commit by user andrea

[helm.git] / weblib / tutorial / chapter2.ma

diff --git a/weblib/tutorial/chapter2.ma b/weblib/tutorial/chapter2.ma
index ad0896d708ce9792ef00ed8cfb19c1bae68b7fe0..d564b7220c03dd5e7348860bf227eb9dffb9595e 100644 (file)
--- a/weblib/tutorial/chapter2.ma
+++ b/weblib/tutorial/chapter2.ma
@@ -77,6 +77,19 @@ For Matita, the task is trivial and we can simply close the goal with // *)
 
 // qed.
 
 
 // qed.
 

+(* COERCIONS *)
+
+inductive bool : Type[0] ≝
+| tt : bool
+| ff : bool.
+
+definition nat_of_bool ≝ λb. match b with 
+[ tt ⇒ \ 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 
+| ff ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 
+].
+
+(* coercion nat_of_bool. ?? *)
+ 

 (* Let us now define the following function: *)
 
 definition twice ≝ λn.\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 n n. 
 (* Let us now define the following function: *)
 
 definition twice ≝ λn.\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 n n. 


@@ -138,29 +151,30 @@ 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
 
 (* 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 is 0 if the input term is even, and 1 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 N × N, 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〉 
+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.*)
 
 let rec div2 n ≝ 
 match n with
 
 We first write down the function, and then discuss it.*)
 
 let rec div2 n ≝ 
 match n with

-[ O ⇒ \ 5a title="Pair construction" 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 href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉
+[ O ⇒ \ 5a title="Pair construction" 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 href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉

 | S a ⇒ \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6
    let p ≝ (div2 a) in
    match (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p) with
 | S a ⇒ \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6
    let p ≝ (div2 a) in
    match (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p) with

-   [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p, \ 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〉
-   | S b ⇒ \ 5a title="Pair construction" 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/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉 
+   [ tt ⇒ \ 5a title="Pair construction" 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/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉 
+   | ff ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉

    ]
 ]. 
 
    ]
 ]. 
 

-(* The function is computed by recursion over the input n. If n is 0, then both quotient
-and remainder are 0. If n = S a, we start computing the half of a, say 〈q,r〉. Then we
-have two cases according to the possible values of b: if b is 0, then we must return
- 〈q,1〉, while if b = 1 (no other case is possible), then we must return 〈S q,0〉.
+(* 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 
 
 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 


@@ -169,7 +183,7 @@ n and a boolean indicating which one between the two conditions A(n) and B(n) is
 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
 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,O〉 in div2.
+witness O and a proof of A(O); this corresponds to returning the pair 〈O,tt〉 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 
 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 


@@ -181,22 +195,21 @@ Let us now prove that our div2 function has the expected behaviour.
 lemma surjective_pairing: ∀A,B.∀p:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 … p〉.
 #A #B * // qed.
 
 lemma surjective_pairing: ∀A,B.∀p:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 … p〉.
 #A #B * // qed.
 

-lemma div2SO: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6q, \ 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〉.
+lemma div2SO: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.

 #n #q #H normalize >H normalize // qed.
 
 #n #q #H normalize >H normalize // qed.
 

-lemma div2S1: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 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/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 q,\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉.
+lemma div2S1: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.

 #n #q #H normalize >H normalize // qed.
 
 #n #q #H normalize >H normalize // qed.
 

- lemma div2_ok: ∀n,q,r. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 →
-   r \ 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 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 
-   r \ 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 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 n \ 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/twice.def(2)"\ 6twice\ 5/a\ 6 q).
+lemma div2_ok: ∀n,q,r. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).

