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_gorup/".
17 include "ordered_set.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_gorup/og_abelian_group.con.
35 record ogroup : Type ≝ {
37 exc_canc_plusr: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
40 notation > "'Ex'≪" non associative with precedence 50 for
41 @{'excedencerewritel}.
43 interpretation "exc_rewl" 'excedencerewritel =
44 (cic:/matita/excedence/exc_rewl.con _ _ _).
46 notation > "'Ex'≫" non associative with precedence 50 for
47 @{'excedencerewriter}.
49 interpretation "exc_rewr" 'excedencerewriter =
50 (cic:/matita/excedence/exc_rewr.con _ _ _).
53 ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
54 intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
55 apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
56 apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
57 apply (Ex≪ (x+0) (opp_inverse ??));
58 apply (Ex≪ (0+x) (plus_comm ???));
59 apply (Ex≪ x (zero_neutral ??));
60 apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
61 apply (Ex≫ (y + (-z + z)) (plus_comm ??z));
62 apply (Ex≫ (y+0) (opp_inverse ??));
63 apply (Ex≫ (0+y) (plus_comm ???));
64 apply (Ex≫ y (zero_neutral ??) L);
67 coercion cic:/matita/ordered_gorup/fexc_plusr.con nocomposites.
69 lemma exc_canc_plusl: ∀G:ogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
70 intros 5 (G x y z L); apply (exc_canc_plusr ??? z);
71 apply (exc_rewl ??? (z+x) (plus_comm ???));
72 apply (exc_rewr ??? (z+y) (plus_comm ???) L);
76 ∀G:ogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
77 intros 5 (G x y z L); apply (exc_canc_plusl ??? (-z));
78 apply (exc_rewl ???? (plus_assoc ??z x));
79 apply (exc_rewr ???? (plus_assoc ??z y));
80 apply (exc_rewl ??? (0+x) (opp_inverse ??));
81 apply (exc_rewr ??? (0+y) (opp_inverse ??));
82 apply (exc_rewl ???? (zero_neutral ??));
83 apply (exc_rewr ???? (zero_neutral ??) L);
86 coercion cic:/matita/ordered_gorup/fexc_plusl.con nocomposites.
89 ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
91 apply (le_rewl ??? (0+x) (zero_neutral ??));
92 apply (le_rewl ??? (x+0) (plus_comm ???));
93 apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
94 apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
95 apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
96 apply (le_rewr ??? (0+y) (zero_neutral ??));
97 apply (le_rewr ??? (y+0) (plus_comm ???));
98 apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
99 apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
100 apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
101 intro H; apply L; clear L; apply (exc_canc_plusr ??? (-z) H);
104 lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
106 apply (plus_cancr_le ??? (-h));
107 apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
108 apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
109 apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
110 apply (le_rewl ??? (f+0) (opp_inverse ??));
111 apply (le_rewl ??? (0+f) (plus_comm ???));
112 apply (le_rewl ??? (f) (zero_neutral ??));
113 apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
114 apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
115 apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
116 apply (le_rewr ??? (g+0) (opp_inverse ??));
117 apply (le_rewr ??? (0+g) (plus_comm ???));
118 apply (le_rewr ??? (g) (zero_neutral ??) H);
121 lemma fle_plusr: ∀G:ogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
122 intros (G f g h H); apply (le_rewl ???? (plus_comm ???));
123 apply (le_rewr ???? (plus_comm ???)); apply fle_plusl; assumption;
127 ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
128 intros 5 (G x y z L);
129 apply (le_rewl ??? (0+x) (zero_neutral ??));
130 apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
131 apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
132 apply (le_rewr ??? (0+y) (zero_neutral ??));
133 apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
134 apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
135 apply (fle_plusl ??? (-z) L);
138 lemma exc_opp_x_zero_to_exc_zero_x:
139 ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
140 intros (G x H); apply (exc_canc_plusr ??? (-x));
141 apply (exc_rewr ???? (plus_comm ???));
142 apply (exc_rewr ???? (opp_inverse ??));
143 apply (exc_rewl ???? (zero_neutral ??) H);
146 lemma le_zero_x_to_le_opp_x_zero:
147 ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
148 intros (G x Px); apply (plus_cancr_le ??? x);
149 apply (le_rewl ??? 0 (opp_inverse ??));
150 apply (le_rewr ??? x (zero_neutral ??) Px);
153 lemma exc_zero_opp_x_to_exc_x_zero:
154 ∀G:ogroup.∀x:G. 0 ≰ -x → x ≰ 0.
155 intros (G x H); apply (exc_canc_plusl ??? (-x));
156 apply (exc_rewr ???? (plus_comm ???));
157 apply (exc_rewl ???? (opp_inverse ??));
158 apply (exc_rewr ???? (zero_neutral ??) H);
161 lemma le_x_zero_to_le_zero_opp_x:
162 ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
163 intros (G x Lx0); apply (plus_cancr_le ??? x);
164 apply (le_rewr ??? 0 (opp_inverse ??));
165 apply (le_rewl ??? x (zero_neutral ??));
170 ∀G:ogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
171 intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2);
172 [right; split; assumption|left;split;[assumption]]
173 apply (plus_cancr_ap ??? y); apply (ap_rewl ???? (zero_neutral ??));
178 ∀G:ogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c.
179 intros (G a b c L H); apply (le_transitive ????? H);
180 apply (plus_cancl_le ??? (-a));
181 apply (le_rewl ??? 0 (opp_inverse ??));
182 apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
183 apply (le_rewr ??? (0 + b) (opp_inverse ??));
184 apply (le_rewr ??? b (zero_neutral ??));