demostrar o automáticamente
     borra x (borra y xs) = borra y (borra x xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "borra x (borra y xs) = borra y (borra x xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 11.2. Demostrar o refutar detalladamente
     borra x (borra y xs) = borra y (borra x xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "borra x (borra y xs) = borra y (borra x xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 12.1. Demostrar o refutar automáticamente
     borraTodas x (borra y xs) = borra y (borraTodas x xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borra y xs) = borra y (borraTodas x xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 12.2. Demostrar o refutar detalladamente
     borraTodas x (borra y xs) = borra y (borraTodas x xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (borra y xs) = borra y (borraTodas x xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 13. Demostrar o refutar:
     borra y (sust x y xs) = borra x xs
  --------------------------------------------------------------------- 
*}

lemma 
  "borra y (sust x y xs) = borra x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 14. Demostrar o refutar:
     borraTodas y (sust x y xs) = borraTodas x xs"
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas y (sust x y xs) = borraTodas x xs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 15.1. Demostrar o refutar automáticamente
     sust x y (borraTodas x zs) = borraTodas x zs
  --------------------------------------------------------------------- 
*}

lemma 
  "sust x y (borraTodas x zs) = borraTodas x zs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 15.2. Demostrar o refutar detalladamente
     sust x y (borraTodas x zs) = borraTodas x zs
  --------------------------------------------------------------------- 
*}

lemma 
  "sust x y (borraTodas x zs) = borraTodas x zs"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 16. Demostrar o refutar:
     sust x y (borraTodas z zs) = borraTodas z (sust x y zs)
  --------------------------------------------------------------------- 
*}

lemma 
  "sust x y (borraTodas z zs) = borraTodas z (sust x y zs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 17. Demostrar o refutar:
     rev (borra x xs) = borra x (rev xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "rev (borra x xs) = borra x (rev xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 18.1. Demostrar o refutar automáticamente
     borraTodas x (xs@ys) = (borraTodas x xs)@(borraTodas x ys)
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (xs@ys) = (borraTodas x xs)@(borraTodas x ys)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 18.2. Demostrar o refutar detalladamente
     borraTodas x (xs@ys) = (borraTodas x xs)@(borraTodas x ys)
  --------------------------------------------------------------------- 
*}

lemma 
  "borraTodas x (xs@ys) = (borraTodas x xs)@(borraTodas x ys)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 19.1. Demostrar o refutar automáticamente
     rev (borraTodas x xs) = borraTodas x (rev xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "rev (borraTodas x xs) = borraTodas x (rev xs)"
oops

text {*
  --------------------------------------------------------------------- 
  Ejercicio 19.2. Demostrar o refutar detalladamente
     rev (borraTodas x xs) = borraTodas x (rev xs)
  --------------------------------------------------------------------- 
*}

lemma 
  "rev (borraTodas x xs) = borraTodas x (rev xs)"
oops

end