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 (**************************************************************************)
17 record semi_lattice_base : Type ≝ {
19 sl_op: sl_carr → sl_carr → sl_carr;
20 sl_op_refl: ∀x.sl_op x x ≈ x;
21 sl_op_comm: ∀x,y:sl_carr. sl_op x y ≈ sl_op y x;
22 sl_op_assoc: ∀x,y,z:sl_carr. sl_op x (sl_op y z) ≈ sl_op (sl_op x y) z;
23 sl_strong_extop: ∀x.strong_ext ? (sl_op x)
26 notation "a \cross b" left associative with precedence 50 for @{ 'op $a $b }.
27 interpretation "semi lattice base operation" 'op a b = (cic:/matita/lattice/sl_op.con _ a b).
29 lemma excess_of_semi_lattice_base: semi_lattice_base → excess.
33 [1: apply mk_excess_dual_smart;
35 apply (mk_excess_base (sl_carr l));
36 [1: apply (λa,b:sl_carr l.a # (a ✗ b));
37 |2: unfold; intros 2 (x H); simplify in H;
38 lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
39 apply (ap_coreflexive ?? H1);
40 |3: unfold; simplify; intros (x y z H1);
41 cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2:
42 lapply (Ap≪ ? (sl_op_comm ???) H2) as H21;
43 lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2;
44 lapply (sl_strong_extop ???? H22); clear H22;
45 left; apply ap_symmetric; assumption;]
46 cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption]
47 right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31;
48 apply (sl_strong_extop ???? H31);]
51 apply apartness_of_excess_base;
53 apply (mk_excess_base (sl_carr l));
54 [1: apply (λa,b:sl_carr l.a # (a ✗ b));
55 |2: unfold; intros 2 (x H); simplify in H;
56 lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
57 apply (ap_coreflexive ?? H1);
58 |3: unfold; simplify; intros (x y z H1);
59 cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2:
60 lapply (Ap≪ ? (sl_op_comm ???) H2) as H21;
61 lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2;
62 lapply (sl_strong_extop ???? H22); clear H22;
63 left; apply ap_symmetric; assumption;]
64 cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption]
65 right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31;
66 apply (sl_strong_extop ???? H31);]
69 |2,3: intros (x y H); assumption;]
72 record semi_lattice : Type ≝ {
74 sl_meet: sl_exc → sl_exc → sl_exc;
75 sl_meet_refl: ∀x.sl_meet x x ≈ x;
76 sl_meet_comm: ∀x,y. sl_meet x y ≈ sl_meet y x;
77 sl_meet_assoc: ∀x,y,z. sl_meet x (sl_meet y z) ≈ sl_meet (sl_meet x y) z;
78 sl_strong_extm: ∀x.strong_ext ? (sl_meet x);
79 sl_le_to_eqm: ∀x,y.x ≤ y → x ≈ sl_meet x y;
80 sl_lem: ∀x,y.(sl_meet x y) ≤ y
83 interpretation "semi lattice meet" 'and a b = (cic:/matita/lattice/sl_meet.con _ a b).
85 lemma sl_feq_ml: ∀ml:semi_lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b).
86 intros (l a b c H); unfold eq in H ⊢ %; unfold Not in H ⊢ %;
87 intro H1; apply H; clear H; apply (sl_strong_extm ???? H1);
90 lemma sl_feq_mr: ∀ml:semi_lattice.∀a,b,c:ml. a ≈ b → (a ∧ c) ≈ (b ∧ c).
92 apply (Eq≈ ? (sl_meet_comm ???)); apply (Eq≈ ?? (sl_meet_comm ???));
93 apply sl_feq_ml; assumption;
98 lemma semi_lattice_of_semi_lattice_base: semi_lattice_base → semi_lattice.
