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/metric_lattice/".
17 include "metric_space.ma".
20 record mlattice_ (R : todgroup) : Type ≝ {
21 ml_mspace_: metric_space R;
23 ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
26 lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
27 intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
28 cases (ml_with_ ? ml); simplify;
29 [apply (metric ? (ml_mspace_ ? ml));|apply (mpositive ? (ml_mspace_ ? ml));
30 |apply (mreflexive ? (ml_mspace_ ? ml));|apply (msymmetric ? (ml_mspace_ ? ml));
31 |apply (mtineq ? (ml_mspace_ ? ml))]
34 coercion cic:/matita/metric_lattice/ml_mspace.con.
36 record is_mlattice (R : todgroup) (ml: mlattice_ R) : Type ≝ {
37 ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
38 ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
41 record mlattice (R : todgroup) : Type ≝ {
42 ml_carr :> mlattice_ R;
43 ml_props:> is_mlattice R ml_carr
46 lemma eq_to_ndlt0: ∀R.∀ml:mlattice R.∀a,b:ml. a ≈ b → ¬ 0 < δ a b.
47 intros (R ml a b E); intro H; apply E; apply (ml_prop1 ?? ml);
51 lemma eq_to_dzero: ∀R.∀ml:mlattice R.∀x,y:ml.x ≈ y → δ x y ≈ 0.
52 intros (R ml x y H); intro H1; apply H; clear H;
53 apply (ml_prop1 ?? ml); split [apply mpositive] apply ap_symmetric;
57 lemma meq_l: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δx y ≈ δz y.
58 intros (R ml x y z); apply le_le_eq;
59 [ apply (le_transitive ???? (mtineq ???y z));
60 apply (le_rewl ??? (0+δz y) (eq_to_dzero ???? H));
61 apply (le_rewl ??? (δz y) (zero_neutral ??)); apply le_reflexive;
62 | apply (le_transitive ???? (mtineq ???y x));
63 apply (le_rewl ??? (0+δx y) (eq_to_dzero ??z x H));
64 apply (le_rewl ??? (δx y) (zero_neutral ??)); apply le_reflexive;]
68 lemma meq_r: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z.
69 intros; apply (eq_trans ???? (msymmetric ??y x));
70 apply (eq_trans ????? (msymmetric ??z y)); apply meq_l; assumption;
74 lemma dap_to_lt: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → 0 < δ x y.
75 intros; split [apply mpositive] apply ap_symmetric; assumption;
78 lemma dap_to_ap: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → x # y.
79 intros (R ml x y H); apply (ml_prop1 ?? ml); split; [apply mpositive;]
80 apply ap_symmetric; assumption;
85 ∀R.∀ml:mlattice R.∀x,y,z:ml. x ≤ y → y ≤ z → δ x z ≈ δ x y + δ y z.
86 intros (R ml x y z Lxy Lyz); apply le_le_eq; [apply mtineq]
87 apply (le_transitive ????? (ml_prop2 ?? ml (y) ??));
88 cut ( δx y+ δy z ≈ δ(y∨x) (y∨z)+ δ(y∧x) (y∧z)); [
89 apply (le_rewr ??? (δx y+ δy z)); [assumption] apply le_reflexive]
90 lapply (le_to_eqm ??? Lxy) as Dxm; lapply (le_to_eqm ??? Lyz) as Dym;
91 lapply (le_to_eqj ??? Lxy) as Dxj; lapply (le_to_eqj ??? Lyz) as Dyj; clear Lxy Lyz;
92 apply (Eq≈ (δ(x∧y) y + δy z)); [apply feq_plusr; apply (meq_l ????? Dxm);]
93 apply (Eq≈ (δ(x∧y) (y∧z) + δy z)); [apply feq_plusr; apply (meq_r ????? Dym);]
94 apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) z)); [apply feq_plusl; apply (meq_l ????? Dxj);]
95 apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) (y∨z))); [apply feq_plusl; apply (meq_r ????? Dyj);]
96 apply (Eq≈ ? (plus_comm ???));
97 apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z))); [apply feq_plusr; apply (meq_l ????? (join_comm ???));]
99 apply (Eq≈ (δ(y∧x) (y∧z))); [apply (meq_l ????? (meet_comm ???));]
103 include "sequence.ma".
104 include "nat/plus.ma".
106 lemma ltwl: ∀a,b,c:nat. b + a < c → a < c.
