X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fre%2Fre.ma;h=40947401f860939ceea720b2363c394a9671aef7;hb=25c634037771dff0138e5e8e3d4378183ff49b86;hp=f8916ee5a5759682c9808c1c4238444780cadbe7;hpb=86b6c02aa918e5e7115f42947428590d5f0a26e6;p=helm.git

diff --git a/helm/software/matita/nlibrary/re/re.ma b/helm/software/matita/nlibrary/re/re.ma
index f8916ee5a..40947401f 100644
--- a/helm/software/matita/nlibrary/re/re.ma
+++ b/helm/software/matita/nlibrary/re/re.ma
@@ -12,295 +12,628 @@
 (*                                                                        *)
 (**************************************************************************)
 
-(*include "logic/connectives.ma".*)
-(*include "logic/equality.ma".*)
 include "datatypes/list.ma".
 include "datatypes/pairs.ma".
+include "arithmetics/nat.ma".
 
-(*include "Plogic/equality.ma".*)
+interpretation "iff" 'iff a b = (iff a b).  
 
-ndefinition word ≝ λS:Type[0].list S.
+nrecord Alpha : Type[1] ≝ { carr :> Type[0];
+   eqb: carr → carr → bool;
+   eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
+}.
+ 
+notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
+interpretation "eqb" 'eqb a b = (eqb ? a b).
+
+ndefinition word ≝ λS:Alpha.list S.
 
-ninductive re (S: Type[0]) : Type[0] ≝
+ninductive re (S: Alpha) : Type[0] ≝
    z: re S
  | e: re S
  | s: S → re S
  | c: re S → re S → re S
  | o: re S → re S → re S
  | k: re S → re S.
+ 
+notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
+notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
+interpretation "star" 'pk a = (k ? a).
+interpretation "or" 'plus a b = (o ? a b).
+           
+notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
+interpretation "cat" 'pc a b = (c ? a b).
 
-(*
-alias symbol "not" (instance 1) = "Clogical not".
-nlemma foo1: ∀S. ¬ (z S = e S). #S; @; #H; ndestruct. nqed.
-nlemma foo2: ∀S,x. ¬ (z S = s S x). #S; #x; @; #H; ndestruct. nqed.
-nlemma foo3: ∀S,x1,x2. ¬ (z S = c S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed.
-nlemma foo4: ∀S,x1,x2. ¬ (z S = o S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed.
-nlemma foo5: ∀S,x. ¬ (z S = k S x). #S; #x; @; #H; ndestruct. nqed.
-nlemma foo6: ∀S,x. ¬ (e S = s S x). #S; #x; @; #H; ndestruct. nqed.
-nlemma foo7: ∀S,x1,x2. ¬ (e S = c S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed.
-nlemma foo8: ∀S,x1,x2. ¬ (e S = o S x1 x2). #S; #x1; #x2; @; #H; ndestruct. nqed.
-nlemma foo9: ∀S,x. ¬ (e S = k S x). #S; #x; @; #H; ndestruct. nqed.
-*)
+(* to get rid of \middot *)
+ncoercion c  : ∀S:Alpha.∀p:re S.  re S →  re S   ≝ c  on _p : re ?  to ∀_:?.?.
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
+interpretation "epsilon" 'epsilon = (e ?).
+
+notation "∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (z ?).
+
+nlet rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝ 
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
+
+nlet rec conjunct (S : Alpha) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
+match l with [ nil ⇒ ? | cons w tl ⇒ r w ∧ conjunct ? tl r ]. napply True. nqed.
+
+ndefinition empty_lang ≝ λS.λw:word S.False.
+notation "{}" non associative with precedence 90 for @{'empty_lang}.
+interpretation "empty lang" 'empty_lang = (empty_lang ?).
+
+ndefinition sing_lang ≝ λS.λx,w:word S.x=w.
+notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.
+interpretation "sing lang" 'sing_lang x = (sing_lang ? x).
+
+ndefinition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.l1 w ∨ l2 w.
+interpretation "union lang" 'union a b = (union ? a b).
+
+ndefinition cat : ∀S,l1,l2,w.Prop ≝ 
+  λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
+interpretation "cat lang" 'pc a b = (cat ? a b).
+
+ndefinition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l. 
+interpretation "star lang" 'pk l = (star ? l).
+
+notation > "𝐋 term 70 E" non associative with precedence 75 for @{in_l ? $E}.
+nlet rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ 
+match r with
+[ z ⇒ {}
+| e ⇒ { [ ] }
+| s x ⇒ { [x] }
+| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
+| o r1 r2 ⇒  𝐋 r1 ∪ 𝐋 r2
+| k r1 ⇒ (𝐋 r1) ^*].
+notation "𝐋 term 70 E" non associative with precedence 75 for @{'in_l $E}.
+interpretation "in_l" 'in_l E = (in_l ? E).
+interpretation "in_l mem" 'mem w l = (in_l ? l w).
+
+notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+interpretation "orb" 'orb a b = (orb a b).
+
+ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
+
+ninductive pitem (S: Alpha) : Type[0] ≝
+   pz: pitem S
+ | pe: pitem S
+ | ps: S → pitem S
+ | pp: S → pitem S
+ | pc: pitem S → pitem S → pitem S
+ | po: pitem S → pitem S → pitem S
+ | pk: pitem S → pitem S.
+ 
+ndefinition pre ≝ λS.pitem S × bool.
+
+interpretation "pstar" 'pk a = (pk ? a).
+interpretation "por" 'plus a b = (po ? a b).
+interpretation "pcat" 'pc a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "ppatom" 'pp a = (pp ? a).
+(* to get rid of \middot *)
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S  ≝ pc on _p : pitem ? to ∀_:?.?.
+interpretation "patom" 'ps a = (ps ? a).
+interpretation "pepsilon" 'epsilon = (pe ?).
+interpretation "pempty" 'empty = (pz ?).
 
