X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter5.ma;h=f4594bc3b019db9e0b2cd641edafebeb01596e0c;hb=d9a1ff8259a7882caa0ffd27282838c00a34cab5;hp=3f58c2c4838344070027c766ecf58744fa39b15b;hpb=05bcc10ec41117f33d2eb6a120df4024ae14b185;p=helm.git

diff --git a/weblib/tutorial/chapter5.ma b/weblib/tutorial/chapter5.ma
index 3f58c2c48..f4594bc3b 100644
--- a/weblib/tutorial/chapter5.ma
+++ b/weblib/tutorial/chapter5.ma
@@ -1,437 +1,247 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "re/lang.ma".
-
-inductive re (S: DeqSet) : 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.
-
-interpretation "re epsilon" 'epsilon = (e ?).
-interpretation "re or" 'plus a b = (o ? a b).
-interpretation "re cat" 'middot a b = (c ? a b).
-interpretation "re star" 'star a = (k ? a).
-
-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 "`∅" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty" 'empty = (z ?).
-
-let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝ 
-match r with
-[ z ⇒ ∅
-| e ⇒ {ϵ}
-| s x ⇒ {[x]}
-| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
-| o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
-| k r1 ⇒ (in_l ? r1) ^*].
-
-notation "\sem{term 19 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).
-
-lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
-// qed.
-
-
-(* pointed items *)
-inductive pitem (S: DeqSet) : 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.
- 
-definition pre ≝ λS.pitem S × bool.
-
-interpretation "pitem star" 'star a = (pk ? a).
-interpretation "pitem or" 'plus a b = (po ? a b).
-interpretation "pitem cat" 'middot 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 "pitem pp" 'pp a = (pp ? a).
-interpretation "pitem ps" 'ps a = (ps ? a).
-interpretation "pitem epsilon" 'epsilon = (pe ?).
-interpretation "pitem empty" 'empty = (pz ?).
-
-let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
- match l with
-  [ pz ⇒ `∅
-  | pe ⇒ ϵ
-  | ps x ⇒ `x
-  | pp x ⇒ `x
-  | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
-  | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
-  | pk E ⇒ (forget ? E)^* ].
-  
-(* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
-interpretation "forget" 'norm a = (forget ? a).
-
-let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝ 
-match r with
-[ pz ⇒ ∅
-| pe ⇒ ∅
-| ps _ ⇒ ∅
-| pp x ⇒ { [x] }
-| pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
-| po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
-| pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^*  ].
-
-interpretation "in_pl" 'in_l E = (in_pl ? E).
-interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
-
-definition in_prl ≝ λS : DeqSet.λp:pre S. 
-  if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
-  
-interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
-interpretation "in_prl" 'in_l E = (in_prl ? E).
-
-lemma sem_pre_true : ∀S.∀i:pitem S. 
-  \sem{〈i,true〉} = \sem{i} ∪ {ϵ}. 
-// qed.
-
-lemma sem_pre_false : ∀S.∀i:pitem S. 
-  \sem{〈i,false〉} = \sem{i}. 
-// qed.
-
-lemma sem_cat: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
-// qed.
-
-lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
-  \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
-// qed.
-
-lemma sem_plus: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
-// qed.
-
-lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w. 
-  \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
-// qed.
-
-lemma sem_star : ∀S.∀i:pitem S.
-  \sem{i^*} = \sem{i} · \sem{|i|}^*.
-// qed.
-
-lemma sem_star_w : ∀S.∀i:pitem S.∀w.
-  \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
-// qed.
-
-lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
-#S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
-
-lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
-#S #e elim e normalize /2/  
-  [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H 
-   >(append_eq_nil …H…) /2/
-  |#r1 #r2 #n1 #n2 % * /2/
-  |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
+(* 
+h1 class="section"Effective searching/h1
+The fact of being able to decide, via a computable boolean function, the 
+equality between elements of a given set is an essential prerequisite for 
+effectively searching an element of that set inside a data structure. In this 
+section we shall define several boolean functions acting on lists of elements in 
+a DeqSet, and prove some of their properties.*)
+
+include "basics/list.ma". 
+include "tutorial/chapter4.ma".
+
+(* The first function we define is an effective version of the membership relation,
+between an element x and a list l. Its definition is a straightforward recursion on
+l.*)
+
+img class="anchor" src="icons/tick.png" id="memb"let rec memb (S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (x:S) (l: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/aspan class="error" title="Parse error: RPAREN expected after [term] (in [arg])"/span S) on l  ≝
+  match l with
+  [ nil ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a
+  | cons a tl ⇒ (x a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a memb S x tl
+  ]span class="error" title="Parse error: NUMBER '1' or [term] or [sym=] expected after [sym=] (in [term])"/spanspan class="error" title="No choices for ID nil"/span.
