Relación 1

header {* 1ª relación de ejercicios *}

theory Relacion_1
imports Main 
begin

text {*
  --------------------------------------------------------------------- 
  Demostrar los siguientes lemas usando sólo las reglas básicas de
  deducción natural de la lógica proposicional.
  --------------------------------------------------------------------- 
*}

lemma I: "A ⟶ A"
oops

lemma "A ∧ B ⟶ B ∧ A"
oops

lemma "A ∧ B ⟶ A ∨ B"
proof
  assume "A ∧ B"
  hence "A" ..
  thus "A ∨ B" ..
qed 

lemma "(A ∨ B) ∨ C ⟶ A ∨ (B ∨ C)"
oops

lemma K: "A ⟶ (B ⟶ A)"
oops

lemma S: "(A ⟶ (B ⟶ C)) ⟶ ((A ⟶ B) ⟶ (A ⟶ C))"
oops

lemma "(A ⟶ B) ⟶ ((B ⟶ C) ⟶ (A ⟶ C))"
proof
  assume 1:"(A ⟶ B)"
  show "((B ⟶ C) ⟶ (A ⟶ C))"
  proof 
    assume 2:"(B ⟶ C)"
    show "(A ⟶ C)"
    proof
      assume 3:"A"
      from 1 3 have 4:"B" by (rule mp)
      from 2 4 show "C" by (rule mp)
    qed
  qed
qed

lemma "¬¬A ⟶ A"
oops

lemma "A ⟶ ¬¬A"
oops

lemma "(¬A ⟶ B) ⟶ (¬B ⟶ A)"
oops

lemma "((A ⟶ B) ⟶ A) ⟶ A"
oops

lemma "A ∨ ¬A"
oops

lemma "(¬(A ∧ B)) = (¬A ∨ ¬B)"
oops

end