-ninductive in_l (S: Type[0]): word S → re S → Prop ≝
-   in_e: in_l S [] (e ?)
- | in_s: ∀x. in_l S [x] (s ? x)
- | in_c: ∀w1,w2,e1,e2. in_l ? w1 e1 → in_l ? w2 e2 → in_l S (w1@w2) (c ? e1 e2)
- | in_o1: ∀w,e1,e2. in_l ? w e1 → in_l S w (o ? e1 e2)
- | in_o2: ∀w,e1,e2. in_l ? w e2 → in_l S w (o ? e1 e2)
- | in_ke: ∀e. in_l S [] (k ? e)
- | in_ki: ∀w1,w2,e. in_l ? w1 e → in_l ? w2 (k ? e) → in_l S (w1@w2) (k ? e).
-
-naxiom in_l_inv_z:
- ∀S,w. ¬ (in_l S w (z ?)).
-(* #S; #w; #H; ninversion H
-  [ #_; #b; ndestruct
-  | #a; #b; #c; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; #h; #i; #l; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; ndestruct
-  | #a; #b; #c; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; #h; #i; ndestruct ]
-nqed. *)
+notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
+nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
+ match l with
+  [ pz ⇒ ∅
+  | pe ⇒ ϵ
+  | ps x ⇒ `x
+  | pp x ⇒ `x
+  | pc E1 E2 ⇒ (|E1| · |E2|)
+  | po E1 E2 ⇒ (|E1| + |E2|)
+  | pk E ⇒ |E|^* ].
+notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
+interpretation "forget" 'forget a = (forget ? a).
+
+notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
+interpretation "fst" 'fst x = (fst ? ? x).
+notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
+interpretation "snd" 'snd x = (snd ? ? x).
+
+notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
+nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ 
+match r with
+[ pz ⇒ {}
+| pe ⇒ {}
+| ps _ ⇒ {}
+| pp x ⇒ { [x] }
+| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋  |r2| ∪ 𝐋\p\ r2
+| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
+| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
+notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'in_pl $E}.
+notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'in_pl $E}.
+interpretation "in_pl" 'in_pl E = (in_pl ? E).
+interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+
+ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
+
+interpretation "epsilon" 'epsilon = (epsilon ?).
+notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
+interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
+
+ndefinition in_prl ≝ λS : Alpha.λp:pre S.  𝐋\p (\fst p) ∪ ϵ (\snd p).
+  
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_pl E = (in_prl ? E).
+
+nlemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct; nqed.
+
+(* lemma 12 *)
+nlemma epsilon_in_true : ∀S.∀e:pre S. [ ] ∈ e ↔ \snd e = true.
+#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//; 
+nnormalize; *; ##[##2:*] nelim e;
+##[ ##1,2: *; ##| #c; *; ##| #c; nnormalize; #; ndestruct; ##| ##7: #p H;
+##| #r1 r2 H G; *; ##[##2: /3/ by or_intror]
+##| #r1 r2 H1 H2; *; /3/ by or_intror, or_introl; ##]
+*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
+nqed.
+
+nlemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ((𝐋\p e) [ ]).
+#S e; nelim e; nnormalize; /2/ by nmk;
+##[ #; @; #; ndestruct;
+##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
+    nrewrite > (append_eq_nil …H…); /2/;
+##| #r1 r2 n1 n2; @; *; /2/;
+##| #r n; @; *; #w1; *; #w2; *; *; #H;     
+    nrewrite > (append_eq_nil …H…); /2/;##]
+nqed.
+
+ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
+notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
+interpretation "oplus" 'oplus a b = (lo ? a b).
 
