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 *)
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15 include "datatypes/pairs.ma".
16 include "datatypes/bool.ma".
17 include "sets/sets.ma".
19 ninductive Admit : CProp[0] ≝ .
22 ninductive list (A:Type[0]) : Type[0] ≝
24 | cons: A -> list A -> list A.
26 nlet rec eq_list (A : setoid) (l1, l2 : list A) on l1 : CProp[0] ≝
28 [ nil ⇒ match l2 return λ_.CProp[0] with [ nil ⇒ ? | _ ⇒ ? ]
29 | cons x xs ⇒ match l2 with [ nil ⇒ ? | cons y ys ⇒ x = y ∧ eq_list ? xs ys]].
30 ##[ napply True|napply False|napply False]nqed.
32 ndefinition LIST : setoid → setoid.
33 #S; @(list S); @(eq_list S); ncases admit; nqed.
35 unification hint 0 ≔ S : setoid;
36 P1 ≟ refl ? (eq0 (LIST S)),
37 P2 ≟ sym ? (eq0 (LIST S)),
38 P3 ≟ trans ? (eq0 (LIST S)),
39 X ≟ mk_setoid (list S) (mk_equivalence_relation ? (eq_list S) P1 P2 P3),
41 (*-----------------------------------------------------------------------*) ⊢
44 notation "hvbox(hd break :: tl)"
45 right associative with precedence 47
48 notation "[ list0 x sep ; ]"
49 non associative with precedence 90
50 for ${fold right @'nil rec acc @{'cons $x $acc}}.
52 notation "hvbox(l1 break @ l2)"
53 right associative with precedence 47
54 for @{'append $l1 $l2 }.
56 interpretation "nil" 'nil = (nil ?).
57 interpretation "cons" 'cons hd tl = (cons ? hd tl).
59 nlet rec append A (l1: list A) l2 on l1 ≝
62 | cons hd tl ⇒ hd :: append A tl l2 ].
64 interpretation "append" 'append l1 l2 = (append ? l1 l2).
66 ntheorem append_nil: ∀A:setoid.∀l:list A.l @ [] = l.
67 #A;#l;nelim l;//; #a;#l1;#IH;nnormalize;/2/;nqed.
69 ndefinition associative ≝ λA:setoid.λf:A → A → A.∀x,y,z.f x (f y z) = f (f x y) z.
71 ninductive one : Type[0] ≝ unit : one.
74 λS:Type[2].λs:S.λT:Type[2].λt:T.λlock:one.
75 match lock return λ_.Type[2] with [ unit ⇒ T].
77 nlet rec lift (S:Type[2]) (s:S) (T:Type[2]) (t:T) (lock:one) on lock : force S s T t lock ≝
78 match lock return λlock.force S s T t lock with [ unit ⇒ t ].
80 ncoercion lift1 : ∀S:Type[1].∀s:S.∀T:Type[1].∀t:T.∀lock:one. force S s T t lock ≝ lift
81 on s : ? to force ?????.
83 ncoercion lift2 : ∀S:Type[2].∀s:S.∀T:Type[2].∀t:T.∀lock:one. force S s T t lock ≝ lift
84 on s : ? to force ?????.
86 unification hint 0 ≔ R : setoid;
87 TR ≟ setoid, MR ≟ (carr R), lock ≟ unit
88 (* ---------------------------------------- *) ⊢
89 setoid ≡ force ?(*Type[0]*) MR TR R lock.
91 unification hint 0 ≔ R : setoid1;
92 TR ≟ setoid1, MR ≟ (carr1 R), lock ≟ unit
93 (* ---------------------------------------- *) ⊢
94 setoid1 ≡ force ? MR TR R lock.
96 ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
97 #A;#x;#y;#z;nelim x[//|#a;#x1;#H;nnormalize;/2/]nqed.
99 interpretation "iff" 'iff a b = (iff a b).
101 ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ refl: eq A x x.
103 nlemma eq_rect_Type0_r':
104 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
105 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
108 nlemma eq_rect_Type0_r:
109 ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
110 #A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); nassumption.
113 nlemma eq_rect_CProp0_r':
114 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
115 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
118 nlemma eq_rect_CProp0_r:
119 ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
120 #A; #a; #P; #p; #x0; #p0; napply (eq_rect_CProp0_r' ??? p0); nassumption.
