X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter4.ma;h=76065b6dbac07ba30ade03af9332d108d71cb997;hb=cd1ea2cc540bf000db797a497314028ce0fda0fa;hp=00b70a0da09ab6eb691b838bb3733d0d161a1c69;hpb=5fbe7da7019bda8fead167c8b1da1b06625551b3;p=helm.git

diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma
index 00b70a0da..76065b6db 100644
--- a/weblib/tutorial/chapter4.ma
+++ b/weblib/tutorial/chapter4.ma
@@ -72,7 +72,7 @@ interpretation "cat lang" 'concat a b = (cat ? a b).
 definition star_lang ≝ λS.λl.λw:a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.a title="exists" href="cic:/fakeuri.def(1)"∃/alw. a href="cic:/matita/tutorial/chapter4/flatten.fix(0,1,4)"flatten/a ? lw a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a w a title="logical and" href="cic:/fakeuri.def(1)"∧/a a href="cic:/matita/tutorial/chapter4/conjunct.fix(0,1,4)"conjunct/a ? lw l. 
 interpretation "star lang" 'kstar l = (star_lang ? l).
 
-(* notation "ℓ term 70 E" non associative with precedence 75 for @{in_l ? $E}. *)
+(* notation "| term 70 E| " non associative with precedence 75 for @{in_l ? $E}. *)
 
 let rec in_l (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (r : a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"re/a S) on r : a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop ≝ 
 match r with
@@ -84,8 +84,8 @@ match r with
  | star r1 ⇒ (in_l ? r1)a title="star lang" href="cic:/fakeuri.def(1)"^/a* 
  ].
 
-notation "ℓ term 70 E" non associative with precedence 75 for @{'in_l $E}.
-interpretation "in_l" 'in_l E = (in_l ? E).
+notation "\sem{E}" non associative with precedence 75 for @{'sem $E}.
+interpretation "in_l" 'sem 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}.
@@ -121,28 +121,21 @@ interpretation "patom" 'pchar a = (pchar ? a).
 interpretation "pepsilon" 'epsilon = (pepsilon ?).
 interpretation "pempty" 'empty = (pzero ?).
 
-notation "\boxv term 19 e \boxv" (* slash boxv *) non associative with precedence 70 for @{forget ? $e}.
+notation "| e |" non associative with precedence 65 for @{forget ? $e}.
 
 let rec forget (S: a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S) on l: a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"re/a S ≝
  match l with
   [ pzero ⇒ a title="empty" href="cic:/fakeuri.def(1)"∅/a
   | pepsilon ⇒ a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a
   | pchar x ⇒ a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"char/a ? x 
-  | ppoint x ⇒ a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"char/a ? x  
-  | pconcat e1 e2 ⇒ │e1│ a title="cat" href="cic:/fakeuri.def(1)"·/a │e2│
-  | por e1 e2 ⇒ │e1│  a title="or" href="cic:/fakeuri.def(1)"+/a │e2│
-  | pstar e ⇒ │e│a title="star" href="cic:/fakeuri.def(1)"^/a*
+  | ppoint x ⇒ a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"char/a ? x 
+  | pconcat e1 e2 ⇒ |e1| a title="cat" href="cic:/fakeuri.def(1)"·/a |e2| 
+  | por e1 e2 ⇒ |e1|  a title="or" href="cic:/fakeuri.def(1)"+/a |e2|
+  | pstar e ⇒ |e|a title="star" href="cic:/fakeuri.def(1)"^/a*
   ].
 
-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 term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
+notation "| e |" non associative with precedence 65 for @{'fmap $e}.
+interpretation "forget" 'fmap a = (forget ? a).
 
 let rec in_pl (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (r : a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S) on r : a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop ≝ 
 match r with
@@ -150,40 +143,612 @@ match r with
   | pepsilon ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a
   | pchar _ ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a
   | ppoint x ⇒ a title="sing lang" href="cic:/fakeuri.def(1)"{/a: xa title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] } 
-  | pconcat pe1 pe2 ⇒ in_pl ? pe1 a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"ℓ/a │pe2│  a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_pl ? pe2 
+  | pconcat pe1 pe2 ⇒ in_pl ? pe1 a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{|pe2|}  a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_pl ? pe2 
   | por pe1 pe2 ⇒ in_pl ? pe1 a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_pl ? pe2
-  | pstar pe ⇒ in_pl ? pe a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"ℓ/a │pe│a title="star" href="cic:/fakeuri.def(1)"^/a*
+  | pstar pe ⇒ in_pl ? pe a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{|pe|}a title="star lang" href="cic:/fakeuri.def(1)"^/a*
   ].
 