-nlemma in_l_inv_e:
- ∀S,w. in_l S w (e ?) → w = [].
- #S; #w; #H; ninversion H
-  [ #a; #b; ndestruct; //
-  | #a; #b; #c; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; #h; #i; #l; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; ndestruct
-  | #a; #b; #c; ndestruct
-  | #a; #b; #c; #d; #e; #f; #g; #h; #i; ndestruct ]
+ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
+   match a with [ mk_pair e1 b1 ⇒
+   match b1 with 
+   [ false ⇒ 〈e1 · \fst b, \snd b〉 
+   | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
+   
+notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
+interpretation "lc" 'lc op a b = (lc ? op a b).
+notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
+
+ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
+   match a with [ mk_pair e1 b1 ⇒
+   match b1 with 
+   [ false ⇒ 〈e1^*, false〉 
+   | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
+
+notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
+interpretation "lk" 'lk op a = (lk ? op a).
+notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
+
+notation > "•" non associative with precedence 60 for @{eclose ?}.
+nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
+ match E with
+  [ pz ⇒ 〈 ∅, false 〉
+  | pe ⇒ 〈 ϵ,  true 〉
+  | ps x ⇒ 〈 `.x, false 〉
+  | pp x ⇒ 〈 `.x, false 〉
+  | po E1 E2 ⇒ •E1 ⊕ •E2
+  | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉 
+  | pk E ⇒ 〈(\fst (•E))^*,true〉].
+notation < "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
+notation > "• x" non associative with precedence 60 for @{'eclose $x}.
+
+ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
+interpretation "reclose" 'eclose x = (reclose ? x).
+
+ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
+notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
+
+nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
+#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
 nqed.
 
-naxiom in_l_inv_s: 
- ∀S,w,x. in_l S w (s ? x) → w = [x].
+nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
+#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
+
+nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
+#S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.
+
+(* theorem 16: 2 *)
+nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
+nwhd in ⊢ (??(??%)?);
+nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
+nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
+nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
+nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
+nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
+nqed.
 
-naxiom in_l_inv_c:
- ∀S,w,E1,E2. in_l S w (c S E1 E2) → ∃w1.∃w2. w = w1@w2 ∧ in_l S w1 E1 ∧ in_l S w2 E2.
+nlemma odotEt : 
+  ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
+#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
 
-ninductive pre (S: Type[0]) : Type[0] ≝
-   pz: pre S
- | pe: pre S
- | ps: S → pre S
- | pp: S → pre S
- | pc: pre S → pre S → pre S
- | po: pre S → pre S → pre S
- | pk: pre S → pre S.
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed.
 
