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 ⇒ True | _ ⇒ False ]
29 | cons x xs ⇒ match l2 with [ nil ⇒ False | cons y ys ⇒ x = y ∧ eq_list ? xs ys]].
31 ndefinition LIST : setoid → setoid.
32 #S; @(list S); @(eq_list S); ncases admit; nqed.
34 unification hint 0 ≔ S : setoid;
35 P1 ≟ refl ? (eq0 (LIST S)),
36 P2 ≟ sym ? (eq0 (LIST S)),
37 P3 ≟ trans ? (eq0 (LIST S)),
38 X ≟ mk_setoid (list S) (mk_equivalence_relation ? (eq_list S) P1 P2 P3),
40 (*-----------------------------------------------------------------------*) ⊢
43 notation "hvbox(hd break :: tl)"
44 right associative with precedence 47
47 notation "[ list0 x sep ; ]"
48 non associative with precedence 90
49 for ${fold right @'nil rec acc @{'cons $x $acc}}.
51 notation "hvbox(l1 break @ l2)"
52 right associative with precedence 47
53 for @{'append $l1 $l2 }.
55 interpretation "nil" 'nil = (nil ?).
56 interpretation "cons" 'cons hd tl = (cons ? hd tl).
58 nlet rec append A (l1: list A) l2 on l1 ≝
61 | cons hd tl ⇒ hd :: append A tl l2 ].
63 interpretation "append" 'append l1 l2 = (append ? l1 l2).
65 ntheorem append_nil: ∀A:setoid.∀l:list A.l @ [] = l.
66 #A;#l;nelim l;//; #a;#l1;#IH;nnormalize;/2/;nqed.
68 ndefinition associative ≝ λA:setoid.λf:A → A → A.∀x,y,z.f x (f y z) = f (f x y) z.
70 ninductive one : Type[0] ≝ unit : one.
73 λS:Type[2].λs:S.λT:Type[2].λt:T.λlock:one.
74 match lock return λ_.Type[2] with [ unit ⇒ T].
76 nlet rec lift (S:Type[2]) (s:S) (T:Type[2]) (t:T) (lock:one) on lock : force S s T t lock ≝
77 match lock return λlock.force S s T t lock with [ unit ⇒ t ].
79 ncoercion lift1 : ∀S:Type[1].∀s:S.∀T:Type[1].∀t:T.∀lock:one. force S s T t lock ≝ lift
80 on s : ? to force ?????.
82 ncoercion lift2 : ∀S:Type[2].∀s:S.∀T:Type[2].∀t:T.∀lock:one. force S s T t lock ≝ lift
83 on s : ? to force ?????.
85 unification hint 0 ≔ R : setoid;
86 TR ≟ setoid, MR ≟ (carr R), lock ≟ unit
87 (* ---------------------------------------- *) ⊢
88 setoid ≡ force ?(*Type[0]*) MR TR R lock.
90 unification hint 0 ≔ R : setoid1;
91 TR ≟ setoid1, MR ≟ (carr1 R), lock ≟ unit
92 (* ---------------------------------------- *) ⊢
93 setoid1 ≡ force ? MR TR R lock.
95 ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
96 #A;#x;#y;#z;nelim x[//|#a;#x1;#H;nnormalize;/2/]nqed.
98 interpretation "iff" 'iff a b = (iff a b).
100 ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ refl: eq A x x.
102 nlemma eq_rect_Type0_r':
103 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
104 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
107 nlemma eq_rect_Type0_r:
108 ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
109 #A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); nassumption.
112 nlemma eq_rect_CProp0_r':
113 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
114 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
117 nlemma eq_rect_CProp0_r:
118 ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
119 #A; #a; #P; #p; #x0; #p0; napply (eq_rect_CProp0_r' ??? p0); nassumption.
122 notation < "a = b" non associative with precedence 45 for @{ 'eqpp $a $b}.
123 interpretation "bool eq" 'eqpp a b = (eq bool a b).
125 ndefinition BOOL : setoid.
126 @bool; @(eq bool); ncases admit.nqed.
128 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
129 alias id "refl" = "cic:/matita/ng/properties/relations/refl.fix(0,1,3)".
