| (pos m) \Rightarrow True
| (neg m) \Rightarrow (le m 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: this alias must disappear: there is a bug in the generation of the .moos *)
+alias symbol "leq" (instance 0) = "integer 'less or equal to'".
definition Zlt : Z \to Z \to Prop \def
\lambda x,y:Z.
| (pos m) \Rightarrow True
| (neg m) \Rightarrow (lt 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: this alias must disappear: there is a bug in the generation of the .moos *)
+alias symbol "lt" (instance 0) = "integer 'less than'".
+
theorem irreflexive_Zlt: irreflexive Z Zlt.
-change with \forall x:Z. Zlt x x \to False.
+change with \forall x:Z. x < x \to False.
intro.elim x.exact H.
-cut (Zlt (neg n) (neg n)) \to False.
+cut neg n < neg n \to False.
apply Hcut.apply H.simplify.apply not_le_Sn_n.
-cut (Zlt (pos n) (pos n)) \to False.
+cut pos n < pos n \to False.
apply Hcut.apply H.simplify.apply not_le_Sn_n.
qed.
| (neg m) \Rightarrow nat_compare m n ]].
theorem Zlt_neg_neg_to_lt:
-\forall n,m:nat. Zlt (neg n) (neg m) \to lt m n.
+\forall n,m:nat. neg n < neg m \to lt m n.
intros.apply H.
qed.
-theorem lt_to_Zlt_neg_neg: \forall n,m:nat.lt m n \to Zlt (neg n) (neg m).
+theorem lt_to_Zlt_neg_neg: \forall n,m:nat.lt m n \to neg n < neg m.
intros.
simplify.apply H.
qed.
theorem Zlt_pos_pos_to_lt:
-\forall n,m:nat. Zlt (pos n) (pos m) \to lt n m.
+\forall n,m:nat. pos n < pos m \to lt n m.
intros.apply H.
qed.
-theorem lt_to_Zlt_pos_pos: \forall n,m:nat.lt n m \to Zlt (pos n) (pos m).
+theorem lt_to_Zlt_pos_pos: \forall n,m:nat.lt n m \to pos n < pos m.
intros.
simplify.apply H.
qed.
theorem Z_compare_to_Prop :
\forall x,y:Z. match (Z_compare x y) with
-[ LT \Rightarrow (Zlt x y)
-| EQ \Rightarrow (eq Z x y)
-| GT \Rightarrow (Zlt y x)].
+[ LT \Rightarrow x < y
+| EQ \Rightarrow x=y
+| GT \Rightarrow y < x].
intros.
elim x. elim y.
simplify.apply refl_eq.
simplify.apply eq_f.exact H.
qed.
-theorem Zlt_to_Zle: \forall x,y:Z. Zlt x y \to Zle (Zsucc x) y.
+theorem Zlt_to_Zle: \forall x,y:Z. x < y \to Zsucc x \leq y.
intros 2.elim x.
-cut (Zlt OZ y) \to (Zle (Zsucc OZ) y).
+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 (Zlt (neg n) y) \to (Zle (Zsucc (neg n)) y).
+cut neg n < y \to Zsucc (neg n) \leq y.
apply Hcut. assumption.elim n.
-cut (Zlt (neg O) y) \to (Zle (Zsucc (neg O)) y).
+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 (Zlt (neg (S n1)) y) \to (Zle (Zsucc (neg (S n1))) y).
+cut neg (S n1) < y \to (Zsucc (neg (S n1))) \leq y.
apply Hcut. assumption.simplify.
elim y.
simplify.exact I.