-nlet rec forget (S: Type[0]) (l : pre S) on l: re S ≝
- match l with
-  [ pz ⇒ z S
-  | pe ⇒ e S
-  | ps x ⇒ s S x
-  | pp x ⇒ s S x
-  | pc E1 E2 ⇒ c S (forget ? E1) (forget ? E2)
-  | po E1 E2 ⇒ o S (forget ? E1) (forget ? E2)
-  | pk E ⇒ k S (forget ? E) ].
-
-ninductive in_pl (S: Type[0]): word S → pre S → Prop ≝
-   in_pp: ∀x. in_pl S [x] (pp S x) 
- | in_pc1: ∀w1,w2,e1,e2. in_pl ? w1 e1 → in_l ? w2 (forget ? e2) →
-            in_pl S (w1@w2) (pc ? e1 e2)
- | in_pc2: ∀w,e1,e2. in_pl ? w e2 → in_pl S w (pc ? e1 e2)
- | in_po1: ∀w,e1,e2. in_pl ? w e1 → in_pl S w (po ? e1 e2)
- | in_po2: ∀w,e1,e2. in_pl ? w e2 → in_pl S w (po ? e1 e2)
- | in_pki: ∀w1,w2,e. in_pl ? w1 e → in_l ? w2 (k ? (forget ? e)) →
-           in_pl S (w1@w2) (pk ? e).
-
-nlet rec eclose (S: Type[0]) (E: pre S) on E ≝
- match E with
-  [ pz ⇒ 〈 false, pz ? 〉
-  | pe ⇒ 〈 true, pe ? 〉
-  | ps x ⇒ 〈 false, pp ? x 〉
-  | pp x ⇒ 〈 false, pp ? x 〉
-  | pc E1 E2 ⇒
-     let E1' ≝ eclose ? E1 in
-     let E1'' ≝ snd … E1' in
-      match fst … E1' with
-       [ true ⇒ 
-          let E2' ≝ eclose ? E2 in
-           〈 fst … E2', pc ? E1'' (snd … E2') 〉
-       | false ⇒ 〈 false, pc ? E1'' E2 〉 ]
-  | po E1 E2 ⇒
-     let E1' ≝ eclose ? E1 in
-     let E2' ≝ eclose ? E2 in
-      〈 fst … E1' ∨ fst … E2', po ? (snd … E1') (snd … E2') 〉
-  | pk E ⇒ 〈 true, pk ? (snd … (eclose S E)) 〉 ].
-
-ntheorem forget_eclose:
- ∀S,E. forget S (snd … (eclose … E)) = forget ? E.
- #S; #E; nelim E; nnormalize; //;
- #p; ncases (fst … (eclose S p)); nnormalize; //.
+nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r). 
+#S p q r; napply extP; #w; nnormalize; @; 
+##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
+##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
 nqed.
 
-ntheorem eclose_true:
- ∀S,E. (* bug refiner se si scambia true con il termine *)
-  true = fst bool (pre S) (eclose S E) → in_l S [] (forget S E).
- #S; #E; nelim E; nnormalize; //
-  [ #H; ncases (?: False); /2/
-  | #x; #H; ncases (?: False); /2/
-  | #x; #H; ncases (?: False); /2/
-  | #w1; #w2; ncases (fst … (eclose S w1)); nnormalize; /3/;
-    #_; #_; #H; ncases (?:False); /2/
-  | #w1; #w2; ncases (fst … (eclose S w1)); nnormalize; /3/]
+nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
+#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
+
+nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 |e1 · e2| =  𝐋 |e1| · 𝐋 |e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
 nqed.
 
-(* to be moved *)
-nlemma eq_append_nil_to_eq_nil1:
- ∀A.∀l1,l2:list A. l1 @ l2 = [] → l1 = [].
- #A; #l1; nelim l1; nnormalize; /2/;
- #x; #tl; #_; #l3; #K; ndestruct.
+nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 |e1 + e2| =  𝐋 |e1| ∪ 𝐋 |e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
+
+nlemma erase_star : ∀S.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed.
+
+ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
+interpretation "substract" 'minus a b = (substract ? a b).
+
+nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
+#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
+
+nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
+#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
+
+nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
+#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
+
+nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
+#S a b w; nnormalize; *; //; nqed.
+
+nlemma erase_bull : ∀S.∀a:pitem S. |\fst (•a)| = |a|.
+#S a; nelim a; // by {};
+##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| · |e2|);
+    nrewrite < IH1; nrewrite < IH2;  
+    nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
+    ncases (•e1); #e3 b; ncases b; nnormalize;
+    ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
+##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| + |e2|);
+    nrewrite < IH2; nrewrite < IH1;
+    nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
+    ncases (•e1); ncases (•e2); //;
+##| #e IH; nchange in ⊢ (???%) with (|e|^* ); nrewrite < IH;
+    nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
 nqed.
 