99 intro slb; apply (mk_semi_lattice (excess_of_semi_lattice_base slb));
100 [1: apply (sl_op slb);
101 |2: intro x; apply (eq_trans (excess_of_semi_lattice_base slb)); [2:
102 apply (sl_op_refl slb);|1:skip] (sl_op slb x x)); ? (sl_op_refl slb x));
104 unfold excess_of_semi_lattice_base; simplify;
109 lapply (ap_rewl (excess_of_semi_lattice_base slb) x ? (sl_op slb x x)
110 (eq_sym (excess_of_semi_lattice_base slb) ?? (sl_op_refl slb x)) t);
111 change in x with (sl_carr slb);
112 apply (Ap≪ (x ✗ x)); (sl_op_refl slb x));
114 whd in H; elim H; clear H;
115 [ apply (ap_coreflexive (excess_of_semi_lattice_base slb) (x ✗ x) t);
117 prelattice (excess_of_directed l_)); [apply (sl_op l_);]
118 unfold excess_of_directed; try unfold apart_of_excess; simplify;
119 unfold excl; simplify;
120 [intro x; intro H; elim H; clear H;
121 [apply (sl_op_refl l_ x);
122 lapply (Ap≫ ? (sl_op_comm ???) t) as H; clear t;
123 lapply (sl_strong_extop l_ ??? H); apply ap_symmetric; assumption
124 | lapply (Ap≪ ? (sl_op_refl ?x) t) as H; clear t;
125 lapply (sl_strong_extop l_ ??? H); apply (sl_op_refl l_ x);
126 apply ap_symmetric; assumption]
127 |intros 3 (x y H); cases H (H1 H2); clear H;
128 [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x y)) H1) as H; clear H1;
129 lapply (sl_strong_extop l_ ??? H) as H1; clear H;
130 lapply (Ap≪ ? (sl_op_comm ???) H1); apply (ap_coreflexive ?? Hletin);
131 |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ y x)) H2) as H; clear H2;
132 lapply (sl_strong_extop l_ ??? H) as H1; clear H;
133 lapply (Ap≪ ? (sl_op_comm ???) H1);apply (ap_coreflexive ?? Hletin);]
134 |intros 4 (x y z H); cases H (H1 H2); clear H;
135 [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x (sl_op l_ y z))) H1) as H; clear H1;
136 lapply (sl_strong_extop l_ ??? H) as H1; clear H;
137 lapply (Ap≪ ? (eq_sym ??? (sl_op_assoc ?x y z)) H1) as H; clear H1;
138 apply (ap_coreflexive ?? H);
139 |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ (sl_op l_ x y) z)) H2) as H; clear H2;
140 lapply (sl_strong_extop l_ ??? H) as H1; clear H;
141 lapply (Ap≪ ? (sl_op_assoc ?x y z) H1) as H; clear H1;
142 apply (ap_coreflexive ?? H);]
143 |intros (x y z H); elim H (H1 H1); clear H;
144 lapply (Ap≪ ? (sl_op_refl ??) H1) as H; clear H1;
145 lapply (sl_strong_extop l_ ??? H) as H1; clear H;
146 lapply (sl_strong_extop l_ ??? H1) as H; clear H1;
147 cases (ap_cotransitive ??? (sl_op l_ y z) H);[left|right|right|left] try assumption;
148 [apply ap_symmetric;apply (Ap≪ ? (sl_op_comm ???));
149 |apply (Ap≫ ? (sl_op_comm ???));
150 |apply ap_symmetric;] assumption;
151 |intros 4 (x y H H1); apply H; clear H; elim H1 (H H);
152 lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
153 lapply (sl_strong_extop l_ ??? H1) as H; clear H1;[2: apply ap_symmetric]
156 cut (sl_op l_ x y ≈ sl_op l_ x (sl_op l_ y y)) as H1;[2:
157 intro; lapply (sl_strong_extop ???? a); apply (sl_op_refl l_ y);
158 apply ap_symmetric; assumption;]
159 lapply (Ap≪ ? (eq_sym ??? H1) H); apply (sl_op_assoc l_ x y y);
164 (* ED(≰,≱) → EB(≰') → ED(≰',≱') *)
165 lemma subst_excess_base: excess_dual → excess_base → excess_dual.
166 intros; apply (mk_excess_dual_smart e1);
169 (* E_(ED(≰,≱),AP(#),c ED = c AP) → ED' → c DE' = c E_ → E_(ED',#,p) *)
170 lemma subst_dual_excess: ∀e:excess_.∀e1:excess_dual.exc_carr e = exc_carr e1 → excess_.
