First experimental version of the declarative proof language for Matita.

[helm.git] / helm / software / matita / tests / decl.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/test/decl".
+
+include "nat/times.ma".
+include "nat/orders.ma".
+
+theorem easy: ∀n,m. n * m = O → n = O ∨ m = O.
+ assume n: nat.
+ assume m: nat.
+ (* base case *)
+ by (refl_eq ? O) we proved (O = O) (trivial).
+ by (or_introl ? ? trivial) we proved (O = O ∨ m = O) (trivial2).
+ by (λ_.trivial2) we proved (O*m=O → O=O ∨ m=O) (base_case).
+ (* inductive case *)
+ we need to prove
+  (∀n1. (n1 * m = O → n1 = O ∨ m = O) → (S n1) * m = O → (S n1) = O ∨ m = O)
+  (inductive_case).
+   assume n1: nat.
+   suppose (n1 * m = O → n1 = O ∨ m = O) (inductive_hyp).
+   (* base case *)
+   by (or_intror ? ? trivial) we proved (S n1 = O ∨ O = O) (pre_base_case2).
+   by (λ_.pre_base_case2) we proved (S n1*O = O → S n1 = O ∨ O = O) (base_case2).
+   (* inductive case *)
+   we need to prove
+    (∀m1. (S n1 * m1 = O → S n1 = O ∨ m1 = O) →
+      (S n1 * S m1 = O → S n1 = O ∨ S m1 = O)) (inductive_hyp2).
+     assume m1: nat.
+     suppose (S n1 * m1 = O → S n1 = O ∨ m1 = O) (useless).
+     suppose (S n1 * S m1 = O) (absurd_hyp).
+     simplify in absurd_hyp.
+     by (sym_eq ? ? ? absurd_hyp) we proved (O = S (m1+n1*S m1)) (absurd_hyp').
+     by (not_eq_O_S ? absurd_hyp') we proved False (the_absurd).
+     by (False_ind ? the_absurd)
+   done.
+   (* the induction *)
+   by (nat_ind (λm.S n1 * m = O → S n1 = O ∨ m = O) base_case2 inductive_hyp2 m)
+ done.
+ (* the induction *)
+ by (nat_ind (λn.n*m=O → n=O ∨ m=O) base_case inductive_case n)
+done.
+qed.
+ 
+theorem easy2: ∀n,m. n * m = O → n = O ∨ m = O.
+ intros 2.
+ elim n 0
+  [ intro;
+    left;
+    reflexivity
+  | intro;
+    elim m 0
+    [ intros;
+      right;
+      reflexivity
+    | intros;
+      simplify in H2;
+      lapply (sym_eq ? ? ? H2);
+      elim (not_eq_O_S ? Hletin)
+    ]
+  ]
+qed.
+      
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