2 include "tutorial/chapter2.ma".
3 include "basics/bool.ma".
5 (* Matita supports polymorphic data types. The most typical case are polymorphic
6 lists, parametric in the type of their elements: *)
8 inductive list (A:Type[0]) : Type[0] ≝
10 | cons: A -> list A -> list A.
12 (* The type notation list A is the type of all lists with elements of type A: it is
13 defined by tow constructors: a polymorphic empty list (nil A) and a cons operation,
14 adding a new head element of type A to a previous list. For instance, (list nat) and
15 and (list bool) are lists of natural numbers and booleans, respectively. But we can
16 also form more complex data typea, like (list (list (nat → nat))), that is a list whose
17 elements are lists of functions from natural number to natural numbers.
19 Typical elements in (list bool) are for instance,
20 nil nat - the empty list of type nat
21 cons nat true (nil nat) - the list containing true
22 cons nat false (cons nat (true (nil nat))) - the list containing false and true
25 Note that both constructos nil and cons are expecting in input the type parameter -
28 We now add a bit of notation, in order to make the syntax more readable. In particular,
29 we would like to write [] instead of (nil A) and a::l instead of (cons A a l), leaving
30 the system the burden to infer A, whenever possible.
33 notation "hvbox(hd break :: tl)"
34 right associative with precedence 47
37 notation "[ list0 x sep ; ]"
38 non associative with precedence 90
39 for ${fold right @'nil rec acc @{'cons $x $acc}}.
41 notation "hvbox(l1 break @ l2)"
42 right associative with precedence 47
43 for @{'append $l1 $l2 }.
45 interpretation "nil" 'nil = (nil ?).
46 interpretation "cons" 'cons hd tl = (cons ? hd tl).
48 (* Let us define a few basic functions over lists. Our first example is the append
49 function, concatenating two lists l1 and l2. The natural way is to proceed by recursion
50 on l1: if it is empty the result is simply l2, while if l1 = hd::tl then we
51 recursively append tl and l2 , and then add hd as first element. Note that the append
52 function itself is polymorphic, and the first argument it takes in input is the type
53 A of the elements of two lists l1 and l2.
54 Moreover, since the append function takes in input several parameters, we must also
55 specify in the its defintion on which one of them we are recurring: in this case l1.
56 If not othewise specified, recursion is supposed to act on the first argument of the
59 let rec append A (l1:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) l2 on l1 ≝
62 | cons hd tl ⇒ hd
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6: append A tl l2 ].
64 interpretation "append" 'append l1 l2 = (append ? l1 l2).
66 (* As usual, the function is executable. For instance, (append A nil l) reduces to
67 l, as shown by the following example: *)
69 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.
70 #A #l normalize // qed.
72 (* Proving that l @ [] = l is just a bit more complex. The situation is exactly the
73 same as for the addition operation of the previous chapter: since append is defined
74 by recutsion over the first argument, the computation of l @ [] is stuck, and we must
75 proceed by induction on l *)
77 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.
78 #A #l (elim l) normalize // qed.
80 (* similarly, we can define the two functions head and tail. Since we can only define
81 total functions, we should decide what to do in case the input list is empty. For tl, it
82 is natural to return the empty list; for hd, we take in input a default element d of type
83 A to return in this case. *)
85 definition head ≝ λA.λl:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.λd:A.
86 match l with [ nil ⇒ d | cons a _ ⇒ a].
88 definition tail ≝ λA.λl:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.
89 match l with [ nil ⇒
\ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] | cons hd tl ⇒ tl].
91 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.
92 #A #a #d #l normalize // qed.
94 (* Problemi con la notazione *)
95 example ex_tail:
\ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"
\ 6tail
\ 5/a
\ 6 ? (
\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"
\ 6cons
\ 5/a
\ 6 ?
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 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 title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6].
98 theorem associative_append:
99 ∀A.∀l1,l2,l3:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A. (l1
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6 l2)
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6 l3
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 l1
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6 (l2
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6 l3).
100 #A #l1 #l2 #l3 (elim l1) normalize // qed.
