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 definition hole ≝ λT:Type.λx:T.x.
185 notation < "\ldots" non associative with precedence 50 for @{'hole}.
186 interpretation "hole" 'hole = (cic:/matita/lattice/hole.con _ _).
191 (* SL(E,M,H2-5(#),H67(≰)) → E' → c E = c E' → H67'(≰') → SL(E,M,p2-5,H67') *)
195 ∀p:exc_ap l = exc_ap e.
196 (∀x,y:e.(le (exc_dual_base e)) x y → x ≈ (?(sl_meet l)) x y) →
197 (∀x,y:e.(le (exc_dual_base e)) ((?(sl_meet l)) x y) y) →
200 change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
202 |intros (l e p H1 H2);
203 apply (mk_semi_lattice e);
204 [ change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
205 cases p; simplify; apply (sl_meet l);
206 |2: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_refl;
207 |3: change in ⊢ (% → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_comm;
208 |4: change in ⊢ (% → % → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_assoc;
209 |5: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_strong_extm;
210 |6: clear H2; apply hole; apply H1;
211 |7: clear H1; apply hole; apply H2;]]
214 lemma excess_of_excess_base: excess_base → excess.
217 [apply (mk_excess_ (mk_excess_dual_smart eb));
218 [apply (apartness_of_excess_base eb);
220 |2,3: intros; assumption]
223 lemma subst_excess_preserves_aprtness:
227 exc_ap l = exc_ap (subst_excess l e p H1 H2).
230 simplify; assumption;
234 lemma subst_excess__preserves_aprtness:
238 exc_ap l = apartness_OF_excess (subst_excess_ l (subst_dual_excess l (subst_excess_base l e) p) H1 H2).
239 intros 3; (unfold subst_excess_; unfold subst_dual_excess; unfold subst_excess_base; unfold exc_ap; unfold mk_excess_dual_smart; simplify);
240 (unfold subst_excess_base in p; unfold mk_excess_dual_smart in p; simplify in p);
245 lemma subst_excess_base_in_excess_:
248 ∀p:exc_carr d = exc_carr eb.
251 apply (subst_dual_excess e_);
252 [apply (subst_excess_base e_ eb);
256 lemma subst_excess_base_in_excess:
259 ∀p:exc_carr d = exc_carr eb.
260 (∀y1,x1:eb. (?(ap_apart d)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) →
261 (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart d)) y2 x2) →
263 [1,3,4:apply Type|2,5:intro f; cases p; apply f]
264 intros (d eb p H1 H2);
265 apply (subst_excess_ d);
266 [apply (subst_excess_base_in_excess_ d eb p);
267 |apply hole; clear H2;
268 change in ⊢ (%→%→?) with (exc_carr eb);
269 change in ⊢ (?→?→?→? (? % ? ?) (? % ? ?)) with eb; intros (y x H3);
270 apply H1; generalize in match H3;
271 unfold ap_apart; unfold subst_excess_base_in_excess_;
272 unfold subst_dual_excess; simplify;
273 generalize in match x;
274 generalize in match y;
275 cases p; simplify; intros; assumption;
276 |apply hole; clear H1;
277 change in ⊢ (%→%→?) with (exc_carr eb);
278 change in ⊢ (?→?→? (? % ? ?) (? % ? ?)→?) with eb; intros (y x H3);
279 unfold ap_apart; unfold subst_excess_base_in_excess_;
280 unfold subst_dual_excess; simplify; generalize in match (H2 ?? H3);
281 generalize in match x; generalize in match y; cases p;
285 lemma tech1: ∀e:excess.
288 exc_ap e = exc_ap_ (subst_excess_base_in_excess e eb p H1 H2).
289 intros (e eb p H1 H2);
290 unfold subst_excess_base_in_excess;
291 unfold subst_excess_; simplify;
292 unfold subst_excess_base_in_excess_;
293 unfold subst_dual_excess; simplify; reflexivity;
298 exc_ap e = exc_ap (mk_excess_ (subst_excess_base e eb) (exc_ap e) p).
299 intros (e eb p);unfold exc_ap; simplify; cases p; simplify; reflexivity;
304 ∀a1,b1,a2,b2,a3,b3,a4,b4,a5,b5.
305 a1=b1 → a2=b2 → a3=b3 → a4=b4 → a5=b5 → mk_apartness a1 a2 a3 a4 a5 = mk_apartness b1 b2 b3 b4 b5.
306 intros; cases H; cases H1; cases H2; cases H3; cases H4; reflexivity;
310 lemma subst_excess_base_in_excess_preserves_apartness:
314 apartness_OF_excess e =
315 apartness_OF_excess (subst_excess_base_in_excess e eb H H1 H2).
316 intros (e eb p H1 H2);
317 unfold subst_excess_base_in_excess;
318 unfold subst_excess_; unfold subst_excess_base_in_excess_;
319 unfold subst_dual_excess; unfold apartness_OF_excess;
320 simplify in ⊢ (? ? ? (? %));
321 rewrite < (tech2 e eb );
327 alias symbol "nleq" = "Excess base excess".
328 lemma subst_excess_base_in_semi_lattice:
331 ∀p:exc_carr sl = exc_carr eb.
