X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter5.ma;h=e5d226c32ca78242f30451d964b12b3d3a0e3238;hb=e593e93ca00c6e9dfb8f0e79cb52684c5b104c3f;hp=a3b88978ba5d9c08b33c23bea8c06d1a31432692;hpb=a2ba04cd90e76937720b16e37cb12a39b46181e3;p=helm.git

diff --git a/weblib/tutorial/chapter5.ma b/weblib/tutorial/chapter5.ma
index a3b88978b..e5d226c32 100644
--- a/weblib/tutorial/chapter5.ma
+++ b/weblib/tutorial/chapter5.ma
@@ -1,10 +1,12 @@
-(* The fact of being able to decide, via a computable boolean function, the 
+(* 
+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 "basics/list.ma". 
 include "tutorial/chapter4.ma".
 
 (* The first function we define is an effective version of the membership relation,
@@ -14,8 +16,8 @@ l.*)
 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)"=/aspan class="error" title="Parse error: NUMBER '1' or [term] or [sym=] expected after [sym=] (in [term])"/span= a) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a memb S x tl
-  ].
+  | cons a tl ⇒ (x a 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).
@@ -51,7 +53,7 @@ qed.
 
 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 #l1 elim l1 normalize [#l2 #H %2 //] 
+#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)"=/a=b) 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. 
 
@@ -103,10 +105,11 @@ lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
   ]
 qed.
 
-(* If we are interested in representing finite sets as lists, is is convenient
-to avoid duplications of elements. The following uniqueb check this property. *)
-
-(*************** unicity test *****************)
+(* 
+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. 
+*)
 
 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 
@@ -124,9 +127,19 @@ let rec unique_append (S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0
      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)":/a:r
   ].
 
-axiom 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. 
+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)"=/a=b)) #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. 
+
+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.
 
 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.
@@ -135,7 +148,9 @@ cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a
 #H >H normalize [@Hind //] >H normalize @Hind //
 qed.
 
-(******************* sublist *******************)
+(*
+h2 class="section"Sublists/h2
+*)
 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.
 
@@ -190,9 +205,9 @@ lemma decidable_sublist:∀S,l1,l2.
   ]
 qed.
 
-(********************* filtering *****************)
+(*h2 class="section"Filtering/h2*)
 
-lemma filter_true: ∀S,f,a,l. 
+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
@@ -210,7 +225,7 @@ qed.
   
 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/filter_true.def(5)"filter_true/a/span/span/ qed.
+/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.
 
 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.
@@ -221,7 +236,9 @@ lemma memb_filter_l: ∀S,f,x,l. (f x a title="leibnitz's equality" href="cic:/
   ]
 qed. 
 
-(********************* exists *****************)
+(*
+h2 class="section"Exists/h2
+*)
 
 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