+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
(* Istruzioni:
http://mowgli.cs.unibo.it/~tassi/exercise-natural_deduction.html
con premesse multiple sono seguite da `[`, `|` e `]`.
Ad esempio:
- apply rule (∧_i (A∨B) ⊥);
+ apply rule (∧#i (A∨B) ⊥);
[ …continua qui il sottoalbero per (A∨B)
| …continua qui il sottoalbero per ⊥
]
scrivete sopra la linea che rappresenta l'applicazione della
regola stessa. Ad esempio:
- aply rule (∨_e (A∨B) [h1] C [h2] C);
+ aply rule (∨#e (A∨B) [h1] C [h2] C);
[ …prova di (A∨B)
| …prova di C sotto l'ipotesi A (di nome h1)
| …prova di C sotto l'ipotesi B (di nome h2)
(* qui inizia l'albero, eseguite passo passo osservando come
si modifica l'albero. *)
apply rule (RAA [H] (⊥)).
-apply rule (¬_e (¬(A ∨ ¬A)) (A ∨ ¬A));
+apply rule (¬#e (¬(A ∨ ¬A)) (A ∨ ¬A));
[ apply rule (discharge [H]).
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬¬A) (¬A));
- [ apply rule (¬_i [K] (⊥)).
- apply rule (¬_e (¬(A ∨ ¬A)) (A ∨ ¬A));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬¬A) (¬A));
+ [ apply rule (¬#i [K] (⊥)).
+ apply rule (¬#e (¬(A ∨ ¬A)) (A ∨ ¬A));
[ (*BEGIN*)apply rule (discharge [H]).(*END*)
- | (*BEGIN*)apply rule (∨_i_r (¬A)).
+ | (*BEGIN*)apply rule (∨#i_r (¬A)).
apply rule (discharge [K]).(*END*)
]
- | (*BEGIN*)apply rule (¬_i [K] (⊥)).
- apply rule (¬_e (¬(A ∨ ¬A)) (A ∨ ¬A));
+ | (*BEGIN*)apply rule (¬#i [K] (⊥)).
+ apply rule (¬#e (¬(A ∨ ¬A)) (A ∨ ¬A));
[ apply rule (discharge [H]).
- | apply rule (∨_i_l (A)).
+ | apply rule (∨#i_l (A)).
apply rule (discharge [K]).
](*END*)
]
theorem ex1 : (C∧G ⇒ E) ⇒ (¬L ⇒ E∨C) ⇒ G ∨ L ⇒ ¬L ⇒ E.
apply rule (prove ((C∧G ⇒ E) ⇒ (¬L ⇒ E∨C) ⇒ G ∨ L ⇒ ¬L ⇒ E));
(*BEGIN*)
-apply rule (⇒_i [h1] ((¬L ⇒ E∨C) ⇒ G ∨ L ⇒ ¬L ⇒ E));
-apply rule (⇒_i [h2] (G ∨ L ⇒ ¬L ⇒ E));
-apply rule (⇒_i [h3] (¬L ⇒ E));
-apply rule (⇒_i [h4] (E));
-apply rule (∨_e (G∨L) [h5] (E) [h6] (E));
+apply rule (⇒#i [h1] ((¬L ⇒ E∨C) ⇒ G ∨ L ⇒ ¬L ⇒ E));
+apply rule (⇒#i [h2] (G ∨ L ⇒ ¬L ⇒ E));
+apply rule (⇒#i [h3] (¬L ⇒ E));
+apply rule (⇒#i [h4] (E));
+apply rule (∨#e (G∨L) [h5] (E) [h6] (E));
[ apply rule (discharge [h3]);
- | apply rule (∨_e (E∨C) [h6] (E) [h7] (E));
- [ apply rule (⇒_e (¬L ⇒ E∨C) (¬L));
+ | apply rule (∨#e (E∨C) [h6] (E) [h7] (E));
+ [ apply rule (⇒#e (¬L ⇒ E∨C) (¬L));
[ apply rule (discharge [h2]);
| apply rule (discharge [h4]);
]
| apply rule (discharge [h6]);
- | apply rule (⇒_e (C∧G ⇒ E) (C∧G));
+ | apply rule (⇒#e (C∧G ⇒ E) (C∧G));
[ apply rule (discharge [h1]);
- | apply rule (∧_i (C) (G));
+ | apply rule (∧#i (C) (G));
[ apply rule (discharge [h7]);
| apply rule (discharge [h5]);
]
]
]
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬L) (L));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬L) (L));
[ apply rule (discharge [h4]);
| apply rule (discharge [h6]);
]
theorem ex2 : (A∧¬B ⇒ C) ⇒ (B∧D ⇒ C) ⇒ (D ⇒ A) ⇒ D ⇒ C.
