X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Forders.ma;h=6ec0c9992a68b18e15976e96e74ef853c2fc04eb;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=b8e92fa2b6987f34cfecbf81841bee405d10ca0a;hpb=9c8bb7e7c2548d2f37e5387cdce45df2b8fc9b43;p=helm.git

diff --git a/helm/matita/library/nat/orders.ma b/helm/matita/library/nat/orders.ma
index b8e92fa2b..6ec0c9992 100644
--- a/helm/matita/library/nat/orders.ma
+++ b/helm/matita/library/nat/orders.ma
@@ -14,7 +14,7 @@
 
 set "baseuri" "cic:/matita/nat/orders".
 
-include "nat/plus.ma".
+include "nat/nat.ma".
 include "higher_order_defs/ordering.ma".
 
 (* definitions *)
@@ -22,98 +22,162 @@ inductive le (n:nat) : nat \to Prop \def
   | le_n : le n n
   | le_S : \forall m:nat. le n m \to le n (S m).
 
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'less or equal to'" 'leq x y = (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'neither less nor equal to'" 'nleq x y =
+  (cic:/matita/logic/connectives/Not.con
+    (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y)).
+
 definition lt: nat \to nat \to Prop \def
-\lambda n,m:nat.le (S n) m.
+\lambda n,m:nat.(S n) \leq m.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'less than'" 'lt x y = (cic:/matita/nat/orders/lt.con x y).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'not less than'" 'nless x y =
+  (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/lt.con x y)).
 
 definition ge: nat \to nat \to Prop \def
-\lambda n,m:nat.le m n.
+\lambda n,m:nat.m \leq n.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'greater or equal to'" 'geq x y = (cic:/matita/nat/orders/ge.con x y).
 
 definition gt: nat \to nat \to Prop \def
-\lambda n,m:nat.lt m n.
+\lambda n,m:nat.m<n.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'greater than'" 'gt x y = (cic:/matita/nat/orders/gt.con x y).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural 'not greater than'" 'ngtr x y =
+  (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/gt.con x y)).
 
 theorem transitive_le : transitive nat le.
-simplify.intros.elim H1.
+unfold transitive.intros.elim H1.
 assumption.
 apply le_S.assumption.
 qed.
 
-theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p
+theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
 \def transitive_le.
 
-theorem le_S_S: \forall n,m:nat. le n m \to le (S n) (S m).
+theorem transitive_lt: transitive nat lt.
+unfold transitive.unfold lt.intros.elim H1.
+apply le_S. assumption.
+apply le_S.assumption.
+qed.
+
+theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
+\def transitive_lt.
+
+theorem le_S_S: \forall n,m:nat. n \leq m \to S n \leq S m.
 intros.elim H.
 apply le_n.
 apply le_S.assumption.
 qed.
 
-theorem le_O_n : \forall n:nat. le O n.
+theorem le_O_n : \forall n:nat. O \leq n.
 intros.elim n.
 apply le_n.apply 
 le_S. assumption.
 qed.
 
-theorem le_n_Sn : \forall n:nat. le n (S n).
+theorem le_n_Sn : \forall n:nat. n \leq S n.
 intros. apply le_S.apply le_n.
 qed.
 
-theorem le_pred_n : \forall n:nat. le (pred n) n.
+theorem le_pred_n : \forall n:nat. pred n \leq n.
 intros.elim n.
 simplify.apply le_n.
 simplify.apply le_n_Sn.
 qed.
 
-theorem le_S_S_to_le : \forall n,m:nat. le (S n) (S m) \to le n m.
-intros.change with le (pred (S n)) (pred (S m)).
-elim H.apply le_n.apply trans_le ? (pred e2).assumption.
+theorem le_S_S_to_le : \forall n,m:nat. S n \leq S m \to n \leq m.
+intros.change with (pred (S n) \leq pred (S m)).
+elim H.apply le_n.apply (trans_le ? (pred n1)).assumption.
 apply le_pred_n.
 qed.
 
-theorem leS_to_not_zero : \forall n,m:nat. (le (S n) m ) \to not_zero m.
+theorem leS_to_not_zero : \forall n,m:nat. S n \leq m \to not_zero m.
 intros.elim H.exact I.exact I.
 qed.
 
 (* not le *)
-theorem not_le_Sn_O: \forall n:nat. Not (le (S n) O).
-intros.simplify.intros.apply leS_to_not_zero ? ? H.
+theorem not_le_Sn_O: \forall n:nat. S n \nleq O.
+intros.unfold Not.simplify.intros.apply (leS_to_not_zero ? ? H).
 qed.
 
