1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/Z/orders".
18 include "nat/orders.ma".
20 definition Zle : Z \to Z \to Prop \def
26 | (pos m) \Rightarrow True
27 | (neg m) \Rightarrow False ]
30 [ OZ \Rightarrow False
31 | (pos m) \Rightarrow n \leq m
32 | (neg m) \Rightarrow False ]
36 | (pos m) \Rightarrow True
37 | (neg m) \Rightarrow m \leq n ]].
39 (*CSC: the URI must disappear: there is a bug now *)
40 interpretation "integer 'less or equal to'" 'leq x y = (cic:/matita/Z/orders/Zle.con x y).
42 definition Zlt : Z \to Z \to Prop \def
47 [ OZ \Rightarrow False
48 | (pos m) \Rightarrow True
49 | (neg m) \Rightarrow False ]
52 [ OZ \Rightarrow False
53 | (pos m) \Rightarrow n<m
54 | (neg m) \Rightarrow False ]
58 | (pos m) \Rightarrow True
59 | (neg m) \Rightarrow m<n ]].
61 (*CSC: the URI must disappear: there is a bug now *)
62 interpretation "integer 'less than'" 'lt x y = (cic:/matita/Z/orders/Zlt.con x y).
64 theorem irreflexive_Zlt: irreflexive Z Zlt.
65 change with \forall x:Z. x < x \to False.
67 cut neg n < neg n \to False.
68 apply Hcut.apply H.simplify.apply not_le_Sn_n.
69 cut pos n < pos n \to False.
70 apply Hcut.apply H.simplify.apply not_le_Sn_n.
73 theorem irrefl_Zlt: irreflexive Z Zlt
76 theorem Zlt_neg_neg_to_lt:
77 \forall n,m:nat. neg n < neg m \to m < n.
81 theorem lt_to_Zlt_neg_neg: \forall n,m:nat.m < n \to neg n < neg m.
86 theorem Zlt_pos_pos_to_lt:
87 \forall n,m:nat. pos n < pos m \to n < m.
91 theorem lt_to_Zlt_pos_pos: \forall n,m:nat.n < m \to pos n < pos m.
96 theorem Zlt_to_Zle: \forall x,y:Z. x < y \to Zsucc x \leq y.
98 cut OZ < y \to Zsucc OZ \leq y.
99 apply Hcut. assumption.simplify.elim y.
102 simplify.apply le_O_n.
103 cut neg n < y \to Zsucc (neg n) \leq y.
104 apply Hcut. assumption.elim n.
105 cut neg O < y \to Zsucc (neg O) \leq y.
106 apply Hcut. assumption.simplify.elim y.
107 simplify.exact I.simplify.apply not_le_Sn_O n1 H2.
109 cut neg (S n1) < y \to (Zsucc (neg (S n1))) \leq y.
110 apply Hcut. assumption.simplify.
113 simplify.apply le_S_S_to_le n2 n1 H3.