X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter2.ma;h=ab758c124f52fafb842e8e3bd4246b61346c2aa6;hb=fc43587e06d59afb5b1c65904372843d928fdfe5;hp=fffe5e1cce96f933c9395dcaacfa42cb2953f89b;hpb=f59274b53205d0ffc399e239de333de106b5c642;p=helm.git

diff --git a/weblib/tutorial/chapter2.ma b/weblib/tutorial/chapter2.ma
index fffe5e1cc..ab758c124 100644
--- a/weblib/tutorial/chapter2.ma
+++ b/weblib/tutorial/chapter2.ma
@@ -158,13 +158,14 @@ qed.
 
 (* Instead of proving the existence of a number corresponding to the half of n, 
 we could be interested in computing it. The best way to do it is to define this 
-division operation together with the remainder, that in our case is just a boolean 
-value: tt if the input term is even, and ff if the input term is odd. Since we 
-must return a pair, we could use a suitably defined record type, or simply a product 
-type nat × bool, defined in the basic library. The product type is just a sort of 
-general purpose record, with standard fields fst and snd, called projections. 
-A pair of values n and m is written (pair … m n) or \langle n,m \rangle - 
-visually rendered as 〈n,m〉 
+division operation together with the remainder, that in our case is just a 
+boolean value: tt if the input term is even, and ff if the input term is odd. 
+Since we must return a pair, we could use a suitably defined record type, or 
+simply a product type nat × bool, defined in the basic library. The product type 
+is just a sort of general purpose record, with standard fields fst and snd, called 
+projections. 
+A pair of values n and m is written (pair … m n) or \langle n,m \rangle - visually 
+rendered as 〈n,m〉 
 
 We first write down the function, and then discuss it.*)
 
@@ -182,20 +183,21 @@ match n with
 (* The function is computed by recursion over the input n. If n is 0, then the 
 quotient is 0 and the remainder is tt. If n = S a, we start computing the half 
 of a, say 〈q,b〉. Then we have two cases according to the possible values of b: 
-if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return 〈S q,tt〉.
-
-It is important to point out the deep, substantial analogy between the algorithm for 
-computing div2 and the the proof of ex_half. In particular ex_half returns a 
-proof of the kind ∃n.A(n)∨B(n): the really informative content in it is the witness
-n and a boolean indicating which one between the two conditions A(n) and B(n) is met.
-This is precisely the quotient-remainder pair returned by div2.
-In both cases we proceed by recurrence (respectively, induction or recursion) over the 
-input argument n. In case n = 0, we conclude the proof in ex_half by providing the
-witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in div2.
-Similarly, in the inductive case n = S a, we must exploit the inductive hypothesis 
-for a (i.e. the result of the recursive call), distinguishing two subcases according 
-to the the two possibilites A(a) or B(a) (i.e. the two possibile values of the remainder 
-for a). The reader is strongly invited to check all remaining details.
+if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return 
+〈S q,tt〉.
+
+It is important to point out the deep, substantial analogy between the algorithm 
+for computing div2 and the the proof of ex_half. In particular ex_half returns a 
+proof of the kind ∃n.A(n)∨B(n): the really informative content in it is the 
+witness n and a boolean indicating which one between the two conditions A(n) and 
+B(n) is met. This is precisely the quotient-remainder pair returned by div2.
+In both cases we proceed by recurrence (respectively, induction or recursion) over 
+the input argument n. In case n = 0, we conclude the proof in ex_half by providing 
+the witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in 
+div2. Similarly, in the inductive case n = S a, we must exploit the inductive 
+hypothesis for a (i.e. the result of the recursive call), distinguishing two subcases 
+according to the the two possibilites A(a) or B(a) (i.e. the two possibile values of 
+the remainder for a). The reader is strongly invited to check all remaining details.
 
 Let us now prove that our div2 function has the expected behaviour.
 *)
@@ -215,7 +217,7 @@ lemma div2_ok: ∀n,q,r. a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"
   |#a #Hind #q #r 
    cut (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a), a href="cic:/matita/basics/types/snd.def(1)"snd/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a)〉) [//] 
    cases (a href="cic:/matita/basics/types/snd.def(1)"snd/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a))
-    [#H >(a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"div2S1/a … H) #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(Hind … H) 
+    [#H >(a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"div2S1/a … H) #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a whd in ⊢ (???%); <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(Hind … H) 
     |#H >(a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"div2SO/a … H) #H1 destruct >a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a @(Hind … H) 
     ]
 qed.
@@ -257,7 +259,8 @@ definition div2Pagain : ∀n.a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,
   |#a * #p #qrspec 
    cut (p a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … p, a href="cic:/matita/basics/types/snd.def(1)"snd/a … p〉) [//] 
    cases (a href="cic:/matita/basics/types/snd.def(1)"snd/a … p)
-    [#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/types/fst.def(1)"fst/a … p),a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"ff/a〉) whd #q #r #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(qrspec … H)
+    [#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/types/fst.def(1)"fst/a … p),a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"ff/a〉) whd #q #r #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a
+     whd in ⊢ (???%); <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(qrspec … H)
     |#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … p,a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"tt/a〉) whd #q #r #H1 destruct >a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a @(qrspec … H) 
   ]
 qed.