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_group/".
19 record pogroup_ : Type ≝ {
20 og_abelian_group_: abelian_group;
21 og_excedence:> excedence;
22 og_with: carr og_abelian_group_ = apart_of_excedence og_excedence
25 lemma og_abelian_group: pogroup_ → abelian_group.
26 intro G; apply (mk_abelian_group G); [1,2,3: rewrite < (og_with G)]
27 [apply (plus (og_abelian_group_ G));|apply zero;|apply opp]
28 unfold apartness_OF_pogroup_; cases (og_with G); simplify;
29 [apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext]
32 coercion cic:/matita/ordered_group/og_abelian_group.con.
34 record pogroup : Type ≝ {
36 canc_plusr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g
40 ∀G:pogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z.
41 intros 5 (G x y z L); apply (canc_plusr_exc ??? (-z));
42 apply (Ex≪ (x + (z + -z)) (plus_assoc ????));
43 apply (Ex≪ (x + (-z + z)) (plus_comm ??z));
44 apply (Ex≪ (x+0) (opp_inverse ??));
45 apply (Ex≪ (0+x) (plus_comm ???));
46 apply (Ex≪ x (zero_neutral ??));
47 apply (Ex≫ (y + (z + -z)) (plus_assoc ????));
48 apply (Ex≫ (y + (-z + z)) (plus_comm ??z));
49 apply (Ex≫ (y+0) (opp_inverse ??));
50 apply (Ex≫ (0+y) (plus_comm ???));
51 apply (Ex≫ y (zero_neutral ??) L);
54 coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites.
56 lemma canc_plusl_exc: ∀G:pogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g.
57 intros 5 (G x y z L); apply (canc_plusr_exc ??? z);
58 apply (exc_rewl ??? (z+x) (plus_comm ???));
59 apply (exc_rewr ??? (z+y) (plus_comm ???) L);
63 ∀G:pogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y.
64 intros 5 (G x y z L); apply (canc_plusl_exc ??? (-z));
65 apply (exc_rewl ???? (plus_assoc ??z x));
66 apply (exc_rewr ???? (plus_assoc ??z y));
67 apply (exc_rewl ??? (0+x) (opp_inverse ??));
68 apply (exc_rewr ??? (0+y) (opp_inverse ??));
69 apply (exc_rewl ???? (zero_neutral ??));
70 apply (exc_rewr ???? (zero_neutral ??) L);
73 coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites.
76 ∀G:pogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
78 apply (le_rewl ??? (0+x) (zero_neutral ??));
79 apply (le_rewl ??? (x+0) (plus_comm ???));
80 apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
81 apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
82 apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
83 apply (le_rewr ??? (0+y) (zero_neutral ??));
84 apply (le_rewr ??? (y+0) (plus_comm ???));
85 apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
86 apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
87 apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
88 intro H; apply L; clear L; apply (canc_plusr_exc ??? (-z) H);
91 lemma fle_plusl: ∀G:pogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
93 apply (plus_cancr_le ??? (-h));
94 apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
95 apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
96 apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
97 apply (le_rewl ??? (f+0) (opp_inverse ??));
98 apply (le_rewl ??? (0+f) (plus_comm ???));
99 apply (le_rewl ??? (f) (zero_neutral ??));
100 apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
101 apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
102 apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
103 apply (le_rewr ??? (g+0) (opp_inverse ??));
104 apply (le_rewr ??? (0+g) (plus_comm ???));
105 apply (le_rewr ??? (g) (zero_neutral ??) H);
108 lemma fle_plusr: ∀G:pogroup. ∀f,g,h:G. f≤g → f+h≤g+h.
109 intros (G f g h H); apply (le_rewl ???? (plus_comm ???));
110 apply (le_rewr ???? (plus_comm ???)); apply fle_plusl; assumption;
114 ∀G:pogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
115 intros 5 (G x y z L);
116 apply (le_rewl ??? (0+x) (zero_neutral ??));
117 apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
118 apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
119 apply (le_rewr ??? (0+y) (zero_neutral ??));
120 apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
121 apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
122 apply (fle_plusl ??? (-z) L);
126 ∀G:pogroup.∀x,y,z:G.z+x < z+y → x < y.