+
+notation < "\memb x l" non associative with precedence 90 for @{'memb $x $l}.
+interpretation "boolean membership" 'memb a l = (memb ? a l).
+
+(* We can now prove several interesing properties for memb:
+- memb_hd: x is a member of x::l
+- memb_cons: if x is a member of l than x is a member of a::l
+- memb_single: if x is a member of [a] then x=a
+- memb_append: if x is a member of l1@l2 then either x is a member of l1
+  or x is a member of l2
+- memb_append_l1: if x is a member of l1 then x is a member of l1@l2
+- memb_append_l2: if x is a member of l2 then x is a member of l1@l2
+- memb_exists: if x is a member of l, than l can decomposed as l1@(x::l2)
+- not_memb_to_not_eq: if x is not a member of l and y is, then x≠y
+- memb_map: if a is a member of l, then (f a) is a member of (map f l)
+- memb_compose: if a is a member of l1 and b is a meber of l2 than
+  (op a b) is a member of (compose op l1 l2)
+*)
+
+img class="anchor" src="icons/tick.png" id="memb_hd"lemma memb_hd: ∀S,a,l. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al) 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 #a #l normalize >(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (a href="cic:/matita/tutorial/chapter4/eqb_true.fix(0,0,4)"eqb_true/a S …) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a S a)) //
+qed.
+
+img class="anchor" src="icons/tick.png" id="memb_cons"lemma memb_cons: ∀S,a,b,l. 
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/aspan class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"/span S a (ba title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al) 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 #a #b #l normalize cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab) normalize // 
+qed.
+
+img class="anchor" src="icons/tick.png" id="memb_single"lemma memb_single: ∀S,a,x. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (xa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a) 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 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x.
+#S #a #x normalize cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax)) #H
+  [#_ >(\P H) // |>H 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/]
+qed.
+
+img class="anchor" src="icons/tick.png" id="memb_append"lemma memb_append: ∀S,a,l1,l2. 
+a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (l1a title="append" href="cic:/fakeuri.def(1)"@/aspan class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"/spanl2) 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l1a 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="logical or" href="cic:/fakeuri.def(1)"∨/a a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l2 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 #a #l1span class="error" title="Parse error: illegal begin of statement"/spanspan class="error" title="Parse error: illegal begin of statement"/span elim l1 normalize [#l2 #H %2 //] 
+#b #tl #Hind #l2 cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab) normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a/span/span/ 
+qed. 
+
+img class="anchor" src="icons/tick.png" id="memb_append_l1"lemma memb_append_l1: ∀S,a,l1,l2. 
+ a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (l1a title="append" href="cic:/fakeuri.def(1)"@/al2) 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 #a #l1 elim l1 normalize
+  [normalize #le #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/
+  |#b #tl #Hind #l2 cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab) normalize /span class="autotactic"2span class="autotrace" trace /span/span/ 
+  ]
+qed. 
+
+img class="anchor" src="icons/tick.png" id="memb_append_l2"lemma memb_append_l2: ∀S,a,l1,l2. 
+ a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l2a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (l1a title="append" href="cic:/fakeuri.def(1)"@/al2) 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 #a #l1 elim l1 normalize //
+#b #tl #Hind #l2 cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab) normalize /span class="autotactic"2span class="autotrace" trace /span/span/ 
+qed. 
+
+img class="anchor" src="icons/tick.png" id="memb_exists"lemma memb_exists: ∀S,a,l.a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aspan class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"/span → a title="exists" href="cic:/fakeuri.def(1)"∃/al1,l2a title="exists" href="cic:/fakeuri.def(1)"./ala title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/al1a title="append" href="cic:/fakeuri.def(1)"@/a(aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al2).
+#S #a #l elim l [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/]
+#b #tl #Hind #H cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … H)
+  [#eqba @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … (a href="cic:/matita/basics/list/list.con(0,1,1)"nil/a S)) @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … tl) >(\P eqba) //
+  |#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
+   @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … (ba title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al1)) @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … l2) >eqtl //
   ]
 qed.
 
