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Diferencia entre revisiones de «Relación 4»

De Lógica matemática y fundamentos (2017-18)
Línea 126: Línea 126:
 
       thus "p ⟶ r" by (rule impI)  
 
       thus "p ⟶ r" by (rule impI)  
 
       qed
 
       qed
 +
lemma ejercicio_4: inmbenber
 +
  assumes 1: "p ⟶ q" and
 +
          2: "q ⟶ r"
 +
  shows "p ⟶ r"
 +
proof -
 +
{ assume 3: "p"
 +
  have 4: "q" using 1 3 by (rule mp)
 +
  have 5: "r" using 2 4 by (rule mp) }
 +
thus "p ⟶ r" by (rule impI)
 +
qed
  
 
text {* ---------------------------------------------------------------  
 
text {* ---------------------------------------------------------------  
Línea 162: Línea 172:
 
   thus  "q ⟶ (p ⟶ r)" by (rule impI)
 
   thus  "q ⟶ (p ⟶ r)" by (rule impI)
 
       qed
 
       qed
 +
 +
lemma ejercicio_5: inmbenber
 +
  assumes 1: "p ⟶ (q ⟶ r)"
 +
  shows  "q ⟶ (p ⟶ r)"
 +
 +
proof -
 +
  {assume 2: "q"
 +
    {assume 3: "p"
 +
      have 4: "q ⟶ r" using 1 3 by (rule mp)
 +
      have 5: "r" using 4 2 by (rule mp) }
 +
    hence " p ⟶ r" by (rule impI) }
 +
  thus  "q ⟶ (p ⟶ r)" by (rule impI)
 +
qed
  
 
text {* ---------------------------------------------------------------  
 
text {* ---------------------------------------------------------------  

Revisión del 17:22 15 mar 2018

chapter {* R4: Deducción natural proposicional *}

theory R4
imports Main 
begin

text {*
  --------------------------------------------------------------------- 
  El objetivo de esta relación es demostrar cada uno de los ejercicios
  usando sólo las reglas básicas de deducción natural de la lógica
  proposicional (sin usar el método auto).

  Las reglas básicas de la deducción natural son las siguientes:
  · conjI:      ⟦P; Q⟧ ⟹ P ∧ Q
  · conjunct1:  P ∧ Q ⟹ P
  · conjunct2:  P ∧ Q ⟹ Q  
  · notnotD:    ¬¬ P ⟹ P
  · notnotI:    P ⟹ ¬¬ P
  · mp:         ⟦P ⟶ Q; P⟧ ⟹ Q 
  · mt:         ⟦F ⟶ G; ¬G⟧ ⟹ ¬F 
  · impI:       (P ⟹ Q) ⟹ P ⟶ Q
  · disjI1:     P ⟹ P ∨ Q
  · disjI2:     Q ⟹ P ∨ Q
  · disjE:      ⟦P ∨ Q; P ⟹ R; Q ⟹ R⟧ ⟹ R 
  · FalseE:     False ⟹ P
  · notE:       ⟦¬P; P⟧ ⟹ R
  · notI:       (P ⟹ False) ⟹ ¬P
  · iffI:       ⟦P ⟹ Q; Q ⟹ P⟧ ⟹ P = Q
  · iffD1:      ⟦Q = P; Q⟧ ⟹ P 
  · iffD2:      ⟦P = Q; Q⟧ ⟹ P
  · ccontr:     (¬P ⟹ False) ⟹ P
  --------------------------------------------------------------------- 
*}

text {*
  Se usarán las reglas notnotI y mt que demostramos a continuación. *}

lemma notnotI: "P ⟹ ¬¬ P"
by auto

lemma mt: "⟦F ⟶ G; ¬G⟧ ⟹ ¬F"
by auto

section {* Implicaciones *}

text {* --------------------------------------------------------------- 
  Ejercicio 1. Demostrar
       p ⟶ q, p ⊢ q
  ------------------------------------------------------------------ *}

lemma ejercicio_1: josrodjim2 carmarria inmbenber
  assumes 1: "p ⟶ q" and
          2: "p"
  shows "q"

proof -
  show 1: "q" using 1 2  by (rule mp)
qed

text {* --------------------------------------------------------------- 
  Ejercicio 2. Demostrar
     p ⟶ q, q ⟶ r, p ⊢ r
  ------------------------------------------------------------------ *}