127 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancl_le; assumption]
128 apply (plus_cancl_ap ???? LE);
132 ∀G:pogroup.∀x,y,z:G.x+z < y+z → x < y.
133 intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancr_le; assumption]
134 apply (plus_cancr_ap ???? LE);
138 lemma exc_opp_x_zero_to_exc_zero_x:
139 ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
140 intros (G x H); apply (canc_plusr_exc ??? (-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:pogroup.∀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 lt_zero_x_to_lt_opp_x_zero:
154 ∀G:pogroup.∀x:G.0 < x → -x < 0.
155 intros (G x Px); apply (plus_cancr_lt ??? x);
156 apply (lt_rewl ??? 0 (opp_inverse ??));
157 apply (lt_rewr ??? x (zero_neutral ??) Px);
160 lemma exc_zero_opp_x_to_exc_x_zero:
161 ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0.
162 intros (G x H); apply (canc_plusl_exc ??? (-x));
163 apply (exc_rewr ???? (plus_comm ???));
164 apply (exc_rewl ???? (opp_inverse ??));
165 apply (exc_rewr ???? (zero_neutral ??) H);
168 lemma le_x_zero_to_le_zero_opp_x:
169 ∀G:pogroup.∀x:G. x ≤ 0 → 0 ≤ -x.
170 intros (G x Lx0); apply (plus_cancr_le ??? x);
171 apply (le_rewr ??? 0 (opp_inverse ??));
172 apply (le_rewl ??? x (zero_neutral ??));
176 lemma lt_x_zero_to_lt_zero_opp_x:
177 ∀G:pogroup.∀x:G. x < 0 → 0 < -x.
178 intros (G x Lx0); apply (plus_cancr_lt ??? x);
179 apply (lt_rewr ??? 0 (opp_inverse ??));
180 apply (lt_rewl ??? x (zero_neutral ??));
186 ∀G:pogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y.
187 intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2);
188 [right; split; assumption|left;split;[assumption]]
189 apply (plus_cancr_ap ??? y); apply (ap_rewl ???? (zero_neutral ??));
194 ∀G:pogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c.
195 intros (G a b c L H); apply (le_transitive ????? H);
196 apply (plus_cancl_le ??? (-a));
197 apply (le_rewl ??? 0 (opp_inverse ??));
198 apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
199 apply (le_rewr ??? (0 + b) (opp_inverse ??));
200 apply (le_rewr ??? b (zero_neutral ??));
205 ∀G:pogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b.
206 intros (G a b L1 L2); apply (le_transitive ???? L1);
207 apply (plus_cancl_le ??? (-a));
208 apply (le_rewl ??? 0 (opp_inverse ??));
209 apply (le_rewr ??? (-a + a + b) (plus_assoc ????));
210 apply (le_rewr ??? (0 + b) (opp_inverse ??));
211 apply (le_rewr ??? b (zero_neutral ??));
216 ∀G:pogroup.∀x,y,z:G.x < y → z + x < z + y.
217 intros (G x y z H); cases H; split; [apply fle_plusl; assumption]
218 apply fap_plusl; assumption;
222 ∀G:pogroup.∀x,y,z:G.x < y → x + z < y + z.
223 intros (G x y z H); cases H; split; [apply fle_plusr; assumption]
224 apply fap_plusr; assumption;
228 lemma ltxy_ltyyxx: ∀G:pogroup.∀x,y:G. y < x → y+y < x+x.
229 intros; apply (lt_transitive ?? (y+x));[2:
230 apply (lt_rewl ???? (plus_comm ???));
231 apply (lt_rewr ???? (plus_comm ???));]
232 apply flt_plusl;assumption;
235 lemma lew_opp : ∀O:pogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c.
