X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter4.ma;h=7f28c5ddb9cb070a6104d375a2944fbd53f26c77;hb=613d8642b1154dde0c026cbdcd96568910198251;hp=bb01c3e8b79c8540a774e7232379bf7ebee374b5;hpb=538cee79ad9754ad46015de1fd34a3ad808f08c7;p=helm.git diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma index bb01c3e8b..7f28c5ddb 100644 --- a/weblib/tutorial/chapter4.ma +++ b/weblib/tutorial/chapter4.ma @@ -1,745 +1,331 @@ +(* +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 *) -include "arithmetics/nat.ma". -include "basics/list.ma". +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). -interpretation "iff" 'iff a b = (iff 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. -record Alpha : Type[1] ≝ { carr :> Type[0]; +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. + +(* 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. + +(* We can also easily prove idempotency for union and intersection *) + +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/re/Alpha.ind(1,0,0)"Alpha/a.a href="cic:/matita/basics/list/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/re/Alpha.ind(1,0,0)"Alpha/a) : Type[0] ≝ - z: re S - | e: re S - | s: S → re S - | c: re S → re S → re S - | o: re S → re S → re S - | k: re S → re S. - -notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}. -notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}. -interpretation "star" 'pk a = (k ? a). -interpretation "or" 'plus a b = (o ? a b). - -notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}. -interpretation "cat" 'pc a b = (c ? 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 @{ 'ps $a}. -interpretation "atom" 'ps a = (s ? a). - -notation "ϵ" non associative with precedence 90 for @{ 'epsilon }. -interpretation "epsilon" 'epsilon = (e ?). - -notation "∅" non associative with precedence 90 for @{ 'empty }. -interpretation "empty" 'empty = (z ?). - -let rec flatten (S : a href="cic:/matita/tutorial/re/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a (a href="cic:/matita/tutorial/re/word.def(3)"word/a S)) on l : a href="cic:/matita/tutorial/re/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/re/Alpha.ind(1,0,0)"Alpha/a) (l : a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a (a href="cic:/matita/tutorial/re/word.def(3)"word/a S)) (r : a href="cic:/matita/tutorial/re/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 ⇒ r w a title="logical and" href="cic:/fakeuri.def(1)"∧/a conjunct ? tl r ]. - -definition empty_lang ≝ λS.λw:a href="cic:/matita/tutorial/re/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_lang = (empty_lang ?). - -definition sing_lang ≝ λS.λx,w:a href="cic:/matita/tutorial/re/word.def(3)"word/a S.xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aw. -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.λl1,l2.λw:a href="cic:/matita/tutorial/re/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/re/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" 'pc a b = (cat ? a b). - -definition star ≝ λS.λl.λw:a href="cic:/matita/tutorial/re/word.def(3)"word/a S.a title="exists" href="cic:/fakeuri.def(1)"∃/alw.a href="cic:/matita/tutorial/re/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/re/conjunct.fix(0,1,4)"conjunct/a ? lw l. -interpretation "star lang" 'pk l = (star ? l). - -notation > "𝐋 term 70 E" non associative with precedence 75 for @{in_l ? $E}. - -let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ -match r with -[ z ⇒ {} -| e ⇒ { [ ] } -| s x ⇒ { [x] } -| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2 -| o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2 -| k r1 ⇒ (𝐋 r1) ^*]. - -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). - -ninductive pitem (S: Alpha) : Type[0] ≝ - pz: pitem S - | pe: pitem S - | ps: S → pitem S - | pp: S → pitem S - | pc: pitem S → pitem S → pitem S - | po: pitem S → pitem S → pitem S - | pk: pitem S → pitem S. - -ndefinition pre ≝ λS.pitem S × bool. - -interpretation "pstar" 'pk a = (pk ? a). -interpretation "por" 'plus a b = (po ? a b). -interpretation "pcat" 'pc a b = (pc ? a b). -notation < ".a" non associative with precedence 90 for @{ 'pp $a}. -notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}. -interpretation "ppatom" 'pp a = (pp ? a). -(* to get rid of \middot *) -ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?. -interpretation "patom" 'ps a = (ps ? a). -interpretation "pepsilon" 'epsilon = (pe ?). -interpretation "pempty" 'empty = (pz ?). - -notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}. -nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝ - match l with - [ pz ⇒ ∅ - | pe ⇒ ϵ - | ps x ⇒ `x - | pp x ⇒ `x - | pc E1 E2 ⇒ (|E1| · |E2|) - | po E1 E2 ⇒ (|E1| + |E2|) - | pk E ⇒ |E|^* ]. -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\p\ term 70 E" non associative with precedence 75 for @{in_pl ? $E}. -nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ -match r with -[ pz ⇒ {} -| pe ⇒ {} -| ps _ ⇒ {} -| pp x ⇒ { [x] } -| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 |r2| ∪ 𝐋\p\ r2 -| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2 -| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ]. -notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'in_pl $E}. -notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'in_pl $E}. -interpretation "in_pl" 'in_pl E = (in_pl ? E). -interpretation "in_pl mem" 'mem w l = (in_pl ? l w). - -ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}. - -interpretation "epsilon" 'epsilon = (epsilon ?). -notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}. -interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b). - -ndefinition in_prl ≝ λS : Alpha.