apply rule (prove ((A∧¬B ⇒ C) ⇒ (B∧D ⇒ C) ⇒ (D ⇒ A) ⇒ D ⇒ C));
(*BEGIN*)
-apply rule (⇒_i [h1] ((B∧D ⇒ C) ⇒ (D ⇒ A) ⇒ D ⇒ C));
-apply rule (⇒_i [h2] ((D ⇒ A) ⇒ D ⇒ C));
-apply rule (⇒_i [h3] (D ⇒ C));
-apply rule (⇒_i [h4] (C));
-apply rule (∨_e (B∨¬B) [h5] (C) [h6] (C));
+apply rule (⇒#i [h1] ((B∧D ⇒ C) ⇒ (D ⇒ A) ⇒ D ⇒ C));
+apply rule (⇒#i [h2] ((D ⇒ A) ⇒ D ⇒ C));
+apply rule (⇒#i [h3] (D ⇒ C));
+apply rule (⇒#i [h4] (C));
+apply rule (∨#e (B∨¬B) [h5] (C) [h6] (C));
[ apply rule (lem 0 EM);
- | apply rule (⇒_e (B∧D⇒C) (B∧D));
+ | apply rule (⇒#e (B∧D⇒C) (B∧D));
[ apply rule (discharge [h2]);
- | apply rule (∧_i (B) (D));
+ | apply rule (∧#i (B) (D));
[ apply rule (discharge [h5]);
| apply rule (discharge [h4]);
]
]
- | apply rule (⇒_e (A∧¬B⇒C) (A∧¬B));
+ | apply rule (⇒#e (A∧¬B⇒C) (A∧¬B));
[ apply rule (discharge [h1]);
- | apply rule (∧_i (A) (¬B));
- [ apply rule (⇒_e (D⇒A) (D));
+ | apply rule (∧#i (A) (¬B));
+ [ apply rule (⇒#e (D⇒A) (D));
[ apply rule (discharge [h3]);
| apply rule (discharge [h4]);
]
theorem ex3: (F ⇒ G∨E) ⇒ (G ⇒ ¬L∨E) ⇒ (L⇒F) ⇒ L ⇒ E.
apply rule (prove ((F ⇒ G∨E) ⇒ (G ⇒ ¬L∨E) ⇒ (L⇒F) ⇒ L ⇒ E));
(*BEGIN*)
-apply rule (⇒_i [h1] ((G ⇒ ¬L∨E) ⇒ (L⇒F) ⇒ L ⇒ E));
-apply rule (⇒_i [h2] ((L⇒F) ⇒ L ⇒ E));
-apply rule (⇒_i [h3] (L ⇒ E));
-apply rule (⇒_i [h4] (E));
-apply rule (∨_e (G∨E) [h5] (E) [h6] (E));
- [ apply rule (⇒_e (F ⇒ G∨E) (F));
+apply rule (⇒#i [h1] ((G ⇒ ¬L∨E) ⇒ (L⇒F) ⇒ L ⇒ E));
+apply rule (⇒#i [h2] ((L⇒F) ⇒ L ⇒ E));
+apply rule (⇒#i [h3] (L ⇒ E));
+apply rule (⇒#i [h4] (E));
+apply rule (∨#e (G∨E) [h5] (E) [h6] (E));
+ [ apply rule (⇒#e (F ⇒ G∨E) (F));
[ apply rule (discharge [h1]);
- | apply rule (⇒_e (L⇒F) (L));
+ | apply rule (⇒#e (L⇒F) (L));
[ apply rule (discharge [h3]);
| apply rule (discharge [h4]);
]
]
- |apply rule (∨_e (¬L∨E) [h7] (E) [h8] (E));
- [ apply rule (⇒_e (G⇒¬L∨E) (G));
+ |apply rule (∨#e (¬L∨E) [h7] (E) [h8] (E));
+ [ apply rule (⇒#e (G⇒¬L∨E) (G));
[ apply rule (discharge [h2]);
| apply rule (discharge [h5]);
]
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬L) (L));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬L) (L));
[ apply rule (discharge [h7]);
| apply rule (discharge [h4]);
]
theorem ex4: ¬(A∧B) ⇒ ¬A∨¬B.