236 intros (O a b c L0 L);
237 apply (le_transitive ????? L);
238 apply (plus_cancl_le ??? (-a));
239 apply (le_rewr ??? 0 (opp_inverse ??));
240 apply (le_rewl ??? (-a+a+-b) (plus_assoc ????));
241 apply (le_rewl ??? (0+-b) (opp_inverse ??));
242 apply (le_rewl ??? (-b) (zero_neutral ?(-b)));
243 apply le_zero_x_to_le_opp_x_zero;
247 lemma ltw_opp : ∀O:pogroup.∀a,b,c:O.0 < b → a < c → a + -b < c.
248 intros (O a b c P L);
249 apply (lt_transitive ????? L);
250 apply (plus_cancl_lt ??? (-a));
251 apply (lt_rewr ??? 0 (opp_inverse ??));
252 apply (lt_rewl ??? (-a+a+-b) (plus_assoc ????));
253 apply (lt_rewl ??? (0+-b) (opp_inverse ??));
254 apply (lt_rewl ??? ? (zero_neutral ??));
255 apply lt_zero_x_to_lt_opp_x_zero;
259 record togroup : Type ≝ {
261 tog_total: ∀x,y:tog_carr.x≰y → y < x
264 lemma lexxyy_lexy: ∀G:togroup. ∀x,y:G. x+x ≤ y+y → x ≤ y.
265 intros (G x y H); intro H1; lapply (tog_total ??? H1) as H2;
266 lapply (ltxy_ltyyxx ??? H2) as H3; lapply (lt_to_excede ??? H3) as H4;
270 lemma eqxxyy_eqxy: ∀G:togroup.∀x,y:G. x + x ≈ y + y → x ≈ y.
271 intros (G x y H); cases (eq_le_le ??? H); apply le_le_eq;
272 apply lexxyy_lexy; assumption;
275 lemma bar: ∀G:abelian_group. ∀x,y:G. 0 # x + y → 0 #x ∨ 0#y.
276 intros; cases (ap_cotransitive ??? y a); [right; assumption]
277 left; apply (plus_cancr_ap ??? y); apply (ap_rewl ???y (zero_neutral ??));
281 lemma pippo: ∀G:pogroup.∀a,b,c,d:G. a < b → c < d → a+c < b + d.
282 intros (G a b c d H1 H2);
283 lapply (flt_plusr ??? c H1) as H3;
284 apply (lt_transitive ???? H3);
285 apply flt_plusl; assumption;
288 lemma pippo2: ∀G:pogroup.∀a,b,c,d:G. a+c ≰ b + d → a ≰ b ∨ c ≰ d.
289 intros (G a b c d H1 H2);
290 cases (exc_cotransitive ??? (a + d) H1); [
291 right; apply (canc_plusl_exc ??? a); assumption]
292 left; apply (canc_plusr_exc ??? d); assumption;
295 lemma pippo3: ∀G:pogroup.∀a,b,c,d:G. a ≤ b → c ≤ d → a+c ≤ b + d.
296 intros (G a b c d H1 H2); intro H3; cases (pippo2 ????? H3);
297 [apply H1|apply H2] assumption;
300 lemma foo: ∀G:togroup.∀x,y:G. 0 ≤ x + y → x < 0 → 0 ≤ y.
301 intros; intro; apply H; lapply (lt_to_excede ??? l);
302 lapply (tog_total ??? e);
303 lapply (tog_total ??? Hletin);
304 lapply (pippo ????? Hletin2 Hletin1);
305 apply (exc_rewl ??? (0+0)); [apply eq_sym; apply zero_neutral]
306 apply lt_to_excede; assumption;
309 lemma pippo4: ∀G:togroup.∀a,b,c,d:G. a+c < b + d → a < b ∨ c < d.
310 intros (G a b c d H1 H2); lapply (lt_to_excede ??? H1);
311 cases (pippo2 ????? Hletin); [left|right] apply tog_total; assumption;