ocaml 3.09 transition

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

diff --git a/helm/matita/library/Z/orders.ma b/helm/matita/library/Z/orders.ma
index bc5ffdb5125f5495b3df6ee878c2cef0e7f04488..c39f693085398eedf4a95a9f15898f05e6cee264 100644 (file)
--- a/helm/matita/library/Z/orders.ma
+++ b/helm/matita/library/Z/orders.ma
@@ -38,6 +38,9 @@ definition Zle : Z \to Z \to Prop \def
 
 (*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.
@@ -60,61 +63,68 @@ definition Zlt : Z \to Z \to Prop \def
     
 (*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.
+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.apply not_le_Sn_n.
-cut pos n < pos n \to False.
-apply Hcut.apply H.simplify.apply not_le_Sn_n.
+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.
 
-(*CSC: qui uso lt perche' ho due istanze diverse di < *)
 theorem Zlt_neg_neg_to_lt: 
-\forall n,m:nat. neg n < neg m \to lt m n.
+\forall n,m:nat. neg n < neg m \to m < n.
 intros.apply H.
 qed.
 
-(*CSC: qui uso lt perche' ho due istanze diverse di < *)
-theorem lt_to_Zlt_neg_neg: \forall n,m:nat.lt m n \to neg n < neg m. 
+theorem lt_to_Zlt_neg_neg: \forall n,m:nat.m < n \to neg n < neg m. 
 intros.
 simplify.apply H.
 qed.
 
-(*CSC: qui uso lt perche' ho due istanze diverse di < *)
 theorem Zlt_pos_pos_to_lt: 
-\forall n,m:nat. pos n < pos m \to lt n m.
+\forall n,m:nat. pos n < pos m \to n < m.
 intros.apply H.
 qed.
 
-(*CSC: qui uso lt perche' ho due istanze diverse di < *)
-theorem lt_to_Zlt_pos_pos: \forall n,m:nat.lt n m \to pos n < pos m. 
+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.
-cut OZ < y \to Zsucc OZ \leq y.
-apply Hcut. assumption.simplify.elim y.
-simplify.exact H1.
-simplify.exact H1.
-simplify.apply le_O_n.
-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.apply not_le_Sn_O n1 H2.
-simplify.exact I.
-cut neg (S n1) < y \to (Zsucc (neg (S n1))) \leq y.
-apply Hcut. assumption.simplify.
-elim y.
-simplify.exact I.
-simplify.apply le_S_S_to_le n2 n1 H3.
-simplify.exact I.
-exact H.
+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.