Relación 4

chapter {* R4: Cuantificadores sobre listas *}

theory R4_Cuantificadores_sobre_listas
imports Main 
begin

text {* 
  --------------------------------------------------------------------- 
  Ejercicio 1. Definir la función 
     todos :: ('a ⇒ bool) ⇒ 'a list ⇒ bool
  tal que (todos p xs) se verifica si todos los elementos de la lista 
  xs cumplen la propiedad p. Por ejemplo, se verifica 
     todos (λx. 1<length x) [[2,1,4],[1,3]]
     ¬todos (λx. 1<length x) [[2,1,4],[3]]

  Nota: La función todos es equivalente a la predefinida list_all. 
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
fun todos :: "('a ⇒ bool) ⇒ 'a list ⇒ bool" where
  "todos p [] = True" |
  "todos p (x#xs) = (p x ∧ todos p xs)"

text {* 
  --------------------------------------------------------------------- 
  Ejercicio 2. Definir la función 
     algunos :: ('a ⇒ bool) ⇒ 'a list ⇒ bool
  tal que (algunos p xs) se verifica si algunos elementos de la lista 
  xs cumplen la propiedad p. Por ejemplo, se verifica 
     algunos (λx. 1<length x) [[2,1,4],[3]]
     ¬algunos (λx. 1<length x) [[],[3]]"

  Nota: La función algunos es equivalente a la predefinida list_ex. 
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
fun algunos  :: "('a ⇒ bool) ⇒ 'a list ⇒ bool" where
  "algunos p [] = False" |
  "algunos p (x#xs) = (p x ∨ algunos p xs)"

text {*
  --------------------------------------------------------------------- 
  Ejercicio 3.1. Demostrar o refutar automáticamente 
     todos (λx. P x ∧ Q x) xs = (todos P xs ∧ todos Q xs)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "todos (λx. P x ∧ Q x) xs = (todos P xs ∧ todos Q xs)"
  by (induct xs) auto

text {*
  --------------------------------------------------------------------- 
  Ejercicio 3.2. Demostrar o refutar detalladamente
     todos (λx. P x ∧ Q x) xs = (todos P xs ∧ todos Q xs)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "todos (λx. P x ∧ Q x) xs = (todos P xs ∧ todos Q xs)" (is "?P P Q xs")
proof (induct xs)
  fix P Q
  show "?P P Q []" by simp
next
  fix a xs
  assume HI: "?P P Q xs"
  have "todos (λx. P x ∧ Q x) (a#xs) = ((P a ∧ Q a) ∧ todos (λx. P x ∧ Q x) xs)" by simp
  also have "... = (P a ∧ Q a ∧ todos P xs ∧ todos Q xs)" using HI by simp
  also have "... = ((P a ∧ todos P xs) ∧ (Q a ∧ todos Q xs))" by (simp add: HOL.conj_comms)
  also have "... = (todos P (a#xs) ∧ (Q a ∧ todos Q xs))" by simp
  finally show "todos (λx. P x ∧ Q x) (a#xs) = (todos P (a#xs) ∧ todos Q (a#xs))" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 4.1. Demostrar o refutar automáticamente
     todos P (x @ y) = (todos P x ∧ todos P y)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "todos P (x @ y) = (todos P x ∧ todos P y)"
  by (induct x) auto

text {*
  --------------------------------------------------------------------- 
  Ejercicio 4.2. Demostrar o refutar detalladamente
     todos P (x @ y) = (todos P x ∧ todos P y)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma todos_append:
  "todos P (x @ y) = (todos P x ∧ todos P y)"
proof (induct x)
  show "todos P ([] @ y) = (todos P [] ∧ todos P y)" by simp
next
  fix a x
  assume HI: "todos P (x @ y) = (todos P x ∧ todos P y)"
  have "todos P ((a#x) @ y) = (P a ∧ todos P (x @ y))" by simp
  also have "... = (P a ∧ todos P x ∧ todos P y)" using HI by simp
  finally show "todos P ((a#x) @ y) = (todos P (a#x) ∧ todos P y)" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 5.1. Demostrar o refutar automáticamente 
     todos P (rev xs) = todos P xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "todos P (rev xs) = todos P xs"
  by (induct xs) (simp_all add: HOL.conj_comms todos_append)

