X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter4.ma;h=7f28c5ddb9cb070a6104d375a2944fbd53f26c77;hb=48c011f52853dd106dbf9cbbd1b9da61277fba3b;hp=00b70a0da09ab6eb691b838bb3733d0d161a1c69;hpb=5fbe7da7019bda8fead167c8b1da1b06625551b3;p=helm.git diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma index 00b70a0da..7f28c5ddb 100644 --- a/weblib/tutorial/chapter4.ma +++ b/weblib/tutorial/chapter4.ma @@ -1,189 +1,331 @@ -include "tutorial/chapter3.ma". +(* +h1 class="section"Naif Set Theory/h1 +*) +include "basics/types.ma". +include "basics/bool.ma". +(* +In this Chapter we shall develop a naif theory of sets represented as +characteristic predicates over some universe codeA/code, that is as objects of type +A→Prop. +For instance the empty set is defined by the always false function: *) + +img class="anchor" src="icons/tick.png" id="empty_set"definition empty_set ≝ λA:Type[0].λa:A.a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a. +notation "\emptyv" non associative with precedence 90 for @{'empty_set}. +interpretation "empty set" 'empty_set = (empty_set ?). + +(* Similarly, a singleton set contaning containing an element a, is defined +by by the characteristic function asserting equality with a *) + +img class="anchor" src="icons/tick.png" id="singleton"definition singleton ≝ λA.λx,a:A.xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aspan class="error" title="Parse error: [term] expected after [sym=] (in [term])"/spana. +(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *) +interpretation "singleton" 'singl x = (singleton ? x). + +(* The membership relation between an element of type A and a set S:A →Prop is +simply the predicate resulting from the application of S to a. +The operations of union, intersection, complement and substraction +are easily defined in terms of the propositional connectives of dijunction, +conjunction and negation *) + +img class="anchor" src="icons/tick.png" id="union"definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a a title="logical or" href="cic:/fakeuri.def(1)"∨/a Q a. +interpretation "union" 'union a b = (union ? a b). + +img class="anchor" src="icons/tick.png" id="intersection"definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a a title="logical and" href="cic:/fakeuri.def(1)"∧/aspan class="error" title="Parse error: [term] expected after [sym∧] (in [term])"/span Q a. +interpretation "intersection" 'intersects a b = (intersection ? a b). + +img class="anchor" src="icons/tick.png" id="complement"definition complement ≝ λU:Type[0].λA:U → Prop.λw.a title="logical not" href="cic:/fakeuri.def(1)"¬/a A w. +interpretation "complement" 'not a = (complement ? a). + +img class="anchor" src="icons/tick.png" id="substraction"definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w a title="logical and" href="cic:/fakeuri.def(1)"∧/a a title="logical not" href="cic:/fakeuri.def(1)"¬/a B w. +interpretation "substraction" 'minus a b = (substraction ? a b). + +(* Finally, we use implication to define the inclusion relation between +sets *) + +img class="anchor" src="icons/tick.png" id="subset"definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a). +interpretation "subset" 'subseteq a b = (subset ? a b). + +(* +h2 class="section"Set Equality/h2 +Two sets are equals if and only if they have the same elements, that is, +if the two characteristic functions are extensionally equivalent: *) + +img class="anchor" src="icons/tick.png" id="eqP"definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a a title="iff" href="cic:/fakeuri.def(1)"↔/aspan class="error" title="Parse error: [term] expected after [sym↔] (in [term])"/span Q a. +notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}. +interpretation "extensional equality" 'eqP a b = (eqP ? a b). + +(* +This notion of equality is different from the intensional equality of +functions; the fact it defines an equivalence relation must be explicitly +proved: *) + +img class="anchor" src="icons/tick.png" id="eqP_sym"lemma eqP_sym: ∀U.∀A,B:U →Prop. + A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A. +#U #A #B #eqAB #a @a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @eqAB qed. + +img class="anchor" src="icons/tick.png" id="eqP_trans"lemma eqP_trans: ∀U.