1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/ordered_groups/".
17 include "ordered_sets.ma".
20 record pre_ogroup : Type ≝ {
21 og_abelian_group_: abelian_group;
22 og_tordered_set:> tordered_set;
23 og_with: carr og_abelian_group_ = og_tordered_set
26 lemma og_abelian_group: pre_ogroup → abelian_group.
27 intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
28 [apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
29 unfold apartness_OF_pre_ogroup; cases (og_with G); simplify;
30 [apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
33 coercion cic:/matita/ordered_groups/og_abelian_group.con.
36 record ogroup : Type ≝ {
38 fle_plusr: ∀f,g,h:og_carr. f≤g → f+h≤g+h
42 ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
44 apply (le_rewl ??? (0+x) (zero_neutral ??));
45 apply (le_rewl ??? (x+0) (plus_comm ???));
46 apply (le_rewl ??? (x+(-z+z))); [apply feq_plusl;apply opp_inverse;]
47 apply (le_rewl ??? (x+(z+ -z))); [apply feq_plusl;apply plus_comm;]
48 apply (le_rewl ??? (x+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
49 apply (le_rewr ??? (0+y) (zero_neutral ??));
50 apply (le_rewr ??? (y+0) (plus_comm ???));
51 apply (le_rewr ??? (y+(-z+z))); [apply feq_plusl;apply opp_inverse;]
52 apply (le_rewr ??? (y+(z+ -z))); [apply feq_plusl;apply plus_comm;]
53 apply (le_rewr ??? (y+z+ -z)); [apply eq_symmetric; apply plus_assoc;]
54 apply (fle_plusr ??? (-z));
58 lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
60 apply (plus_cancr_le ??? (-h));
61 apply (le_rewl ??? (f+h+ -h)); [apply feq_plusr;apply plus_comm;]
62 apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
63 apply (le_rewl ??? (f+(-h+h))); [apply feq_plusl;apply plus_comm;]
64 apply (le_rewl ??? (f+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
65 apply (le_rewl ??? (0+f) (plus_comm ???));
66 apply (le_rewl ??? (f) (eq_symmetric ??? (zero_neutral ??)));
67 apply (le_rewr ??? (g+h+ -h)); [apply feq_plusr;apply plus_comm;]
68 apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
69 apply (le_rewr ??? (g+(-h+h))); [apply feq_plusl;apply plus_comm;]
70 apply (le_rewr ??? (g+0)); [apply feq_plusl; apply eq_symmetric; apply opp_inverse]
71 apply (le_rewr ??? (0+g) (plus_comm ???));
72 apply (le_rewr ??? (g) (eq_symmetric ??? (zero_neutral ??)));
77 ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
79 apply (le_rewl ??? (0+x) (zero_neutral ??));
80 apply (le_rewl ??? ((-z+z)+x)); [apply feq_plusr;apply opp_inverse;]
81 apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
82 apply (le_rewr ??? (0+y) (zero_neutral ??));
83 apply (le_rewr ??? ((-z+z)+y)); [apply feq_plusr;apply opp_inverse;]
84 apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
85 apply (fle_plusl ??? (-z));
90 lemma le_zero_x_to_le_opp_x_zero:
91 ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
92 intros (G x Px); apply (plus_cancr_le ??? x);
93 apply (le_rewl ??? 0 (eq_symmetric ??? (opp_inverse ??)));
94 apply (le_rewr ??? x (eq_symmetric ??? (zero_neutral ??)));
98 lemma le_x_zero_to_le_zero_opp_x:
99 ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
100 intros (G x Lx0); apply (plus_cancr_le ??? x);
101 apply (le_rewr ??? 0 (eq_symmetric ??? (opp_inverse ??)));
102 apply (le_rewl ??? x (eq_symmetric ??? (zero_neutral ??)));