-(* lemma 12 *)
-lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
-#S * #i #b cases b // normalize #H @False_ind /2/ 
-qed.
-
-lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
-#S * #i #b #btrue normalize in btrue; >btrue %2 // 
-qed.
-
-definition lo ≝ λS:DeqSet.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).
-
-lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
-// qed.
-
-definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
-  match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
+img class="anchor" src="icons/tick.png" id="not_memb_to_not_eq"lemma not_memb_to_not_eq: ∀S,a,b,l. 
+ a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a → a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S b l a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a.
+#S #a #b #l cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab)) // 
+#eqab >(\P eqab) #H >H #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/
+qed. 
  
-notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}.
-interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
-
-lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
-  A = B → A =1 B. 
-#S #A #B #H >H /2/ qed.
-
-lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
-  \sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
-#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
->sem_pre_true >sem_cat >sem_pre_true /2/ 
+img class="anchor" src="icons/tick.png" id="memb_map"lemma memb_map: ∀S1,S2,f,a,l. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S1 a la title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → 
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S2 (f a) (a href="cic:/matita/basics/list/map.fix(0,3,1)"map/a … f l) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+#S1 #S2 #f #a #l elim l normalize [//]
+#x #tl #memba cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax))
+  [#eqx >eqx >(\P eqx) >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … (f x))) normalize //
+  |#eqx >eqx cases (f aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/af x) normalize /span class="autotactic"2span class="autotrace" trace /span/span/
+  ]
 qed.
- 
-definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
-  match e1 with 
-  [ mk_Prod i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◂ (bcast ? i2)) 
-    | false ⇒ 〈i1 · i2,false〉
-    ]
-  ].
-        
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
-  match e with 
-  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
-
-notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
-interpretation "lc" 'lc op a b = (lc ? op a b).
-
-definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S.
-  match e with 
-  [ mk_Prod i1 b1 ⇒
-    match b1 with 
-    [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
-    |false ⇒ 〈i1^*,false〉
-    ]
-  ]. 
-
-(* 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 ?}.
-
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
- match i with
-  [ pz ⇒ 〈 `∅, false 〉
-  | pe ⇒ 〈 ϵ,  true 〉
-  | ps x ⇒ 〈 `.x, false〉
-  | pp x ⇒ 〈 `.x, false 〉
-  | po i1 i2 ⇒ •i1 ⊕ •i2
-  | pc i1 i2 ⇒ •i1 ▸ i2
-  | pk i ⇒ 〈(\fst (•i))^*,true〉].
-  
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
-interpretation "eclose" 'eclose x = (eclose ? x).
-
-lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 + i2) = •i1 ⊕ •i2.
-// qed.
 
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▸ i2.
-// qed.
-
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
-  •i^* = 〈(\fst(•i))^*,true〉.
-// qed.
-
-definition reclose ≝ λS. lift S (eclose S). 
-interpretation "reclose" 'eclose x = (reclose ? x).
-
-(* theorem 16: 2 *)
-lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
-  \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. 
-#S * #i1 #b1 * #i2 #b2 #w %
-  [cases b1 cases b2 normalize /2/ * /3/ * /3/
-  |cases b1 cases b2 normalize /2/ * /3/ * /3/
+img class="anchor" src="icons/tick.png" id="memb_compose"lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.   
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S1 a1 l1 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S2 a2 l2 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S3 (op a1 a2) (a href="cic:/matita/basics/list/compose.def(2)"compose/a S1 S2 S3 op l1 l2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+#S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
+#x #tl #Hind #l2 #memba1 #memba2 cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … memba1)
+  [#eqa1 >(\P eqa1) @a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"memb_append_l1/a @a href="cic:/matita/tutorial/chapter5/memb_map.def(5)"memb_map/a // 
+  |#membtl @a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"memb_append_l2/a @Hind //
   ]
 qed.
 
