X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Forders.ma;h=d592ed0afa4951d6d080109771eab40923568984;hb=33a02e0b639217093eb63f30169aaa6ac8c78907;hp=5389418b5237d7bde8cf4a9424bba777a5b38a62;hpb=3eff4cc36820df9faddb3cb16390717851db499c;p=helm.git

diff --git a/helm/matita/library/nat/orders.ma b/helm/matita/library/nat/orders.ma
index 5389418b5..d592ed0af 100644
--- a/helm/matita/library/nat/orders.ma
+++ b/helm/matita/library/nat/orders.ma
@@ -22,22 +22,25 @@ 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).
-alias symbol "leq" (instance 0) = "natural 'less or equal to'".
 
 definition lt: nat \to nat \to Prop \def
 \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).
 
 definition ge: nat \to nat \to Prop \def
 \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.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).
 
 theorem transitive_le : transitive nat le.
@@ -49,6 +52,15 @@ qed.
 theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
 \def transitive_le.
 
+theorem transitive_lt: transitive nat lt.
+simplify.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.
@@ -92,10 +104,19 @@ apply H.assumption.
 apply le_S_S_to_le.assumption.
 qed.
 
-(* ??? this needs not le *)
-theorem S_pred: \forall n:nat.O<n \to n=S (pred n).
-intro.elim n.apply False_ind.exact not_le_Sn_O O H.
-apply eq_f.apply pred_Sn.
+(* 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.simplify.apply le_S_S.assumption.
+qed.
+
+(* not eq *)
+theorem lt_to_not_eq : \forall n,m:nat. n<m \to \lnot (n=m).
+simplify.intros.cut (le (S n) m) \to False.
+apply Hcut.assumption.rewrite < H1.
+apply not_le_Sn_n.
 qed.
 
 (* le vs. lt *)
@@ -105,6 +126,11 @@ apply le_S. apply le_n.
 apply le_S. assumption.
 qed.
 
+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. \lnot (n \leq m) \to m<n.
 intros 2.
 apply nat_elim2 (\lambda n,m.\lnot (n \leq m) \to m<n).
@@ -120,6 +146,19 @@ 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.simplify.
+apply le_S_S.assumption.
+qed.
+
 (* le elimination *)
 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
 intro.elim n.reflexivity.
@@ -143,6 +182,29 @@ 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.simplify.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.simplify.
+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.
@@ -166,3 +228,75 @@ intros 2.simplify.intro.elim H.
 left.apply le_S_S.assumption.
 right.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.
+simplify.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. apply H.
+qed.
+
+theorem increasing_to_le: \forall f:nat \to nat. (increasing f) 
+\to \forall m:nat. ex nat (\lambda 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.
+apply H.
+qed.
+
+theorem increasing_to_le2: \forall f:nat \to nat. (increasing f) 
+\to \forall m:nat. (f O) \le m \to 
+ex nat (\lambda 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.