102 (* Problemi con la notazione *)
104 ∀A.∀a:A.∀l,l1:
\ 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(a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l1)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 (l
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6 (
\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"
\ 6cons
\ 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 l1.
107 (* Other typical functions over lists are those computing the length
108 of a list, and the function returning the nth element *)
110 let rec length (A:Type[0]) (l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) on l ≝
112 [ nil ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"
\ 6O
\ 5/a
\ 6
113 | cons a tl ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"
\ 6S
\ 5/a
\ 6 (length A tl)].
115 let rec nth n (A:Type[0]) (l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) (d:A) ≝
117 [O ⇒
\ 5a href="cic:/matita/tutorial/chapter3/hd.def(1)"
\ 6hd
\ 5/a
\ 6 A l d
118 |S m ⇒ nth m A (
\ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"
\ 6tail
\ 5/a
\ 6 A l) d].
120 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.
123 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.
126 (* Proving that the length of l1@l2 is the sum of the lengths of l1
127 and l2 just requires a trivial induction on the first list. *)
129 lemma length_add: ∀A.∀l1,l2:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.
130 \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"
\ 6length
\ 5/a
\ 6 ? (l1
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6l2)
\ 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/chapter3/length.fix(0,1,1)"
\ 6length
\ 5/a
\ 6 ? l1) (
\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"
\ 6length
\ 5/a
\ 6 ? l2).
131 #A #l1 elim l1 normalize // qed.
133 (* Let us come to a more interesting question. How can we prove that the empty list is
134 different from any list with at least one element, that is from any list of the kind (a::l)?
135 We start defining a simple predicate stating if a list is empty or not. The predicate
136 is computed by inspection over the list *)
138 definition is_nil: ∀A:Type[0].
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A → Prop ≝
139 λA.λl.match l with [ nil ⇒ l
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] | cons hd tl ⇒ (l
\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"
\ 6≠
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6])].
141 (* Next we need a simple result about negation: if you wish to prove ¬P you are
142 authorized to add P to your hypothesis: *)
144 lemma neg_aux : ∀P:Prop. (P →
\ 5a title="logical not" href="cic:/fakeuri.def(1)"
\ 6¬
\ 5/a
\ 6P) →
\ 5a title="logical not" href="cic:/fakeuri.def(1)"
\ 6¬
\ 5/a
\ 6P.
145 #P #PtonegP % /3/ qed.
147 theorem diff_cons_nil:
148 ∀A:Type[0].∀l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.∀a:A. a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l
\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"
\ 6≠
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6].
149 #A #l #a @
\ 5a href="cic:/matita/tutorial/chapter3/neg_aux.def(3)"
\ 6neg_aux
\ 5/a
\ 6 #Heq
150 (* we start assuming the new hypothesis Heq of type a::l = [] using neg_aux.
151 Next we use the change tactic to pass from the current goal a::l≠ [] to the expression
152 is_nil a::l, convertible with it. *)
153 (change with (
\ 5a href="cic:/matita/tutorial/chapter3/is_nil.def(1)"
\ 6is_nil
\ 5/a
\ 6 ? (a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l)))
154 (* Now, we rewrite with Heq, obtaining (is_nil A []), that reduces to the trivial
158 (* As an application of the previous result let us prove that l1@l2 is empty if and
159 only if both l1 and l2 are empty. The idea is to proceed by cases on l1: if l1=[] the
160 statement is trivial; on the other side, if l1 = a::tl, then the hypothesis
161 (a::tl)@l2 = [] is absurd, hence we can prove anything from it. When we know we can
162 prove both A and ¬A, a sensible way to proceed is to apply False_ind: ∀P.False → P to the
163 current goal, that breaks down to prove False, and then absurd: ∀A:Prop. A → ¬A → False
164 to reduce to the contradictory cases. Usually, you may invoke automation to take care
165 to solve the absurd case. *)
167 lemma nil_to_nil: ∀A.∀l1,l2:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 \ 5span style="text-decoration: underline;"
\ 6\ 5/span
\ 6A.
168 l1
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6l2
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] → l1
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6]
\ 5a title="logical and" href="cic:/fakeuri.def(1)"
\ 6∧
\ 5/a
\ 6 l2
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6].