text {*
  --------------------------------------------------------------------- 
  Ejercicio 5.2. Demostrar o refutar detalladamente
     todos P (rev xs) = todos P xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "todos P (rev xs) = todos P xs"
proof (induct xs)
  show "todos P (rev []) = todos P []" by simp
next
  fix a xs
  assume HI: "todos P (rev xs) = todos P xs"
  have "todos P (rev (a#xs)) = todos P ((rev xs) @ [a])" by simp
  also have "... = (todos P (rev xs) ∧ todos P [a])" by (simp add: todos_append)
  also have "... = (todos P xs ∧ todos P [a])" using HI by simp
  also have "... = (todos P xs ∧ P a)" by simp
  also have "... = (P a ∧ todos P xs)" by (simp add: HOL.conj_comms)
  finally show "todos P (rev (a#xs)) = todos P (a#xs)" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 6. Demostrar o refutar:
    algunos (λx. P x ∧ Q x) xs = (algunos P xs ∧ algunos Q xs)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos (λx. P x ∧ Q x) xs = (algunos P xs ∧ algunos Q xs)"
  nitpick
  oops

(* Contraejemplo *)
value "algunos (λx. even x ∧ odd x) [1, 2::nat] ≠
  algunos even [1, 2::nat] ∧ algunos odd [1, 2::nat]"

text {*
  --------------------------------------------------------------------- 
  Ejercicio 7.1. Demostrar o refutar automáticamente 
     algunos P (map f xs) = algunos (P ∘ f) xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P (map f xs) = algunos (P o f) xs"
  by (induct xs) auto

text {*
  --------------------------------------------------------------------- 
  Ejercicio 7.2. Demostrar o refutar detalladamente
     algunos P (map f xs) = algunos (P ∘ f) xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P (map f xs) = algunos (P o f) xs"
proof (induct xs)
  show "algunos P (map f []) = algunos (P ∘ f) []" by simp
next
  fix a xs
  assume HI: "algunos P (map f xs) = algunos (P ∘ f) xs"
  have "algunos P (map f (a#xs)) = (P (f a) ∨ algunos P (map f xs))" by simp
  also have "... = (P (f a) ∨ algunos (P ∘ f) xs)" using HI by simp
  also have "... = ((P ∘ f) a ∨ algunos (P ∘ f) xs)" by simp
  finally show "algunos P (map f (a#xs)) = algunos (P ∘ f) (a#xs)" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 8.1. Demostrar o refutar automáticamente 
     algunos P (xs @ ys) = (algunos P xs ∨ algunos P ys)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P (xs @ ys) = (algunos P xs ∨ algunos P ys)"
  by (induct xs) auto

text {*
  --------------------------------------------------------------------- 
  Ejercicio 8.2. Demostrar o refutar detalladamente
     algunos P (xs @ ys) = (algunos P xs ∨ algunos P ys)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma algunos_append:
  "algunos P (xs @ ys) = (algunos P xs ∨ algunos P ys)"
proof (induct xs)
  show "algunos P ([] @ ys) = (algunos P [] ∨ algunos P ys)" by simp
next
  fix a xs
  assume HI: "algunos P (xs @ ys) = (algunos P xs ∨ algunos P ys)"
  have "algunos P ((a#xs) @ ys) = (P a ∨ algunos P (xs @ ys))" by simp
  also have "... = (P a ∨ algunos P xs ∨ algunos P ys)" using HI by simp
  finally show  "algunos P ((a#xs) @ ys) = (algunos P (a#xs) ∨ algunos P ys)" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 9.1. Demostrar o refutar automáticamente
     algunos P (rev xs) = algunos P xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P (rev xs) = algunos P xs"
  by (induct xs) (simp_all add: HOL.disj_comms algunos_append)