130 unification hint 0 ≔ ;
131 P1 ≟ refl ? (eq0 BOOL),
132 P2 ≟ sym ? (eq0 BOOL),
133 P3 ≟ trans ? (eq0 BOOL),
134 X ≟ mk_setoid bool (mk_equivalence_relation ? (eq bool) P1 P2 P3)
135 (*-----------------------------------------------------------------------*) ⊢
138 nrecord Alpha : Type[1] ≝ {
140 eqb: acarr → acarr → bool;
141 eqb_true: ∀x,y. (eqb x y = true) = (x = y)
144 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
145 interpretation "eqb" 'eqb a b = (eqb ? a b).
147 ninductive re (S: Type[0]) : Type[0] ≝
151 | c: re S → re S → re S
152 | o: re S → re S → re S
155 naxiom eq_re : ∀S:Alpha.re S → re S → CProp[0].
156 ndefinition RE : Alpha → setoid.
157 #A; @(re A); @(eq_re A); ncases admit. nqed.
159 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
160 alias id "carr" = "cic:/matita/ng/sets/setoids/carr.fix(0,0,1)".
161 unification hint 0 ≔ A : Alpha;
162 P1 ≟ refl ? (eq0 (RE A)),
163 P2 ≟ sym ? (eq0 (RE A)),
164 P3 ≟ trans ? (eq0 (RE A)),
165 X ≟ mk_setoid (re A) (mk_equivalence_relation ? (eq_re A) P1 P2 P3),
167 (*-----------------------------------------------------------------------*) ⊢
170 notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
171 notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
172 interpretation "star" 'pk a = (k ? a).
173 interpretation "or" 'plus a b = (o ? a b).
175 notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
176 interpretation "cat" 'pc a b = (c ? a b).
178 (* to get rid of \middot *)
179 ncoercion c : ∀S.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
181 notation < "a" non associative with precedence 90 for @{ 'ps $a}.
182 notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
183 interpretation "atom" 'ps a = (s ? a).
185 notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
186 interpretation "epsilon" 'epsilon = (e ?).
188 notation "0" non associative with precedence 90 for @{ 'empty }.
189 interpretation "empty" 'empty = (z ?).
191 notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
192 notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(list $S) }.
194 nlet rec flatten S (l : list (list S)) on l : list S ≝
195 match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
197 nlet rec conjunct S (l : list (list S)) (L : lang S) on l: CProp[0] ≝
198 match l with [ nil ⇒ True | cons w tl ⇒ w ∈ L ∧ conjunct ? tl L ].
201 ndefinition empty_set : ∀A.Ω^A ≝ λA.{ w | False }.
202 notation "∅" non associative with precedence 90 for @{'emptyset}.
203 interpretation "empty set" 'emptyset = (empty_set ?).
206 notation "{}" non associative with precedence 90 for @{'empty_lang}.
207 interpretation "empty lang" 'empty_lang = (empty_lang ?).
210 ndefinition sing_lang : ∀A:setoid.∀x:A.Ω^A ≝ λS.λx.{ w | x = w }.
211 interpretation "sing lang" 'singl x = (sing_lang ? x).
213 interpretation "subset construction with type" 'comprehension t \eta.x =
216 ndefinition cat : ∀A:setoid.∀l1,l2:lang A.lang A ≝
217 λS.λl1,l2.{ w ∈ list S | ∃w1,w2.w =_0 w1 @ w2 ∧ w1 ∈ l1 ∧ w2 ∈ l2}.
218 interpretation "cat lang" 'pc a b = (cat ? a b).
220 ndefinition star : ∀A:setoid.∀l:lang A.lang A ≝
221 λS.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}.
222 interpretation "star lang" 'pk l = (star ? l).
224 notation > "𝐋 term 70 E" non associative with precedence 75 for @{L_re ? $E}.
225 nlet rec L_re (S : Alpha) (r : re S) on r : lang S ≝
230 | c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
231 | o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2
233 notation "𝐋 term 70 E" non associative with precedence 75 for @{'L_re $E}.
234 interpretation "in_l" 'L_re E = (L_re ? E).
236 notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
237 ndefinition orb ≝ λa,b:bool. match a with [ true ⇒ true | _ ⇒ b ].
238 interpretation "orb" 'orb a b = (orb a b).
240 ninductive pitem (S: Type[0]) : Type[0] ≝
245 | pc: pitem S → pitem S → pitem S
246 | po: pitem S → pitem S → pitem S
247 | pk: pitem S → pitem S.
