commit by user andrea

[helm.git] / weblib / tutorial / chapter3.ma

diff --git a/weblib/tutorial/chapter3.ma b/weblib/tutorial/chapter3.ma
index f7ad9155715878b850f9441dc9a1df70d4aaa716..3d7eb7a56f60b61b6e5a137787c71e94f72316f9 100644 (file)
--- a/weblib/tutorial/chapter3.ma
+++ b/weblib/tutorial/chapter3.ma
@@ -59,14 +59,14 @@ function.*)
 let rec append A (l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝ 
   match l1 with
   [ nil ⇒  l2
-  | cons hd tl ⇒  hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: append A tl l2 ].
+  | cons hd tl ⇒  hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5span class="error" title="Parse error: [sym:] expected after [sym:] (in [term])"\ 6\ 5/span\ 6: append A tl l2 ].
 
 interpretation "append" 'append l1 l2 = (append ? l1 l2).
 
 (* As usual, the function is executable. For instance, (append A nil l) reduces to
 l, as shown by the following example: *)
 
-example nil_append: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+example nil_append: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6] \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
 #A #l normalize // qed.
 
 (* Proving that l @ [] = l is just a bit more complex. The situation is exactly 
@@ -74,7 +74,7 @@ the same as for the addition operation of the previous chapter: since append is
 defined by recutsion over the first argument, the computation of l @ [] is stuck, 
 and we must proceed by induction on l *) 
 
-lemma append_nil: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+lemma append_nil: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
 #A #l (elim l) normalize // qed.
 
 (* similarly, we can define the two functions head and tail. Since we can only define
@@ -88,7 +88,7 @@ definition head ≝ λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0
 definition tail ≝  λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
   match l with [ nil ⇒  \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] | cons hd tl ⇒  tl].
 
-example ex_head: ∀A.∀a,d,l. \ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"\ 6head\ 5/a\ 6 A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) d \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
+example ex_head: ∀A.∀a,d,l. \ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"\ 6head\ 5/a\ 6 A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) d \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 a.
 #A #a #d #l normalize // qed.
 
 (* Problemi con la notazione *)
@@ -123,7 +123,7 @@ let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0
 example ex_length: \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
 normalize // qed.
 
-example ex_nth: \ 5a href="cic:/matita/tutorial/chapter3/nth.fix(0,0,2)"\ 6nth\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])) \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+example ex_nth: \ 5a href="cic:/matita/tutorial/chapter3/nth.fix(0,0,2)"\ 6nth\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6])) \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
 normalize // qed.
 
 (* Proving that the length of l1@l2 is the sum of the lengths of l1
@@ -196,7 +196,7 @@ otherwise. We use an if_then_else function included from bool.ma to this purpose
 
 definition filter ≝ 
   λT.λp:T → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
-  \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0.\ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? (p x) (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l0) l0) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
+  \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0. if p x then x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l0 else l0) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
 
 (* Here are a couple of simple lemmas on the behaviour of the filter function. 
 It is often convenient to state such lemmas, in order to be able to use rewriting
@@ -265,7 +265,7 @@ that essentially allow you to iterate on every subset of a given enumerated
  let rec fold (A,B:Type[0]) (op:B→B→B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l:B ≝  
  match l with 
   [ nil ⇒ b 
-  | cons a l ⇒ \ 5a href="cic:/matita/basics/bool/if_then_else.def(1)"\ 6if_then_else\ 5/a\ 6 ? (p a) (op (f a) (fold A B op b p f l))
+  | cons a l ⇒ if p a then op (f a) (fold A B op b p f l) else
       (fold A B op b p f l)].
 
 (* It is also important to spend a few time to introduce some fancy notation