∀A,B,C:U →Prop. + A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C. +#U #A #B #C #eqAB #eqBC #a @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a // qed. + +(* For the same reason, we must also prove that all the operations behave well +with respect to eqP: *) + +img class="anchor" src="icons/tick.png" id="eqP_union_r"lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. + A a title="extensional equality" href="cic:/fakeuri.def(1)"=/aspan class="error" title="Parse error: NUMBER '1' or [term] expected after [sym=] (in [term])"/span1 C → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="union" href="cic:/fakeuri.def(1)"∪/a B. +#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_or_r.def(2)"iff_or_r/a @eqAB qed. + +img class="anchor" src="icons/tick.png" id="eqP_union_l"lemma eqP_union_l: ∀U.∀A,B,C:U →Prop. + B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a C. +#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_or_l.def(2)"iff_or_l/a @eqBC qed. + +img class="anchor" src="icons/tick.png" id="eqP_intersect_r"lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop. + A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="intersection" href="cic:/fakeuri.def(1)"∩/a B. +#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed. + +img class="anchor" src="icons/tick.png" id="eqP_intersect_l"lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop. + B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="intersection" href="cic:/fakeuri.def(1)"∩/aspan class="error" title="Parse error: [term] expected after [sym∩] (in [term])"/span B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C. +#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a @eqBC qed. + +img class="anchor" src="icons/tick.png" id="eqP_substract_r"lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop. + A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="substraction" href="cic:/fakeuri.def(1)"-/a B. +#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed. + +img class="anchor" src="icons/tick.png" id="eqP_substract_l"lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop. + B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="substraction" href="cic:/fakeuri.def(1)"-/a C. +#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/iff_not.def(4)"iff_not/a/span/span/ qed. + +(* +h2 class="section"Simple properties of sets/h2 +We can now prove several properties of the previous set-theoretic operations. +In particular, union is commutative and associative, and the empty set is an +identity element: *) + +img class="anchor" src="icons/tick.png" id="union_empty_r"lemma union_empty_r: ∀U.∀A:U→Prop. + A a title="union" href="cic:/fakeuri.def(1)"∪/a a title="empty set" href="cic:/fakeuri.def(1)"∅/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A. +#U #A #w % [* // normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace /span/span/ | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/] +qed. + +img class="anchor" src="icons/tick.png" id="union_comm"lemma union_comm : ∀U.∀A,B:U →Prop. + A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="union" href="cic:/fakeuri.def(1)"∪/a A. +#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ qed. + +img class="anchor" src="icons/tick.png" id="union_assoc"lemma union_assoc: ∀U.∀A,B,C:U → Prop. + A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="union" href="cic:/fakeuri.def(1)"∪/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="union" href="cic:/fakeuri.def(1)"∪/a C). +#S #A #B #C #w % [* [* /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ ] | * [/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/ | * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/] +qed. -(* As a simple application of lists, let us now consider strings of characters -over a given alphabet Alpha. We shall assume to have a decidable equality between -characters, that is a (computable) function eqb associating a boolean value true -or false to each pair of characters; eqb is correct, in the sense that (eqb x y) -if and only if (x = y). The type Alpha of alphabets is hence defined by the -following record *) +(* In the same way we prove commutativity and associativity for set +interesection *) + +img class="anchor" src="icons/tick.png" id="cap_comm"lemma cap_comm : ∀U.∀A,B:U →Prop. + A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="intersection" href="cic:/fakeuri.def(1)"∩/a A. +#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed. + +img class="anchor" src="icons/tick.png" id="cap_assoc"lemma cap_assoc: ∀U.