lemma ejercicio_2: josrodjim2 carmarria inmbenber
  assumes 1:  "p ⟶ q" and
          2:  "q ⟶ r" and
          3:  "p" 
  shows "r"


proof-
  have 4: "q" using 1 3 by (rule mp)
 show  "r" using 2 4  by (rule mp)
qed


text {* --------------------------------------------------------------- 
  Ejercicio 3. Demostrar
     p ⟶ (q ⟶ r), p ⟶ q, p ⊢ r
  ------------------------------------------------------------------ *}

lemma ejercicio_3: josrodjim2 carmarria inmbenber
  assumes 1: "p ⟶ (q ⟶ r)" and
          2: "p ⟶ q" and
          3:  "p"
  shows "r"

proof-
  have 4: "q" using 2 3 by (rule mp)
  have 5: "(q ⟶ r)" using 1 3  by (rule mp)
  show "r" using 5 4 by (rule mp)

qed

text {* --------------------------------------------------------------- 
  Ejercicio 4. Demostrar
     p ⟶ q, q ⟶ r ⊢ p ⟶ r
  ------------------------------------------------------------------ *}

lemma ejercicio_4: josrodjim2
  assumes 1: "p ⟶ q" and
          2: "q ⟶ r" 
  shows "p ⟶ r"

proof-
  {  assume 3: "p"
  have 4: "q" using 1 3 by (rule mp)
  have 5: "r" using 2 4 by (rule mp)}
  hence 6: "p ⟶ r"  using 3 5  by (rule impI)
  show "p⟶r" using 6 by this

oops

lemma ejercicio_4: carmarria
  assumes 1: "p ⟶ q" and
          2: "q ⟶ r" 
  shows "p ⟶ r"
proof -
 { assume 3: p 
   have 4: "q" using 1 3 by (rule mp)
   have 5: "r" using 2 4 by (rule mp)
 }
      thus "p ⟶ r" by (rule impI) 
      qed
lemma ejercicio_4: inmbenber
  assumes 1: "p ⟶ q" and
          2: "q ⟶ r" 
  shows "p ⟶ r"
proof -
 { assume 3: "p" 
   have 4: "q" using 1 3 by (rule mp)
   have 5: "r" using 2 4 by (rule mp) }
 thus "p ⟶ r" by (rule impI) 
qed

text {* --------------------------------------------------------------- 
  Ejercicio 5. Demostrar
     p ⟶ (q ⟶ r) ⊢ q ⟶ (p ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_5: josrodjim2 (NO DA ERROR, PERO SI ALERTA)
  assumes "p ⟶ (q ⟶ r)" 
  shows   "q ⟶ (p ⟶ r)"

proof -

  { assume 2: "q"
    {assume 3: "p"
      have 4: "q ⟶r" using 1 3 by (rule mp)
      have 5: "r" using 4 2 by (rule mp)}
    hence 6: "p⟶r" using 3 5 by (rule impI)}
    hence 7: "q ⟶ (p ⟶ r)" using 2 6 by (rule impI)
    show "q ⟶ (p ⟶ r)" using 7 by this
 
oops

lemma ejercicio_5: carmarria 
  assumes 1: "p ⟶ (q ⟶ r)" 
  shows   "q ⟶ (p ⟶ r)"

proof - 
  {assume 2: q
    {assume 3: p 
      have 4: "q ⟶ r" using 1 3 by (rule mp)
      have 5: "r" using 4 2 by (rule mp)
    }
    hence " p ⟶ r" by (rule impI)
  }
  thus  "q ⟶ (p ⟶ r)" by (rule impI)
      qed

lemma ejercicio_5: inmbenber
  assumes 1: "p ⟶ (q ⟶ r)" 
  shows   "q ⟶ (p ⟶ r)"

proof - 
  {assume 2: "q"
    {assume 3: "p" 
      have 4: "q ⟶ r" using 1 3 by (rule mp)
      have 5: "r" using 4 2 by (rule mp) }
    hence " p ⟶ r" by (rule impI) }
  thus  "q ⟶ (p ⟶ r)" by (rule impI)
qed