-(* to be moved *)
-nlemma eq_append_nil_to_eq_nil2:
- ∀A.∀l1,l2:list A. l1 @ l2 = [] → l2 = [].
- #A; #l1; nelim l1; nnormalize; /2/;
- #x; #tl; #_; #l3; #K; ndestruct.
+nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+#S p; ncases p; //; nqed.
+
+nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p. 
+#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
+*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
+napply Hw2; nqed.
+
+(* theorem 16: 1 → 3 *)
+nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
+      𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| → 
+         𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
+#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
+##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2); 
+    nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
+    nchange in ⊢ (??%?) with (?∪?); 
+    nchange in ⊢ (??(??%?)?) with (?∪?);
+    nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
+    nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
+    nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 |e2'|); ##[##2:
+      nchange with (𝐋\p 〈e2'',b2''〉 =  𝐋\p e2' ∪ 𝐋 |e2'|); 
+      ngeneralize in match th1;
+      nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
+    nrewrite > (eta_lp ? e2); 
+    nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
+    nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);       
+    nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
+    nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
+##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
+    nchange in match (𝐋\p ?) with (?∪?);
+    nchange in match (𝐋\p (e1'·?)) with (?∪?);
+    nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
+    nrewrite > (cup0…); 
+    nrewrite > (cupA…); //;##]
 nqed.
 
-ntheorem in_l_empty_c:
- ∀S,E1,E2. in_l S [] (c … E1 E2) → in_l S [] E2.
- #S; #E1; #E2; #H; ninversion H
-  [ #_; #H2; ndestruct
-  | #x; #K; ndestruct
-  | #w1; #w2; #E1'; #E2'; #H1; #H2; #H3; #H4; #H5; #H6;
-    nrewrite < H5; nlapply (eq_append_nil_to_eq_nil2 … w1 w2 ?); //;
-    ndestruct; //
-  | #w; #E1'; #E2'; #H1; #H2; #H3; #H4; ndestruct
-  | #w; #E1'; #E2'; #H1; #H2; #H3; #H4; ndestruct
-  | #E; #_; #K; ndestruct 
-  | #w1; #w2; #w3; #H1; #H2; #H3; #H4; #H5; #H6; ndestruct ]
+nlemma sub_dot_star : 
+  ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
+#S X b; napply extP; #w; @;
+##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+    *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
+    @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+    @; //; napply (subW … sube);
+##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
+    #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
+    ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
+        @; ncases b in H1; #H1; 
+        ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
+                nrewrite > (associative_append ? w' w1 w2);
+                nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
+           ##| ncases w' in Pw';
+               ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
+               ##| #x xs Px; @(x::xs); @(w1@w2); 
+                   nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
+                   @wl'; @; //; ##] ##]
+        ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
+            nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
+            ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
+                nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
+                @[]; @; //;
+            ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //] 
+                @; //; @; //; @; *;##]##]##] 
 nqed.
 
