1 include "basics/types.ma".
3 (* Most of the types we have seen so far are enumerated types, composed by a
4 finite set of alternatives, and records, composed by tuples of heteregoneous
5 elements. A more interesting case of type definition is when some of the rules
6 defining its elements are recursive, i.e. they allow the formation of more
7 elements of the type in terms of the already defined ones. The most typical case
8 is provided by the natural numbers, that can be defined as the smallest set
9 generated by a constant 0 and a successor function from natural numbers to natural
12 inductive nat : Type[0] ≝
16 (* The two terms O and S are called constructors: they define the signature of the
17 type, whose objects are the elements freely generated by means of them. So,
18 examples of natural numbers are O, S O, S (S O), S (S (S O)) and so on.
20 The language of Matita allows the definition of well founded recursive functions
21 over inductive types; in order to guarantee termination of recursion you are only
22 allowed to make recursive calls on structurally smaller arguments than the ones
23 you received in input. Most mathematical functions can be naturally defined in this
24 way. For instance, the sum of two natural numbers can be defined as follows *)
29 | S a ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"
\ 6S
\ 5/a
\ 6 (add a m)
32 (* It is worth to observe that the previous algorithm works by recursion over the
33 first argument. This means that, for instance, (add O x) will reduce to x, as
34 expected, but (add x O) is stuck.
35 How can we prove that, for a generic x, (add x O) = x? The mathematical tool to do
36 it is called induction. The induction principle states that, given a property P(n)
37 over natural numbers, if we prove P(0) and prove that, for any m, P(m) implies P(S m),
38 than we can conclude P(n) for any n.
40 The elim tactic, allow you to apply induction in a very simple way. If your goal is
41 P(n), the invocation of
43 will break down your task to prove the two subgoals P(0) and ∀m.P(m) → P(S m).
45 Let us apply it to our case *)
47 lemma add_0: ∀a.
\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"
\ 6add
\ 5/a
\ 6 a
\ 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 a.
50 (* If you stop the computation here, you will see on the right the two subgoals
52 - ∀x. add x 0 = x → add (S x) O = S x
53 After normalization, both goals are trivial.
58 (* In a similar way, it is convenient to state a lemma about the behaviour of
59 add when the second argument is not zero. *)
61 lemma add_S : ∀a,b.
\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"
\ 6add
\ 5/a
\ 6 a (
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"
\ 6S
\ 5/a
\ 6 b)
\ 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/add.fix(0,0,1)"
\ 6add
\ 5/a
\ 6 a b).
63 (* In the same way as before, we proceed by induction over a. *)
65 #a #b elim a normalize //
68 (* We are now in the position to prove the commutativity of the sum *)
70 theorem add_comm : ∀a,b.
\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"
\ 6add
\ 5/a
\ 6 a b
\ 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 b a.
73 (* We have two sub goals:
75 G2: ∀x.(∀b. add x b = add b x) → ∀b. S (add x b) = add b (S x).
76 G1 is just our lemma add_O. For G2, we start introducing x and the induction
77 hypothesis IH; then, the goal is proved by rewriting using add_S and IH.
78 For Matita, the task is trivial and we can simply close the goal with // *)
84 inductive bool : Type[0] ≝
88 definition nat_of_bool ≝ λb. match b with
89 [ 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
90 | ff ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"
\ 6O
\ 5/a
\ 6
93 (* coercion nat_of_bool. ?? *)
95 (* Let us now define the following function: *)
97 definition twice ≝ λn.
\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"
\ 6add
\ 5/a
\ 6 n n.
99 (* We are interested to prove that for any natural number n there exists a natural
100 number m that is the integer half of n. This will give us the opportunity to
101 introduce new connectives and quantifiers and, later on, to make some interesting
102 consideration on proofs and computations. *)
104 theorem ex_half: ∀n.
\ 5a title="exists" href="cic:/fakeuri.def(1)"
\ 6∃
\ 5/a
\ 6m. 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 m
\ 5a title="logical or" 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 m).
107 (* We proceed by induction on n, that breaks down to the following goals:
108 G1: ∃m.O = add O O ∨ O = S (add m m)
109 G2: ∀x.(∃m. x = add m m ∨ x = S (add m m))→ ∃m. S x = add m m ∨ S x = S (add m m)
110 The only way we have to prove an existential goal is by exhibiting the witness,
111 that in the case of first goal is O. We do it by apply the term called ex_intro
112 instantiated by the witness. Then, it is clear that we must follow the left branch
113 of the disjunction. One way to do it is by applying the term or_introl, that is
114 the first constructor of the disjunction. However, remembering the names of
115 constructors can be annyoing: we can invoke the application of the n-th
116 constructor of an inductive type (inferred by the current goal) by typing %n. At
117 this point we are left with the subgoal O = add O O, that is closed by
118 computation. It is worth to observe that invoking automation at depth /3/ would
119 also automatically close G1.
121 [@(
\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"
\ 6ex_intro
\ 5/a
\ 6 …
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"
\ 6O
\ 5/a
\ 6) %1 //
123 (* The case of G2 is more complex. We should start introducing x and the
125 IH: ∃m. x = add m m ∨ x = S (add m m)
126 At this point we should assume the existence of m enjoying the inductive
127 hypothesis. To eliminate the existential from the context we can just use the
128 case tactic. This situation where we introduce something into the context and
129 immediately eliminate it by case analysis is so frequent that Matita provides a
130 convenient shorthand: you can just type a single "*".
