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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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "nat/compare.ma".
+include "nat/times.ma".
+
+inductive pos: Set \def
+| one : pos
+| next : pos \to pos.
+
+inductive int: Set \def
+| positive: nat \to int
+| negative : nat \to int.
+
+inductive empty : Set \def .
+
+let rec pos2nat x \def 
+  match x with  
+  [ one \Rightarrow (S O)
+  | (next z) \Rightarrow S (pos2nat z)].
+
+definition nat2int \def \lambda x. positive x.
+
+coercion cic:/matita/tests/coercions/pos2nat.con.
+
+coercion cic:/matita/tests/coercions/nat2int.con.
+
+definition fst \def \lambda x,y:int.x.
+
+theorem a: fst O one = fst (positive O) (next one).
+reflexivity.
+qed.
+
+definition double: 
+  \forall f:int \to int. pos \to int 
+\def 
+  \lambda f:int \to int. \lambda x : pos .f (nat2int x).
+  
+definition double1: 
+  \forall f:int \to int. pos \to int 
+\def 
+  \lambda f:int \to int. \lambda x : pos .f (pos2nat x).
+
+definition double2: 
+  \forall f:int \to int. pos \to int 
+\def 
+  \lambda f:int \to int. \lambda x : pos .f (nat2int (pos2nat x)).
+
+(* This used to test eq_f as a coercion. However, posing both eq_f and sym_eq
+   as coercions made the qed time of some TPTP problems reach infty.
+   Thus eq_f is no longer a coercion (nor is sym_eq).
+theorem coercion_svelta : \forall T,S:Type.\forall f:T \to S.\forall x,y:T.x=y \to f y = f x.
+  intros.
+  apply ((\lambda h:f y = f x.h) H).
+qed.
+*)
+
+variant pos2nat' : ? \def pos2nat.
+
+inductive initial: Set \def iii : initial.
+
+definition i2pos: ? \def \lambda x:initial.one.
+
+coercion cic:/matita/tests/coercions/i2pos.con.
+
+coercion cic:/matita/tests/coercions/pos2nat'.con.
+
+inductive listn (A:Type) : nat \to Type \def
+ | Nil : listn A O
+ | Next : \forall n.\forall l:listn A n.\forall a:A.listn A (S n).
+ 
+definition if : \forall A:Type.\forall b:bool.\forall a,c:A.A \def
+  \lambda A,b,a,c.
+  match b with
+  [ true \Rightarrow a
+  | false \Rightarrow c].  
+ 
+let rec ith (A:Type) (n,m:nat) (dummy:A) (l:listn A n) on l \def
+  match l with
+  [ Nil \Rightarrow dummy
+  | (Next w l x) \Rightarrow if A (eqb w m) x (ith A w m dummy l)].  
+
+definition listn2function: 
+  \forall A:Type.\forall dummy:A.\forall n.listn A n \to nat \to A
+\def
+  \lambda A,dummy,n,l,m.ith A n m dummy l.
+  
+definition natlist2map: ? \def listn2function nat O.
+  
+coercion cic:/matita/tests/coercions/natlist2map.con 1.
+definition map:  \forall n:nat.\forall l:listn nat n. nat \to nat \def
+  \lambda n:nat.\lambda l:listn nat n.\lambda m:nat.l m.
+  
+definition church: nat \to nat \to nat \def times.
+
+coercion cic:/matita/tests/coercions/church.con 1.
+lemma foo0 : ∀n:nat. n n = n * n.
+intros; reflexivity;
+qed.
+lemma foo01 : ∀n:nat. n n n = n * n * n.
+intros; reflexivity;
+qed.
+
+definition mapmult:  \forall n:nat.\forall l:listn nat n. nat \to nat \to nat \def
+  \lambda n:nat.\lambda l:listn nat n.\lambda m,o:nat.
+  l (m m) o (o o o).
+  
+lemma foo : ∀n:nat. n n n n n n = n * n * n * n * n * n.
+intros; reflexivity;
+qed.
+
+axiom f : nat → nat.
+
+lemma foo1 : ∀n:nat. f n n = f n * n.
+
+axiom T0 : Type.
+axiom T1 : Type.
+axiom T2 : Type.
+axiom T3 : Type.
+
+axiom c1 : T0 -> T1.
+axiom c2 : T1 -> T2.
+axiom c3 : T2 -> T3.
+axiom c4 : T2 -> T1.
+
+coercion cic:/matita/tests/coercions/c1.con.
+coercion cic:/matita/tests/coercions/c2.con.
+coercion cic:/matita/tests/coercions/c3.con.
+coercion cic:/matita/tests/coercions/c4.con.
+
+
+
+  
+  
+ 
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