text {* --------------------------------------------------------------- 
  Ejercicio 6. Demostrar
     p ⟶ (q ⟶ r) ⊢ (p ⟶ q) ⟶ (p ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_6:josrodjim2
  assumes 1: "p ⟶ (q ⟶ r)" 
  shows   "(p ⟶ q) ⟶ (p ⟶ r)"

proof- 

  {assume 2: "p⟶q" 
    {assume 3:  "p"
      have 4: "q⟶r" using 1 3 by (rule mp)
      have 5: "q" using 2 3 by (rule mp)
      have 6: "r" using 4 5 by (rule mp)}
      hence 7: "p⟶r" using 3 6 by (rule impI)}
      hence  8: "(p ⟶ q) ⟶ (p ⟶ r)" using 2 7 by (rule impI)
      show  "(p ⟶ q) ⟶ (p ⟶ r)" using 8 by this
oops

lemma ejercicio_6: carmarria
  assumes 1: "p ⟶ (q ⟶ r)" 
  shows   "(p ⟶ q) ⟶ (p ⟶ r)"
proof -
  {assume 2: "p ⟶ q"
    {assume 3: "p"
      have 4: "q" using 2 3 by (rule mp)
      have 5: "q ⟶ r" using 1 3 by (rule mp)
      have 6: "r" using 5 4 by (rule mp)
    }
    hence 7: "p ⟶ r" by (rule impI)
  }
  thus "(p ⟶ q) ⟶ (p ⟶ r)" by (rule impI)
      qed

text {* --------------------------------------------------------------- 
  Ejercicio 7. Demostrar
     p ⊢ q ⟶ p
  ------------------------------------------------------------------ *}

lemma ejercicio_7: (NO SE QUE ESTA MAL)
  assumes "p"  
  shows   "q ⟶ p"

proof -

  { assume 2:  "q"
    have 3: "p" using 1
 }
  hence  4: "q⟶p" using  2 3   by (rule impI)

show "q⟶p" using 4  by this
    
oops

lemma ejercicio_7: carmarria
  assumes 1: "p"  
  shows   "q ⟶ p"
proof -
  {assume 2: q
    have 3: "p" using 1 by this}
  thus "q ⟶ p" by (rule impI)
      qed
      
text {* --------------------------------------------------------------- 
  Ejercicio 8. Demostrar
     ⊢ p ⟶ (q ⟶ p)
  ------------------------------------------------------------------ *}

lemma ejercicio_8: carmarria
  "p ⟶ (q ⟶ p)"

proof -
  {assume 1: p
    {assume 2: q
      have 3: p using 1 by this
    }
    hence 4: "q ⟶ p" by (rule impI)
  }
  thus "p ⟶ (q ⟶ p)" by (rule impI)
      qed
text {* --------------------------------------------------------------- 
  Ejercicio 9. Demostrar
     p ⟶ q ⊢ (q ⟶ r) ⟶ (p ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_9: carmarria
  assumes 1: "p ⟶ q" 
  shows   "(q ⟶ r) ⟶ (p ⟶ r)"
proof -
  {assume 2: "q ⟶ r"
    {assume 3: "p"
      have 4: "q" using 1 3 by (rule mp)
      have 5: "r" using 2 4 by (rule mp)
    }
    hence 6: "p ⟶ r" by (rule impI)
  }
  thus 7: "(q ⟶ r) ⟶ (p ⟶ r)" by (rule impI)
      qed

text {* --------------------------------------------------------------- 
  Ejercicio 10. Demostrar
     p ⟶ (q ⟶ (r ⟶ s)) ⊢ r ⟶ (q ⟶ (p ⟶ s))
  ------------------------------------------------------------------ *}

lemma ejercicio_10: carmarria
  assumes 1: "p ⟶ (q ⟶ (r ⟶ s))" 
  shows   "r ⟶ (q ⟶ (p ⟶ s))"

proof-
  {assume 2: r
    {assume 3: q
      {assume 4: p
        have 5: "q ⟶ (r ⟶ s)" using 1 4 by (rule mp)
        have 6: "r ⟶ s" using 5 3 by (rule mp)
        have 7: s using 6 2 by (rule mp)
      }
      hence 8: "p ⟶ s" by (rule impI)
    }
    hence 9: "q ⟶ (p ⟶ s)" by (rule impI)
  }
  thus  "r ⟶ (q ⟶ (p ⟶ s))" by (rule impI)
      qed