∀A,B,C:U→Prop. + A a title="intersection" href="cic:/fakeuri.def(1)"∩/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C. +#U #A #B #C #w % [ * #Aw * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ span class="autotactic"span class="autotrace"/span/span| * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ ] +qed. -interpretation "iff" 'iff a b = (iff a b). +(* We can also easily prove idempotency for union and intersection *) -record Alpha : Type[1] ≝ { carr :> Type[0]; +img class="anchor" src="icons/tick.png" id="union_idemp"lemma union_idemp: ∀U.∀A:U →Prop. + A a title="union" href="cic:/fakeuri.def(1)"∪/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A. +#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/] qed. + +img class="anchor" src="icons/tick.png" id="cap_idemp"lemma cap_idemp: ∀U.∀A:U →Prop. + A a title="intersection" href="cic:/fakeuri.def(1)"∩/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A. +#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] qed. + +(* We conclude our examples with a couple of distributivity theorems, and a +characterization of substraction in terms of interesection and complementation. *) + +img class="anchor" src="icons/tick.png" id="distribute_intersect"lemma distribute_intersect : ∀U.∀A,B,C:U→Prop. + (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C). +#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] +qed. + +img class="anchor" src="icons/tick.png" id="distribute_substract"lemma distribute_substract : ∀U.∀A,B,C:U→Prop. + (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="substraction" href="cic:/fakeuri.def(1)"-/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="substraction" href="cic:/fakeuri.def(1)"-/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="substraction" href="cic:/fakeuri.def(1)"-/a C). +#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] +qed. + +img class="anchor" src="icons/tick.png" id="substract_def"lemma substract_def:∀U.∀A,B:U→Prop. Aa title="substraction" href="cic:/fakeuri.def(1)"-/aB a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a a title="complement" href="cic:/fakeuri.def(1)"¬/aB. +#U #A #B #w normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ +qed. + +(* +h2 class="section"Bool vs. Prop/h2 +In several situation it is important to assume to have a decidable equality +between elements of a set U, namely a boolean function eqb: U→U→bool such that +for any pair of elements a and b in U, (eqb x y) is true if and only if x=y. +A set equipped with such an equality is called a DeqSet: *) + +img class="anchor" src="icons/tick.png" id="DeqSet"record DeqSet : Type[1] ≝ { carr :> Type[0]; eqb: carr → carr → a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a; eqb_true: ∀x,y. (eqb x y a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) a title="iff" href="cic:/fakeuri.def(1)"↔/a (x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a y) }. - + +(* We use the notation == to denote the decidable equality, to distinguish it +from the propositional equality. In particular, a term of the form a==b is a +boolean, while a=b is a proposition. *) + notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }. interpretation "eqb" 'eqb a b = (eqb ? a b). -definition word ≝ λS: a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"list/a S. +(* +h2 class="section"Small Scale Reflection/h2 +It is convenient to have a simple way to reflect a proof of the fact +that (eqb a b) is true into a proof of the proposition (a = b); to this aim, +we introduce two operators "\P" and "\b". *) -inductive re (S: a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) : Type[0] ≝ - zero: re S - | epsilon: re S - | char: S → re S - | concat: re S → re S → re S - | or: re S → re S → re S - | star: re S → re S. - -(* notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}. *) -notation "a ^ *" non associative with precedence 90 for @{ 'kstar $a}. -interpretation "star" 'kstar a = (star ? a). -interpretation "or" 'plus a b = (or ? a b). - -notation "a · b" non associative with precedence 60 for @{ 'concat $a $b}. -interpretation "cat" 'concat a b = (concat ? a b). - -(* to get rid of \middot -coercion c : ∀S:Alpha.