1 include "tutorial/chapter2.ma".
2 include "basics/bool.ma".
4 (* Matita supports polymorphic data types. The most typical case are polymorphic
5 lists, parametric in the type of their elements: *)
7 inductive list (A:Type[0]) : Type[0] ≝
9 | cons: A -> list A -> list A.
11 (* The type notation list A is the type of all lists with elements of type A: it is
12 defined by two constructors: a polymorphic empty list (nil A) and a cons operation,
13 adding a new head element of type A to a previous list. For instance, (list nat) and
14 and (list bool) are lists of natural numbers and booleans, respectively. But we can
15 also form more complex data types, like (list (list (nat → nat))), that is a list whose
16 elements are lists of functions from natural numbers to natural numbers.
18 Typical elements in (list bool) are for instance,
19 nil nat - the empty list of type nat
20 cons nat true (nil nat) - the list containing true
21 cons nat false (cons nat (true (nil nat))) - the list containing false and true
24 Note that both constructos nil and cons are expecting in input the type parameter -
27 We now add a bit of notation, in order to make the syntax more readable. In particular,
28 we would like to write [] instead of (nil A) and a::l instead of (cons A a l), leaving
29 the system the burden to infer A, whenever possible.
32 notation "hvbox(hd break :: tl)"
33 right associative with precedence 47
36 notation "[ list0 x sep ; ]"
37 non associative with precedence 90
38 for ${fold right @'nil rec acc @{'cons $x $acc}}.
40 notation "hvbox(l1 break @ l2)"
41 right associative with precedence 47
42 for @{'append $l1 $l2 }.
44 interpretation "nil" 'nil = (nil ?).
45 interpretation "cons" 'cons hd tl = (cons ? hd tl).
47 (* Let us define a few basic functions over lists. Our first example is the append
48 function, concatenating two lists l1 and l2. The natural way is to proceed by recursion
49 on l1: if it is empty the result is simply l2, while if l1 = hd::tl then we
50 recursively append tl and l2 , and then add hd as first element. Note that the append
51 function itself is polymorphic, and the first argument it takes in input is the type
52 A of the elements of two lists l1 and l2.
53 Moreover, since the append function takes in input several parameters, we must also
54 specify in the its defintion on which one of them we are recurring: in this case l1.
55 If not othewise specified, recursion is supposed to act on the first argument of the
58 let rec append A (l1:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A) l2 on l1 ≝
61 | cons hd tl ⇒ hd
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6: append A tl l2 ].
63 interpretation "append" 'append l1 l2 = (append ? l1 l2).
65 (* As usual, the function is executable. For instance, (append A nil l) reduces to
66 l, as shown by the following example: *)
68 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.
69 #A #l normalize // qed.
71 (* Proving that l @ [] = l is just a bit more complex. The situation is exactly the
72 same as for the addition operation of the previous chapter: since append is defined
73 by recutsion over the first argument, the computation of l @ [] is stuck, and we must
74 proceed by induction on l *)
76 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.
77 #A #l (elim l) normalize // qed.
79 (* similarly, we can define the two functions head and tail. Since we can only define
80 total functions, we should decide what to do in case the input list is empty. For tl, it
81 is natural to return the empty list; for hd, we take in input a default element d of type
82 A to return in this case. *)
84 definition head ≝ λA.λl:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.λd:A.
85 match l with [ nil ⇒ d | cons a _ ⇒ a].
87 definition tail ≝ λA.λl:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.
88 match l with [ nil ⇒
\ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6] | cons hd tl ⇒ tl].
90 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.
91 #A #a #d #l normalize // qed.
93 (* Problemi con la notazione *)
94 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].
97 theorem associative_append:
98 ∀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).
99 #A #l1 #l2 #l3 (elim l1) normalize // qed.
101 (* Problemi con la notazione *)
102 lemma a_append: ∀A.∀a.∀l:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A. (a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:
\ 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 a
\ 5a title="cons" href="cic:/fakeuri.def(1)"
\ 6:
\ 5/a
\ 6:l.