123 notation < "a = b" non associative with precedence 45 for @{ 'eqpp $a $b}.
124 interpretation "bool eq" 'eqpp a b = (eq bool a b).
126 ndefinition BOOL : setoid.
127 @bool; @(eq bool); ncases admit.nqed.
129 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
130 alias id "refl" = "cic:/matita/ng/properties/relations/refl.fix(0,1,3)".
131 unification hint 0 ≔ ;
132 P1 ≟ refl ? (eq0 BOOL),
133 P2 ≟ sym ? (eq0 BOOL),
134 P3 ≟ trans ? (eq0 BOOL),
135 X ≟ mk_setoid bool (mk_equivalence_relation ? (eq bool) P1 P2 P3)
136 (*-----------------------------------------------------------------------*) ⊢
139 nrecord Alpha : Type[1] ≝ {
141 eqb: acarr → acarr → bool;
142 eqb_true: ∀x,y. (eqb x y = true) = (x = y)
145 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
146 interpretation "eqb" 'eqb a b = (eqb ? a b).
148 ninductive re (S: Type[0]) : Type[0] ≝
152 | c: re S → re S → re S
153 | o: re S → re S → re S
156 naxiom eq_re : ∀S:Alpha.re S → re S → CProp[0].
157 ndefinition RE : Alpha → setoid.
158 #A; @(re A); @(eq_re A); ncases admit. nqed.
160 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
161 alias id "carr" = "cic:/matita/ng/sets/setoids/carr.fix(0,0,1)".
162 unification hint 0 ≔ A : Alpha;
163 P1 ≟ refl ? (eq0 (RE A)),
164 P2 ≟ sym ? (eq0 (RE A)),
165 P3 ≟ trans ? (eq0 (RE A)),
166 X ≟ mk_setoid (re A) (mk_equivalence_relation ? (eq_re A) P1 P2 P3),
168 (*-----------------------------------------------------------------------*) ⊢
171 notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
172 notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
173 interpretation "star" 'pk a = (k ? a).
174 interpretation "or" 'plus a b = (o ? a b).
176 notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
177 interpretation "cat" 'pc a b = (c ? a b).
179 (* to get rid of \middot *)
180 ncoercion c : ∀S.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
182 notation < "a" non associative with precedence 90 for @{ 'ps $a}.
183 notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
184 interpretation "atom" 'ps a = (s ? a).
186 notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
187 interpretation "epsilon" 'epsilon = (e ?).
189 notation "0" non associative with precedence 90 for @{ 'empty }.
190 interpretation "empty" 'empty = (z ?).
192 notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
193 notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(list $S) }.
195 nlet rec flatten S (l : list (list S)) on l : list S ≝
196 match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
198 nlet rec conjunct S (l : list (list S)) (L : lang S) on l: CProp[0] ≝
199 match l with [ nil ⇒ True | cons w tl ⇒ w ∈ L ∧ conjunct ? tl L ].
202 ndefinition empty_set : ∀A.Ω^A ≝ λA.{ w | False }.
203 notation "∅" non associative with precedence 90 for @{'emptyset}.
204 interpretation "empty set" 'emptyset = (empty_set ?).
207 notation "{}" non associative with precedence 90 for @{'empty_lang}.
208 interpretation "empty lang" 'empty_lang = (empty_lang ?).
211 ndefinition sing_lang : ∀A:setoid.∀x:A.Ω^A ≝ λS.λx.{ w | x = w }.
212 interpretation "sing lang" 'singl x = (sing_lang ? x).
214 interpretation "subset construction with type" 'comprehension t \eta.x =
217 ndefinition cat : ∀A:setoid.∀l1,l2:lang A.lang A ≝
218 λS.λl1,l2.{ w ∈ list S | ∃w1,w2.w =_0 w1 @ w2 ∧ w1 ∈ l1 ∧ w2 ∈ l2}.
219 interpretation "cat lang" 'pc a b = (cat ? a b).
221 ndefinition star : ∀A:setoid.∀l:lang A.lang A ≝
222 λS.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}.
223 interpretation "star lang" 'pk l = (star ? l).
225 notation > "𝐋 term 70 E" non associative with precedence 75 for @{L_re ? $E}.
226 nlet rec L_re (S : Alpha) (r : re S) on r : lang S ≝
231 | c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
232 | o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2
234 notation "𝐋 term 70 E" non associative with precedence 75 for @{'L_re $E}.