169 #A #l1 cases l1 normalize /2/ #a #tl #l2 #H @
\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"
\ 6False_ind
\ 5/a
\ 6 /2/ qed.
171 (* Let us come to some important, higher order, polymorphic functionals
172 acting over lists. A typical example is the map function, taking a function
173 f:A → B, a list l = [a1; a2; ... ; an] and returning the list
174 [f a1; f a2; ... ; f an]. *)
176 let rec map (A,B:Type[0]) (f: A → B) (l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) on l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 B ≝
177 match l with [ nil ⇒
\ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] | cons x tl ⇒ f x
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6: (map A B f tl)].
179 (* Another major example is the fold function, that taken a list
180 l = [a1; a2; ... ;an], a base value b:B, and a function f: A → B → B returns
181 (f a1 (f a2 (... (f an b)...))). *)
183 let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) on l :B ≝
184 match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
186 (* As an example of application of foldr, let us use it to define a filter function
187 that given a list l: list A and a boolean test p:A → bool returns the sublist of elements
188 satisfying the test. In this case, the result type B of foldr should be (list A), the base
189 value is [], and f: A → list A →list A is the function that taken x and l returns x::l, if
190 x satisfies the test, and l otherwise. We use an if_then_else function included from
191 bbol.ma to this purpose. *)
194 λT.λp:T →
\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"
\ 6bool
\ 5/a
\ 6.
195 \ 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].
197 (* Here are a couple of simple lemmas on the behaviour of the filter function.
198 It is often convenient to state such lemmas, in order to be able to use rewriting
199 as an alternative to reduction in proofs: reduction is a bit difficult to control.
202 lemma filter_true : ∀A,l,a,p. p a
\ 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 →
203 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"
\ 6filter
\ 5/a
\ 6 A p (a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:
\ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"
\ 6filter
\ 5/a
\ 6 A p l.
204 #A #l #a #p #pa (elim l) normalize >pa // qed.
206 lemma filter_false : ∀A,l,a,p. p a
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 →
207 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"
\ 6filter
\ 5/a
\ 6 A p (a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"
\ 6filter
\ 5/a
\ 6 A p l.
208 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
210 (* As another example, let us redefine the map function using foldr. The
211 result type B is (list B), the base value b is [], and the fold function
212 of type A → list B → list B is the function mapping a and l to (f a)::l.
215 definition map_again ≝ λA,B,f,l.
\ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"
\ 6foldr
\ 5/a
\ 6 A (
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 B) (λa,l.f a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l)
\ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] l.
217 (* Can we prove that map_again is "the same" as map? We should first of all
218 clarify in which sense we expect the two functions to be equal. Equality in
219 Matita has an intentional meaning: it is the smallest predicate induced by
220 convertibility, i.e. syntactical equality up to normalization. From an
221 intentional point of view, map and map_again are not functions, but programs,
222 and they are clearly different. What we would like to say is that the two
223 programs behave in the same way: this is a different, extensional equality
224 that can be defined in the following way. *)
226 definition ExtEq ≝ λA,B:Type[0].λf,g:A→B.∀a:A.f a
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 g a.
228 (* Proving that map and map_again are extentionally equal in the
229 previous sense can be proved by a trivial structural induction on the list *)
231 lemma eq_maps: ∀A,B,f.
\ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"
\ 6ExtEq
\ 5/a
\ 6 ?? (
\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"
\ 6map
\ 5/a
\ 6 A B f) (
\ 5a href="cic:/matita/tutorial/chapter3/map_again.def(2)"
\ 6map_again
\ 5/a
\ 6 A B f).
232 #A #B #f #n (elim n) normalize // qed.
234 (* Let us make another remark about extensional equality. It is clear that,
235 if f is extensionally equal to g, then (map A B f) is extensionally equal to
236 (map A B g). Let us prove it. *)
238 theorem eq_map : ∀A,B,f,g.
\ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"
\ 6ExtEq
\ 5/a
\ 6 A B f g →
\ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"
\ 6ExtEq
\ 5/a
\ 6 ?? (
\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"
\ 6map
\ 5/a
\ 6 \ 5span style="text-decoration: underline;"
\ 6\ 5/span
\ 6A B f) (
\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"
\ 6map
\ 5/a
\ 6 A B g).