249 ndefinition pre ≝ λS.pitem S × bool.
251 notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
252 interpretation "fst" 'fst x = (fst ? ? x).
253 notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
254 interpretation "snd" 'snd x = (snd ? ? x).
256 interpretation "pstar" 'pk a = (pk ? a).
257 interpretation "por" 'plus a b = (po ? a b).
258 interpretation "pcat" 'pc a b = (pc ? a b).
259 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
260 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
261 interpretation "ppatom" 'pp a = (pp ? a).
262 (* to get rid of \middot *)
263 ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
264 interpretation "patom" 'ps a = (ps ? a).
265 interpretation "pepsilon" 'epsilon = (pe ?).
266 interpretation "pempty" 'empty = (pz ?).
268 notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
269 nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
275 | pc E1 E2 ⇒ (|E1| · |E2|)
276 | po E1 E2 ⇒ (|E1| + |E2|)
278 notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
279 interpretation "forget" 'forget a = (forget ? a).
281 notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{L_pi ? $E}.
282 nlet rec L_pi (S : Alpha) (r : pitem S) on r : lang S ≝
288 | pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 |r2| ∪ 𝐋\p\ r2
289 | po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
290 | pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
291 notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'L_pi $E}.
292 notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'L_pi $E}.
293 interpretation "in_pl" 'L_pi E = (L_pi ? E).
295 unification hint 0 ≔ S,a,b;
298 (* -------------------------------------------- *) ⊢
299 eq_list S a b ≡ eq_rel L (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).
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.
329 (* desiderata : Σ(λx.H‡#)
330 ottenute : Σ H (λx,H.H‡#) -- quindi monoriscrittura. H toplevel permette inferenza di y e z in Sig
333 notation "∑" non associative with precedence 90 for @{Sig ?????}.
335 nlemma test : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
336 ∀x,y:setoid1_of_setoid S.x =_1 y → (Ex S (λw.ee x w ∧ True)) =_1 (Ex S (λw.ee y w ∧ True)).
338 napply (.=_1 (∑ E (λw,H.(H ╪_1 #)╪_1 #))).
342 nlemma test2 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
343 ∀x,y:setoid1_of_setoid S.x =_1 y →
344 (True ∧ (Ex S (λw.ee x w ∧ True))) =_1 (True ∧ (Ex S (λw.ee y w ∧ True))).
346 napply (.=_1 #‡(∑ E (λw,H.(H ╪_1 #) ╪_1 #))).
350 nlemma ex_setoid : ∀S:setoid.(S ⇒_1 CPROP) → setoid.
351 #T P; @ (Ex T (λx:T.P x)); @;
352 ##[ #H1 H2; ncases H1; ncases H2; #x Px y Py; napply (x = y);
353 ##| #H; napply (?:?); ncases H; nelim H; nnormalize;
355 nlemma test3 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
356 ∀x,y:setoid1_of_setoid S.x =_1 y →
357 ((Ex S (λw.ee x w ∧ True) ∨ True)) =_1 ((Ex S (λw.ee y w ∧ True) ∨ True)).
360 napply (.=_1 (∑ E (λw,H.(H ╪_1 #) ╪_1 #)) ╪_1 #).
364 ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
365 #S r; @(𝐋\p r); #w1 w2 E; nelim r;
369 ##| #x; @; #H; nchange in H with ([?] =_0 ?); ##[ napply ((.=_0 H) E); ##]
370 napply ((.=_0 H) E^-1);
372 nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
373 nchange in match (w2 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
376 nlapply (.=_1 (∑ E (λx,H.?))╪_1 #);
377 napply (.=_1 (∑ E (λx,H.#))╪_1 #);
380 nchange in match (w2 ∈ 𝐋\p (?·?)) with (?∨?);
384 //; napply (trans ?? ??? H E);
385 napply (trans (list S) (eq0 (LIST S)) ? [?] ?(*w1 [x] w2*) H E);
386 nlapply (trans (list S) (eq0 (LIST S))).
388 napply (.= H); nnormalize; nlapply (trans ? [x] w1 w2 E H); napply (.= ?) [napply (w1 = [x])] ##[##2: napply #; napply sym1; napply refl1 ]
390 ndefinition epsilon ≝
391 λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
393 interpretation "epsilon" 'epsilon = (epsilon ?).