171 intros (e e1 p); apply (mk_excess_ e1 e); cases p; reflexivity;
174 (* E(E_,H1,H2) → E_' → H1' → H2' → E(E_',H1',H2') *)
175 alias symbol "nleq" = "Excess excess_".
176 lemma subst_excess_: ∀e:excess. ∀e1:excess_.
177 (∀y,x:e1. y # x → y ≰ x ∨ x ≰ y) →
178 (∀y,x:e1.y ≰ x ∨ x ≰ y → y # x) →
180 intros (e e1 H1 H2); apply (mk_excess e1 H1 H2);
183 (* SL(E,M,H2-5(#),H67(≰)) → E' → c E = c E' → H67'(≰') → SL(E,M,p2-5,H67') *)
187 ∀p:exc_ap l = exc_ap e.
188 (∀x,y:e.(le (exc_dual_base e)) x y → x ≈ (?(sl_meet l)) x y) →
189 (∀x,y:e.(le (exc_dual_base e)) ((?(sl_meet l)) x y) y) →
192 change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
194 |intros (l e p H1 H2);
195 apply (mk_semi_lattice e);
196 [ change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
197 cases p; simplify; apply (sl_meet l);
198 |2: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_refl;
199 |3: change in ⊢ (% → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_comm;
200 |4: change in ⊢ (% → % → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_assoc;
201 |5: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_strong_extm;
202 |6: clear H2; apply H1;
203 |7: clear H1; apply H2;]]
206 lemma excess_of_excess_base: excess_base → excess.
209 [apply (mk_excess_ (mk_excess_dual_smart eb));
210 [apply (apartness_of_excess_base eb);
212 |2,3: intros; assumption]
215 lemma subst_excess_base_in_semi_lattice:
218 ∀p:exc_carr sl = exc_carr eb.
219 (∀y1,x1:eb. (?(ap_apart sl)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) →
220 (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart sl)) y2 x2) →
221 (∀x3,y3:eb.(le eb) x3 y3 → (?(ap_apart sl)) x3 ((?(sl_meet sl)) x3 y3)) →
222 (∀x4,y4:eb.(le eb) ((?(sl_meet sl)) x4 y4) y4) →
224 [2,3,7,9,10: apply Type|4:apply (exc_carr eb);
225 |1,5,6,8,11: intro f; cases p; apply f;]
226 intros (sl eb H H1 H2 H3 H4); apply (subst_excess sl);
227 [apply (subst_excess_ sl);
228 [apply (subst_dual_excess sl);
229 [apply (subst_excess_base sl eb);
231 | (*clear H2 H3 H4;*)
232 change in ⊢ (% -> % -> ?) with (exc_carr eb);
233 unfold subst_excess_base; unfold mk_excess_dual_smart;
234 unfold subst_dual_excess; simplify in ⊢ (?→?→?→%);
235 (unfold exc_ap; simplify in ⊢ (?→?→? % ? ?→?));
236 simplify; intros (x y H2); apply H1;
237 generalize in match H2;
238 generalize in match x as x;
239 generalize in match y as y; (*clear H1 H2 x y;*)
240 change in ⊢ (?→?→match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?→?)
241 with (Type_OF_semi_lattice sl);
242 change in ⊢ (?→?→match match ? return λ_:?.(λ_:? ? % ?.? ? % ?) with [_⇒? ? %] return ? with [_⇒?] ??→?) with (Type_OF_semi_lattice sl);
243 cases H; intros; assumption; (* se faccio le clear... BuM! *)
248 record lattice_ : Type ≝ {
249 latt_mcarr:> semi_lattice;
250 latt_jcarr_: semi_lattice;
251 (*latt_with1: latt_jcarr_ = subst latt_jcarr (exc_dual_dual latt_mcarr)*)
252 (* latt_with1: (subst_excess_
255 (excess_dual_OF_excess (sl_exc latt_jcarr_))
256 (excess_base_OF_excess (sl_exc latt_mcarr))))) =
260 latt_with1: excess_base_OF_excess1 (sl_exc latt_jcarr_) = excess_base_OF_excess (sl_exc latt_mcarr);
261 latt_with2: excess_base_OF_excess (sl_exc latt_jcarr_) = excess_base_OF_excess1 (sl_exc latt_mcarr);
262 latt_with3: apartness_OF_excess (sl_exc latt_jcarr_) = apartness_OF_excess (sl_exc latt_mcarr)
267 lemma latt_jcarr : lattice_ → semi_lattice.