241 (* the relevant point is that we cannot proceed by rewriting f with g via
242 eqfg, here. Rewriting only works with Matita intensional equality, while here
243 we are dealing with a different predicate, defined by the user. The right way
244 to proceed is to unfold the definition of ExtEq, and work by induction on l,
245 as usual when we want to prove extensional equality between functions over
246 inductive types; again the rest of the proof is trivial. *)
248 #l (elim l) normalize // qed.
250 (**************************** BIGOPS *******************************)
252 (* Building a library of basic functions, it is important to achieve a
253 good degree of abstraction and generality, in order to be able to reuse
254 suitable instances of the same function in different context. This has not
255 only the obvious benefit of factorizing code, but especially to avoid
256 repeating proofs of generic properties over and over again.
257 A really convenient tool is the following combination of fold and filter,
258 that essentially allow you to iterate on every subset of a given enumerated
259 (finite) type, represented as a list. *)
261 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)"
\ 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 ≝
264 | 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))
265 (fold A B op b p f l)].
267 (* It is also important to spend a few time to introduce some fancy notation
268 for these iterators. *)
270 notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
272 for @{'fold $op $nil (λ${ident i}. $p) (λ${ident i}. $f) $l}.
274 notation "\fold [ op , nil ]_{ident i ∈ l } f"
276 for @{'fold $op $nil (λ${ident i}.true) (λ${ident i}. $f) $l}.
278 interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
281 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a
\ 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 →
282 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l| p i} (f i)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6
283 op (f a)
\ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ l| p i} (f i).
284 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
287 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
288 p a
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 →
\ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l| p i} (f i)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6
289 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ l| p i} (f i).
290 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
293 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
294 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ l| p i} (f i)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6
295 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ (
\ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"
\ 6filter
\ 5/a
\ 6 A p l)} (f i).
296 #A #B #a #l #p #op #nil #f elim l //
297 #a #tl #Hind cases(
\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"
\ 6true_or_false
\ 5/a
\ 6 (p a)) #pa
298 [ >
\ 5a href="cic:/matita/tutorial/chapter3/filter_true.def(3)"
\ 6filter_true
\ 5/a
\ 6 // >
\ 5a href="cic:/matita/tutorial/chapter3/fold_true.def(3)"
\ 6fold_true
\ 5/a
\ 6 // >
\ 5a href="cic:/matita/tutorial/chapter3/fold_true.def(3)"
\ 6fold_true
\ 5/a
\ 6 //
299 | >
\ 5a href="cic:/matita/tutorial/chapter3/filter_false.def(3)"
\ 6filter_false
\ 5/a
\ 6 // >
\ 5a href="cic:/matita/tutorial/chapter3/fold_false.def(3)"
\ 6fold_false
\ 5/a
\ 6 // ]
302 record Aop (A:Type[0]) (nil:A) : Type[0] ≝
304 nill:∀a. op nil a
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 a;
305 nilr:∀a. op a nil
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 a;
306 assoc: ∀a,b,c.op a (op b c)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 op (op a b) c
309 theorem fold_sum: ∀A,B. ∀I,J:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.∀nil.∀op:
\ 5a href="cic:/matita/tutorial/chapter3/Aop.ind(1,0,2)"
\ 6Aop
\ 5/a
\ 6 B nil.∀f:A → B.
310 op (
\ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ I} (f i)) (
\ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ J} (f i))
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6
311 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ (I
\ 5a title="append" href="cic:/fakeuri.def(1)"
\ 6@
\ 5/a
\ 6J)} (f i).
312 #A #B #I #J #nil #op #f (elim I) normalize
313 [>
\ 5a href="cic:/matita/tutorial/chapter3/nill.fix(0,2,2)"
\ 6nill
\ 5/a
\ 6//|#a #tl #Hind <
\ 5a href="cic:/matita/tutorial/chapter3/assoc.fix(0,2,2)"
\ 6assoc
\ 5/a
\ 6 //]