-ntheorem eclose_true':
- ∀S,E. (* bug refiner se si scambia true con il termine *)
-  in_l S [] (forget S E) → true = fst bool (pre S) (eclose S E).
- #S; #E; nelim E; nnormalize; //
-  [ #H; ncases (?:False); /2/
-  |##2,3: #x; #H; ncases (?:False); nlapply (in_l_inv_s ??? H); #K; ndestruct
-  | #E1; #E2; ncases (fst … (eclose S E1)); nnormalize
-     [ #H1; #H2; #H3; ninversion H3; /3/;
-   ##| #H1; #H2; #H3; ninversion H3
-        [ #_; #K; ndestruct
-        | #x; #K; ndestruct
-        | #w1; #w2; #E1'; #E2'; #H4; #H5; #K1; #K2; #K3; #K4; ndestruct;
-          napply H1; nrewrite < (eq_append_nil_to_eq_nil1 … w1 w2 ?); //
-        | #w1; #E1'; #E2'; #H4; #H5; #H6; #H7; ndestruct
-        | #w1; #E1'; #E2'; #H4; #H5; #H6; #H7; ndestruct
-        | #E'; #_; #K; ndestruct
-        | #a; #b; #c; #d; #e; #f; #g; #h; #i; ndestruct ]##]
-##| #E1; ncases (fst … (eclose S E1)); nnormalize; //;
-    #E2; #H1; #H2; #H3; ninversion H3
-     [ #_; #K; ndestruct
-     | #w; #_; #K; ndestruct
-     | #a; #b; #c; #d; #e; #f; #g; #h; #i; #l; ndestruct
-     | #w; #E1'; #E2'; #H1'; #H2'; #H3'; #H4; ndestruct;
-       ncases (?: False); napply (absurd ?? (not_eq_true_false …));
-       /2/
-     | #w; #E1'; #E2'; #H1'; #H2'; #H3'; #H4; ndestruct; /2/
-     | #a; #b; #c; ndestruct
-     | #a; #b; #c; #d; #e; #f; #g; #h; #i; ndestruct]##]
-nqed.     
+(* theorem 16: 1 *)
+alias symbol "pc" (instance 13) = "cat lang".
+alias symbol "in_pl" (instance 23) = "in_pl".
+alias symbol "in_pl" (instance 5) = "in_pl".
+alias symbol "eclose" (instance 21) = "eclose".
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 𝐋 |e|.
+#S e; nelim e; //;
+  ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
+  ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
+  ##| #e1 e2 IH1 IH2;  
+      nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
+      nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
+      nrewrite > (IH1 …); nrewrite > (cup_dotD …);
+      nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
+      nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
+      nrewrite < (erase_dot …); nrewrite < (cupA …); //;
+  ##| #e1 e2 IH1 IH2;
+      nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
+      nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
+      nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
+      nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …); 
+      nrewrite < (erase_plus …); //.
+  ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
+      nchange in match (𝐋\p ?) with  (𝐋\p 〈e'^*,true〉);
+      nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
+      nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+      nrewrite > (erase_bull…e);
+      nrewrite > (erase_star …);
+      nrewrite > (?: 𝐋\p e' =  𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2:
+        nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH; 
+        ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH; 
+            nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
+        ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
+      nrewrite > (cup_dotD…); nrewrite > (cupA…); 
+      nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //;
+      nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##]
+ nqed.
+
+(* theorem 16: 3 *)      
+nlemma odot_dot: 
+  ∀S.∀e1,e2: pre S.  𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
+#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
+
+nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} =  𝐋 e^*.
+#S e; napply extP; #w; nnormalize; @;
+##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2; 
+    *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
+    nrewrite < defw; nrewrite < defw2; @; //; @;//;
+##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
+    #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
+    @; /2/; @xs; /2/;##]
+ nqed.
+
+nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
+#S e; @[]; /2/; nqed.
+
+nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
+#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
+
+nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
+#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
+
+nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
+  ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
+#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
+##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
+nqed.
 
-(*
-ntheorem eclose_superset:
- ∀S,E.
-  ∀w. in_l S w (forget … E) ∨ in_pl ? w E → 
-       let E' ≝ eclose … E in
-       in_pl ? w (snd … E') ∨ fst … E' = true ∧ w = [].
- #S; #E; #w; *
-  [ ngeneralize in match w; nelim E; nnormalize
-     [ #w'; #H; ncases (? : False); /2/
-   ##| #w'; #H; @2; @; //; napply in_l_inv_e; //; (* auto non va *)
-   ##|##3,4: #x; #w'; #H; @1; nrewrite > (in_l_inv_s … H); //;
-   ##| #E1; #E2; #H1; #H2; #w'; #H3;
-       ncases (in_l_inv_c … H3); #w1; *; #w2; *; *; #H4; #H5; #H6;
-       ncases (fst … (eclose S E1)) in H1 H2 ⊢ %; nnormalize
-        [ #H1; #H2; ncases (H1 … H5); ncases (H2 … H6)
-          [ #K1; #K2; nrewrite > H4; /3/;
-        ##| *; #_; #K1; #K2; nrewrite > H4; /3/;
-        ##| #K1; *; #_; #K2; nrewrite > H4; @1; nrewrite > K2;
-            /3/ ]
-   
-    @2; @; //; ninversion H; //;
-##| #H; nwhd; @1; (* manca intro per letin*)
-    (* LEMMA A PARTE? *) (* manca clear E' *)
-    nelim H; nnormalize; /2/
-     [ #w1; #w2; #p; ncases (fst … (eclose S p));
-       nnormalize; /2/
-     | #w; #p; ncases (fst … (eclose S p));
-       nnormalize; /2/ ]
+(* theorem 16: 4 *)      
+nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
+#S p; ncases p; #e b; ncases b;
+##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
+    nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
+    nchange in ⊢ (??%?) with (?∪?);
+    nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+    nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
+      nlapply (bull_cup ? e); #bc;
+      nchange in match (𝐋\p (•e)) in bc with (?∪?);
+      nchange in match b' in bc with b';
+      ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
+      nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
+    nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
+    nrewrite > (sub_dot_star…);
+    nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;    
+##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (cup0…);
+    nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
+    nrewrite < (cup0 ? (𝐋\p e)); //;##]
 nqed.
-*)
 