106 ∀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.
109 (* Other typical functions over lists are those computing the length
110 of a list, and the function returning the nth element *)
112 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 ≝
114 [ nil ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"
\ 6O
\ 5/a
\ 6
115 | cons a tl ⇒
\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"
\ 6S
\ 5/a
\ 6 (length A tl)].
117 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) ≝
119 [O ⇒
\ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"
\ 6head
\ 5/a
\ 6 A l d
120 |S m ⇒ nth m A (
\ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"
\ 6tail
\ 5/a
\ 6 A l) d].
122 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.
125 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.
128 (* Proving that the length of l1@l2 is the sum of the lengths of l1
129 and l2 just requires a trivial induction on the first list. *)
131 lemma length_add: ∀A.∀l1,l2:
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A.
132 \ 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).
133 #A #l1 elim l1 normalize // qed.
135 (* Let us come to a more interesting question. How can we prove that the empty list is
136 different from any list with at least one element, that is from any list of the kind (a::l)?
137 We start defining a simple predicate stating if a list is empty or not. The predicate
138 is computed by inspection over the list *)
140 definition is_nil: ∀A:Type[0].
\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"
\ 6list
\ 5/a
\ 6 A → Prop ≝
141 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])].
143 (* Next we need a simple result about negation: if you wish to prove ¬P you are
144 authorized to add P to your hypothesis: *)
146 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.
147 #P #PtonegP % /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/absurd.def(2)"
\ 6absurd
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ qed.
149 theorem diff_cons_nil:
150 ∀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].
151 #A #l #a @
\ 5a href="cic:/matita/tutorial/chapter3/neg_aux.def(3)"
\ 6neg_aux
\ 5/a
\ 6 #Heq
152 (* we start assuming the new hypothesis Heq of type a::l = [] using neg_aux.
153 Next we use the change tactic to pass from the current goal a::l≠ [] to the expression
154 is_nil a::l, convertible with it. *)
155 (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)))
156 (* Now, we rewrite with Heq, obtaining (is_nil A []), that reduces to the trivial
160 (* As an application of the previous result let us prove that l1@l2 is empty if and
161 only if both l1 and l2 are empty. The idea is to proceed by cases on l1: if l1=[] the
162 statement is trivial; on the other side, if l1 = a::tl, then the hypothesis
163 (a::tl)@l2 = [] is absurd, hence we can prove anything from it. When we know we can
164 prove both A and ¬A, a sensible way to proceed is to apply False_ind: ∀P.False → P to the
165 current goal, that breaks down to prove False, and then absurd: ∀A:Prop. A → ¬A → False
166 to reduce to the contradictory cases. Usually, you may invoke automation to take care
167 to solve the absurd case. *)
169 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.
170 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].
171 #A #l1 cases l1 normalize /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"
\ 6conj
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ #a #tl #l2 #H @
\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"
\ 6False_ind
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/absurd.def(2)"
\ 6absurd
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ qed.
173 (* Let us come to some important, higher order, polymorphic functionals
174 acting over lists. A typical example is the map function, taking a function
175 f:A → B, a list l = [a1; a2; ... ; an] and returning the list
176 [f a1; f a2; ... ; f an]. *)
178 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 ≝
179 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)].
181 (* Another major example is the fold function, that taken a list
182 l = [a1; a2; ... ;an], a base value b:B, and a function f: A → B → B returns
183 (f a1 (f a2 (... (f an b)...))). *)
185 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 ≝
186 match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
188 (* As an example of application of foldr, let us use it to define a filter function
189 that given a list l: list A and a boolean test p:A → bool returns the sublist of elements
190 satisfying the test. In this case, the result type B of foldr should be (list A), the base
191 value is [], and f: A → list A →list A is the function that taken x and l returns x::l, if
192 x satisfies the test, and l otherwise. We use an if_then_else function included from
193 bbol.ma to this purpose. *)
196 λT.λp:T →
\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"
\ 6bool
\ 5/a
\ 6.