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?. *) - -(* notation < "a" non associative with precedence 90 for @{ 'ps $a}. *) -notation "` term 90 a" non associative with precedence 90 for @{ 'atom $a}. -interpretation "atom" 'atom a = (char ? a). - -notation "ϵ" non associative with precedence 90 for @{ 'epsilon }. -interpretation "epsilon" 'epsilon = (epsilon ?). - -notation "∅" (* slash emptyv *) non associative with precedence 90 for @{ 'empty }. -interpretation "empty" 'empty = (zero ?). - -let rec flatten (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"list/a (a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S)) on l : a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S ≝ -match l with [ nil ⇒ a title="nil" href="cic:/fakeuri.def(1)"[/a ] | cons w tl ⇒ w a title="append" href="cic:/fakeuri.def(1)"@/a flatten ? tl ]. - -let rec conjunct (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"list/a (a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S)) (L :a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop) on l: Prop ≝ -match l with [ nil ⇒ a href="cic:/matita/basics/logic/True.ind(1,0,0)"True/a | cons w tl ⇒ L w a title="logical and" href="cic:/fakeuri.def(1)"∧/a conjunct ? tl L ]. - -definition empty_lang ≝ λS.λw:a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a. -(* notation "{}" non associative with precedence 90 for @{'empty_lang}. *) -interpretation "empty lang" 'empty = (empty_lang ?). - -definition sing_lang ≝ λS.λx,w:a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a w. -notation "{: x }" non associative with precedence 90 for @{'sing_lang $x}. -interpretation "sing lang" 'sing_lang x = (sing_lang ? x). - -definition union : ∀S,L1,L2,w.Prop ≝ λS,L1,L2.λw: a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.L1 w a title="logical or" href="cic:/fakeuri.def(1)"∨/a L2 w. -interpretation "union lang" 'union a b = (union ? a b). - -definition cat : ∀S,l1,l2,w.Prop ≝ - λS.λl1,l2.λw:a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.a title="exists" href="cic:/fakeuri.def(1)"∃/aw1,w2.w1 a title="append" href="cic:/fakeuri.def(1)"@/a w2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a w a title="logical and" href="cic:/fakeuri.def(1)"∧/a l1 w1 a title="logical and" href="cic:/fakeuri.def(1)"∧/a l2 w2. -interpretation "cat lang" 'concat a b = (cat ? a b). - -definition star_lang ≝ λS.λl.λw:a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S.a title="exists" href="cic:/fakeuri.def(1)"∃/alw. a href="cic:/matita/tutorial/chapter4/flatten.fix(0,1,4)"flatten/a ? lw a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a w a title="logical and" href="cic:/fakeuri.def(1)"∧/a a href="cic:/matita/tutorial/chapter4/conjunct.fix(0,1,4)"conjunct/a ? lw l. -interpretation "star lang" 'kstar l = (star_lang ? l). - -(* notation "ℓ term 70 E" non associative with precedence 75 for @{in_l ? $E}. *) - -let rec in_l (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (r : a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"re/a S) on r : a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop ≝ -match r with - [ zero ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a - | epsilon ⇒ a title="sing lang" href="cic:/fakeuri.def(1)"{/a: a title="nil" href="cic:/fakeuri.def(1)"[/a] } - | char x ⇒ a title="sing lang" href="cic:/fakeuri.def(1)"{/a: xa title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] } - | concat r1 r2 ⇒ in_l ? r1 a title="cat lang" href="cic:/fakeuri.def(1)"·/a in_l ? r2 - | or r1 r2 ⇒ in_l ? r1 a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_l ? r2 - | star r1 ⇒ (in_l ? r1)a title="star lang" href="cic:/fakeuri.def(1)"^/a* - ]. - -notation "ℓ term 70 E" non associative with precedence 75 for @{'in_l $E}. -interpretation "in_l" 'in_l E = (in_l ? E). -interpretation "in_l mem" 'mem w l = (in_l ? l w). - -notation "a ∨ b" left associative with precedence 30 for @{'orb $a $b}. -interpretation "orb" 'orb a b = (orb a b). - -(* ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f]. -notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. -notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. -interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f). *) - -inductive pitem (S: a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) : Type[0] ≝ - | pzero: pitem S - | pepsilon: pitem S - | pchar: S → pitem S - | ppoint: S → pitem S - | pconcat: pitem S → pitem S → pitem S - | por: pitem S → pitem S → pitem S - | pstar: pitem S → pitem S. +notation "\P H" non associative with precedence 90 + for @{(proj1 … (eqb_true ???) $H)}. + +notation "\b H" non associative with precedence 90 + for @{(proj2 … (eqb_true ???) $H)}. + +(* If H:eqb a b = true, then \P H: a = b, and conversely if h:a = b, then +\b h: eqb a b = true. Let us see an example of their use: the following +statement asserts that we can reflect a proof that eqb a b is false into +a proof of the proposition a ≠ b. *) + +img class="anchor" src="icons/tick.png" id="eqb_false"lemma eqb_false: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a,b:S. + (a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a ? a b) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="iff" href="cic:/fakeuri.def(1)"↔/a a a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a b. + +(* We start the proof introducing the hypothesis, and then split the "if" and +"only if" cases *) -definition pre ≝ λS.a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S a title="Product" href="cic:/fakeuri.def(1)"×/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"bool/a. - -interpretation "pstar" 'kstar a = (pstar ? a). -interpretation "por" 'plus a b = (por ? a b). -interpretation "pcat" 'concat a b = (pconcat ? a b). - -notation "• a" non associative with precedence 90 for @{ 'ppoint $a}. -(* notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}. *) - -interpretation "ppatom" 'ppoint a = (ppoint ? a). -(* to get rid of \middot -ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?. *) -interpretation "patom" 'pchar a = (pchar ? a). -interpretation "pepsilon" 'epsilon = (pepsilon ?). -interpretation "pempty" 'empty = (pzero ?). - -notation "\boxv term 19 e \boxv" (* slash boxv *) non associative with precedence 70 for @{forget ? $e}. - -let rec forget (S: a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S) on l: a href="cic:/matita/tutorial/chapter4/re.ind(1,0,1)"re/a S ≝ - match l with - [ pzero ⇒ a title="empty" href="cic:/fakeuri.def(1)"∅/a - | pepsilon ⇒ a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a - | pchar x ⇒ a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"char/a ? x - | ppoint x ⇒ a href="cic:/matita/tutorial/chapter4/re.con(0,3,1)"char/a ? x - | pconcat e1 e2 ⇒ │e1│ a title="cat" href="cic:/fakeuri.def(1)"·/a │e2│ - | por e1 e2 ⇒ │e1│ a title="or" href="cic:/fakeuri.def(1)"+/a │e2│ - | pstar e ⇒ │e│a title="star" href="cic:/fakeuri.def(1)"^/a* - ]. - -notation "│ term 19 e │" non associative with precedence 70 for @{'forget $e}. -interpretation "forget" 'forget a = (forget ? a). - -notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}. -interpretation "fst" 'fst x = (fst ? ? x). -notation "\snd term 90 x" non associative with precedence 90 for @{'snd $x}. -interpretation "snd" 'snd x = (snd ? ? x). - -notation "ℓ term 70 E" non associative with precedence 75 for @{in_pl ? $E}. - -let rec in_pl (S : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a) (r : a href="cic:/matita/tutorial/chapter4/pitem.ind(1,0,1)"pitem/a S) on r : a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop ≝ -match r with - [ pzero ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a - | pepsilon ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a - | pchar _ ⇒ a title="empty lang" href="cic:/fakeuri.def(1)"∅/a - | ppoint x ⇒ a title="sing lang" href="cic:/fakeuri.def(1)"{/a: xa title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] } - | pconcat pe1 pe2 ⇒ in_pl ? pe1 a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"ℓ/a │pe2│ a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_pl ? pe2 - | por pe1 pe2 ⇒ in_pl ? pe1 a title="union lang" href="cic:/fakeuri.def(1)"∪/a in_pl ? pe2 - | pstar pe ⇒ in_pl ? pe a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"ℓ/a │pe│a title="star" href="cic:/fakeuri.def(1)"^/a* - ]. - -interpretation "in_pl" 'in_l E = (in_pl ? E). -interpretation "in_pl mem" 'mem w l = (in_pl ? l w). - -definition eps: ∀S:a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"bool/a → a href="cic:/matita/tutorial/chapter4/word.def(3)"word/a S → Prop - ≝ λS,b. a href="cic:/matita/basics/bool/if_then_else.def(1)"if_then_else/a ? b a title="sing lang" href="cic:/fakeuri.def(1)"{/a: a title="nil" href="cic:/fakeuri.def(1)"[/a] } a title="empty lang" href="cic:/fakeuri.def(1)"∅/a. - -(* interpretation "epsilon" 'epsilon = (epsilon ?). *) -notation "ϵ _ b" non associative with precedence 90 for @{'app_epsilon $b}. -interpretation "epsilon lang" 'app_epsilon b = (eps ? b). - -definition in_prl ≝ λS : a href="cic:/matita/tutorial/chapter4/Alpha.ind(1,0,0)"Alpha/a.