 #n elim n
 #n elim n

-  [#q #r normalize #H destruct %1 /2/
-  |#a #Hind #q #r cases (Hind … (\ 5a href="cic:/matita/tutorial/chapter2/surjective_pairing.def(3)"\ 6surjective_pairing\ 5/a\ 6 …)) * #eqr #eqa
-    [>(\ 5a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"\ 6div2SO\ 5/a\ 6 a (\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a))) // #H %2 destruct <eqa /2/
-    |>(\ 5a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"\ 6div2S1\ 5/a\ 6 a (\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a))) // #H %1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 <eqa /2/
+  [#q #r normalize #H destruct //
+  |#a #Hind #q #r 
+   cut (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a), \ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a)〉) [//] 
+   cases (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a))
+    [#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"\ 6div2S1\ 5/a\ 6 … H) #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(Hind … H) 
+    |#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"\ 6div2SO\ 5/a\ 6 … H) #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(Hind … H) 

     ]
     ]

-  ]

 qed.
 
 (* There is still another possibility, however, namely to mix the program and its 
 qed.
 
 (* There is still another possibility, however, namely to mix the program and its 


@@ -219,27 +232,31 @@ record Sub (A:Type[0]) (P:A → Prop) : Type[0] ≝
   {witness: A; 
    proof: P witness}.
 
   {witness: A; 
    proof: P witness}.
 

- definition div2Spec ≝ λn.λp.∀q,r. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 →
-  r \ 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 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 
-  r \ 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 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 n \ 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/twice.def(2)"\ 6twice\ 5/a\ 6 q).
-
-(* We can now construct a function from n to {p|div2Spec n p} by composing the objects
+definition qr_spec ≝ λn.λp.∀q,r. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
+  
+(* We can now construct a function from n to {p|qr_spec n p} by composing the objects

 we already have *)
 
 we already have *)
 

-definition div2P: ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6 Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/div2Spec.def(3)"\ 6div2Spec\ 5/a\ 6 n) ≝ λn.
+definition div2P: ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6 Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6\ 5/span\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n) ≝ λn.

  \ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n) (\ 5a href="cic:/matita/tutorial/chapter2/div2_ok.def(4)"\ 6div2_ok\ 5/a\ 6 n).
 
 (* But we can also try do directly build such an object *)
 
  \ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n) (\ 5a href="cic:/matita/tutorial/chapter2/div2_ok.def(4)"\ 6div2_ok\ 5/a\ 6 n).
 
 (* But we can also try do directly build such an object *)
 

-definition div2Pbis : ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/div2Spec.def(3)"\ 6div2Spec\ 5/a\ 6 n).
+definition div2Pagain : ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n).

 #n elim n
 #n elim n

-  [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" 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 href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉) normalize #q #r #H %1 destruct /2/
-  |#a * #p #spec cases (spec … (\ 5a href="cic:/matita/tutorial/chapter2/surjective_pairing.def(3)"\ 6surjective_pairing\ 5/a\ 6 …)) * #eqr #eqa
-    [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 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〉) #q #r #H %2 destruct /2/ 
-    |@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" 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/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6〉) #q #r #H %1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 /2/
-    ]
+  [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" 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 href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) normalize #q #r #H destruct //
+  |#a * #p #qrspec 
+   cut (p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p〉) [//] 
+   cases (\ 5a href="cic:/matita/basics/types/snd.def(1)"\ 6snd\ 5/a\ 6 … p)
+    [#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" 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/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) whd #q #r #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(qrspec … H)
+    |#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.def(1)"\ 6fst\ 5/a\ 6 … p,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉) whd #q #r #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(qrspec … H) 

   ]
   ]

+qed.

 
 

+example quotient7: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2P.def(5)"\ 6div2P\ 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(0,2,0)"\ 6S\ 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(0,2,0)"\ 6S\ 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(0,2,0)"\ 6S\ 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 title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" 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(0,2,0)"\ 6S\ 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/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.
+// qed.

 
 

-lemma foo: True. 
-\ 5pre\ 6 \ 5/pre\ 6
\ No newline at end of file
+example quotient8: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2P.def(5)"\ 6div2P\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 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 title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 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/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.
+// qed. 
+\ 5pre\ 6\ 5pre\ 6 \ 5/pre\ 6\ 5/pre\ 6
\ No newline at end of file