- renamed ocaml/ to components/

[helm.git] / helm / matita / library / Z / orders.ma

diff --git a/helm/matita/library/Z/orders.ma b/helm/matita/library/Z/orders.ma
deleted file mode 100644 (file)
index c39f693..0000000
--- a/helm/matita/library/Z/orders.ma
+++ /dev/null
@@ -1,130 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||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/Z/orders".
-
-include "Z/z.ma".
-include "nat/orders.ma".
-
-definition Zle : Z \to Z \to Prop \def
-\lambda x,y:Z.
-  match x with
-  [ OZ \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow True
-    | (pos m) \Rightarrow True
-    | (neg m) \Rightarrow False ]
-  | (pos n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow False
-    | (pos m) \Rightarrow n \leq m
-    | (neg m) \Rightarrow False ]
-  | (neg n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow True
-    | (pos m) \Rightarrow True
-    | (neg m) \Rightarrow m \leq n ]].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer 'less or equal to'" 'leq x y = (cic:/matita/Z/orders/Zle.con x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer 'neither less nor equal to'" 'nleq x y =
-  (cic:/matita/logic/connectives/Not.con (cic:/matita/Z/orders/Zle.con x y)).
-
-definition Zlt : Z \to Z \to Prop \def
-\lambda x,y:Z.
-  match x with
-  [ OZ \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow False
-    | (pos m) \Rightarrow True
-    | (neg m) \Rightarrow False ]
-  | (pos n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow False
-    | (pos m) \Rightarrow n<m
-    | (neg m) \Rightarrow False ]
-  | (neg n) \Rightarrow 
-    match y with 
-    [ OZ \Rightarrow True
-    | (pos m) \Rightarrow True
-    | (neg m) \Rightarrow m<n ]].
-    
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer 'less than'" 'lt x y = (cic:/matita/Z/orders/Zlt.con x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer 'not less than'" 'nless x y =
-  (cic:/matita/logic/connectives/Not.con (cic:/matita/Z/orders/Zlt.con x y)).
-
-theorem irreflexive_Zlt: irreflexive Z Zlt.
-change with (\forall x:Z. x < x \to False).
-intro.elim x.exact H.
-cut (neg n < neg n \to False).
-apply Hcut.apply H.simplify.unfold lt.apply not_le_Sn_n.
-cut (pos n < pos n \to False).
-apply Hcut.apply H.simplify.unfold lt.apply not_le_Sn_n.
-qed.
-
-theorem irrefl_Zlt: irreflexive Z Zlt
-\def irreflexive_Zlt.
-
-theorem Zlt_neg_neg_to_lt: 
-\forall n,m:nat. neg n < neg m \to m < n.
-intros.apply H.
-qed.
-
-theorem lt_to_Zlt_neg_neg: \forall n,m:nat.m < n \to neg n < neg m. 
-intros.
-simplify.apply H.
-qed.
-
-theorem Zlt_pos_pos_to_lt: 
-\forall n,m:nat. pos n < pos m \to n < m.
-intros.apply H.
-qed.
-
-theorem lt_to_Zlt_pos_pos: \forall n,m:nat.n < m \to pos n < pos m. 
-intros.
-simplify.apply H.
-qed.
-
-theorem Zlt_to_Zle: \forall x,y:Z. x < y \to Zsucc x \leq y.
-intros 2.
-elim x.
-(* goal: x=OZ *)
-  cut (OZ < y \to Zsucc OZ \leq y).
-    apply Hcut. assumption.
-    simplify.elim y. 
-      simplify.exact H1.
-      simplify.apply le_O_n.
-      simplify.exact H1.
-(* goal: x=pos *)      
-  exact H.
-(* goal: x=neg *)      
-  cut (neg n < y \to Zsucc (neg n) \leq y).
-    apply Hcut. assumption.
-    elim n.
-      cut (neg O < y \to Zsucc (neg O) \leq y).
-        apply Hcut. assumption.
-        simplify.elim y.
-          simplify.exact I.
-          simplify.exact I.
-          simplify.apply (not_le_Sn_O n1 H2).
-      cut (neg (S n1) < y \to (Zsucc (neg (S n1))) \leq y).
-        apply Hcut. assumption.simplify.
-        elim y.
-          simplify.exact I.
-          simplify.exact I.
-          simplify.apply (le_S_S_to_le n2 n1 H3).
-qed.