apply rule (prove (¬(A∧B) ⇒ ¬A∨¬B));
(*BEGIN*)
-apply rule (⇒_i [h1] (¬A∨¬B));
-apply rule (∨_e (A ∨ ¬A) [h2] ((¬A∨¬B)) [h3] ((¬A∨¬B)));
+apply rule (⇒#i [h1] (¬A∨¬B));
+apply rule (∨#e (A ∨ ¬A) [h2] ((¬A∨¬B)) [h3] ((¬A∨¬B)));
[ apply rule (lem 0 EM);
- | apply rule (∨_e (B ∨ ¬B) [h4] ((¬A∨¬B)) [h5] ((¬A∨¬B)));
+ | apply rule (∨#e (B ∨ ¬B) [h4] ((¬A∨¬B)) [h5] ((¬A∨¬B)));
[ apply rule (lem 0 EM);
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬(A∧B)) (A∧B));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬(A∧B)) (A∧B));
[ apply rule (discharge [h1]);
- | apply rule (∧_i (A) (B));
+ | apply rule (∧#i (A) (B));
[ apply rule (discharge [h2]);
| apply rule (discharge [h4]);
]
]
- | apply rule (∨_i_r (¬B));
+ | apply rule (∨#i_r (¬B));
apply rule (discharge [h5]);
]
- | apply rule (∨_i_l (¬A));
+ | apply rule (∨#i_l (¬A));
apply rule (discharge [h3]);
]
(*END*)
theorem ex5: ¬(A∨B) ⇒ (¬A ∧ ¬B).
apply rule (prove (¬(A∨B) ⇒ (¬A ∧ ¬B)));
(*BEGIN*)
-apply rule (⇒_i [h1] (¬A ∧ ¬B));
-apply rule (∨_e (A∨¬A) [h2] (¬A ∧ ¬B) [h3] (¬A ∧ ¬B));
+apply rule (⇒#i [h1] (¬A ∧ ¬B));
+apply rule (∨#e (A∨¬A) [h2] (¬A ∧ ¬B) [h3] (¬A ∧ ¬B));
[ apply rule (lem 0 EM);
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬(A∨B)) (A∨B));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬(A∨B)) (A∨B));
[ apply rule (discharge [h1]);
- | apply rule (∨_i_l (A));
+ | apply rule (∨#i_l (A));
apply rule (discharge [h2]);
]
- | apply rule (∨_e (B∨¬B) [h10] (¬A ∧ ¬B) [h11] (¬A ∧ ¬B));
+ | apply rule (∨#e (B∨¬B) [h10] (¬A ∧ ¬B) [h11] (¬A ∧ ¬B));
[ apply rule (lem 0 EM);
- | apply rule (⊥_e (⊥));
- apply rule (¬_e (¬(A∨B)) (A∨B));
+ | apply rule (⊥#e (⊥));
+ apply rule (¬#e (¬(A∨B)) (A∨B));
[ apply rule (discharge [h1]);
- | apply rule (∨_i_r (B));
+ | apply rule (∨#i_r (B));
apply rule (discharge [h10]);
]
- | apply rule (∧_i (¬A) (¬B));
+ | apply rule (∧#i (¬A) (¬B));
[ apply rule (discharge [h3]);
|apply rule (discharge [h11]);
]
theorem ex6: ¬A ∧ ¬B ⇒ ¬(A∨B).
apply rule (prove (¬A ∧ ¬B ⇒ ¬(A∨B)));
(*BEGIN*)
-apply rule (⇒_i [h1] (¬(A∨B)));
-apply rule (¬_i [h2] (⊥));
-apply rule (∨_e (A∨B) [h3] (⊥) [h3] (⊥));
+apply rule (⇒#i [h1] (¬(A∨B)));
+apply rule (¬#i [h2] (⊥));
+apply rule (∨#e (A∨B) [h3] (⊥) [h3] (⊥));
[ apply rule (discharge [h2]);
- | apply rule (¬_e (¬A) (A));
- [ apply rule (∧_e_l (¬A∧¬B));
+ | apply rule (¬#e (¬A) (A));
+ [ apply rule (∧#e_l (¬A∧¬B));
apply rule (discharge [h1]);
| apply rule (discharge [h3]);
]
- | apply rule (¬_e (¬B) (B));
- [ apply rule (∧_e_r (¬A∧¬B));
+ | apply rule (¬#e (¬B) (B));
+ [ apply rule (∧#e_r (¬A∧¬B));
apply rule (discharge [h1]);
| apply rule (discharge [h3]);
]
theorem ex7: ((A ⇒ ⊥) ⇒ ⊥) ⇒ A.
apply rule (prove (((A ⇒ ⊥) ⇒ ⊥) ⇒ A));
(*BEGIN*)
-apply rule (⇒_i [h1] (A));
+apply rule (⇒#i [h1] (A));
apply rule (RAA [h2] (⊥));
-apply rule (⇒_e ((A⇒⊥)⇒⊥) (A⇒⊥));
+apply rule (⇒#e ((A⇒⊥)⇒⊥) (A⇒⊥));
[ apply rule (discharge [h1]);
- | apply rule (⇒_i [h3] (⊥));
- apply rule (¬_e (¬A) (A));
+ | apply rule (⇒#i [h3] (⊥));
+ apply rule (¬#e (¬A) (A));
[ apply rule (discharge [h2]);
| apply rule (discharge [h3]);
]