more structured and indented version of nat.ma

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

diff --git a/helm/matita/library/nat/nat.ma b/helm/matita/library/nat/nat.ma
index ff41ba62e25e2844b18240ddf80ba36a0bc52cdf..0156a98cadea4856ff10761233958ad107516c24 100644 (file)
--- a/helm/matita/library/nat/nat.ma
+++ b/helm/matita/library/nat/nat.ma
@@ -1,5 +1,5 @@
 (**************************************************************************)
-(*       ___                                                               *)
+(*       ___                                                                 *)
 (*      ||M||                                                             *)
 (*      ||A||       A project by Andrea Asperti                           *)
 (*      ||T||                                                             *)
@@ -14,8 +14,6 @@
 
 set "baseuri" "cic:/matita/nat/nat".
 
-include "logic/equality.ma".
-include "logic/connectives.ma".
 include "higher_order_defs/functions.ma".
 
 inductive nat : Set \def
@@ -23,66 +21,86 @@ inductive nat : Set \def
   | S : nat \to nat.
 
 definition pred: nat \to nat \def
-\lambda n:nat. match n with
-[ O \Rightarrow  O
-| (S p) \Rightarrow p ].
+ \lambda n:nat. match n with
+ [ O \Rightarrow  O
+ | (S p) \Rightarrow p ].
 
-theorem pred_Sn : \forall n:nat.
-(eq nat n (pred (S n))).
-intros; reflexivity.
+theorem pred_Sn : \forall n:nat.n=(pred (S n)).
+ intros. reflexivity.
 qed.
 
 theorem injective_S : injective nat nat S.
-simplify.
-intros.
-rewrite > pred_Sn.
-rewrite > pred_Sn y.
-apply eq_f. assumption.
+ unfold injective.
+ intros.
+ rewrite > pred_Sn.
+ rewrite > (pred_Sn y).
+ apply eq_f. assumption.
 qed.
 
-theorem inj_S : \forall n,m:nat. 
-(eq nat (S n) (S m)) \to (eq nat n m)
-\def injective_S.
+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. 
-Not (eq nat n m) \to Not (eq nat (S n) (S m)).
-intros. simplify. intros.
-apply H. apply injective_S. assumption.
+ \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.
+ \lambda n: nat.
   match n with
   [ O \Rightarrow False
   | (S p) \Rightarrow True ].
 
-theorem not_eq_O_S : \forall n:nat. Not (eq nat O (S n)).
-intros. simplify. intros.
-cut (not_zero O).
-exact Hcut.
-rewrite > H.exact I.
+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. Not (eq nat n (S n)).
-intros.elim n.
-apply not_eq_O_S.
-apply not_eq_S.assumption.
+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.
+ \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.
-intros.apply H2. apply H3.
+ \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.