λp:pre S. 𝐋\p (\fst p) ∪ ϵ (\snd p). +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)}. -interpretation "in_prl mem" 'mem w l = (in_prl ? l w). -interpretation "in_prl" 'in_pl E = (in_prl ? E). - -nlemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ]. -#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct; nqed. - -(* lemma 12 *) -nlemma epsilon_in_true : ∀S.∀e:pre S. [ ] ∈ e ↔ \snd e = true. -#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//; -nnormalize; *; ##[##2:*] nelim e; -##[ ##1,2: *; ##| #c; *; ##| #c; nnormalize; #; ndestruct; ##| ##7: #p H; -##| #r1 r2 H G; *; ##[##2: /3/ by or_intror] -##| #r1 r2 H1 H2; *; /3/ by or_intror, or_introl; ##] -*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//; -nqed. - -nlemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ((𝐋\p e) [ ]). -#S e; nelim e; nnormalize; /2/ by nmk; -##[ #; @; #; ndestruct; -##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H; - nrewrite > (append_eq_nil …H…); /2/; -##| #r1 r2 n1 n2; @; *; /2/; -##| #r n; @; *; #w1; *; #w2; *; *; #H; - nrewrite > (append_eq_nil …H…); /2/;##] -nqed. - -ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉. -notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}. -interpretation "oplus" 'oplus a b = (lo ? a b). - -ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S. - match a with [ mk_pair e1 b1 ⇒ - match b1 with - [ false ⇒ 〈e1 · \fst b, \snd b〉 - | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]]. - -notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}. -interpretation "lc" 'lc op a b = (lc ? op a b). -notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}. - -ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S. - match a with [ mk_pair e1 b1 ⇒ - match b1 with - [ false ⇒ 〈e1^*, false〉 - | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]]. - -notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}. -interpretation "lk" 'lk op a = (lk ? op a). -notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}. - -notation > "•" non associative with precedence 60 for @{eclose ?}. -nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝ - match E with - [ pz ⇒ 〈 ∅, false 〉 - | pe ⇒ 〈 ϵ, true 〉 - | ps x ⇒ 〈 `.x, false 〉 - | pp x ⇒ 〈 `.x, false 〉 - | po E1 E2 ⇒ •E1 ⊕ •E2 - | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉 - | pk E ⇒ 〈(\fst (•E))^*,true〉]. -notation < "• x" non associative with precedence 60 for @{'eclose $x}. -interpretation "eclose" 'eclose x = (eclose ? x). -notation > "• x" non associative with precedence 60 for @{'eclose $x}. - -ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉. -interpretation "reclose" 'eclose x = (reclose ? x). - -ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w. -notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}. -notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}. -interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b). - -naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q. - -nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S] -#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *; -nqed. - -nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c). -#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed. - -nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a. -#S a b; napply extP; #w; @; *; nnormalize; /2/; nqed. - -(* theorem 16: 2 *) -nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2. -#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2; -nwhd in ⊢ (??(??%)?); -nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2)); -nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2)); -nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …); -nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …); -nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //; -nqed. - -nlemma odotEt : - ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉. -#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed. - -nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed. - -nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r). -#S p q r; napply extP; #w; nnormalize; @; -##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj; -##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##] -nqed. - -nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p. -#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed. - -nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 |e1 · e2| = 𝐋 |e1| · 𝐋 |e2|. -#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj; -nqed. - -nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 |e1 + e2| = 𝐋 |e1| ∪ 𝐋 |e2|. -#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed. - -nlemma erase_star : ∀S.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed. - -ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w. -interpretation "substract" 'minus a b = (substract ? a b). - -nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}. -#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed. - -nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a. -#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed. - -nlemma subK : ∀S.∀a:word S → Prop. a - a = {}. -#S a; napply extP; #w; nnormalize; @; *; /2/; nqed. - -nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w. -#S a b w; nnormalize; *; //; nqed. - -nlemma erase_bull : ∀S.∀a:pitem S. |\fst (•a)| = |a|. -#S a; nelim a; // by {}; -##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| · |e2|); - nrewrite < IH1; nrewrite < IH2; - nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉)); - ncases (•e1); #e3 b; ncases b; nnormalize; - ##[ ncases (•e2); //; ##| nrewrite > IH2; //] -##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| + |e2|); - nrewrite < IH2; nrewrite < IH1; - nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2)); - ncases (•e1); ncases (•e2); //; -##| #e IH; nchange in ⊢ (???