235 interpretation "in_l" 'L_re E = (L_re ? E).
237 notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
238 ndefinition orb ≝ λa,b:bool. match a with [ true ⇒ true | _ ⇒ b ].
239 interpretation "orb" 'orb a b = (orb a b).
241 ninductive pitem (S: Type[0]) : Type[0] ≝
246 | pc: pitem S → pitem S → pitem S
247 | po: pitem S → pitem S → pitem S
248 | pk: pitem S → pitem S.
250 ndefinition pre ≝ λS.pitem S × bool.
252 notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
253 interpretation "fst" 'fst x = (fst ? ? x).
254 notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
255 interpretation "snd" 'snd x = (snd ? ? x).
257 interpretation "pstar" 'pk a = (pk ? a).
258 interpretation "por" 'plus a b = (po ? a b).
259 interpretation "pcat" 'pc a b = (pc ? a b).
260 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
261 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
262 interpretation "ppatom" 'pp a = (pp ? a).
263 (* to get rid of \middot *)
264 ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
265 interpretation "patom" 'ps a = (ps ? a).
266 interpretation "pepsilon" 'epsilon = (pe ?).
267 interpretation "pempty" 'empty = (pz ?).
269 notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
270 nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
276 | pc E1 E2 ⇒ (|E1| · |E2|)
277 | po E1 E2 ⇒ (|E1| + |E2|)
279 notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
280 interpretation "forget" 'forget a = (forget ? a).
282 notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{L_pi ? $E}.
283 nlet rec L_pi (S : Alpha) (r : pitem S) on r : lang S ≝
289 | pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 |r2| ∪ 𝐋\p\ r2
290 | po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
291 | pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
292 notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'L_pi $E}.
293 notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'L_pi $E}.
294 interpretation "in_pl" 'L_pi E = (L_pi ? E).
296 unification hint 0 ≔ S,a,b;
298 (* -------------------------------------------- *) ⊢
299 eq_list S a b ≡ eq_rel (list S) (eq0 R) a b.
301 notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
302 notation > "B ⇒_1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
303 notation "B ⇒\sub 0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
304 notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
306 interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
307 interpretation "unary morphism 1" 'umorph1 A B = (unary_morphism1 A B).
309 ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid on _s: setoid to setoid1.
311 nlemma exists_is_morph: (* BUG *) ∀S,T:setoid.∀P: S ⇒_1 (T ⇒_1 (CProp[0]:?)).
312 ∀y,z:S.y =_0 z → (Ex T (P y)) = (Ex T (P z)).
314 ##[ *; #x Px; @x; alias symbol "refl" (instance 4) = "refl".
315 alias symbol "prop2" (instance 2) = "prop21".
316 napply (. E^-1‡#); napply Px;
317 ##| *; #x Px; @x; napply (. E‡#); napply Px;##]
320 ndefinition ex_morph : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
321 #S; @; ##[ #P; napply (Ex ? P); ##| #P1 P2 E; @;
322 *; #x; #H; @ x; nlapply (E x x ?); //; *; /2/;
325 nlemma Sig: ∀S,T:setoid.∀P: S → (T → CProp[0]).
326 ∀y,z:T.y = z → (∀x.y=z → P x y = P x z) → (Ex S (λx.P x y)) =_1 (Ex S (λx.P x z)).
327 #S T P y z Q E; @; *; #x Px; @x; nlapply (E x Q); *; /2/; nqed.
330 nlemma test : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
331 ∀x,y:setoid1_of_setoid S.x =_1 y → (Ex S (λw.ee x w ∧ True)) =_1 (Ex S (λw.ee y w ∧ True)).
333 alias symbol "trans" (instance 1) = "trans1".
334 alias symbol "refl" (instance 5) = "refl".
335 alias symbol "prop2" (instance 3) = "prop21".
336 alias symbol "trans" (instance 1) = "trans1".
337 alias symbol "prop2" (instance 3) = "prop21".
339 eseguire senza modificare per ottenrere degli alias, fare back di 1 passo
340 e ri-eseguire. Se riesegue senza aggiungere altri alias allora hai gli
341 alias giusti (ma se fate back di più passi, gli alias non vanno più bene...).
