commit by user andrea

diff --git a/weblib/basics/pts.ma b/weblib/basics/pts.ma
index 1b87a2d0b18e1dc2d5e16d228f4a9d5bf78fc79e..b7d9a5da9deb592e73670182de1459befd569321 100644 (file)
--- a/weblib/basics/pts.ma
+++ b/weblib/basics/pts.ma
@@ -19,23 +19,3 @@ universe constraint Type[1] < Type[2].
 universe constraint Type[2] < Type[3].
 universe constraint Type[3] < Type[4].
 
-(*inductive True : Prop ≝ I : True.
-
-(*lemma fa : ∀X:Prop.X → X.
-#X #p //
-qed.
-
-(* check fa*)
-
-lemma ggr ≝ fa.*)
-
-inductive False : Prop ≝ .
-
-inductive bool : Prop ≝ True : bool | false : bool.
- 
-inductive eq (A:Type[1]) (x:A) : A → Prop ≝
-    refl: eq A x x.
-
-lemma provable_True : <A href="cic:/matita/basics/pts/True.ind(1,0,0)">True</A> → eq Prop True True.
-#H %
-qed.*)
\ No newline at end of file

diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma
index cf7f4aafce387300d1191b7300a5b7bcc4ca5d4c..6cb68782b99077fa38cb47f0ad02c9362981189b 100644 (file)
--- 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:\ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6lw. \ 5a href="cic:/matita/tutorial/chapter4/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 ? lw \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/conjunct.fix(0,1,4)"\ 6conjunct\ 5/a\ 6 ? 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 : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝ 
 match r with
@@ -84,7 +84,7 @@ match r with
  | star r1 ⇒ (in_l ? r1)\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* 
  ].
 
-notation "ℓ term 70 E" non associative with precedence 75 for @{'in_l $E}.
+notation "\sem{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).
 
@@ -121,69 +121,71 @@ 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: \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on l: \ 5a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S ≝
  match l with
   [ pzero ⇒ \ 5a title="empty" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
   | pepsilon ⇒ \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6
   | pchar x ⇒ \ 5a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"\ 6char\ 5/a\ 6 ? x 
-  | ppoint x ⇒ \ 5a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"\ 6char\ 5/a\ 6 ? x  
-  | pconcat e1 e2 ⇒ │e1│ \ 5a title="cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 │e2│
-  | por e1 e2 ⇒ │e1│  \ 5a title="or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 │e2│
-  | pstar e ⇒ │e│\ 5a title="star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
+  | ppoint x ⇒ \ 5a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"\ 6char\ 5/a\ 6 ? x 
+  | pconcat e1 e2 ⇒ |e1| \ 5a title="cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 |e2| 
+  | por e1 e2 ⇒ |e1|  \ 5a title="or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 |e2|
+  | pstar e ⇒ |e|\ 5a title="star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
   ].
 
-notation "│ term 19 e │" non associative with precedence 70 for @{'forget $e}.
-interpretation "forget" 'forget a = (forget ? a).
+notation "| e |" non associative with precedence 65 for @{'fmap $e}.
+interpretation "forget" 'fmap a = (forget ? a).
 
+lemma foo:∀a. 'fst a = 0. 
+ 
 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 "\sem{E}" non associative with precedence 75 for @{in_pl ? $E}. *)
 
-let rec in_pl (S : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝ 
+let rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ 
 match r with
-  [ pzero ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6 
-  | pepsilon ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
-  | pchar _ ⇒ \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
-  | ppoint x ⇒ \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] } 
-  | pconcat pe1 pe2 ⇒ in_pl ? pe1 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 │pe2│  \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 in_pl ? pe2 
-  | por pe1 pe2 ⇒ in_pl ? pe1 \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 in_pl ? pe2
-  | pstar pe ⇒ in_pl ? pe \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 │pe│\ 5a title="star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*
+  [ pzero ⇒ ∅ 
+  | pepsilon ⇒ ∅
+  | pchar _ ⇒ ∅
+  | ppoint x ⇒ {: x::[] } 
+  | pconcat pe1 pe2 ⇒ in_pl ? pe1 · \sem{|pe2|}  ∪ in_pl ? pe2 
+  | por pe1 pe2 ⇒ in_pl ? pe1 ∪ in_pl ? pe2
+  | pstar pe ⇒ in_pl ? pe · \sem{|pe|}^*
   ].
 
 interpretation "in_pl" 'in_l E = (in_pl ? E).
 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
 
-definition eps: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter4/word.def(3)"\ 6word\ 5/a\ 6 S → Prop 
-  ≝ λS,b. \ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? b \ 5a title="sing lang" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6: \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] } \ 5a title="empty lang" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
+definition eps: ∀S:Alpha.bool → word S → Prop 
+  ≝ λS,b. if_then_else ? b {: [] } ∅.
 
 (* 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 : \ 5a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"\ 6Alpha\ 5/a\ 6.λp:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S.  \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 (\ 5a title="fst" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="union lang" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="epsilon lang" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6_(\ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
+definition in_prl ≝ λS : Alpha.λp:pre S.  ℓ (\fst p) ∪ ϵ_(\snd 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:\ 5a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 ((\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 pi) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
-#S #pi (elim pi) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
-  [#pi1 #pi2 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * #w1 * #w2 * * #appnil 
-   cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
-  |#p11 #p12 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
-  |#pi #H % * #w1 * #w2 * * #appnil (cases (\ 5a href="cic:/matita/tutorial/chapter3/nil_to_nil.def(5)"\ 6nil_to_nil\ 5/a\ 6 … appnil)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+lemma not_epsilon_lp :∀S.∀pi:pitem S. ¬ ((ℓ pi) []).
+#S #pi (elim pi) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace nmk\ 5/span\ 6\ 5/span\ 6/
+  [#pi1 #pi2 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace absurd\ 5/span\ 6\ 5/span\ 6/ * #w1 * #w2 * * #appnil 
+   cases (nil_to_nil … appnil) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace absurd\ 5/span\ 6\ 5/span\ 6/
+  |#p11 #p12 #H1 #H2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace absurd\ 5/span\ 6\ 5/span\ 6/
+  |#pi #H % * #w1 * #w2 * * #appnil (cases (nil_to_nil … appnil)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace absurd\ 5/span\ 6\ 5/span\ 6/
   ]
 qed.
 
-lemma if_true_epsilon: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → ((\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6ℓ\ 5/a\ 6 e) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+lemma if_true_epsilon: ∀S.∀e:pre S. \snd e = true → ((ℓ e) []).
 #S #e #H %2 >H // qed.
 
-lemma if_epsilon_true : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter4/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e → \ 5a title="snd" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S * #pi #b * [normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /2/] cases b normalize // @False_ind
+lemma if_epsilon_true : ∀S.∀e:pre S. [ ] ∈ e → \snd e = true.
+#S * #pi #b * [normalize #abs @False_ind /2/] cases b normalize // @False_ind
 qed.
 
 definition lor ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.