-interpretation "in_pl" 'in_l E = (in_pl ? E).
+interpretation "in_pl" 'sem E = (in_pl ? E).
 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
 
-definition eps: ∀S:a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"bool/a → a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop 
+definition eps: ∀S:a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a → a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop 
   ≝ λS,b. a href="cic:/matita/basics/bool/if_then_else.def(1)"if_then_else/a ? b a title="sing lang" href="cic:/fakeuri.def(1)"{/a: a title="nil" href="cic:/fakeuri.def(1)"[/a] } a title="empty lang" href="cic:/fakeuri.def(1)"∅/a.
 
-(* interpretation "epsilon" 'epsilon = (epsilon ?). *)
 notation "ϵ _ b" non associative with precedence 90 for @{'app_epsilon $b}.
 interpretation "epsilon lang" 'app_epsilon b = (eps ? b).
 
-definition in_prl ≝ λS : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.λp:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S.  a title="in_pl" href="cic:/fakeuri.def(1)"ℓ/a (a title="fst" href="cic:/fakeuri.def(1)"\fst/a p) a title="union lang" href="cic:/fakeuri.def(1)"∪/a a title="epsilon lang" href="cic:/fakeuri.def(1)"ϵ/a_(a title="snd" href="cic:/fakeuri.def(1)"\snd/a p).
+definition in_prl ≝ λS : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.λp:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S.  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p} a title="union lang" href="cic:/fakeuri.def(1)"∪/a a title="epsilon lang" href="cic:/fakeuri.def(1)"ϵ/a_(a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p).
   
 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
-interpretation "in_prl" 'in_l E = (in_prl ? E).
-
-lemma not_epsilon_lp :∀S.∀pi:a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S. a title="logical not" href="cic:/fakeuri.def(1)"¬/a ((a title="in_pl" href="cic:/fakeuri.def(1)"ℓ/a pi) a title="nil" href="cic:/fakeuri.def(1)"[/a]).
-#S #pi (elim pi) normalize /2/
-  [#pi1 #pi2 #H1 #H2 % * /2/ * #w1 * #w2 * * #appnil 
-   cases (a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"nil_to_nil/a … appnil) /2/
-  |#p11 #p12 #H1 #H2 % * /2/
-  |#pi #H % * #w1 * #w2 * * #appnil (cases (a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"nil_to_nil/a … appnil)) /2/
+interpretation "in_prl" 'sem E = (in_prl ? E).
+
+lemma not_epsilon_lp :∀S.∀pi:a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S.a title="logical not" href="cic:/fakeuri.def(1)"\neg/a(a title="nil" href="cic:/fakeuri.def(1)"[/a] a title="in_pl mem" href="cic:/fakeuri.def(1)"∈/a pi).
+#S #pi (elim pi) normalize /  span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a/span/span/
+  [#pi1 #pi2 #H1 #H2 % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ * #w1 * #w2 * * #appnil 
+   cases (a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"nil_to_nil/a … appnil) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+  |#p11 #p12 #H1 #H2 % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+  |#pi #H % * #w1 * #w2 * * #appnil (cases (a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"nil_to_nil/a … appnil)) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
   ]
 qed.
 
-lemma if_true_epsilon: ∀S.∀e:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="snd" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → ((a title="in_prl" href="cic:/fakeuri.def(1)"ℓ/a e) a title="nil" href="cic:/fakeuri.def(1)"[/a]).
+lemma if_true_epsilon: ∀S.∀e:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → (a title="nil" href="cic:/fakeuri.def(1)"[/a] a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e).
 #S #e #H %2 >H // qed.
 
-lemma if_epsilon_true : ∀S.∀e:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="nil" href="cic:/fakeuri.def(1)"[/a ] a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e → a title="snd" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
-#S * #pi #b * [#abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/] cases b normalize // @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a
+lemma if_epsilon_true : ∀S.∀e:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="nil" href="cic:/fakeuri.def(1)"[/a ] a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e → a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+#S * #pi #b * [normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/] cases b normalize // @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a
 qed.
 