269 apply mk_semi_lattice;
272 [apply (mk_excess_dual_smart l);
275 |unfold mk_excess_dual_smart; simplify;
276 intros (x y H); cases (ap2exc ??? H); [right|left] assumption;
277 |unfold mk_excess_dual_smart; simplify;
278 intros (x y H);cases H; apply exc2ap;[right|left] assumption;]]
279 unfold mk_excess_dual_smart; simplify;
280 [1: change with ((λx.ap_carr x) l → (λx.ap_carr x) l → (λx.ap_carr x) l);
281 simplify; unfold apartness_OF_lattice_;
282 cases (latt_with3 l); apply (sl_meet (latt_jcarr_ l));
283 |2: change in ⊢ (%→?) with ((λx.ap_carr x) l); simplify;
284 unfold apartness_OF_lattice_;
285 cases (latt_with3 l); apply (sl_meet_refl (latt_jcarr_ l));
286 |3: change in ⊢ (%→%→?) with ((λx.ap_carr x) l); simplify; unfold apartness_OF_lattice_;
287 cases (latt_with3 l); apply (sl_meet_comm (latt_jcarr_ l));
288 |4: change in ⊢ (%→%→%→?) with ((λx.ap_carr x) l); simplify; unfold apartness_OF_lattice_;
289 cases (latt_with3 l); apply (sl_meet_assoc (latt_jcarr_ l));
290 |5: change in ⊢ (%→%→%→?) with ((λx.ap_carr x) l); simplify; unfold apartness_OF_lattice_;
291 cases (latt_with3 l); apply (sl_strong_extm (latt_jcarr_ l));
294 unfold excess_base_OF_lattice_;
295 change in ⊢ (?→?→? ? (% ? ?) ?)
296 with (match latt_with3 l
300 .(λmatched:eq apartness (apartness_OF_semi_lattice (latt_jcarr_ l)) right_1
301 .ap_carr right_1→ap_carr right_1→ap_carr right_1)
303 [refl_eq⇒sl_meet (latt_jcarr_ l)]
306 change in ⊢ (?→?→? ? ((?:%->%->%) ? ?) ?)
307 with ((λx.exc_carr x) (excess_base_OF_semi_lattice (latt_mcarr l)));
308 unfold excess_base_OF_lattice_ in ⊢ (?→?→? ? ((?:%->%->%) ? ?) ?);
309 simplify in ⊢ (?→?→? ? ((?:%->%->%) ? ?) ?);
310 change in ⊢ (?→?→? ? (% ? ?) ?) with
311 (match refl_eq ? (excess__OF_semi_lattice (latt_mcarr l)) in eq
312 return (λR.λE:eq ? (excess_base_OF_semi_lattice (latt_mcarr l)) R.R → R → R)
314 (match latt_with3 l in eq
317 .(λmatched:eq apartness (apartness_OF_semi_lattice (latt_jcarr_ l)) right
318 .ap_carr right→ap_carr right→ap_carr right))
319 with [refl_eq⇒ sl_meet (latt_jcarr_ l)]
321 exc_carr (excess_base_OF_semi_lattice (latt_mcarr l))
322 →exc_carr (excess_base_OF_semi_lattice (latt_mcarr l))
323 →exc_carr (excess_base_OF_semi_lattice (latt_mcarr l))
326 generalize in ⊢ (?→?→? ? (match % return ? with [_⇒?] ? ?) ?);
327 unfold excess_base_OF_lattice_ in ⊢ (? ? ? %→?);
328 cases (latt_with1 l);
329 change in ⊢ (?→?→?→? ? (match ? return ? with [_⇒(?:%→%->%)] ? ?) ?)