-theorem not_le_Sn_n: \forall n:nat. Not (le (S n) n).
-intros.elim n.apply not_le_Sn_O.simplify.intros.cut le (S e1) e1.
+theorem not_le_Sn_n: \forall n:nat. S n \nleq n.
+intros.elim n.apply not_le_Sn_O.unfold Not.simplify.intros.cut (S n1 \leq n1).
 apply H.assumption.
 apply le_S_S_to_le.assumption.
 qed.
 
+(* le to lt or eq *)
+theorem le_to_or_lt_eq : \forall n,m:nat. 
+n \leq m \to n < m \lor n = m.
+intros.elim H.
+right.reflexivity.
+left.unfold lt.apply le_S_S.assumption.
+qed.
+
+(* not eq *)
+theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
+unfold Not.intros.cut ((le (S n) m) \to False).
+apply Hcut.assumption.rewrite < H1.
+apply not_le_Sn_n.
+qed.
+
 (* le vs. lt *)
-theorem lt_to_le : \forall n,m:nat. lt n m \to le n m.
-simplify.intros.elim H.
+theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
+simplify.intros.unfold lt in H.elim H.
 apply le_S. apply le_n.
 apply le_S. assumption.
 qed.
 
-theorem not_le_to_lt: \forall n,m:nat. Not (le n m) \to lt m n.
+theorem lt_S_to_le : \forall n,m:nat. n < S m \to n \leq m.
+simplify.intros.
+apply le_S_S_to_le.assumption.
+qed.
+
+theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
 intros 2.
-apply nat_elim2 (\lambda n,m.Not (le n m) \to lt m n).
-intros.apply absurd (le O n1).apply le_O_n.assumption.
-simplify.intros.apply le_S_S.apply le_O_n.
-simplify.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
+apply (nat_elim2 (\lambda n,m.n \nleq m \to m<n)).
+intros.apply (absurd (O \leq n1)).apply le_O_n.assumption.
+unfold Not.unfold lt.intros.apply le_S_S.apply le_O_n.
+unfold Not.unfold lt.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
 assumption.
 qed.
 
-theorem lt_to_not_le: \forall n,m:nat. lt n m \to Not (le m n).
-simplify.intros 3.elim H.
-apply not_le_Sn_n n H1.
+theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
+unfold Not.unfold lt.intros 3.elim H.
+apply (not_le_Sn_n n H1).
 apply H2.apply lt_to_le. apply H3.
 qed.
 
+theorem not_lt_to_le: \forall n,m:nat. Not (lt n m) \to le m n.
+simplify.intros.
+apply lt_S_to_le.
+apply not_le_to_lt.exact H.
+qed.
+
+theorem le_to_not_lt: \forall n,m:nat. le n m \to Not (lt m n).
+intros.
+change with (Not (le (S m) n)).
+apply lt_to_not_le.unfold lt.
+apply le_S_S.assumption.
+qed.
+
 (* le elimination *)
-theorem le_n_O_to_eq : \forall n:nat. (le n O) \to (eq nat O n).
+theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
 intro.elim n.reflexivity.
 apply False_ind.
-(* non si applica not_le_Sn_O *)
-apply  (not_le_Sn_O ? H1).
+apply not_le_Sn_O.
+goal 17. apply H1.
 qed.
 
-theorem le_n_O_elim: \forall n:nat.le n O \to \forall P: nat \to Prop.
+theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
 P O \to P n.
 intro.elim n.
 assumption.
@@ -121,35 +185,128 @@ apply False_ind.
 apply  (not_le_Sn_O ? H1).
 qed.
 
-theorem le_n_Sm_elim : \forall n,m:nat.le n (S m) \to 
-\forall P:Prop. (le (S n) (S m) \to P) \to (eq nat n (S m) \to P) \to P.
+theorem le_n_Sm_elim : \forall n,m:nat.n \leq S m \to 
+\forall P:Prop. (S n \leq S m \to P) \to (n=S m \to P) \to P.
 intros 4.elim H.
 apply H2.reflexivity.
 apply H3. apply le_S_S. assumption.
 qed.
 