-lemma odot_true : 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = i1 ◂ (•i2).
-// qed.
-
-lemma odot_true_bis : 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
-#S #i1 #i2 normalize cases (•i2) // qed.
-
-lemma odot_false: 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,false〉 ▸ i2 = 〈i1 · i2, false〉.
-// qed.
+(* 
+h2 class="section"Unicity/h2
+If we are interested in representing finite sets as lists, is is convenient
+to avoid duplications of elements. The following uniqueb check this property. 
+*)
 
-lemma LcatE : ∀S.∀e1,e2:pitem S.
-  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
-// qed.
-
-lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
-// qed.
-
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
-  |i1 + i2| = |i1| + |i2|.
-// qed.
-
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
-// qed.
-
-lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
-#S #i elim i // 
-  [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
-    cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
-  | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
-    cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //  
-  | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
-  ]
-qed.
-  
-lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
-/2/ qed.
+img class="anchor" src="icons/tick.png" id="uniqueb"let rec uniqueb (S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) l on l : a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a ≝
+  match l with 
+  [ nil ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a
+  | cons a tl ⇒ a href="cic:/matita/basics/bool/notb.def(1)"notb/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a tl) a title="boolean and" href="cic:/fakeuri.def(1)"∧/a uniqueb S tl
+  ].
 
-(* theorem 16: 1 → 3 *)
-lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
-   \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
-   \sem{e1 ▸ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
-#S * #i1 #b1 #i2 cases b1
-  [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
-  |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
-   >erase_bull @eqP_trans [|@(eqP_union_l … H)]
-    @eqP_trans [|@eqP_union_l[|@union_comm ]]
-    @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
-  ]
-qed.
+(* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
 
-lemma sem_fst: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
-#S * #i * 
-  [>sem_pre_true normalize in ⊢ (??%?); #w % 
-    [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
-  |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
-  ]
-qed.
+img class="anchor" src="icons/tick.png" id="unique_append"let rec unique_append (S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (l1,l2: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a S) on l1 ≝
+  match l1 with
+  [ nil ⇒ l2
+  | cons a tl ⇒ 
+     let r ≝ unique_append S tl l2 in
+     if a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a r then r else aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/ar
+  ].
 
-lemma item_eps: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
-#S #i #w % 
-  [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
-  |* //
+img class="anchor" src="icons/tick.png" id="memb_unique_append"lemma memb_unique_append: ∀S,a,l1,l2. 
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S l1 l2) 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l1a 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="logical or" href="cic:/fakeuri.def(1)"∨/a a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l2 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 #a #l1 elim l1 normalize [#l2 #H %2 //] 
+#b #tl #Hind #l2 cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab)) #Hab >Hab normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a/span/span/
+cases (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S b (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S tl l2)) normalize 
+  [@Hind | >Hab normalize @Hind]   
+qed. 
+
+img class="anchor" src="icons/tick.png" id="unique_append_elim"lemma unique_append_elim: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀P: S → Prop.∀l1,l2. 
+  (∀x. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x l1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aspan class="error" title="Parse error: NUMBER '1' or [term] or [sym=] expected after [sym=] (in [term])"/span a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → P x) → (∀x. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x l2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → P x) →
+    ∀x. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S l1 l2) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → P x. 
+#S #P #l1 #l2 #Hl1 #Hl2 #x #membx cases (a href="cic:/matita/tutorial/chapter5/memb_unique_append.def(6)"memb_unique_append/aspan class="error" title="No choices for ID memb_unique_append"/span … membx) /span class="autotactic"2span class="autotrace" trace /span/span/ 
+qed.
+
+img class="anchor" src="icons/tick.png" id="unique_append_unique"lemma unique_append_unique: ∀S,l1,l2. a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"uniqueb/a S l2 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 href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"uniqueb/a S (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S l1 l2) 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 #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
+cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S tl l2))) 
+#H >H normalize [@Hind //] >H normalize @Hind //
+qed.
+
+(*
+h2 class="section"Sublists/h2
+*)
+img class="anchor" src="icons/tick.png" id="sublist"definition sublist ≝ 
+  λS,l1,l2.∀a. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l1 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+
+img class="anchor" src="icons/tick.png" id="sublist_length"lemma sublist_length: ∀S,l1,l2. 
+ a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"uniqueb/a S l1 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 href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S l1 l2 → a title="norm" href="cic:/fakeuri.def(1)"|/al1a title="norm" href="cic:/fakeuri.def(1)"|/a a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"≤/a a title="norm" href="cic:/fakeuri.def(1)"|/al2a title="norm" href="cic:/fakeuri.def(1)"|/a.
+#S #l1 elim l1 // 
+#a #tl #Hind #l2 #unique #sub
+cut (a title="exists" href="cic:/fakeuri.def(1)"∃/al3,l4a title="exists" href="cic:/fakeuri.def(1)"./al2a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/al3a title="append" href="cic:/fakeuri.def(1)"@/a(aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/al4)) [@a href="cic:/matita/tutorial/chapter5/memb_exists.def(5)"memb_exists/a @sub //]
+* #l3 * #l4 #eql2 >eql2 >a href="cic:/matita/basics/list/length_append.def(2)"length_append/a normalize 
+applyS a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"le_S_S/a <a href="cic:/matita/basics/list/length_append.def(2)"length_append/a @Hind [@(a href="cic:/matita/basics/bool/andb_true_r.def(4)"andb_true_r/a … unique)]
+>eql2 in sub; #sub #x #membx 
+cases (a href="cic:/matita/tutorial/chapter5/memb_append.def(5)"memb_append/a … (sub x (a href="cic:/matita/basics/bool/orb_true_r2.def(3)"orb_true_r2/a … membx)))
+  [#membxl3 @a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"memb_append_l1/a //
+  |#membxal4 cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … membxal4)
+    [#eqxa @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a lapply (a href="cic:/matita/basics/bool/andb_true_l.def(4)"andb_true_l/a … unique)
+     <(\P eqxa) >membx normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ |#membxl4 @a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"memb_append_l2/a //
+    ]
   ]
 qed.
-  
-lemma sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
- \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
-#S #e #i #A #seme
-@eqP_trans [|@sem_fst]
-@eqP_trans [||@eqP_union_r [|@eqP_sym @item_eps]]
-@eqP_trans [||@distribute_substract] 
-@eqP_substract_r //
-qed.
 