342 ora (m x w) e True possono essere sostituiti da ?, se invece
343 si toglie anche l'∧, allora un super bug si scatena (meta contesto locale
344 scazzato, con y al posto di x, unificata con se stessa ma col contesto
345 locale corretto...). lo stesso (o simile) bug salta fuori se esegui
346 senza gli alias giusti con ? al posto di True o (m x w).
348 bug a parte, pare inferisca tutto lui...
349 la E astratta nella prova è solo per fargli inferire x e y, se Sig
350 lo si riformula in modo più naturale (senza y=z) allora bisogna passare
351 x e y esplicitamente. *)
352 napply (.= (Sig ? S (λw,x.(m x w) ∧ True) ?? E (λw,E.(E‡#)‡#)));
355 napply (λw:S.(.= ((E‡#)‡#)) #); ##[##2: napply w| napply m. #H; napply H;
359 let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
362 nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
365 nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
367 ncheck (exists_is_morph (LIST S) (LIST S) ? ?? (E‡#)).
368 nletin xxx ≝ (exists_is_morph); (LIST S)); (LIST S) ee x y E);
370 nchange with (F x = F y);
371 nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
380 Σ(λx.(#‡E)‡#) : ∃x.x = w ∧ m → ∃x.x = w2 ∧ m
381 λx.(#‡E)‡# : ∀x.x = w ∧ m → x = w2 ∧ m
387 ------------------------------
388 ex (λx.g w x) ==?== fun11 F w
390 x ⊢ fun11 ?h ≟ λw. ?g w x
391 ?G ≟ morphism_leibniz (T → S) ∘ ?h
392 ------------------------------
393 (λw. (λx:T.?g w x)) ==?== fun11 ?G
396 x ⊢ fun11 ?h ==?== λw. eq x w ∧ m [w]
397 (λw,x.eq x w ∧ m[w]) ==?== fun11 ?G
398 ex (λx.?g w x) ==?== ex (λx.x = w ∧ m[w])
400 ndefinition ex_morph : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
401 #S; @; ##[ #P; napply (Ex ? P); ##| ncases admit. ##] nqed.
403 ndefinition ex_morph1 : ∀S:setoid. (S ⇒_1 CPROP) ⇒_1 CPROP.
404 #S; @; ##[ #P; napply (Ex ? (λx.P); ##| ncases admit. ##] nqed.
408 ∀ee: (setoid1_of_setoid (list S)) ⇒_1 (setoid1_of_setoid (list S)) ⇒_1 CPROP.
410 let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
413 nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
417 nchange with (F x = F y);
418 nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
424 nlemma d : ∀S:Alpha.∀ee: (setoid1_of_setoid (list S)) ⇒_1 (setoid1_of_setoid (list S)) ⇒_1 CPROP.∀x,y:list S.x = y →
425 let form ≝ comp1_unary_morphisms ??? (ex_morph (list S)) ee in
428 nletin F ≝ (comp1_unary_morphisms ??? (ex_morph (list S)) ee);
432 nchange with (F x = F y);
433 nchange in E with (eq_rel1 ? (eq1 (setoid1_of_setoid (LIST S))) x y);
439 nlemma d : ∀S:Alpha.(setoid1_of_setoid (list S)) ⇒_1 CPROP.
440 #S; napply (comp1_unary_morphisms ??? (ex_morph (list S)) ?);
446 ndefinition comp_ex : ∀X,S,T,K.∀P:X ⇒_1 (S ⇒_1 T).∀Pc : (S ⇒_1 T) ⇒_1 K. X ⇒_1 K.
449 ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
450 #S r; @(𝐋\p r); #w1 w2 E; nelim r; /2/;
451 ##[ #x; @; #H; ##[ nchange in H ⊢ % with ([?]=?); napply ((.= H) E)]
452 nchange in H ⊢ % with ([?]=?); napply ((.= H) E^-1);
454 nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
456 napply (.= (Eexists ?? ? w1 w2 E)‡#);
459 nchange in match (w2 ∈ 𝐋\p (?·?)) with (?∨?);
463 //; napply (trans ?? ??? H E);
464 napply (trans (list S) (eq0 (LIST S)) ? [?] ?(*w1 [x] w2*) H E);
465 nlapply (trans (list S) (eq0 (LIST S))).
467 napply (.= H); nnormalize; nlapply (trans ? [x] w1 w2 E H); napply (.= ?) [napply (w1 = [x])] ##[##2: napply #; napply sym1; napply refl1 ]
469 ndefinition epsilon ≝
470 λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
472 interpretation "epsilon" 'epsilon = (epsilon ?).