text {*
  --------------------------------------------------------------------- 
  Ejercicio 9.2. Demostrar o refutar detalladamente
     algunos P (rev xs) = algunos P xs
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P (rev xs) = algunos P xs"
proof (induct xs)
  show "algunos P (rev []) = algunos P []" by simp
next
  fix a xs
  assume HI: "algunos P (rev xs) = algunos P xs"
  have "algunos P (rev (a#xs)) = algunos P (rev xs @ [a])" by simp
  also have "... = (algunos P (rev xs) ∨ algunos P [a])" by (simp add: algunos_append)
  also have "... = (algunos P xs ∨ algunos P [a])" using HI by simp
  also have "... = (algunos P xs ∨ P a)" by simp
  also have "... = (P a ∨ algunos P xs)" by (simp add: HOL.disj_comms)
  finally show "algunos P (rev (a#xs)) = algunos P (a#xs)" by simp
qed

text {*
  --------------------------------------------------------------------- 
  Ejercicio 10. Encontrar un término no trivial Z tal que sea cierta la 
  siguiente ecuación:
     algunos (λx. P x ∨ Q x) xs = Z
  y demostrar la equivalencia de forma automática y detallada.
  --------------------------------------------------------------------- 
*}

text {*
  --------------------------------------------------------------------- 
  Ejercicio 11.1. Demostrar o refutar automáticamente
     algunos P xs = (¬ todos (λx. (¬ P x)) xs)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P xs = (¬ todos (λx. (¬ P x)) xs)"
  by (induct xs) auto

text {*
  --------------------------------------------------------------------- 
  Ejercicio 11.2. Demostrar o refutar datalladamente
     algunos P xs = (¬ todos (λx. (¬ P x)) xs)
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "algunos P xs = (¬ todos (λx. (¬ P x)) xs)"
proof (induct xs)
  have "algunos P [] = False" by simp
  also have "... = (¬ todos (λx. (¬ P x)) [])" by simp
  finally show "algunos P [] = (¬ todos (λx. (¬ P x)) [])" by simp
next
  fix a xs
  assume HI: "algunos P xs = (¬ todos (λx. (¬ P x)) xs)"
  have "(¬ todos (λx. (¬ P x)) (a#xs)) = (¬ ((¬ P a) ∧ todos (λx. (¬ P x)) xs))" by simp
  also have "... = (P a ∨ ¬ todos (λx. (¬ P x)) xs)" by simp
  also have "... = (P a ∨ algunos P xs)" using HI by simp
  finally show "algunos P (a#xs) = (¬ todos (λx. (¬ P x)) (a#xs))" by simp
qed
     
text {*
  --------------------------------------------------------------------- 
  Ejercicio 12. Definir la funcion primitiva recursiva 
     estaEn :: 'a ⇒ 'a list ⇒ bool
  tal que (estaEn x xs) se verifica si el elemento x está en la lista
  xs. Por ejemplo, 
     estaEn (2::nat) [3,2,4] = True
     estaEn (1::nat) [3,2,4] = False
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
fun estaEn :: "'a ⇒ 'a list ⇒ bool" where
  "estaEn x [] = False" |
  "estaEn x (y#xs) = (x=y ∨ estaEn x xs)"

text {*
  --------------------------------------------------------------------- 
  Ejercicio 13. Expresar la relación existente entre estaEn y algunos. 
  Demostrar dicha relación de forma automática y detallada.
  --------------------------------------------------------------------- 
*}

(* pabalagon *)
lemma "estaEn x xs = algunos (λa. x=a) xs"
  by (induct xs) auto

(* pabalagon *)
lemma "estaEn x xs = algunos (λa. x=a) xs"
proof (induct xs)
  show "estaEn x [] = algunos (λa. x=a) []" by simp
next
  fix y xs
  assume HI: "estaEn x xs = algunos (λa. x=a) xs"
  have "estaEn x (y#xs) = (x=y ∨ estaEn x xs)" by simp
  also have "... = (x=y ∨ algunos (λa. x=a) xs)" using HI by simp
  finally show "estaEn x (y#xs) = algunos (λa. x=a) (y#xs)" by simp
qed

end