107 intros 3 (x y z); elim y (H z IH H); [apply H]
108 apply IH; apply lt_S_to_lt; apply H;
111 lemma ltwr: ∀a,b,c:nat. a + b < c → a < c.
112 intros 3 (x y z); rewrite > sym_plus; apply ltwl;
117 λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
119 notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
121 interpretation "tends to" 'tends s x =
122 (cic:/matita/sequence/tends0.con _ s).
125 axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
127 alias symbol "leq" = "ordered sets less or equal than".
128 alias symbol "and" = "constructive and".
129 alias symbol "exists" = "constructive exists (Type)".
131 ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.
132 (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
133 ∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
134 tends0 ? (d2s ? ml xn x).
135 intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
137 alias num (instance 0) = "natural number".
138 elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha;
139 elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
140 constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
141 lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
142 elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
143 elim (H n3) (H7 H8); clear H H1 H2;
144 lapply (le_to_eqm ??? H7) as Dxm; lapply (le_to_eqj ??? H7) as Dym;
146 cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x); [2:
147 apply (le_transitive ???? (mtineq ???? (an n3)));
148 lapply (le_mtri ????? H7 H8);
149 lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? Hletin);
150 cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))); [2:
151 apply (Eq≈ (0 + δ(an n3) (xn n3)) ? (zero_neutral ??));
152 apply (Eq≈ (δ(an n3) (xn n3) + 0) ? (plus_comm ???));
153 apply (Eq≈ (δ(an n3) (xn n3) + (-δ(xn n3) (bn n3) + δ(xn n3) (bn n3))) ? (opp_inverse ??));
154 apply (Eq≈ (δ(an n3) (xn n3) + (δ(xn n3) (bn n3) + -δ(xn n3) (bn n3))) ? (plus_comm ?? (δ(xn n3) (bn n3))));
155 apply (Eq≈ ? ? (eq_sym ??? (plus_assoc ????))); assumption;] clear Hletin1;
156 apply (le_rewl ??? ( δ(an n3) (xn n3)+ δ(an n3) x));[
157 apply feq_plusr; apply msymmetric;]
158 apply (le_rewl ??? (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x));[
159 apply feq_plusr; assumption;]
160 clear Hcut Hletin Dym Dxm H8 H7 H6 H5 H4 H3;
161 apply (le_rewl ??? (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x));[
162 apply feq_plusr; apply plus_comm;]
163 apply (le_rewl ??? (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????));
164 apply (le_rewl ??? ((δ(an n3) (bn n3)+δ(an n3) x)+- δ(xn n3) (bn n3)) (plus_comm ???));
165 apply lew_opp; [apply mpositive] apply fle_plusr;
166 apply (le_rewr ???? (plus_comm ???));
167 apply (le_rewr ??? ( δ(an n3) x+ δx (bn n3)) (msymmetric ????));
170 apply (lt_le_transitive ????? (mpositive ????));
171 split; elim He; [apply le_zero_x_to_le_opp_x_zero; assumption;]
173 left; apply exc_zero_opp_x_to_exc_x_zero;
174 apply (Ex≫ ? (eq_opp_opp_x_x ??));assumption;]
175 right; apply exc_opp_x_zero_to_exc_zero_x;
176 apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
177 clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
178 apply (le_lt_transitive ???? ? (core1 ?? He));
179 apply (le_transitive ???? Hcut);
180 apply (le_transitive ?? (e/3+ δ(an n3) x+ δ(an n3) x)); [
181 repeat apply fle_plusr; cases H6; assumption;]
182 apply (le_transitive ?? (e/3+ e/2 + δ(an n3) x)); [
183 apply fle_plusr; apply fle_plusl; cases H4; assumption;]
184 apply (le_transitive ?? (e/3+ e/2 + e/2)); [
185 apply fle_plusl; cases H4; assumption;]
190 (* 3.17 conclusione: δ x y ≈ 0 *)
191 (* 3.20 conclusione: δ x y ≈ 0 *)
193 strong_sup x ≝ ∀n. s n ≤ x ∧ ∀y x ≰ y → ∃n. s n ≰ y
194 strong_sup_zoli x ≝ ∀n. s n ≤ x ∧ ∄y. y#x ∧ y ≤ x
196 (* 3.22 sup debole (più piccolo dei maggioranti) *)
197 (* 3.23 conclusion: δ x sup(...) ≈ 0 *)
198 (* 3.25 vero nel reticolo e basta (niente δ) *)
199 (* 3.36 conclusion: δ x y ≈ 0 *)