text {* --------------------------------------------------------------- 
  Ejercicio 11. Demostrar
     ⊢ (p ⟶ (q ⟶ r)) ⟶ ((p ⟶ q) ⟶ (p ⟶ r))
  ------------------------------------------------------------------ *}

lemma ejercicio_11: carmarria
  "(p ⟶ (q ⟶ r)) ⟶ ((p ⟶ q) ⟶ (p ⟶ r))"
proof - 
  {assume 1: "p ⟶ (q ⟶ r)"
    {assume 2: "p ⟶ q"
      {assume 3: p
        have 4: q using 2 3 by (rule mp)
        have 5: "q ⟶ r" using 1 3 by (rule mp)
        have 6: r using 5 4 by (rule mp)
      }
      hence 7: "p ⟶ r" by (rule impI)
    }
    hence 8: "(p ⟶ q) ⟶ (p ⟶ r)" by (rule impI)
  }
  thus "(p ⟶ (q ⟶ r)) ⟶ ((p ⟶ q) ⟶ (p ⟶ r))" by (rule impI)
qed

text {* --------------------------------------------------------------- 
  Ejercicio 12. Demostrar
     (p ⟶ q) ⟶ r ⊢ p ⟶ (q ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_12: carmarria
  assumes 1: "(p ⟶ q) ⟶ r" 
  shows   "p ⟶ (q ⟶ r)"
proof -
  {assume 2: p
    {assume 3: q
      {assume 4: p
        have 5: q using 3 by this
      }
      hence 6: "p ⟶ q" by (rule impI)
      have 7: r using 1 6 by (rule mp)
    }
    hence 8: "q ⟶ r" by (rule impI)
  }
  thus "p ⟶ (q ⟶ r)" by (rule impI)
qed

section {* Conjunciones *}

text {* --------------------------------------------------------------- 
  Ejercicio 13. Demostrar
     p, q ⊢  p ∧ q
  ------------------------------------------------------------------ *}

lemma ejercicio_13:
  assumes "p"
          "q" 
  shows "p ∧ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 14. Demostrar
     p ∧ q ⊢ p
  ------------------------------------------------------------------ *}

lemma ejercicio_14:
  assumes "p ∧ q"  
  shows   "p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 15. Demostrar
     p ∧ q ⊢ q
  ------------------------------------------------------------------ *}

lemma ejercicio_15:
  assumes "p ∧ q" 
  shows   "q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 16. Demostrar
     p ∧ (q ∧ r) ⊢ (p ∧ q) ∧ r
  ------------------------------------------------------------------ *}

lemma ejercicio_16:
  assumes "p ∧ (q ∧ r)"
  shows   "(p ∧ q) ∧ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 17. Demostrar
     (p ∧ q) ∧ r ⊢ p ∧ (q ∧ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_17:
  assumes "(p ∧ q) ∧ r" 
  shows   "p ∧ (q ∧ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 18. Demostrar
     p ∧ q ⊢ p ⟶ q
  ------------------------------------------------------------------ *}

lemma ejercicio_18:
  assumes "p ∧ q" 
  shows   "p ⟶ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 19. Demostrar
     (p ⟶ q) ∧ (p ⟶ r) ⊢ p ⟶ q ∧ r   
  ------------------------------------------------------------------ *}

lemma ejercicio_19:
  assumes "(p ⟶ q) ∧ (p ⟶ r)" 
  shows   "p ⟶ q ∧ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 20. Demostrar
     p ⟶ q ∧ r ⊢ (p ⟶ q) ∧ (p ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_20:
  assumes "p ⟶ q ∧ r" 
  shows   "(p ⟶ q) ∧ (p ⟶ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 21. Demostrar
     p ⟶ (q ⟶ r) ⊢ p ∧ q ⟶ r
  ------------------------------------------------------------------ *}

lemma ejercicio_21:
  assumes "p ⟶ (q ⟶ r)" 
  shows   "p ∧ q ⟶ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 22. Demostrar
     p ∧ q ⟶ r ⊢ p ⟶ (q ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_22:
  assumes "p ∧ q ⟶ r" 
  shows   "p ⟶ (q ⟶ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 23. Demostrar
     (p ⟶ q) ⟶ r ⊢ p ∧ q ⟶ r
  ------------------------------------------------------------------ *}