+definition lor ≝ λS:a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.λa,b:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a a a title="por" href="cic:/fakeuri.def(1)"+/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a b,a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a a a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a b〉.
+
+notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
+interpretation "oplus" 'oplus a b = (lor ? a b).
+
+definition item_concat: ∀S:a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S → a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S → a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S ≝
+ λS,i,e.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai a title="pcat" href="cic:/fakeuri.def(1)"·/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e, a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a e〉.
+
+definition lcat: ∀S:Alpha.∀bcast:(∀S:Alpha.pre S →pre S).pre S → pre S → pre S
+ ≝ λS:Alpha.λbcast:∀S:Alpha.pre S → pre S.λe1,e2:pre S.
+  match e1 with [ mk_pair i1 b1 ⇒ e2]. 
+
+  match e1 with [ _ => e1].
+
+ match b1 with 
+   [ false ⇒ e2 | _ => e1 ]].
 
+ | true ⇒ item_concat S i1 (bcast S e2)
+   ]
+].
+   
+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.
+
+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.
+
+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.
+
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed.
+
+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.
+
+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.
+
+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.
+
+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.
+
+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.
+
+(* 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.
+
+(* 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.
+
+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.     
+
+(* 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 〉
+  | 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: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.
+
+
+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' ⇒ w' ↦* (x ↦ \snd 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 (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 ?
+  〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
+  〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
+ @; //; #x; ncases x
+  [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+  | #y; ncases y
+     [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+     | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
+nqed.
+
+(*
+nlet corec foo (a: unit):
+ equiv NAT
+  (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
+  (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
+≝ ?.
+ @;
+  ##[ nnormalize; //
+  ##| #x; ncases x
+       [ nnormalize in ⊢ (??%%);
+         nnormalize in foo: (? → ??%%);
+         @; //; #y; ncases y
+           [ nnormalize in ⊢ (??%%); napply foo_nop
+           | #y; ncases y
+              [ nnormalize in ⊢ (??%%);
+                
+            ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
+     ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
+  ##]
+nqed.
+*)
+
+ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
+ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
+ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
+
+
+nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
+ nnormalize in match test3;
+ nnormalize;
+//;
+nqed.
+
+(**********************************************************)
+
+ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
+   der_z: der S a (z S) (z S)
+ | der_e: der S a (e S) (z S)
+ | der_s1: der S a (s S a) (e ?)
+ | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
+ | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
+            der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
+ | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
+            der S a (c ? e1 e2) (c ? e1' e2)
+ | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
+    der S a (o ? e1 e2) (o ? e1' e2').
+
+nlemma eq_rect_CProp0_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+nqed.
+
+nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
+
+naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
+(* #S; #r1; #r2; #w; nelim r1
+  [ #K; ninversion K
+  | #H1; #H2; napply (in_c ? []); //
+  | (* tutti casi assurdi *) *)
+
+ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
+   in_l_empty1: ∀E.in_l S [] E → in_l' S [] E 
+ | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
+
+ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
+   mk_eq_re: ∀E1,E2.
+    (in_l S [] E1 → in_l S [] E2) →
+    (in_l S [] E2 → in_l S [] E1) →
+    (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
+      eq_re S E1 E2.
+
+(* serve il lemma dopo? *)
+ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
+ #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
+  [ #r; #K (* ok *)
+  | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
+
+(* IL VICEVERSA NON VALE *)
+naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
+(* #S; #w; #E; #H; nelim H
+  [ //
+  | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
+  ]
+nqed. *)
+
+ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
+ #S; #a; #E; #E'; #w; #H; nelim H
+  [##1,2: #H1; ninversion H1
+     [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
+     |##2,9: #X; #Y; #K; ncases (?:False); /2/
+     |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+     |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     |##6,13: #x; #y; #K; ncases (?:False); /2/
+     |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
+##| #H1; ninversion H1
+     [ //
+     | #X; #Y; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     | #x; #y; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #H1; #H2; #H3; ninversion H3
+     [ #_; #K; ncases (?:False); /2/
+     | #X; #Y; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+     | #x; #y; #K; ncases (?:False); /2/
+     | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;
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