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+Require Export ZArith.
+Require Export ZArithRing.
+
+(* Cette tactique a pour objectif de remplacer toute instance
+ de (POS (xI e)) ou de (POS (xO e)) par
+ 2*(POS e)+1 ou 2*(POS e), pour rendre les expressions plus
+ à même d'être utilisées par Ring, lorsque ces expressions contiennent
+ des variables de type positive. *)
+Tactic Definition compute_POS :=
+ (Match Context With
+ | [|- [(POS (xI ?1))]] -> Let v = ?1 In
+ (Match v With
+ | [xH] ->
+ (Fail 1)
+ |_->
+ Rewrite (POS_xI v))
+ | [ |- [(POS (xO ?1))]] -> Let v = ?1 In
+ Match v With
+ |[xH]->
+ (Fail 1)
+ |[?]->
+ Rewrite (POS_xO v)).
+
+Goal (x:positive)(POS (xI (xI x)))=`4*(POS x)+3`.
+Intros.
+Repeat compute_POS.
+Ring.
+Qed.