394 notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
395 interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
397 ndefinition L_pr ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
399 interpretation "L_pr" 'L_pi E = (L_pr ? E).
401 nlemma append_eq_nil : ∀S:setoid.∀w1,w2:list S. w1 @ w2 = [ ] → w1 = [ ].
402 #S w1; ncases w1; //. nqed.
405 nlemma epsilon_in_true : ∀S:Alpha.∀e:pre S. [ ] ∈ 𝐋\p e = (\snd e = true).
406 #S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//;
407 *; ##[##2:*] nelim e;
408 ##[ ##1,2: *; ##| #c; *; ##| #c; *| ##7: #p H;
409 ##| #r1 r2 H G; *; ##[##2: nassumption; ##]
410 ##| #r1 r2 H1 H2; *; /2/ by {}]
411 *; #w1; *; #w2; *; *;
412 ##[ #defw1 H1 foo; napply H; napply (. #‡#); (append_eq_nil … defw1)^-1‡#);
414 nrewrite > (append_eq_nil ? … w1 w2 …); /3/ by {};//;
417 nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
418 #S e; nelim e; nnormalize; /2/ by nmk;
419 ##[ #; @; #; ndestruct;
420 ##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
421 nrewrite > (append_eq_nil …H…); /2/;
422 ##| #r1 r2 n1 n2; @; *; /2/;
423 ##| #r n; @; *; #w1; *; #w2; *; *; #H;
424 nrewrite > (append_eq_nil …H…); /2/;##]
427 ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
428 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
429 interpretation "oplus" 'oplus a b = (lo ? a b).
431 ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
432 match a with [ mk_pair e1 b1 ⇒
434 [ false ⇒ 〈e1 · \fst b, \snd b〉
435 | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
437 notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
438 interpretation "lc" 'lc op a b = (lc ? op a b).
439 notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
441 ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
442 match a with [ mk_pair e1 b1 ⇒
444 [ false ⇒ 〈e1^*, false〉
445 | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
447 notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
448 interpretation "lk" 'lk op a = (lk ? op a).
449 notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
451 notation > "•" non associative with precedence 60 for @{eclose ?}.
452 nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
456 | ps x ⇒ 〈 `.x, false 〉
457 | pp x ⇒ 〈 `.x, false 〉
458 | po E1 E2 ⇒ •E1 ⊕ •E2
459 | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉
460 | pk E ⇒ 〈(\fst (•E))^*,true〉].
461 notation < "• x" non associative with precedence 60 for @{'eclose $x}.
462 interpretation "eclose" 'eclose x = (eclose ? x).
463 notation > "• x" non associative with precedence 60 for @{'eclose $x}.
465 ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
466 interpretation "reclose" 'eclose x = (reclose ? x).
468 ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
469 notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
470 notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
471 interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
473 naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
475 nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
476 #S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
479 nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
480 #S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
482 nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
483 #S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.
486 nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
487 #S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
488 nwhd in ⊢ (??(??%)?);
489 nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
490 nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
491 nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
492 nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
493 nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
497 ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
498 #S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
500 nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
502 nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
503 #S p q r; napply extP; #w; nnormalize; @;
504 ##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
505 ##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
508 nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
509 #S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
511 nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| = 𝐋 .|e1| · 𝐋 .|e2|.
512 #S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
515 nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| = 𝐋 .|e1| ∪ 𝐋 .|e2|.
516 #S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
518 nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
520 ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
521 interpretation "substract" 'minus a b = (substract ? a b).
523 nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
524 #S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
526 nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
527 #S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
529 nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
530 #S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
532 nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
533 #S a b w; nnormalize; *; //; nqed.
535 nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
536 #S a; nelim a; // by {};
537 ##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
538 nrewrite < IH1; nrewrite < IH2;
539 nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
540 ncases (•e1); #e3 b; ncases b; nnormalize;
541 ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
542 ##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
543 nrewrite < IH2; nrewrite < IH1;
544 nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
545 ncases (•e1); ncases (•e2); //;
546 ##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
547 nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
550 nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
551 #S p; ncases p; //; nqed.