330 with ((λx.ap_carr x) (latt_mcarr l));
331 simplify in ⊢ (?→?→?→? ? (match ? return ? with [_⇒(?:%→%->%)] ? ?) ?);
332 cases (latt_with3 l);
334 change in ⊢ (? ? % ?→?) with ((λx.ap_carr x) l);
335 simplify in ⊢ (% → ?);
336 change in ⊢ (?→?→?→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?)
337 with ((λx.ap_carr x) (apartness_OF_lattice_ l));
338 unfold apartness_OF_lattice_;
339 cases (latt_with3 l); simplify;
340 change in ⊢ (? ? ? %→%→%→?) with ((λx.exc_carr x) l);
341 unfold excess_base_OF_lattice_;
342 cases (latt_with1 l); simplify;
343 change in \vdash (? -> % -> % -> ?) with (exc_carr (excess_base_OF_semi_lattice (latt_jcarr_ l)));
344 change in ⊢ ((? ? % ?)→%→%→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?)
345 with ((λx.exc_carr x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)));
348 unfold excess_base_OF_semi_lattice1;
349 unfold excess_base_OF_excess1;
350 unfold excess_base_OF_excess_1;
354 change in ⊢ (?→?→? ? (% ? ?) ?) with
355 (match refl_eq ? (Type_OF_lattice_ l) in eq
356 return (λR.λE:eq ? (Type_OF_lattice_ l) R.R → R → R)
358 match latt_with3 l in eq
361 .(λmatched:eq apartness (apartness_OF_semi_lattice (latt_jcarr_ l)) right
362 .ap_carr right→ap_carr right→ap_carr right))
363 with [refl_eq⇒ sl_meet (latt_jcarr_ l)]
365 generalize in ⊢ (?→?→? ? (match % return ? with [_⇒?] ? ?) ?);
366 change in ⊢ (? ? % ?→?) with ((λx.ap_carr x) l);
367 simplify in ⊢ (% → ?);
368 change in ⊢ (?→?→?→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?)
369 with ((λx.ap_carr x) (apartness_OF_lattice_ l));
370 unfold apartness_OF_lattice_;
371 cases (latt_with3 l); simplify;
372 change in ⊢ (? ? ? %→%→%→?) with ((λx.exc_carr x) l);
373 unfold excess_base_OF_lattice_;
374 cases (latt_with1 l); simplify;
375 change in \vdash (? -> % -> % -> ?) with (exc_carr (excess_base_OF_semi_lattice (latt_jcarr_ l)));
376 change in ⊢ ((? ? % ?)→%→%→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?)
377 with ((λx.exc_carr x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)));
380 change in ⊢ (?→?→%) with (le (mk_excess_base
381 ((λx.exc_carr x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)))
382 ((λx.exc_excess x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)))
383 ((λx.exc_coreflexive x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)))
384 ((λx.exc_cotransitive x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)))
389 .(λE:eq Type (exc_carr (excess_base_OF_semi_lattice1 (latt_jcarr_ l))) R
392 [refl_eq⇒sl_meet (latt_jcarr_ l)] x y) y);
393 simplify in ⊢ (?→?→? (? % ???) ? ?);
394 change in ⊢ (?→?→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?)
395 with ((λx.exc_carr x) (excess_base_OF_semi_lattice1 (latt_jcarr_ l)));
396 simplify in ⊢ (?→?→? ? (match ? return λ_:?.(λ_:? ? % ?.?) with [_⇒?] ? ?) ?);
397 lapply (match H in eq return
398 λright.λe:eq ? (exc_carr (excess_base_OF_semi_lattice1 (latt_jcarr_ l))) right.
403 (mk_excess_base right ???)
407 λR:Type.(λE:eq Type (exc_carr (excess_base_OF_semi_lattice1 (latt_jcarr_ l))) R.R→R→R)
409 [refl_eq⇒sl_meet (latt_jcarr_ l)] x y) y
410 with [refl_eq ⇒ ?]) as XX;
411 [cases e; apply (exc_excess (latt_jcarr_ l));
412 |unfold;cases e;simplify;apply (exc_coreflexive (latt_jcarr_ l));
413 |unfold;cases e;simplify;apply (exc_cotransitive (latt_jcarr_ l));
417 simplify; apply (sl_lem);
424 coercion cic:/matita/lattice/latt_jcarr.con.