lemma ejercicio_23:
  assumes "(p ⟶ q) ⟶ r" 
  shows   "p ∧ q ⟶ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 24. Demostrar
     p ∧ (q ⟶ r) ⊢ (p ⟶ q) ⟶ r
  ------------------------------------------------------------------ *}

lemma ejercicio_24:
  assumes "p ∧ (q ⟶ r)" 
  shows   "(p ⟶ q) ⟶ r"
oops

section {* Disyunciones *}

text {* --------------------------------------------------------------- 
  Ejercicio 25. Demostrar
     p ⊢ p ∨ q
  ------------------------------------------------------------------ *}

lemma ejercicio_25:
  assumes "p"
  shows   "p ∨ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 26. Demostrar
     q ⊢ p ∨ q
  ------------------------------------------------------------------ *}

lemma ejercicio_26:
  assumes "q"
  shows   "p ∨ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 27. Demostrar
     p ∨ q ⊢ q ∨ p
  ------------------------------------------------------------------ *}

lemma ejercicio_27:
  assumes "p ∨ q"
  shows   "q ∨ p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 28. Demostrar
     q ⟶ r ⊢ p ∨ q ⟶ p ∨ r
  ------------------------------------------------------------------ *}

lemma ejercicio_28:
  assumes "q ⟶ r" 
  shows   "p ∨ q ⟶ p ∨ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 29. Demostrar
     p ∨ p ⊢ p
  ------------------------------------------------------------------ *}

lemma ejercicio_29:
  assumes "p ∨ p"
  shows   "p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 30. Demostrar
     p ⊢ p ∨ p
  ------------------------------------------------------------------ *}

lemma ejercicio_30:
  assumes "p" 
  shows   "p ∨ p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 31. Demostrar
     p ∨ (q ∨ r) ⊢ (p ∨ q) ∨ r
  ------------------------------------------------------------------ *}

lemma ejercicio_31:
  assumes "p ∨ (q ∨ r)" 
  shows   "(p ∨ q) ∨ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 32. Demostrar
     (p ∨ q) ∨ r ⊢ p ∨ (q ∨ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_32:
  assumes "(p ∨ q) ∨ r" 
  shows   "p ∨ (q ∨ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 33. Demostrar
     p ∧ (q ∨ r) ⊢ (p ∧ q) ∨ (p ∧ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_33:
  assumes "p ∧ (q ∨ r)" 
  shows   "(p ∧ q) ∨ (p ∧ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 34. Demostrar
     (p ∧ q) ∨ (p ∧ r) ⊢ p ∧ (q ∨ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_34:
  assumes "(p ∧ q) ∨ (p ∧ r)" 
  shows   "p ∧ (q ∨ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 35. Demostrar
     p ∨ (q ∧ r) ⊢ (p ∨ q) ∧ (p ∨ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_35:
  assumes "p ∨ (q ∧ r)" 
  shows   "(p ∨ q) ∧ (p ∨ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 36. Demostrar
     (p ∨ q) ∧ (p ∨ r) ⊢ p ∨ (q ∧ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_36:
  assumes "(p ∨ q) ∧ (p ∨ r)"
  shows   "p ∨ (q ∧ r)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 37. Demostrar
     (p ⟶ r) ∧ (q ⟶ r) ⊢ p ∨ q ⟶ r
  ------------------------------------------------------------------ *}

lemma ejercicio_37:
  assumes "(p ⟶ r) ∧ (q ⟶ r)" 
  shows   "p ∨ q ⟶ r"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 38. Demostrar
     p ∨ q ⟶ r ⊢ (p ⟶ r) ∧ (q ⟶ r)
  ------------------------------------------------------------------ *}

lemma ejercicio_38:
  assumes "p ∨ q ⟶ r" 
  shows   "(p ⟶ r) ∧ (q ⟶ r)"
oops

section {* Negaciones *}

text {* --------------------------------------------------------------- 
  Ejercicio 39. Demostrar
     p ⊢ ¬¬p
  ------------------------------------------------------------------ *}

lemma ejercicio_39:
  assumes "p"
  shows   "¬¬p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 40. Demostrar
     ¬p ⊢ p ⟶ q
  ------------------------------------------------------------------ *}

lemma ejercicio_40:
  assumes "¬p" 
  shows   "p ⟶ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 41. Demostrar
     p ⟶ q ⊢ ¬q ⟶ ¬p
  ------------------------------------------------------------------ *}