λp:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="in_pl" href="cic:/fakeuri.def(1)"ℓ/a (a title="fst" href="cic:/fakeuri.def(1)"\fst/a p) a title="union lang" href="cic:/fakeuri.def(1)"∪/a a title="epsilon lang" href="cic:/fakeuri.def(1)"ϵ/a_(a title="snd" href="cic:/fakeuri.def(1)"\snd/a p). +#S #a #b % #H + +(* The latter is easily reduced to prove the goal true=false under the assumption +H1: a = b *) + [@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"not_eq_true_false/a) #H1 + +(* since by assumption H false is equal to (a==b), by rewriting we obtain the goal +true=(a==b) that is just the boolean version of H1 *) + + H // qed. +notation "\bf H" non associative with precedence 90 + for @{(proj2 … (eqb_false ???) $H)}. -lemma if_epsilon_true : ∀S.∀e:a href="cic:/matita/tutorial/chapter4/pre.def(1)"pre/a S. a title="nil" href="cic:/fakeuri.def(1)"[/a ] a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e → a title="snd" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. -#S * #pi #b * [#abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /2/] cases b normalize // @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a +(* The following statement proves that propositional equality in a +DeqSet is decidable in the traditional sense, namely either a=b or a≠b *) + + img class="anchor" src="icons/tick.png" id="dec_eq"lemma dec_eq: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a,b:S. a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b a title="logical or" href="cic:/fakeuri.def(1)"∨/a a a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a b. +#S #a #b cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a ? a b)) #H + [%1 @(\P H) | %2 @(\Pf H)] qed. +(* +h2 class="section"Unification Hints/h2 +A simple example of a set with a decidable equality is bool. We first define +the boolean equality beqb, that is just the xand function, then prove that +beqb b1 b2 is true if and only if b1=b2, and finally build the type DeqBool by +instantiating the DeqSet record with the previous information *) + +img class="anchor" src="icons/tick.png" id="beqb"definition beqb ≝ λb1,b2. + match b1 with [ true ⇒ b2 | false ⇒ a href="cic:/matita/basics/bool/notb.def(1)"notb/a b2]. + +notation < "a == b" non associative with precedence 45 for @{beqb $a $b }. + +img class="anchor" src="icons/tick.png" id="beqb_true"lemma beqb_true: ∀b1,b2. a href="cic:/matita/basics/logic/iff.def(1)"iff/a (a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a b1 b2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) (b1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b2). +#b1 #b2 cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ +qed. + +img class="anchor" src="icons/tick.png" id="DeqBool"definition DeqBool ≝ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a. + +(* At this point, we would expect to be able to prove things like the +following: for any boolean b, if b==false is true then b=false. +Unfortunately, this would not work, unless we declare b of type +DeqBool (change the type in the following statement and see what +happens). *) + +img class="anchor" src="icons/tick.png" id="exhint"example exhint: ∀b:a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"DeqBool/a. (ba title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → ba title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a. +#b #H @(\P H) +qed. + +(* The point is that == expects in input a pair of objects whose type must be the +carrier of a DeqSet; bool is indeed the carrier of DeqBool, but the type inference +system has no knowledge of it (it is an information that has been supplied by the +user, and stored somewhere in the library). More explicitly, the type inference +inference system, would face an unification problem consisting to unify bool +against the carrier of something (a metavaribale) and it has no way to synthetize +the answer. To solve this kind of situations, matita provides a mechanism to hint +the system the expected solution. A unification hint is a kind of rule, whose rhd +is the unification problem, containing some metavariables X1, ..., Xn, and whose +left hand side is the solution suggested to the system, in the form of equations +Xi=Mi. The hint is accepted by the system if and only the solution is correct, that +is, if it is a unifier for the given problem. +To make an example, in the previous case, the