-(* theorem 16: 1 *)
-theorem sem_bull: ∀S:DeqSet. ∀e:pitem S.  \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
-#S #e elim e 
-  [#w normalize % [/2/ | * //]
-  |/2/ 
-  |#x normalize #w % [ /2/ | * [@False_ind | //]]
-  |#x normalize #w % [ /2/ | * // ] 
-  |#i1 #i2 #IH1 #IH2 >eclose_dot
-   @eqP_trans [|@odot_dot_aux //] >sem_cat 
-   @eqP_trans
-     [|@eqP_union_r
-       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
-   @eqP_trans [|@union_assoc]
-   @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l //
-  |#i1 #i2 #IH1 #IH2 >eclose_plus
-   @eqP_trans [|@sem_oplus] >sem_plus >erase_plus 
-   @eqP_trans [|@(eqP_union_l … IH2)]
-   @eqP_trans [|@eqP_sym @union_assoc]
-   @eqP_trans [||@union_assoc] @eqP_union_r
-   @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_trans [||@eqP_union_l [|@union_comm]]
-   @eqP_trans [||@union_assoc] /2/
-  |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
-   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@sem_fst_aux //]]]
-   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
-   @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
-   @eqP_sym @star_fix_eps 
+img class="anchor" src="icons/tick.png" id="sublist_unique_append_l1"lemma sublist_unique_append_l1: 
+  ∀S,l1,l2. a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S l1 (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S l1 l2).
+#S #l1 elim l1 normalize [#l2 #S #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/]
+#x #tl #Hind #l2 #a 
+normalize cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax)) #eqax >eqax 
+[<(\P eqax) cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S tl l2)))
+  [#H >H normalize // | #H >H normalize >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … a)) //]
+|cases (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S tl l2)) normalize 
+  [/span class="autotactic"2span class="autotrace" trace /span/span/ |>eqax normalize /span class="autotactic"2span class="autotrace" trace /span/span/]
+]
+qed.
+
+img class="anchor" src="icons/tick.png" id="sublist_unique_append_l2"lemma sublist_unique_append_l2: 
+  ∀S,l1,l2. a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S l2 (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S l1 l2).
+#S #l1 elim l1 [normalize //] #x #tl #Hind normalize 
+#l2 #a cases (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a S tl l2)) normalize
+[@Hind | cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ax) normalize // @Hind]
+qed.
+
+img class="anchor" src="icons/tick.png" id="decidable_sublist"lemma decidable_sublist:∀S,l1,l2. 
+  (a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S l1 l2) a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="logical not" href="cic:/fakeuri.def(1)"¬/a(a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S l1 l2).
+#S #l1 #l2 elim l1 
+  [%1 #a normalize in ⊢ (%→?); #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/
+  |#a #tl * #subtl 
+    [cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l2)) #memba
+      [%1 whd #x #membx cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … membx)
+        [#eqax >(\P eqax) // |@subtl]
+      |%2 @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … (a href="cic:/matita/basics/bool/eqnot_to_noteq.def(4)"eqnot_to_noteq/a … a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a memba)) #H1 @H1 @a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"memb_hd/a
+      ]
+    |%2 @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … subtl) #H1 #x #H2 @H1 @a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"memb_cons/a //
+    ] 
   ]
 qed.
 
-definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
-  lift S (lc S eclose e).
-
-notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
+(*h2 class="section"Filtering/h2*)
 
-interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-
-lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b. 
-  〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
-#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) // 
-qed.
-
-lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
-  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
-// 
-qed.
+img class="anchor" src="icons/tick.png" id="memb_filter_true"lemma memb_filter_true: ∀S,f,a,l. 
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/basics/list/filter.def(2)"filter/a S f l) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → f a 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 #f #a #l elim l [normalize #H @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/]
+#b #tl #Hind cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (f b)) #H
+normalize >H normalize [2:@Hind]
+cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab)) #eqab
+  [#_ >(\P eqab) // | >eqab normalize @Hind]
+qed. 
   
-lemma erase_odot:∀S.∀e1,e2:pre S.
-  |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
-#S * #i1 * * #i2 #b2 // >odot_true_b // 
+img class="anchor" src="icons/tick.png" id="memb_filter_memb"lemma memb_filter_memb: ∀S,f,a,l. 
+  a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a (a href="cic:/matita/basics/list/filter.def(2)"filter/a S f l) 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S a l 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 #f #a #l elim l [normalize //]
+#b #tl #Hind normalize (cases (f b)) normalize 
+cases (aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab) normalize // @Hind
 qed.
-
-lemma ostar_true: ∀S.∀i:pitem S.
-  〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
-// qed.
-
-lemma ostar_false: ∀S.∀i:pitem S.
-  〈i,false〉^⊛ = 〈i^*, false〉.
-// qed.
   
-lemma erase_ostar: ∀S.∀e:pre S.
-  |\fst (e^⊛)| = |\fst e|^*.
-#S * #i * // qed.
-
-lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
-  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }.
-#S #e1 #i 
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//]
-#H >H cases (e1 ▸ i) #i1 #b1 cases b1 
-  [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l /2/ 
-  |/2/
+img class="anchor" src="icons/tick.png" id="memb_filter"lemma memb_filter: ∀S,f,l,x. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? f l) 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x l 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="logical and" href="cic:/fakeuri.def(1)"∧/a (f x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a).
+/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a, a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"memb_filter_memb/a, a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"memb_filter_true/a/span/span/ qed.
+
+img class="anchor" src="icons/tick.png" id="memb_filter_l"lemma memb_filter_l: ∀S,f,x,l. (f x 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x l 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S x (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? f l) 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 #f #x #l #fx elim l normalize //
+#b #tl #Hind cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (xa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ab)) #eqxb
+  [<(\P eqxb) >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … x)) >fx normalize >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … x)) normalize //
+  |>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
   ]
-qed.
-
-lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
-  e1 ⊙ 〈i,false〉 = e1 ▸ i.
-#S #e1 #i  
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ false〉) [//]
-cases (e1 ▸ i) #i1 #b1 cases b1 #H @H
-qed.
+qed. 
 
-lemma sem_odot: 
-  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
-#S #e1 * #i2 * 
-  [>sem_pre_true 
-   @eqP_trans [|@sem_odot_true]
-   @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
-  |>sem_pre_false >eq_odot_false @odot_dot_aux //
-  ]
-qed.
+(*
+h2 class="section"Exists/h2
+*)
 
-(* theorem 16: 4 *)      
-theorem sem_ostar: ∀S.∀e:pre S. 
-  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
-#S * #i #b cases b
-  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
-   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]]
-   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
-   @eqP_trans [||@eqP_sym @distr_cat_r]
-   @eqP_trans [|@union_assoc] @eqP_union_l
-   @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps 
-  |>sem_pre_false >sem_pre_false >sem_star /2/
-  ]
-qed.
-  
+img class="anchor" src="icons/tick.png" id="exists"let rec exists (A:Type[0]) (p:A → a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a) (l:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a A) on l : a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a ≝
+match l with
+[ nil ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a
+| cons h t ⇒ a href="cic:/matita/basics/bool/orb.def(1)"orb/a (p h) (exists A p t)
+].
\ No newline at end of file