131 The star symbol should be reminiscent of an explosion: the idea is that you have
132 a structured hypothesis, and you ask to explode it into its constituents. In the
133 case of the existential, it allows to pass from a goal of the shape
135 to a goal of the shape
139 (* At this point we are left with a new goal with the following shape
140 G3: ∀m. x = add m m ∨ x = S (add m m) → ....
141 We should introduce m, the hypothesis H: x = add m m ∨ x = S (add m m), and
142 then reason by cases on this hypothesis. It is the same situation as before:
143 we explode the disjunctive hypothesis into its possible consituents. In the case
144 of a disjunction, the * tactic allows to pass from a goal of the form
146 to two subgoals of the form
150 (* In the first subgoal, we are under the assumption that x = add m m. The half
151 of (S x) is hence m, and we have to prove the right branch of the disjunction.
152 In the second subgoal, we are under the assumption that x = S (add m m). The halh
153 of (S x) is hence (S m), and have to follow the left branch of the disjunction.
155 [@(
\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"
\ 6ex_intro
\ 5/a
\ 6 … m) /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ | @(
\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"
\ 6ex_intro
\ 5/a
\ 6 … (
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"
\ 6S
\ 5/a
\ 6 m)) normalize /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"
\ 6or_introl
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
159 (* Instead of proving the existence of a number corresponding to the half of n,
160 we could be interested in computing it. The best way to do it is to define this
161 division operation together with the remainder, that in our case is just a
162 boolean value: tt if the input term is even, and ff if the input term is odd.
163 Since we must return a pair, we could use a suitably defined record type, or
164 simply a product type nat × bool, defined in the basic library. The product type
165 is just a sort of general purpose record, with standard fields fst and snd, called
167 A pair of values n and m is written (pair … m n) or \langle n,m \rangle - visually
170 We first write down the function, and then discuss it.*)
174 [ 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〉
175 | S a ⇒
\ 5span style="text-decoration: underline;"
\ 6\ 5/span
\ 6
177 match (
\ 5a href="cic:/matita/basics/types/snd.def(1)"
\ 6snd
\ 5/a
\ 6 … p) with
178 [ 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〉
179 | 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〉
183 (* The function is computed by recursion over the input n. If n is 0, then the
184 quotient is 0 and the remainder is tt. If n = S a, we start computing the half
185 of a, say 〈q,b〉. Then we have two cases according to the possible values of b:
186 if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return
189 It is important to point out the deep, substantial analogy between the algorithm
190 for computing div2 and the the proof of ex_half. In particular ex_half returns a
191 proof of the kind ∃n.A(n)∨B(n): the really informative content in it is the
192 witness n and a boolean indicating which one between the two conditions A(n) and
193 B(n) is met. This is precisely the quotient-remainder pair returned by div2.
194 In both cases we proceed by recurrence (respectively, induction or recursion) over
195 the input argument n. In case n = 0, we conclude the proof in ex_half by providing
196 the witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in
197 div2. Similarly, in the inductive case n = S a, we must exploit the inductive
198 hypothesis for a (i.e. the result of the recursive call), distinguishing two subcases
199 according to the the two possibilites A(a) or B(a) (i.e. the two possibile values of
200 the remainder for a). The reader is strongly invited to check all remaining details.
202 Let us now prove that our div2 function has the expected behaviour.
205 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〉.
208 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〉.
209 #n #q #H normalize >H normalize // qed.
211 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〉.
212 #n #q #H normalize >H normalize // qed.
214 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).
216 [#q #r normalize #H destruct //
218 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)〉) [//]
219 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))
220 [#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)
221 |#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)
225 (* There is still another possibility, however, namely to mix the program and its
226 specification into a single entity. The idea is to refine the output type of the
227 div2 function: it should not be just a generic pair 〈q,r〉 of natural numbers but a
228 specific pair satisfying the specification of the function. In other words, we need
229 the possibility to define, for a type A and a property P over A, the subset type
230 {a:A|P(a)} of all elements a of type A that satisfy the property P. Subset types
231 are just a particular case of the so called dependent types, that is types that
232 can depend over arguments (such as arrays of a specified length, taken as a
233 parameter).These kind of types are quite unusual in traditional programming
234 languages, and their study is one of the new frontiers of the current research on
237 There is nothing special in a subset type {a:A|P(a)}: it is just a record composed
238 by an element of a of type A and a proof of P(a). The crucial point is to have a
239 language reach enough to comprise proofs among its expressions.
242 record Sub (A:Type[0]) (P:A → Prop) : Type[0] ≝
246 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).
248 (* We can now construct a function from n to {p|qr_spec n p} by composing the objects
251 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.
252 \ 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).
254 (* But we can also try do directly build such an object *)
256 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).
258 [@(
\ 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 //
260 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〉) [//]
261 cases (
\ 5a href="cic:/matita/basics/types/snd.def(1)"
\ 6snd
\ 5/a
\ 6 … p)
262 [#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)
263 |#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)
267 example quotient7:
\ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"
\ 6witness
\ 5/a
\ 6 … (
\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"
\ 6div2Pagain
\ 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〉.
270 example quotient8:
\ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"
\ 6witness
\ 5/a
\ 6 … (
\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"
\ 6div2Pagain
\ 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))))))
271 \ 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〉.
273 \ 5pre
\ 6\ 5pre
\ 6 \ 5/pre
\ 6\ 5/pre
\ 6