553 nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
554 #S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
555 *; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
558 (* theorem 16: 1 → 3 *)
559 nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
560 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| →
561 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
562 #S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
563 ##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
564 nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
565 nchange in ⊢ (??%?) with (?∪?);
566 nchange in ⊢ (??(??%?)?) with (?∪?);
567 nchange in match (𝐋\p 〈?,?〉) with (?∪?);
568 nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
569 nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
570 nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
571 nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 .|e2'|);
572 ngeneralize in match th1;
573 nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
574 nrewrite > (eta_lp ? e2);
575 nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
576 nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);
577 nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
578 nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
579 ##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
580 nchange in match (𝐋\p ?) with (?∪?);
581 nchange in match (𝐋\p (e1'·?)) with (?∪?);
582 nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
584 nrewrite > (cupA…); //;##]
587 nlemma sub_dot_star :
588 ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
589 #S X b; napply extP; #w; @;
590 ##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
591 *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
592 @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
593 @; //; napply (subW … sube);
594 ##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
595 #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
596 ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
597 @; ncases b in H1; #H1;
598 ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
599 nrewrite > (associative_append ? w' w1 w2);
600 nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
601 ##| ncases w' in Pw';
602 ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
603 ##| #x xs Px; @(x::xs); @(w1@w2);
604 nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
606 ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
607 nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
608 ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
609 nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
611 ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
612 @; //; @; //; @; *;##]##]##]
616 alias symbol "pc" (instance 13) = "cat lang".
617 alias symbol "in_pl" (instance 23) = "in_pl".
618 alias symbol "in_pl" (instance 5) = "in_pl".
619 alias symbol "eclose" (instance 21) = "eclose".
620 ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 .|e|.
622 ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
623 ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
625 nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
626 nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
627 nrewrite > (IH1 …); nrewrite > (cup_dotD …);
628 nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
629 nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
630 nrewrite < (erase_dot …); nrewrite < (cupA …); //;
632 nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
633 nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
634 nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
635 nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
636 nrewrite < (erase_plus …); //.
637 ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
638 nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
639 nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
640 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
641 nrewrite > (erase_bull…e);
642 nrewrite > (erase_star …);
643 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b')); ##[##2:
644 nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
645 ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
646 nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
647 ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
648 nrewrite > (cup_dotD…); nrewrite > (cupA…);
649 nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
650 nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
655 ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
656 #S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
658 nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
659 #S e; napply extP; #w; nnormalize; @;
660 ##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
661 *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
662 nrewrite < defw; nrewrite < defw2; @; //; @;//;
663 ##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
664 #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
668 nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
669 #S e; @[]; /2/; nqed.
671 nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
672 #S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
674 nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
675 #S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
677 nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
678 ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
679 #S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
680 ##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
684 nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^*.
685 #S p; ncases p; #e b; ncases b;
686 ##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
687 nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
688 nchange in ⊢ (??%?) with (?∪?);
689 nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
690 nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b' )); ##[##2:
691 nlapply (bull_cup ? e); #bc;
692 nchange in match (𝐋\p (•e)) in bc with (?∪?);
693 nchange in match b' in bc with b';
694 ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
695 nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
696 nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
697 nrewrite > (sub_dot_star…);
698 nchange in match (𝐋\p 〈?,?〉) with (?∪?);
699 nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
700 ##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
702 nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 .|e|^* );
703 nrewrite < (cup0 ? (𝐋\p e)); //;##]
706 nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
711 | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
712 | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
713 | k e1 ⇒ pk ? (pre_of_re ? e1)].
715 nlemma notFalse : ¬False. @; //; nqed.
717 nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
718 #S A; nnormalize; napply extP; #w; @; ##[##2: *]
719 *; #w1; *; #w2; *; *; //; nqed.
721 nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
722 #S e; nelim e; ##[##1,2,3: //]
723 ##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
724 nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
725 ##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
726 nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
727 ##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
728 nrewrite > H1; napply dot0; ##]
731 nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 .|pre_of_re S e| = 𝐋 e.
733 ##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
734 nrewrite < H1; nrewrite < H2; //
735 ##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
736 nrewrite < H1; nrewrite < H2; //
737 ##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
742 nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
743 #S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
744 nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
747 nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
748 #S f g H; nrewrite > H; //; nqed.
751 ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ .|e|.
753 ##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
754 nrewrite > defsnde; #H;
755 nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
760 notation > "\move term 90 x term 90 E"
761 non associative with precedence 60 for @{move ? $x $E}.
762 nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
766 | ps y ⇒ 〈 `y, false 〉
767 | pp y ⇒ 〈 `y, x == y 〉
768 | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
769 | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
770 | pk e ⇒ (\move x e)^⊛ ].