unification problem is bool = carr X, +and the hint is to take X= mk_DeqSet bool beqb true. The hint is correct, since +bool is convertible with (carr (mk_DeqSet bool beb true)). *) + +unification hint 0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"≔/a ; + X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a +(* ---------------------------------------- *) ⊢ + a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a ≡ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a X. + +unification hint 0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"≔/a b1,b2:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a; + X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a +(* ---------------------------------------- *) ⊢ + a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a b1 b2 ≡ a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a X b1 b2. + +(* After having provided the previous hints, we may rewrite example exhint +declaring b of type bool. *) + +img class="anchor" src="icons/tick.png" id="exhint1"example exhint1: ∀b:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a. (b a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → b a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a. +#b #H @(\P H) +qed. + +(* The cartesian product of two DeqSets is still a DeqSet. To prove +this, we must as usual define the boolen equality function, and prove +it correctly reflects propositional equality. *) + +img class="anchor" src="icons/tick.png" id="eq_pairs"definition eq_pairs ≝ + λA,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λp1,p2:Aa title="Product" href="cic:/fakeuri.def(1)"×/aB.(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p1 a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p2) a title="boolean and" href="cic:/fakeuri.def(1)"∧/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p1 a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p2). + +img class="anchor" src="icons/tick.png" id="eq_pairs_true"lemma eq_pairs_true: ∀A,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀p1,p2:Aa title="Product" href="cic:/fakeuri.def(1)"×/aB. + a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a A B p1 p2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a a title="iff" href="cic:/fakeuri.def(1)"↔/a p1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a p2. +#A #B * #a1 #b1 * #a2 #b2 % + [#H cases (a href="cic:/matita/basics/bool/andb_true.def(5)"andb_true/a …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) // + |#H destruct normalize >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … a2)) >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … b2)) // + ] +qed. +img class="anchor" src="icons/tick.png" id="DeqProd"definition DeqProd ≝ λA,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a. + a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a (Aa title="Product" href="cic:/fakeuri.def(1)"×/aB) (a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a A B) (a href="cic:/matita/tutorial/chapter4/eq_pairs_true.def(6)"eq_pairs_true/a A B). + +(* Having an unification problem of the kind T1×T2 = carr X, what kind +of hint can we give to the system? We expect T1 to be the carrier of a +DeqSet C1, T2 to be the carrier of a DeqSet C2, and X to be DeqProd C1 C2. +This is expressed by the following hint: *) + +unification hint 0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"≔/a C1,C2; + T1 ≟ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a C1, + T2 ≟ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a C2, + X ≟ a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"DeqProd/a C1 C2 +(* ---------------------------------------- *) ⊢ + T1a title="Product" href="cic:/fakeuri.def(1)"×/aT2 ≡ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a X. + +unification hint 0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"≔/a T1,T2,p1,p2; + X ≟ a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"DeqProd/a T1 T2 +(* ---------------------------------------- *) ⊢ + a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a T1 T2 p1 p2 ≡ a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a X p1 p2. + +img class="anchor" src="icons/tick.png" id="hint2"example hint2: ∀b1,b2. + a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ab1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ab1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. +#b1 #b2 #H @(\P H). \ No newline at end of file