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
41 ∀G:ogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
42 intros 5 (G x y z L); apply (exc_canc_plusr ??? (-z));
43 apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
44 apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
45 apply (Ex≪ (x+0) (opp_inverse ??));
46 apply (Ex≪ (0+x) (plus_comm ???));
47 apply (Ex≪ x (zero_neutral ??));
48 apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
49 apply (Ex≫ (y + (-z + z)) (plus_comm ??z));
50 apply (Ex≫ (y+0) (opp_inverse ??));
51 apply (Ex≫ (0+y) (plus_comm ???));
52 apply (Ex≫ y (zero_neutral ??) L);
55 coercion cic:/matita/ordered_gorup/fexc_plusr.con nocomposites.
57 lemma exc_canc_plusl: ∀G:ogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
58 intros 5 (G x y z L); apply (exc_canc_plusr ??? z);
59 apply (exc_rewl ??? (z+x) (plus_comm ???));
60 apply (exc_rewr ??? (z+y) (plus_comm ???) L);
64 ∀G:ogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
65 intros 5 (G x y z L); apply (exc_canc_plusl ??? (-z));
66 apply (exc_rewl ???? (plus_assoc ??z x));
67 apply (exc_rewr ???? (plus_assoc ??z y));
68 apply (exc_rewl ??? (0+x) (opp_inverse ??));
69 apply (exc_rewr ??? (0+y) (opp_inverse ??));
70 apply (exc_rewl ???? (zero_neutral ??));
71 apply (exc_rewr ???? (zero_neutral ??) L);
74 coercion cic:/matita/ordered_gorup/fexc_plusl.con nocomposites.
77 ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
79 apply (le_rewl ??? (0+x) (zero_neutral ??));
80 apply (le_rewl ??? (x+0) (plus_comm ???));
81 apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
82 apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
83 apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
84 apply (le_rewr ??? (0+y) (zero_neutral ??));
85 apply (le_rewr ??? (y+0) (plus_comm ???));
86 apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
87 apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
88 apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
89 intro H; apply L; clear L; apply (exc_canc_plusr ??? (-z) H);
92 lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
94 apply (plus_cancr_le ??? (-h));
95 apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
96 apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
97 apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
98 apply (le_rewl ??? (f+0) (opp_inverse ??));
99 apply (le_rewl ??? (0+f) (plus_comm ???));
100 apply (le_rewl ??? (f) (zero_neutral ??));
101 apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
102 apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
103 apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
104 apply (le_rewr ??? (g+0) (opp_inverse ??));
105 apply (le_rewr ??? (0+g) (plus_comm ???));
106 apply (le_rewr ??? (g) (zero_neutral ??) H);
109 lemma fle_plusr: ∀G:ogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
110 intros (G f g h H); apply (le_rewl ???? (plus_comm ???));
111 apply (le_rewr ???? (plus_comm ???)); apply fle_plusl; assumption;
115 ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
116 intros 5 (G x y z L);
117 apply (le_rewl ??? (0+x) (zero_neutral ??));
118 apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
119 apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
120 apply (le_rewr ??? (0+y) (zero_neutral ??));
121 apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
122 apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
123 apply (fle_plusl ??? (-z) L);
127 ∀G:ogroup.∀x,y,z:G.z+x < z+y → x < y.
128 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancl_le; assumption]
129 apply (plus_cancl_ap ???? LE);
133 ∀G:ogroup.∀x,y,z:G.x+z < y+z → x < y.
134 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancr_le; assumption]
135 apply (plus_cancr_ap ???? LE);
139 lemma exc_opp_x_zero_to_exc_zero_x:
140 ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
141 intros (G x H); apply (exc_canc_plusr ??? (-x));
142 apply (exc_rewr ???? (plus_comm ???));
143 apply (exc_rewr ???? (opp_inverse ??));
144 apply (exc_rewl ???? (zero_neutral ??) H);
147 lemma le_zero_x_to_le_opp_x_zero:
148 ∀G:ogroup.∀x:G.0 ≤ x → -x ≤ 0.
149 intros (G x Px); apply (plus_cancr_le ??? x);
150 apply (le_rewl ??? 0 (opp_inverse ??));
151 apply (le_rewr ??? x (zero_neutral ??) Px);
154 lemma lt_zero_x_to_lt_opp_x_zero:
155 ∀G:ogroup.∀x:G.0 < x → -x < 0.
156 intros (G x Px); apply (plus_cancr_lt ??? x);
157 apply (lt_rewl ??? 0 (opp_inverse ??));
158 apply (lt_rewr ??? x (zero_neutral ??) Px);
161 lemma exc_zero_opp_x_to_exc_x_zero:
162 ∀G:ogroup.∀x:G. 0 ≰ -x → x ≰ 0.
163 intros (G x H); apply (exc_canc_plusl ??? (-x));
164 apply (exc_rewr ???? (plus_comm ???));
165 apply (exc_rewl ???? (opp_inverse ??));
166 apply (exc_rewr ???? (zero_neutral ??) H);
169 lemma le_x_zero_to_le_zero_opp_x:
170 ∀G:ogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
171 intros (G x Lx0); apply (plus_cancr_le ??? x);
172 apply (le_rewr ??? 0 (opp_inverse ??));
173 apply (le_rewl ??? x (zero_neutral ??));
177 lemma lt_x_zero_to_lt_zero_opp_x:
178 ∀G:ogroup.∀x:G. x < 0 → 0 < -x.
179 intros (G x Lx0); apply (plus_cancr_lt ??? x);
180 apply (lt_rewr ??? 0 (opp_inverse ??));
181 apply (lt_rewl ??? x (zero_neutral ??));
187 ∀G:ogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
188 intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2);
189 [right; split; assumption|left;split;[assumption]]
190 apply (plus_cancr_ap ??? y); apply (ap_rewl ???? (zero_neutral ??));
195 ∀G:ogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c.
196 intros (G a b c L H); apply (le_transitive ????? H);
197 apply (plus_cancl_le ??? (-a));
198 apply (le_rewl ??? 0 (opp_inverse ??));
199 apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
200 apply (le_rewr ??? (0 + b) (opp_inverse ??));
201 apply (le_rewr ??? b (zero_neutral ??));
206 ∀G:ogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
207 intros (G a b L1 L2); apply (le_transitive ???? L1);
208 apply (plus_cancl_le ??? (-a));
209 apply (le_rewl ??? 0 (opp_inverse ??));
210 apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
211 apply (le_rewr ??? (0 + b) (opp_inverse ??));
212 apply (le_rewr ??? b (zero_neutral ??));