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+(**************************************************************************)
+(*       ___	                                                              *)
+(*      ||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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "higher_order_defs/functions.ma".
+
+theorem esempio: \forall A,B,C:Prop.(A \to B \to C) \to (A \to B)
+\to A \to C.
+
+
+
+inductive nat : Set \def
+  | O : nat
+  | S : nat \to nat.
+
+definition pred: nat \to nat \def
+ \lambda n:nat. match n with
+ [ O \Rightarrow  O
+ | (S p) \Rightarrow p ].
+
+theorem pred_Sn : \forall n:nat.n=(pred (S n)).
+ intros. simplify. reflexivity.
+qed.
+
+theorem injective_S : injective nat nat S.
+ unfold injective.
+ intros.
+ rewrite > pred_Sn.
+ rewrite > (pred_Sn y).
+ apply eq_f. assumption.
+qed.
+
+theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m \def
+ injective_S.
+
+theorem not_eq_S  : \forall n,m:nat. 
+ \lnot n=m \to S n \neq S m.
+ intros. unfold Not. intros.
+ apply H. apply injective_S. assumption.
+qed.
+
+definition not_zero : nat \to Prop \def
+ \lambda n: nat.
+  match n with
+  [ O \Rightarrow False
+  | (S p) \Rightarrow True ].
+
+theorem not_eq_O_S : \forall n:nat. O \neq S n.
+ intros. unfold Not. intros.
+ cut (not_zero O).
+ exact Hcut.
+ rewrite > H.exact I.
+qed.
+
+theorem not_eq_n_Sn : \forall n:nat. n \neq S n.
+ intros.elim n.
+ apply not_eq_O_S.
+ apply not_eq_S.assumption.
+qed.
+
+theorem nat_case:
+ \forall n:nat.\forall P:nat \to Prop. 
+  P O \to  (\forall m:nat. P (S m)) \to P n.
+intros.elim n
+  [ assumption
+  | apply H1 ]
+qed.
+
+theorem nat_case1:
+ \forall n:nat.\forall P:nat \to Prop. 
+  (n=O \to P O) \to  (\forall m:nat. (n=(S m) \to P (S m))) \to P n.
+intros 2; elim n
+  [ apply H;reflexivity
+  | apply H2;reflexivity ]
+qed.
+
+theorem nat_elim2 :
+ \forall R:nat \to nat \to Prop.
+  (\forall n:nat. R O n) 
+  \to (\forall n:nat. R (S n) O) 
+  \to (\forall n,m:nat. R n m \to R (S n) (S m))
+  \to \forall n,m:nat. R n m.
+intros 5;elim n 
+  [ apply H
+  | apply (nat_case m)
+    [ apply H1
+    | intro; apply H2; apply H3 ] ]
+qed.
+
+theorem decidable_eq_nat : \forall n,m:nat.decidable (n=m).
+ intros.unfold decidable.
+ apply (nat_elim2 (\lambda n,m.(Or (n=m) ((n=m) \to False))))
+ [ intro; elim n1
+   [ left; reflexivity
+   | right; apply not_eq_O_S ]
+ | intro; right; intro; apply (not_eq_O_S n1); apply sym_eq; assumption
+ | intros; elim H
+   [ left; apply eq_f; assumption
+   | right; intro; apply H1; apply inj_S; assumption ] ]
+qed.