473 notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
474 interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
476 ndefinition L_pr ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
478 interpretation "L_pr" 'L_pi E = (L_pr ? E).
480 nlemma append_eq_nil : ∀S:setoid.∀w1,w2:list S. w1 @ w2 = [ ] → w1 = [ ].
481 #S w1; ncases w1; //. nqed.
484 nlemma epsilon_in_true : ∀S:Alpha.∀e:pre S. [ ] ∈ 𝐋\p e = (\snd e = true).
485 #S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//;
486 *; ##[##2:*] nelim e;
487 ##[ ##1,2: *; ##| #c; *; ##| #c; *| ##7: #p H;
488 ##| #r1 r2 H G; *; ##[##2: nassumption; ##]
489 ##| #r1 r2 H1 H2; *; /2/ by {}]
490 *; #w1; *; #w2; *; *;
491 ##[ #defw1 H1 foo; napply H; napply (. #‡#); (append_eq_nil … defw1)^-1‡#);
493 nrewrite > (append_eq_nil ? … w1 w2 …); /3/ by {};//;
496 nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
497 #S e; nelim e; nnormalize; /2/ by nmk;
498 ##[ #; @; #; ndestruct;
499 ##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
500 nrewrite > (append_eq_nil …H…); /2/;
501 ##| #r1 r2 n1 n2; @; *; /2/;
502 ##| #r n; @; *; #w1; *; #w2; *; *; #H;
503 nrewrite > (append_eq_nil …H…); /2/;##]
506 ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
507 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
508 interpretation "oplus" 'oplus a b = (lo ? a b).
510 ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
511 match a with [ mk_pair e1 b1 ⇒
513 [ false ⇒ 〈e1 · \fst b, \snd b〉
514 | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
516 notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
517 interpretation "lc" 'lc op a b = (lc ? op a b).
518 notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
520 ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
521 match a with [ mk_pair e1 b1 ⇒
523 [ false ⇒ 〈e1^*, false〉
524 | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
526 notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
527 interpretation "lk" 'lk op a = (lk ? op a).
528 notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
530 notation > "•" non associative with precedence 60 for @{eclose ?}.
531 nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
535 | ps x ⇒ 〈 `.x, false 〉
536 | pp x ⇒ 〈 `.x, false 〉
537 | po E1 E2 ⇒ •E1 ⊕ •E2
538 | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉
539 | pk E ⇒ 〈(\fst (•E))^*,true〉].
540 notation < "• x" non associative with precedence 60 for @{'eclose $x}.
541 interpretation "eclose" 'eclose x = (eclose ? x).
542 notation > "• x" non associative with precedence 60 for @{'eclose $x}.
544 ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
545 interpretation "reclose" 'eclose x = (reclose ? x).
547 ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
548 notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
549 notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
550 interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
552 naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
554 nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
555 #S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
558 nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
559 #S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
561 nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
562 #S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.
565 nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
566 #S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
567 nwhd in ⊢ (??(??%)?);
568 nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
569 nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
570 nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
571 nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
572 nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
576 ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
577 #S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
579 nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
581 nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
582 #S p q r; napply extP; #w; nnormalize; @;
583 ##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
584 ##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
587 nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
588 #S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
590 nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| = 𝐋 .|e1| · 𝐋 .|e2|.
591 #S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
594 nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| = 𝐋 .|e1| ∪ 𝐋 .|e2|.
595 #S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
597 nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
599 ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
600 interpretation "substract" 'minus a b = (substract ? a b).
602 nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
603 #S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
605 nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
606 #S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
608 nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
609 #S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
611 nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
612 #S a b w; nnormalize; *; //; nqed.
614 nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
615 #S a; nelim a; // by {};
616 ##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
617 nrewrite < IH1; nrewrite < IH2;
618 nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
619 ncases (•e1); #e3 b; ncases b; nnormalize;
620 ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
621 ##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
622 nrewrite < IH2; nrewrite < IH1;
623 nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
624 ncases (•e1); ncases (•e2); //;
625 ##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
626 nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
629 nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
630 #S p; ncases p; //; nqed.