lemma ejercicio_41:
  assumes "p ⟶ q"
  shows   "¬q ⟶ ¬p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 42. Demostrar
     p∨q, ¬q ⊢ p
  ------------------------------------------------------------------ *}

lemma ejercicio_42:
  assumes "p∨q"
          "¬q" 
  shows   "p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 42. Demostrar
     p ∨ q, ¬p ⊢ q
  ------------------------------------------------------------------ *}

lemma ejercicio_43:
  assumes "p ∨ q"
          "¬p" 
  shows   "q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 40. Demostrar
     p ∨ q ⊢ ¬(¬p ∧ ¬q)
  ------------------------------------------------------------------ *}

lemma ejercicio_44:
  assumes "p ∨ q" 
  shows   "¬(¬p ∧ ¬q)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 45. Demostrar
     p ∧ q ⊢ ¬(¬p ∨ ¬q)
  ------------------------------------------------------------------ *}

lemma ejercicio_45:
  assumes "p ∧ q" 
  shows   "¬(¬p ∨ ¬q)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 46. Demostrar
     ¬(p ∨ q) ⊢ ¬p ∧ ¬q
  ------------------------------------------------------------------ *}

lemma ejercicio_46:
  assumes "¬(p ∨ q)" 
  shows   "¬p ∧ ¬q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 47. Demostrar
     ¬p ∧ ¬q ⊢ ¬(p ∨ q)
  ------------------------------------------------------------------ *}

lemma ejercicio_47:
  assumes "¬p ∧ ¬q" 
  shows   "¬(p ∨ q)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 48. Demostrar
     ¬p ∨ ¬q ⊢ ¬(p ∧ q)
  ------------------------------------------------------------------ *}

lemma ejercicio_48:
  assumes "¬p ∨ ¬q"
  shows   "¬(p ∧ q)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 49. Demostrar
     ⊢ ¬(p ∧ ¬p)
  ------------------------------------------------------------------ *}

lemma ejercicio_49:
  "¬(p ∧ ¬p)"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 50. Demostrar
     p ∧ ¬p ⊢ q
  ------------------------------------------------------------------ *}

lemma ejercicio_50:
  assumes "p ∧ ¬p" 
  shows   "q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 51. Demostrar
     ¬¬p ⊢ p
  ------------------------------------------------------------------ *}

lemma ejercicio_51:
  assumes "¬¬p"
  shows   "p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 52. Demostrar
     ⊢ p ∨ ¬p
  ------------------------------------------------------------------ *}

lemma ejercicio_52:
  "p ∨ ¬p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 53. Demostrar
     ⊢ ((p ⟶ q) ⟶ p) ⟶ p
  ------------------------------------------------------------------ *}

lemma ejercicio_53:
  "((p ⟶ q) ⟶ p) ⟶ p"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 54. Demostrar
     ¬q ⟶ ¬p ⊢ p ⟶ q
  ------------------------------------------------------------------ *}

lemma ejercicio_54:
  assumes "¬q ⟶ ¬p"
  shows   "p ⟶ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 55. Demostrar
     ¬(¬p ∧ ¬q) ⊢ p ∨ q
  ------------------------------------------------------------------ *}

lemma ejercicio_55:
  assumes "¬(¬p ∧ ¬q)"
  shows   "p ∨ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 56. Demostrar
     ¬(¬p ∨ ¬q) ⊢ p ∧ q
  ------------------------------------------------------------------ *}

lemma ejercicio_56:
  assumes "¬(¬p ∨ ¬q)" 
  shows   "p ∧ q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 57. Demostrar
     ¬(p ∧ q) ⊢ ¬p ∨ ¬q
  ------------------------------------------------------------------ *}

lemma ejercicio_57:
  assumes "¬(p ∧ q)"
  shows   "¬p ∨ ¬q"
oops

text {* --------------------------------------------------------------- 
  Ejercicio 58. Demostrar
     ⊢ (p ⟶ q) ∨ (q ⟶ p)
  ------------------------------------------------------------------ *}

lemma ejercicio_58:
  "(p ⟶ q) ∨ (q ⟶ p)"
oops

end
Obtenido de «https://www.glc.us.es/~jalonso/LMF2018/index.php?title=Relación_4&oldid=113»