-nrecord decidable : Type[1] ≝
- { carr :> Type[0];
-   eqb: carr → carr → bool;
-   eqb_true: ∀x,y. eqb x y = true → x=y;
-   eqb_false: ∀x,y. eqb x y = false → x≠y
- }.
+nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝ 
+  match e with 
+  [ z ⇒ pz ?
+  | e ⇒ pe ?
+  | s x ⇒ ps ? x
+  | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
+  | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
+  | k e1 ⇒ pk ? (pre_of_re ? e1)].
+
+nlemma notFalse : ¬False. @; //; nqed.
+
+nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
+#S A; nnormalize; napply extP; #w; @; ##[##2: *]
+*; #w1; *; #w2; *; *; //; nqed.
+
+nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
+#S e; nelim e; ##[##1,2,3: //]
+##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
+    nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
+##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
+    nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
+##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
+    nrewrite > H1; napply dot0; ##]
+nqed.
+
+nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
+#S A; nelim A; //; 
+##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
+    nrewrite < H1; nrewrite < H2; //
+##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
+    nrewrite < H1; nrewrite < H2; //
+##| #e1 H1; nchange in match (𝐋  (e1^* )) with ((𝐋 e1)^* );
+    nrewrite < H1; //]
+nqed.     
 
-nlet rec move (S: decidable) (x:S) (E: pre S) on E ≝
+(* corollary 17 *)
+nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
+#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
+nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
+nqed.
+
+nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
+#S f g H; nrewrite > H; //; nqed.
+ 
+(* corollary 18 *)
+ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
+#S e; @;
+##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
+    nrewrite > defsnde; #H; 
+    nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
+
+STOP
+
+notation > "\move term 90 x term 90 E" 
+non associative with precedence 60 for @{move ? $x $E}.
+nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
  match E with
-  [ pz ⇒ 〈 false, pz ? 〉
-  | pe ⇒ 〈 false, pe ? 〉
-  | ps y ⇒ 〈false, ps ? y 〉
-  | pp y ⇒ 〈 eqb … x y, ps ? y 〉
-  | pc E1 E2 ⇒
-     let E1' ≝ move ? x E1 in
-     let E2' ≝ move ? x E2 in
-     let E1'' ≝ snd … E1' in
-     let E2'' ≝ snd ?? E2' in
-      match fst … E1' with
-       [ true =>
-          let E2''' ≝ eclose S E2'' in
-           〈 fst … E2' ∨ fst … E2''', pc ? E1'' (snd … E2''') 〉
-       | false ⇒ 〈 fst … E2', pc ? E1'' E2'' 〉 ]
-  | po E1 E2 ⇒
-     let E1' ≝ move ? x E1 in
-     let E2' ≝ move ? x E2 in
-      〈 fst … E1' ∨ fst … E2', po ? (snd … E1') (snd … E2') 〉
-  | pk E ⇒
-     let E' ≝ move S x E in
-     let E'' ≝ snd bool (pre S) E' in
-      match fst … E' with
-       [ true ⇒ 〈 true, pk ? (snd … (eclose … E'')) 〉
-       | false ⇒ 〈 false, pk ? E'' 〉 ]].
+  [ pz ⇒ 〈 ∅, false 〉
+  | pe ⇒ 〈 ϵ, false 〉
+  | ps y ⇒ 〈 `y, false 〉
+  | pp y ⇒ 〈 `y, x == y 〉
+  | po e1 e2 ⇒ \move x e1 ⊕ \move x e2 
+  | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
+  | pk e ⇒ (\move x e)^⊛ ].
+notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
+notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
+interpretation "move" 'move x E = (move ? x E).
+
+ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
+interpretation "rmove" 'move x E = (rmove ? x E).
+
+nlemma XXz :  ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+
+nlemma XXe :  ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+nlemma XXze :  ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ)  → False.
+#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
+nqed.
+
+
+naxiom in_move_cat:
+ ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) → 
+   (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
+#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
+ncases e1 in H; ncases e2;
+##[##1: *; ##[*; nnormalize; #; ndestruct] 
+   #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+   nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##|##2: *; ##[*; nnormalize; #; ndestruct] 
+   #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+   nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##| #r; *; ##[ *; nnormalize; #; ndestruct] 
+   #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+   ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
+   nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
+##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
+   #H; ninversion H; nnormalize; #; ndestruct; 
+   ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
+##| #r1 r2; *; ##[ *; #defw]
+    ...
+nqed.
 
