(* *)
(**************************************************************************)
-include "demo/natural_deduction_support.ma".
+include "didactic/support/natural_deduction.ma".
lemma RAA_to_EM : A ∨ ¬ A.
- apply (prove (A ∨ ¬ A));
+ apply rule (prove (A ∨ ¬ A));
- apply (RAA [H] ⊥);
- apply (¬_e (¬A) A);
- [ apply (¬_i [H1] ⊥);
- apply (¬_e (¬(A∨¬A)) (A∨¬A));
- [ apply [H];
- | apply (∨_i_l A);
- apply [H1];
+ apply rule (RAA [H] ⊥);
+ apply rule (¬#e (¬A) A);
+ [ apply rule (¬#i [H1] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_l A);
+ apply rule (discharge [H1]);
]
- | apply (RAA [H2] ⊥);
- apply (¬_e (¬(A∨¬A)) (A∨¬A));
- [ apply [H];
- | apply (∨_i_r (¬A));
- apply [H2];
+ | apply rule (RAA [H2] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_r (¬A));
+ apply rule (discharge [H2]);
]
]
qed.
lemma RA_to_EM1 : A ∨ ¬ A.
- apply (prove (A ∨ ¬ A));
+ apply rule (prove (A ∨ ¬ A));
- apply (RAA [H] ⊥);
- apply (¬_e (¬¬A) (¬A));
- [ apply (¬_i [H2] ⊥);
- apply (¬_e (¬(A∨¬A)) (A∨¬A));
- [ apply [H];
- | apply (∨_i_r (¬A));
- apply [H2];
+ apply rule (RAA [H] ⊥);
+ apply rule (¬#e (¬¬A) (¬A));
+ [ apply rule (¬#i [H2] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_r (¬A));
+ apply rule (discharge [H2]);
]
- | apply (¬_i [H1] ⊥);
- apply (¬_e (¬(A∨¬A)) (A∨¬A));
- [ apply [H];
- | apply (∨_i_l A);
- apply [H1];
+ | apply rule (¬#i [H1] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_l A);
+ apply rule (discharge [H1]);
]
]
qed.
-lemma ex0 : (A ⇒ ⊥) ⇒ A ⇒ B ∧ ⊤.
-
- apply (prove ((A ⇒ ⊥) ⇒ A ⇒ B∧⊤));
-
- apply (⇒_i [H] (A ⇒ B∧⊤));
- apply (⇒_i [H1] (B∧⊤));
- apply (∧_i B ⊤);
- [ apply (⊥_e ⊥);
- apply (⇒_e (A ⇒ ⊥) A);
- [ apply [H];
- | apply [H1];
- ]
- | apply (⊤_i);
- ]
-qed.
-
lemma ex1 : (A ⇒ E) ∨ B ⇒ A ∧ C ⇒ (E ∧ C) ∨ B.
- apply (prove ((A⇒E)∨B⇒A∧C⇒E∧C∨B));
+ apply rule (prove ((A⇒E)∨B⇒A∧C⇒E∧C∨B));
- apply (⇒_i [H] (A∧C⇒E∧C∨B));
- apply (⇒_i [K] (E∧C∨B));
- apply (∨_e ((A⇒E)∨B) [C1] (E∧C∨B) [C2] (E∧C∨B));
-[ apply [H];
-| apply (∨_i_l (E∧C));
- apply (∧_i E C);
- [ apply (⇒_e (A⇒E) A);
- [ apply [C1];
- | apply (∧_e_l (A∧C)); apply [K];
+ apply rule (⇒#i [H] (A∧C⇒E∧C∨B));
+ apply rule (⇒#i [K] (E∧C∨B));
+ apply rule (∨#e ((A⇒E)∨B) [C1] (E∧C∨B) [C2] (E∧C∨B));
+[ apply rule (discharge [H]);
+| apply rule (∨#i_l (E∧C));
+ apply rule (∧#i E C);
+ [ apply rule (⇒#e (A⇒E) A);
+ [ apply rule (discharge [C1]);
+ | apply rule (∧#e_l (A∧C)); apply rule (discharge [K]);
]
- | apply (∧_e_r (A∧C)); apply [K];
+ | apply rule (∧#e_r (A∧C)); apply rule (discharge [K]);
]
-| apply (∨_i_r B); apply [C2];
+| apply rule (∨#i_r B); apply rule (discharge [C2]);
]
qed.
-lemma dmg : ¬(A ∨ B) ⇒ ¬A ∧ ¬B.
-
- apply (prove (¬(A ∨ B) ⇒ ¬A ∧ ¬B));
- apply (⇒_i [H] (¬A ∧ ¬B));
-
- apply (¬_e (¬A) A);
-
-
-
-
-(*
-lemma ex2: ΠN:Type.ΠR:N→N→CProp.
-
- (∀a:N.∀b:N.R a b ⇒ R b a) ⇒ ∀z:N.(∃x.R x z) ⇒ ∃y. R z y.
-
- intros (N R);apply (prove ((∀a:N.∀b:N.R a b ⇒ R b a) ⇒ ∀z:N.(∃x.R x z) ⇒ ∃y. R z y));
-
- apply (⇒_i [H] (∀z:N.(∃x:N.R x z)⇒∃y:N.R z y));
- apply (∀_i [z] ((∃x:N.R x z)⇒∃y:N.R z y));
- apply (⇒_i [H2] (∃y:N.R z y));
- apply (∃_e (∃x:N.R x z) [n] [H3] (∃y:N.R z y));
- [ apply [H2]
- | apply (∃_i n (R z n));
- apply (⇒_e (R n z ⇒ R z n) (R n z));
- [ apply (∀_e (∀b:N.R n b ⇒ R b n) z);
- apply (∀_e (∀a:N.∀b:N.R a b ⇒ R b a) n);
- apply [H]
- | apply [H3]
- ]
- ]
-qed.
-*)
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