197 \ 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].
199 (* Here are a couple of simple lemmas on the behaviour of the filter function.
200 It is often convenient to state such lemmas, in order to be able to use rewriting
201 as an alternative to reduction in proofs: reduction is a bit difficult to control.
204 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 →
205 \ 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.
206 #A #l #a #p #pa (elim l) normalize >pa // qed.
208 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 →
209 \ 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.
210 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
212 (* As another example, let us redefine the map function using foldr. The
213 result type B is (list B), the base value b is [], and the fold function
214 of type A → list B → list B is the function mapping a and l to (f a)::l.
217 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.
219 (* Can we prove that map_again is "the same" as map? We should first of all
220 clarify in which sense we expect the two functions to be equal. Equality in
221 Matita has an intentional meaning: it is the smallest predicate induced by
222 convertibility, i.e. syntactical equality up to normalization. From an
223 intentional point of view, map and map_again are not functions, but programs,
224 and they are clearly different. What we would like to say is that the two
225 programs behave in the same way: this is a different, extensional equality
226 that can be defined in the following way. *)
228 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.
230 (* Proving that map and map_again are extentionally equal in the
231 previous sense can be proved by a trivial structural induction on the list *)
233 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).
234 #A #B #f #n (elim n) normalize // qed.
236 (* Let us make another remark about extensional equality. It is clear that,
237 if f is extensionally equal to g, then (map A B f) is extensionally equal to
238 (map A B g). Let us prove it. *)
240 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).
243 (* the relevant point is that we cannot proceed by rewriting f with g via
244 eqfg, here. Rewriting only works with Matita intensional equality, while here
245 we are dealing with a different predicate, defined by the user. The right way
246 to proceed is to unfold the definition of ExtEq, and work by induction on l,
247 as usual when we want to prove extensional equality between functions over
248 inductive types; again the rest of the proof is trivial. *)
250 #l (elim l) normalize // qed.
252 (**************************** BIGOPS *******************************)
254 (* Building a library of basic functions, it is important to achieve a
255 good degree of abstraction and generality, in order to be able to reuse
256 suitable instances of the same function in different context. This has not
257 only the obvious benefit of factorizing code, but especially to avoid
258 repeating proofs of generic properties over and over again.
259 A really convenient tool is the following combination of fold and filter,
260 that essentially allow you to iterate on every subset of a given enumerated
261 (finite) type, represented as a list. *)
263 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 ≝
266 | 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))
267 (fold A B op b p f l)].
269 (* It is also important to spend a few time to introduce some fancy notation
270 for these iterators. *)
272 notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
274 for @{'fold $op $nil (λ${ident i}. $p) (λ${ident i}. $f) $l}.
276 notation "\fold [ op , nil ]_{ident i ∈ l } f"
278 for @{'fold $op $nil (λ${ident i}.true) (λ${ident i}. $f) $l}.
280 interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
283 ∀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 →
284 \ 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
285 op (f a)
\ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ l| p i} (f i).
286 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
289 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
290 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
291 \ 5a title="\fold" href="cic:/fakeuri.def(1)"
\ 6\fold
\ 5/a
\ 6[op,nil]_{i ∈ l| p i} (f i).
292 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
295 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
296 \ 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
297 \ 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).
298 #A #B #a #l #p #op #nil #f elim l //
299 #a #tl #Hind cases(
\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"
\ 6true_or_false
\ 5/a
\ 6 (p a)) #pa
300 [ >
\ 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 //
301 | >
\ 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 // ]
304 record Aop (A:Type[0]) (nil:A) : Type[0] ≝
306 nill:∀a. op nil a
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 a;
307 nilr:∀a. op a nil
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 a;
308 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
311 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.
312 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
313 \ 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).
314 #A #B #I #J #nil #op #f (elim I) normalize
315 [>
\ 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 //]