632 nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
633 #S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
634 *; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
637 (* theorem 16: 1 → 3 *)
638 nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
639 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| →
640 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
641 #S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
642 ##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
643 nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
644 nchange in ⊢ (??%?) with (?∪?);
645 nchange in ⊢ (??(??%?)?) with (?∪?);
646 nchange in match (𝐋\p 〈?,?〉) with (?∪?);
647 nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
648 nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
649 nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
650 nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 .|e2'|);
651 ngeneralize in match th1;
652 nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
653 nrewrite > (eta_lp ? e2);
654 nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
655 nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);
656 nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
657 nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
658 ##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
659 nchange in match (𝐋\p ?) with (?∪?);
660 nchange in match (𝐋\p (e1'·?)) with (?∪?);
661 nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
663 nrewrite > (cupA…); //;##]
666 nlemma sub_dot_star :
667 ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
668 #S X b; napply extP; #w; @;
669 ##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
670 *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
671 @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
672 @; //; napply (subW … sube);
673 ##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
674 #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
675 ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
676 @; ncases b in H1; #H1;
677 ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
678 nrewrite > (associative_append ? w' w1 w2);
679 nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
680 ##| ncases w' in Pw';
681 ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
682 ##| #x xs Px; @(x::xs); @(w1@w2);
683 nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
685 ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
686 nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
687 ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
688 nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
690 ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
691 @; //; @; //; @; *;##]##]##]
695 alias symbol "pc" (instance 13) = "cat lang".
696 alias symbol "in_pl" (instance 23) = "in_pl".
697 alias symbol "in_pl" (instance 5) = "in_pl".
698 alias symbol "eclose" (instance 21) = "eclose".
699 ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 .|e|.
701 ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
702 ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
704 nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
705 nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
706 nrewrite > (IH1 …); nrewrite > (cup_dotD …);
707 nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
708 nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
709 nrewrite < (erase_dot …); nrewrite < (cupA …); //;
711 nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
712 nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
713 nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
714 nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
715 nrewrite < (erase_plus …); //.
716 ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
717 nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
718 nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
719 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
720 nrewrite > (erase_bull…e);
721 nrewrite > (erase_star …);
722 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b')); ##[##2:
723 nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
724 ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
725 nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
726 ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
727 nrewrite > (cup_dotD…); nrewrite > (cupA…);
728 nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
729 nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
734 ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
735 #S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
737 nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
738 #S e; napply extP; #w; nnormalize; @;
739 ##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
740 *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
741 nrewrite < defw; nrewrite < defw2; @; //; @;//;
742 ##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
743 #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
747 nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
748 #S e; @[]; /2/; nqed.
750 nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
751 #S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
753 nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
754 #S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
756 nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
757 ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
758 #S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
759 ##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
763 nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^*.
764 #S p; ncases p; #e b; ncases b;
765 ##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
766 nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
767 nchange in ⊢ (??%?) with (?∪?);
768 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
769 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b' )); ##[##2:
770 nlapply (bull_cup ? e); #bc;
771 nchange in match (𝐋\p (•e)) in bc with (?∪?);
772 nchange in match b' in bc with b';
773 ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
774 nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
775 nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
776 nrewrite > (sub_dot_star…);
777 nchange in match (𝐋\p 〈?,?〉) with (?∪?);
778 nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
779 ##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
781 nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 .|e|^* );
782 nrewrite < (cup0 ? (𝐋\p e)); //;##]
785 nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
790 | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
791 | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
792 | k e1 ⇒ pk ? (pre_of_re ? e1)].
794 nlemma notFalse : ¬False. @; //; nqed.
796 nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
797 #S A; nnormalize; napply extP; #w; @; ##[##2: *]
798 *; #w1; *; #w2; *; *; //; nqed.
800 nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
801 #S e; nelim e; ##[##1,2,3: //]
802 ##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
803 nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
804 ##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
805 nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
806 ##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
807 nrewrite > H1; napply dot0; ##]
810 nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 .|pre_of_re S e| = 𝐋 e.
812 ##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
813 nrewrite < H1; nrewrite < H2; //
814 ##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
815 nrewrite < H1; nrewrite < H2; //
816 ##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
821 nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
822 #S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
823 nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
826 nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
827 #S f g H; nrewrite > H; //; nqed.
830 ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ .|e|.
832 ##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
833 nrewrite > defsnde; #H;
834 nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
839 notation > "\move term 90 x term 90 E"
840 non associative with precedence 60 for @{move ? $x $E}.