332 (∀y1,x1:eb. (?(ap_apart sl)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) →
333 (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart sl)) y2 x2) →
334 (∀x3,y3:eb.(le eb) x3 y3 → (?(eq sl)) x3 ((?(sl_meet sl)) x3 y3)) →
335 (∀x4,y4:eb.(le eb) ((?(sl_meet sl)) x4 y4) y4) →
337 [2:apply Prop|3,7,9,10:apply Type|4:apply (exc_carr eb)|1,5,6,8,11:intro f; cases p; apply f;]
338 intros (sl eb H H1 H2 H3 H4);
339 apply (subst_excess sl);
340 [apply (subst_excess_base_in_excess sl eb H H1 H2);
341 |apply subst_excess_base_in_excess_preserves_apartness;
342 |change in ⊢ (%→%→?) with ((λx.ap_carr x) (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)); simplify;
344 generalize in match (H3 ?? LE);
345 generalize in match H1 as H1;generalize in match H2 as H2;
346 generalize in match x as x; generalize in match y as y;
349 (reduce in H ⊢ %; cases H; simplify; intros; assumption);
352 cases (subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2); simplify;
353 change in x:(%) with (exc_carr eb);
354 change in y:(%) with (exc_carr eb);
355 generalize in match OK; generalize in match x as x; generalize in match y as y;
356 cases H; simplify; (* funge, ma devo fare 2 rewrite in un colpo solo *)
362 record lattice_ : Type ≝ {
363 latt_mcarr:> semi_lattice;
364 latt_jcarr_: semi_lattice;
365 W1:?; W2:?; W3:?; W4:?; W5:?;
366 latt_with1: latt_jcarr_ = subst_excess_base_in_semi_lattice latt_jcarr_
367 (excess_base_OF_semi_lattice latt_mcarr) W1 W2 W3 W4 W5
370 lemma latt_jcarr : lattice_ → semi_lattice.
371 intro l; apply (subst_excess_base_in_semi_lattice (latt_jcarr_ l) (excess_base_OF_semi_lattice (latt_mcarr l)) (W1 l) (W2 l) (W3 l) (W4 l) (W5 l));
374 coercion cic:/matita/lattice/latt_jcarr.con.
376 interpretation "Lattice meet" 'and a b =
377 (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _) a b).
379 interpretation "Lattice join" 'or a b =
380 (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _) a b).
382 record lattice : Type ≝ {
383 latt_carr:> lattice_;
384 absorbjm: ∀f,g:latt_carr. (f ∨ (f ∧ g)) ≈ f;
385 absorbmj: ∀f,g:latt_carr. (f ∧ (f ∨ g)) ≈ f
388 notation "'meet'" non associative with precedence 50 for @{'meet}.
389 notation "'meet_refl'" non associative with precedence 50 for @{'meet_refl}.
390 notation "'meet_comm'" non associative with precedence 50 for @{'meet_comm}.
391 notation "'meet_assoc'" non associative with precedence 50 for @{'meet_assoc}.
392 notation "'strong_extm'" non associative with precedence 50 for @{'strong_extm}.
393 notation "'le_to_eqm'" non associative with precedence 50 for @{'le_to_eqm}.
394 notation "'lem'" non associative with precedence 50 for @{'lem}.
395 notation "'join'" non associative with precedence 50 for @{'join}.
396 notation "'join_refl'" non associative with precedence 50 for @{'join_refl}.
397 notation "'join_comm'" non associative with precedence 50 for @{'join_comm}.
398 notation "'join_assoc'" non associative with precedence 50 for @{'join_assoc}.
399 notation "'strong_extj'" non associative with precedence 50 for @{'strong_extj}.
400 notation "'le_to_eqj'" non associative with precedence 50 for @{'le_to_eqj}.
401 notation "'lej'" non associative with precedence 50 for @{'lej}.
403 interpretation "Lattice meet" 'meet = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _)).
404 interpretation "Lattice meet_refl" 'meet_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_mcarr.con _)).
405 interpretation "Lattice meet_comm" 'meet_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_mcarr.con _)).
406 interpretation "Lattice meet_assoc" 'meet_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_mcarr.con _)).
407 interpretation "Lattice strong_extm" 'strong_extm = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_mcarr.con _)).
408 interpretation "Lattice le_to_eqm" 'le_to_eqm = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_mcarr.con _)).
409 interpretation "Lattice lem" 'lem = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_mcarr.con _)).
410 interpretation "Lattice join" 'join = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _)).
411 interpretation "Lattice join_refl" 'join_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_jcarr.con _)).
412 interpretation "Lattice join_comm" 'join_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_jcarr.con _)).
413 interpretation "Lattice join_assoc" 'join_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_jcarr.con _)).
414 interpretation "Lattice strong_extm" 'strong_extj = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_jcarr.con _)).
415 interpretation "Lattice le_to_eqj" 'le_to_eqj = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_jcarr.con _)).
416 interpretation "Lattice lej" 'lej = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_jcarr.con _)).
418 notation "'feq_jl'" non associative with precedence 50 for @{'feq_jl}.
419 notation "'feq_jr'" non associative with precedence 50 for @{'feq_jr}.
420 notation "'feq_ml'" non associative with precedence 50 for @{'feq_ml}.
421 notation "'feq_mr'" non associative with precedence 50 for @{'feq_mr}.
422 interpretation "Lattice feq_jl" 'feq_jl = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_jcarr.con _)).
423 interpretation "Lattice feq_jr" 'feq_jr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_jcarr.con _)).
424 interpretation "Lattice feq_ml" 'feq_ml = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_mcarr.con _)).
425 interpretation "Lattice feq_mr" 'feq_mr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_mcarr.con _)).
428 interpretation "Lattive meet le" 'leq a b =
429 (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice1.con _) a b).
431 interpretation "Lattive join le (aka ge)" 'geq a b =
432 (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice.con _) a b).
434 (* these coercions help unification, handmaking a bit of conversion
437 lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a.
438 intros(l a b H); apply H;
441 lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b.
442 intros(l a b H); apply H;
445 coercion cic:/matita/lattice/le_to_ge.con nocomposites.
446 coercion cic:/matita/lattice/ge_to_le.con nocomposites.