426 interpretation "Lattice meet" 'and a b =
427 (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _) a b).
429 interpretation "Lattice join" 'or a b =
430 (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _) a b).
432 record lattice : Type ≝ {
433 latt_carr:> lattice_;
434 absorbjm: ∀f,g:latt_carr. (f ∨ (f ∧ g)) ≈ f;
435 absorbmj: ∀f,g:latt_carr. (f ∧ (f ∨ g)) ≈ f
438 notation "'meet'" non associative with precedence 50 for @{'meet}.
439 notation "'meet_refl'" non associative with precedence 50 for @{'meet_refl}.
440 notation "'meet_comm'" non associative with precedence 50 for @{'meet_comm}.
441 notation "'meet_assoc'" non associative with precedence 50 for @{'meet_assoc}.
442 notation "'strong_extm'" non associative with precedence 50 for @{'strong_extm}.
443 notation "'le_to_eqm'" non associative with precedence 50 for @{'le_to_eqm}.
444 notation "'lem'" non associative with precedence 50 for @{'lem}.
445 notation "'join'" non associative with precedence 50 for @{'join}.
446 notation "'join_refl'" non associative with precedence 50 for @{'join_refl}.
447 notation "'join_comm'" non associative with precedence 50 for @{'join_comm}.
448 notation "'join_assoc'" non associative with precedence 50 for @{'join_assoc}.
449 notation "'strong_extj'" non associative with precedence 50 for @{'strong_extj}.
450 notation "'le_to_eqj'" non associative with precedence 50 for @{'le_to_eqj}.
451 notation "'lej'" non associative with precedence 50 for @{'lej}.
453 interpretation "Lattice meet" 'meet = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _)).
454 interpretation "Lattice meet_refl" 'meet_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_mcarr.con _)).
455 interpretation "Lattice meet_comm" 'meet_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_mcarr.con _)).
456 interpretation "Lattice meet_assoc" 'meet_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_mcarr.con _)).
457 interpretation "Lattice strong_extm" 'strong_extm = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_mcarr.con _)).
458 interpretation "Lattice le_to_eqm" 'le_to_eqm = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_mcarr.con _)).
459 interpretation "Lattice lem" 'lem = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_mcarr.con _)).
460 interpretation "Lattice join" 'join = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _)).
461 interpretation "Lattice join_refl" 'join_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_jcarr.con _)).
462 interpretation "Lattice join_comm" 'join_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_jcarr.con _)).
463 interpretation "Lattice join_assoc" 'join_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_jcarr.con _)).
464 interpretation "Lattice strong_extm" 'strong_extj = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_jcarr.con _)).
465 interpretation "Lattice le_to_eqj" 'le_to_eqj = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_jcarr.con _)).
466 interpretation "Lattice lej" 'lej = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_jcarr.con _)).
468 notation "'feq_jl'" non associative with precedence 50 for @{'feq_jl}.
469 notation "'feq_jr'" non associative with precedence 50 for @{'feq_jr}.
470 notation "'feq_ml'" non associative with precedence 50 for @{'feq_ml}.
471 notation "'feq_mr'" non associative with precedence 50 for @{'feq_mr}.
472 interpretation "Lattice feq_jl" 'feq_jl = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_jcarr.con _)).
473 interpretation "Lattice feq_jr" 'feq_jr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_jcarr.con _)).
474 interpretation "Lattice feq_ml" 'feq_ml = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_mcarr.con _)).
475 interpretation "Lattice feq_mr" 'feq_mr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_mcarr.con _)).
478 interpretation "Lattive meet le" 'leq a b =
479 (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice1.con _) a b).
481 interpretation "Lattive join le (aka ge)" 'geq a b =
482 (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice.con _) a b).
484 (* these coercions help unification, handmaking a bit of conversion
487 lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a.
488 intros(l a b H); apply H;
491 lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b.
492 intros(l a b H); apply H;
495 coercion cic:/matita/lattice/le_to_ge.con nocomposites.
496 coercion cic:/matita/lattice/ge_to_le.con nocomposites.