841 nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
845 | ps y ⇒ 〈 `y, false 〉
846 | pp y ⇒ 〈 `y, x == y 〉
847 | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
848 | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
849 | pk e ⇒ (\move x e)^⊛ ].
850 notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
851 notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
852 interpretation "move" 'move x E = (move ? x E).
854 ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
855 interpretation "rmove" 'move x E = (rmove ? x E).
857 nlemma XXz : ∀S:Alpha.∀w:word S. w .∈ ∅ → False.
858 #S w abs; ninversion abs; #; ndestruct;
862 nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
863 #S w abs; ninversion abs; #; ndestruct;
866 nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
867 #S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
872 ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
873 (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
874 #S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
875 ncases e1 in H; ncases e2;
876 ##[##1: *; ##[*; nnormalize; #; ndestruct]
877 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
878 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
879 ##|##2: *; ##[*; nnormalize; #; ndestruct]
880 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
881 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
882 ##| #r; *; ##[ *; nnormalize; #; ndestruct]
883 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
884 ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
885 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
886 ##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
887 #H; ninversion H; nnormalize; #; ndestruct;
888 ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
889 ##| #r1 r2; *; ##[ *; #defw]
894 ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
895 #S E; ncases E; #r b; nelim r;
897 ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
898 #H; ninversion H; #; ndestruct;
899 ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
900 #H; ninversion H; #; ndestruct;##]
901 ##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
902 *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
903 ##|#a c w; @; nnormalize;
904 ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
905 #H; ninversion H; #; ndestruct;
906 ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
907 #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
908 ##|#r1 r2 H1 H2 a w; @;
909 ##[ #H; ncases (in_move_cat … H);
910 ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
911 ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
912 nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
921 notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
922 nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
925 | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
927 ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
929 ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
931 ∀E1,E2: bool × (pre S).
933 (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
936 ndefinition NAT: decidable.
940 include "hints_declaration.ma".
942 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
943 unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
945 ninductive unit: Type[0] ≝ I: unit.
947 nlet corec foo_nop (b: bool):
949 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
950 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
952 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
954 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
955 | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
959 nlet corec foo (a: unit):
961 (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
962 (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
967 [ nnormalize in ⊢ (??%%);
968 nnormalize in foo: (? → ??%%);
970 [ nnormalize in ⊢ (??%%); napply foo_nop
972 [ nnormalize in ⊢ (??%%);
974 ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
975 ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
980 ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
981 ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
982 ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
985 nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
986 nnormalize in match test3;
991 (**********************************************************)
993 ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
994 der_z: der S a (z S) (z S)
995 | der_e: der S a (e S) (z S)
996 | der_s1: der S a (s S a) (e ?)
997 | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
998 | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
999 der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
1000 | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
1001 der S a (c ? e1 e2) (c ? e1' e2)
1002 | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
1003 der S a (o ? e1 e2) (o ? e1' e2').
1005 nlemma eq_rect_CProp0_r:
1006 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
1007 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
1010 nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
1012 naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
1013 (* #S; #r1; #r2; #w; nelim r1
1015 | #H1; #H2; napply (in_c ? []); //
1016 | (* tutti casi assurdi *) *)
1018 ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
1019 in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
1020 | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
1022 ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
1024 (in_l S [] E1 → in_l S [] E2) →
1025 (in_l S [] E2 → in_l S [] E1) →
1026 (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
1029 (* serve il lemma dopo? *)
1030 ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
1031 #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
1033 | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
1035 (* IL VICEVERSA NON VALE *)
1036 naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
1037 (* #S; #w; #E; #H; nelim H
1039 | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
1043 ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
1044 #S; #a; #E; #E'; #w; #H; nelim H
1045 [##1,2: #H1; ninversion H1
1046 [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
1047 |##2,9: #X; #Y; #K; ncases (?:False); /2/
1048 |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
1049 |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1050 |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1051 |##6,13: #x; #y; #K; ncases (?:False); /2/
1052 |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
1053 ##| #H1; ninversion H1
1055 | #X; #Y; #K; ncases (?:False); /2/
1056 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
1057 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1058 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1059 | #x; #y; #K; ncases (?:False); /2/
1060 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
1061 ##| #H1; #H2; #H3; ninversion H3
1062 [ #_; #K; ncases (?:False); /2/
1063 | #X; #Y; #K; ncases (?:False); /2/
1064 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
1065 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1066 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
1067 | #x; #y; #K; ncases (?:False); /2/
1068 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
1069 ##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;