+(* lt and le trans *)
+theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
+intros.elim H1.
+assumption.unfold lt.apply le_S.assumption.
+qed.
+
+theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
+intros 4.elim H.
+assumption.apply H2.unfold lt.
+apply lt_to_le.assumption.
+qed.
+
+theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
+intros.apply (le_to_lt_to_lt O n).
+apply le_O_n.assumption.
+qed.
+
+theorem lt_O_n_elim: \forall n:nat. lt O n \to 
+\forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
+intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
+apply H2.
+qed.
+
 (* other abstract properties *)
 theorem antisymmetric_le : antisymmetric nat le.
-simplify.intros 2.
-apply nat_elim2 (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
+unfold antisymmetric.intros 2.
+apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
 intros.apply le_n_O_to_eq.assumption.
-intros.apply False_ind.apply not_le_Sn_O ? H.
+intros.apply False_ind.apply (not_le_Sn_O ? H).
 intros.apply eq_f.apply H.
 apply le_S_S_to_le.assumption.
 apply le_S_S_to_le.assumption.
 qed.
 
-theorem antisym_le: \forall n,m:nat. le n m \to le m n \to eq nat n m
+theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
 \def antisymmetric_le.
 
-theorem decidable_le: \forall n,m:nat. decidable (le n m).
+theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
 intros.
-apply nat_elim2 (\lambda n,m.decidable (le n m)).
-intros.simplify.left.apply le_O_n.
-intros.simplify.right.exact not_le_Sn_O n1.
-intros 2.simplify.intro.elim H.
+apply (nat_elim2 (\lambda n,m.decidable (n \leq m))).
+intros.unfold decidable.left.apply le_O_n.
+intros.unfold decidable.right.exact (not_le_Sn_O n1).
+intros 2.unfold decidable.intro.elim H.
 left.apply le_S_S.assumption.
-right.intro.apply H1.apply le_S_S_to_le.assumption.
+right.unfold Not.intro.apply H1.apply le_S_S_to_le.assumption.
 qed.
 
+theorem decidable_lt: \forall n,m:nat. decidable (n < m).
+intros.exact (decidable_le (S n) m).
+qed.
+
+(* well founded induction principles *)
 
+theorem nat_elim1 : \forall n:nat.\forall P:nat \to Prop. 
+(\forall m.(\forall p. (p \lt m) \to P p) \to P m) \to P n.
+intros.cut (\forall q:nat. q \le n \to P q).
+apply (Hcut n).apply le_n.
+elim n.apply (le_n_O_elim q H1).
+apply H.
+intros.apply False_ind.apply (not_le_Sn_O p H2).
+apply H.intros.apply H1.
+cut (p < S n1).
+apply lt_S_to_le.assumption.
+apply (lt_to_le_to_lt p q (S n1) H3 H2).
+qed.
+
+(* some properties of functions *)
+
+definition increasing \def \lambda f:nat \to nat. 
+\forall n:nat. f n < f (S n).
+
+theorem increasing_to_monotonic: \forall f:nat \to nat.
+increasing f \to monotonic nat lt f.
+unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
+apply (trans_le ? (f n1)).
+assumption.apply (trans_le ? (S (f n1))).
+apply le_n_Sn.
+apply H.
+qed.
+
+theorem le_n_fn: \forall f:nat \to nat. (increasing f) 
+\to \forall n:nat. n \le (f n).
+intros.elim n.
+apply le_O_n.
+apply (trans_le ? (S (f n1))).
+apply le_S_S.apply H1.
+simplify in H. unfold increasing in H.unfold lt in H.apply H.
+qed.
+
+theorem increasing_to_le: \forall f:nat \to nat. (increasing f) 
+\to \forall m:nat. \exists i. m \le (f i).
+intros.elim m.
+apply (ex_intro ? ? O).apply le_O_n.
+elim H1.
+apply (ex_intro ? ? (S a)).
+apply (trans_le ? (S (f a))).
+apply le_S_S.assumption.
+simplify in H.unfold increasing in H.unfold lt in H.
+apply H.
+qed.
+
+theorem increasing_to_le2: \forall f:nat \to nat. (increasing f) 
+\to \forall m:nat. (f O) \le m \to 
+\exists i. (f i) \le m \land m <(f (S i)).
+intros.elim H1.
+apply (ex_intro ? ? O).
+split.apply le_n.apply H.
+elim H3.elim H4.
+cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
+elim Hcut.
+apply (ex_intro ? ? a).
+split.apply le_S. assumption.assumption.
+apply (ex_intro ? ? (S a)).
+split.rewrite < H7.apply le_n.
+rewrite > H7.
+apply H.
+apply le_to_or_lt_eq.apply H6.
+qed.