ocaml 3.09 transition

[helm.git] / helm / matita / library / nat / exp.ma

diff --git a/helm/matita/library/nat/exp.ma b/helm/matita/library/nat/exp.ma
index 6fcd9da0ec646620f36b8546af0d54f1854dad2c..11d84f74ca7deb3b676007693f72b585c0547435 100644 (file)
--- a/helm/matita/library/nat/exp.ma
+++ b/helm/matita/library/nat/exp.ma
@@ -14,33 +14,84 @@
 
 set "baseuri" "cic:/matita/nat/exp".
 
-include "nat/times.ma". 
+include "nat/div_and_mod.ma".
 
 let rec exp n m on m\def 
  match m with 
  [ O \Rightarrow (S O)
  | (S p) \Rightarrow (times n (exp n p)) ].
 
+interpretation "natural exponent" 'exp a b = (cic:/matita/nat/exp/exp.con a b).
+
 theorem exp_plus_times : \forall n,p,q:nat. 
-eq nat (exp n (plus p q)) (times (exp n p) (exp n q)).
+n \sup (p + q) = (n \sup p) * (n \sup q).
 intros.elim p.
 simplify.rewrite < plus_n_O.reflexivity.
 simplify.rewrite > H.symmetry.
 apply assoc_times.
 qed.
 
-theorem exp_n_O : \forall n:nat. eq nat (S O) (exp n O).
+theorem exp_n_O : \forall n:nat. S O = n \sup O.
 intro.simplify.reflexivity.
 qed.
 
-theorem exp_n_SO : \forall n:nat. eq nat n (exp n (S O)).
+theorem exp_n_SO : \forall n:nat. n = n \sup (S O).
 intro.simplify.rewrite < times_n_SO.reflexivity.
 qed.
 
 theorem exp_exp_times : \forall n,p,q:nat. 
-eq nat (exp (exp n p) q) (exp n (times p q)).
+(n \sup p) \sup q = n \sup (p * q).
 intros.
 elim q.simplify.rewrite < times_n_O.simplify.reflexivity.
 simplify.rewrite > H.rewrite < exp_plus_times.
 rewrite < times_n_Sm.reflexivity.
-qed.
\ No newline at end of file
+qed.
+
+theorem lt_O_exp: \forall n,m:nat. O < n \to O < n \sup m. 
+intros.elim m.simplify.unfold lt.apply le_n.
+simplify.unfold lt.rewrite > times_n_SO.
+apply le_times.assumption.assumption.
+qed.
+
+theorem lt_m_exp_nm: \forall n,m:nat. (S O) < n \to m < n \sup m.
+intros.elim m.simplify.unfold lt.reflexivity.
+simplify.unfold lt.
+apply (trans_le ? ((S(S O))*(S n1))).
+simplify.
+rewrite < plus_n_Sm.apply le_S_S.apply le_S_S.
+rewrite < sym_plus.
+apply le_plus_n.
+apply le_times.assumption.assumption.
+qed.
+
+theorem exp_to_eq_O: \forall n,m:nat. (S O) < n 
+\to n \sup m = (S O) \to m = O.
+intros.apply antisym_le.apply le_S_S_to_le.
+rewrite < H1.change with (m < n \sup m).
+apply lt_m_exp_nm.assumption.
+apply le_O_n.
+qed.
+
+theorem injective_exp_r: \forall n:nat. (S O) < n \to 
+injective nat nat (\lambda m:nat. n \sup m).
+simplify.intros 4.
+apply (nat_elim2 (\lambda x,y.n \sup x = n \sup y \to x = y)).
+intros.apply sym_eq.apply (exp_to_eq_O n).assumption.
+rewrite < H1.reflexivity.
+intros.apply (exp_to_eq_O n).assumption.assumption.
+intros.apply eq_f.
+apply H1.
+(* esprimere inj_times senza S *)
+cut (\forall a,b:nat.O < n \to n*a=n*b \to a=b).
+apply Hcut.simplify.unfold lt.apply le_S_S_to_le. apply le_S. assumption.
+assumption.
+intros 2.
+apply (nat_case n).
+intros.apply False_ind.apply (not_le_Sn_O O H3).
+intros.
+apply (inj_times_r m1).assumption.
+qed.
+
+variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
+p \sup n = p \sup m \to n = m \def
+injective_exp_r.