-(*
 ntheorem move_ok:
- ∀S:decidable.∀E,a,w.
-  in_pl S w (snd … (move S a E)) → in_pl S (a::w) E.
- #S; #E; #a; #w;
+ ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E. 
+#S E; ncases E; #r b; nelim r;
+##[##1,2: #a w; @; 
+   ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
+      #H; ninversion H; #; ndestruct;
+   ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
+       #H; ninversion H; #; ndestruct;##]
+##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
+   *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
+##|#a c w; @; nnormalize; 
+   ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2;  nrewrite > (eqb_t … ca); @; ##]
+       #H; ninversion H; #; ndestruct;
+   ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
+              #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
+##|#r1 r2 H1 H2 a w; @;
+   ##[ #H; ncases (in_move_cat … H);
+      ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
+          ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good; 
+          nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
+      ##|
+      ...
+##|
+##|
+##]
 nqed.
-*)
 
-nlet rec move_star S w E on w ≝
+
+notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
+nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
  match w with
   [ nil ⇒ E
-  | cons x w' ⇒ move_star S w' (move S x (snd … E))].
+  | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
 
-ndefinition in_moves ≝ λS,w,E. fst … (move_star S w E).
+ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
 
 ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
  mk_equiv:
   ∀E1,E2: bool × (pre S).
-   fst ?? E1  = fst ?? E2 →
-    (∀x. equiv S (move S x (snd … E1)) (move S x (snd … E2))) →
+   \fst E1  = \fst E2 →
+    (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
      equiv S E1 E2.
 
 ndefinition NAT: decidable.
  @ nat eqb; /2/.
 nqed.
 
+include "hints_declaration.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
+
 ninductive unit: Type[0] ≝ I: unit.
 
 nlet corec foo_nop (b: bool):
- equiv NAT
+ equiv ?
   〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
   〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
  @; //; #x; ncases x
@@ -332,32 +665,12 @@ nlet corec foo (a: unit):
 nqed.
 *)
 
-notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
-notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
-interpretation "star" 'pk a = (pk ? a).
-                      
-notation "❨a|b❩" non associative with precedence 90 for @{ 'po $a $b}.
-interpretation "or" 'po a b = (po ? a b).
-           
-notation < "a b" non associative with precedence 60 for @{ 'pc $a $b}.
-notation > "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
-interpretation "cat" 'pc a b = (pc ? a b).
-
-notation < "a" non associative with precedence 90 for @{ 'pp $a}.
-notation > "` term 90 a" non associative with precedence 90 for @{ 'pp $a}.
-interpretation "atom" 'pp a = (pp ? a).
-
-(* to get rid of \middot *)
-ncoercion rex_concat : ∀S:Type[0].∀p:pre S. pre S → pre S  ≝ pc
-on _p : pre ? to ∀_:?.?.
-(* we could also get rid of ` with a coercion from nat → pre nat *) 
+ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
+ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
+ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
 
-ndefinition test1 ≝ ❨ `0 | `1 ❩^* `0.
-ndefinition test2 ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
-ndefinition test3 ≝ (`0 (`0`1)^* `1)^*.
 
-nlemma foo: in_moves NAT
-  [0;0;1;0;1;1] (eclose ? test3) = true.
+nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
  nnormalize in match test3;
  nnormalize;
 //;