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[helm.git] / matita / contribs / library_auto / auto / nat / plus.ma

diff --git a/matita/contribs/library_auto/auto/nat/plus.ma b/matita/contribs/library_auto/auto/nat/plus.ma
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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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/plus".
+
+include "auto/nat/nat.ma".
+
+let rec plus n m \def 
+ match n with 
+ [ O \Rightarrow m
+ | (S p) \Rightarrow S (plus p m) ].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural plus" 'plus x y = (cic:/matita/library_autobatch/nat/plus/plus.con x y).
+
+theorem plus_n_O: \forall n:nat. n = n+O.
+intros.elim n
+[ autobatch
+  (*simplify.
+  reflexivity*)
+| autobatch
+  (*simplify.
+  apply eq_f.
+  assumption.*)
+]
+qed.
+
+theorem plus_n_Sm : \forall n,m:nat. S (n+m) = n+(S m).
+intros.elim n
+[ autobatch
+  (*simplify.
+  reflexivity.*)
+| simplify.
+  autobatch
+  (*
+  apply eq_f.
+  assumption.*)]
+qed.
+
+theorem sym_plus: \forall n,m:nat. n+m = m+n.
+intros.elim n
+[ autobatch
+  (*simplify.
+  apply plus_n_O.*)
+| simplify.
+  autobatch
+  (*rewrite > H.
+  apply plus_n_Sm.*)]
+qed.
+
+theorem associative_plus : associative nat plus.
+unfold associative.intros.elim x
+[autobatch
+ (*simplify.
+ reflexivity.*)
+|simplify.
+ autobatch
+ (*apply eq_f.
+ assumption.*)
+]
+qed.
+
+theorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
+\def associative_plus.
+
+theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.n+m).
+intro.simplify.intros 2.elim n
+[ exact H
+| autobatch
+  (*apply H.apply inj_S.apply H1.*)
+]
+qed.
+
+theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
+\def injective_plus_r.
+
+theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.n+m).
+intro.simplify.intros.autobatch.
+(*apply (injective_plus_r m).
+rewrite < sym_plus.
+rewrite < (sym_plus y).
+assumption.*)
+qed.
+
+theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
+\def injective_plus_l.