771 notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
772 notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
773 interpretation "move" 'move x E = (move ? x E).
775 ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
776 interpretation "rmove" 'move x E = (rmove ? x E).
778 nlemma XXz : ∀S:Alpha.∀w:word S. w .∈ ∅ → False.
779 #S w abs; ninversion abs; #; ndestruct;
783 nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
784 #S w abs; ninversion abs; #; ndestruct;
787 nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
788 #S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
793 ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
794 (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
795 #S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
796 ncases e1 in H; ncases e2;
797 ##[##1: *; ##[*; nnormalize; #; ndestruct]
798 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
799 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
800 ##|##2: *; ##[*; nnormalize; #; ndestruct]
801 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
802 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
803 ##| #r; *; ##[ *; nnormalize; #; ndestruct]
804 #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
805 ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
806 nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
807 ##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
808 #H; ninversion H; nnormalize; #; ndestruct;
809 ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
810 ##| #r1 r2; *; ##[ *; #defw]
815 ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
816 #S E; ncases E; #r b; nelim r;
818 ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
819 #H; ninversion H; #; ndestruct;
820 ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
821 #H; ninversion H; #; ndestruct;##]
822 ##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
823 *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
824 ##|#a c w; @; nnormalize;
825 ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
826 #H; ninversion H; #; ndestruct;
827 ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
828 #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
829 ##|#r1 r2 H1 H2 a w; @;
830 ##[ #H; ncases (in_move_cat … H);
831 ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
832 ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
833 nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
842 notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
843 nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
846 | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
848 ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
850 ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
852 ∀E1,E2: bool × (pre S).
854 (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
857 ndefinition NAT: decidable.
861 include "hints_declaration.ma".
863 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
864 unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
866 ninductive unit: Type[0] ≝ I: unit.
868 nlet corec foo_nop (b: bool):
870 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
871 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
873 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
875 [ nnormalize in ⊢ (??%%); napply (foo_nop false)
876 | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
880 nlet corec foo (a: unit):
882 (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
883 (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
888 [ nnormalize in ⊢ (??%%);
889 nnormalize in foo: (? → ??%%);
891 [ nnormalize in ⊢ (??%%); napply foo_nop
893 [ nnormalize in ⊢ (??%%);
895 ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
896 ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
901 ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
902 ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
903 ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
906 nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
907 nnormalize in match test3;
912 (**********************************************************)
914 ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
915 der_z: der S a (z S) (z S)
916 | der_e: der S a (e S) (z S)
917 | der_s1: der S a (s S a) (e ?)
918 | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
919 | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
920 der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
921 | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
922 der S a (c ? e1 e2) (c ? e1' e2)
923 | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
924 der S a (o ? e1 e2) (o ? e1' e2').
926 nlemma eq_rect_CProp0_r:
927 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
928 #A; #a; #x; #p; ncases p; #P; #H; nassumption.
931 nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
933 naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
934 (* #S; #r1; #r2; #w; nelim r1
936 | #H1; #H2; napply (in_c ? []); //
937 | (* tutti casi assurdi *) *)
939 ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
940 in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
941 | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
943 ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
945 (in_l S [] E1 → in_l S [] E2) →
946 (in_l S [] E2 → in_l S [] E1) →
947 (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
950 (* serve il lemma dopo? *)
951 ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
952 #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
954 | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
956 (* IL VICEVERSA NON VALE *)
957 naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
958 (* #S; #w; #E; #H; nelim H
960 | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
964 ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
965 #S; #a; #E; #E'; #w; #H; nelim H
966 [##1,2: #H1; ninversion H1
967 [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
968 |##2,9: #X; #Y; #K; ncases (?:False); /2/
969 |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
970 |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
971 |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
972 |##6,13: #x; #y; #K; ncases (?:False); /2/
973 |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
974 ##| #H1; ninversion H1
976 | #X; #Y; #K; ncases (?:False); /2/
977 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
978 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
979 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
980 | #x; #y; #K; ncases (?:False); /2/
981 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
982 ##| #H1; #H2; #H3; ninversion H3
983 [ #_; #K; ncases (?:False); /2/
984 | #X; #Y; #K; ncases (?:False); /2/
985 | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
986 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
987 | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
988 | #x; #y; #K; ncases (?:False); /2/
989 | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
990 ##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;