%) with (|e|^* ); nrewrite < IH; - nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##] -nqed. - -nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉. -#S p; ncases p; //; nqed. - -nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p. -#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##] -*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1; -napply Hw2; nqed. - -(* theorem 16: 1 → 3 *) -nlemma odot_dot_aux : ∀S.∀e1,e2: pre S. - 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| → - 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2. -#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1'; -##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2); - nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2))); - nchange in ⊢ (??%?) with (?∪?); - nchange in ⊢ (??(??%?)?) with (?∪?); - nchange in match (𝐋\p 〈?,?〉) with (?∪?); - nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…); - nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…); - nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 |e2'|); ##[##2: - nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 |e2'|); - ngeneralize in match th1; - nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##] - nrewrite > (eta_lp ? e2); - nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2'); - nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…); - nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…); - nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //; -##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉; - nchange in match (𝐋\p ?) with (?∪?); - nchange in match (𝐋\p (e1'·?)) with (?∪?); - nchange in match (𝐋\p 〈e1',?〉) with (?∪?); - nrewrite > (cup0…); - nrewrite > (cupA…); //;##] -nqed. - -nlemma sub_dot_star : - ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*. -#S X b; napply extP; #w; @; -##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //] - *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj; - @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //; - @; //; napply (subW … sube); -##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //] - #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *; - ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2; - @; ncases b in H1; #H1; - ##[##2: nrewrite > (sub0…); @w'; @(w1@w2); - nrewrite > (associative_append ? w' w1 w2); - nrewrite > defwl'; @; ##[@;//] @(wl'); @; //; - ##| ncases w' in Pw'; - ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //; - ##| #x xs Px; @(x::xs); @(w1@w2); - nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct] - @wl'; @; //; ##] ##] - ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl'); - nrewrite < (wlnil); nrewrite > (append_nil…); ncases b; - ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]); - nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct] - @[]; @; //; - ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //] - @; //; @; //; @; *;##]##]##] -nqed. - -(* theorem 16: 1 *) -alias symbol "pc" (instance 13) = "cat lang". -alias symbol "in_pl" (instance 23) = "in_pl". -alias symbol "in_pl" (instance 5) = "in_pl". -alias symbol "eclose" (instance 21) = "eclose". -ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 |e|. -#S e; nelim e; //; - ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror; - ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *; - ##| #e1 e2 IH1 IH2; - nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉); - nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2); - nrewrite > (IH1 …); nrewrite > (cup_dotD …); - nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …); - nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …); - nrewrite < (erase_dot …); nrewrite < (cupA …); //; - ##| #e1 e2 IH1 IH2; - nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …); - nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …); - nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…); - nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …); - nrewrite < (erase_plus …); //. - ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH; - nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉); - nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]}); - nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* ); - nrewrite > (erase_bull…e); - nrewrite > (erase_star …); - nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2: - nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH; - ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH; - nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//; - ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##] - nrewrite > (cup_dotD…); nrewrite > (cupA…); - nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //; - nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##] - nqed. - -(* theorem 16: 3 *) -nlemma odot_dot: - ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2. -#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed. - -nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*. -#S e; napply extP; #w; nnormalize; @; -##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2; - *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl); - nrewrite < defw; nrewrite < defw2; @; //; @;//; -##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //] - #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw; - @; /2/; @xs; /2/;##] - nqed. - -nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*. -#S e; @[]; /2/; nqed. - -nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l. -#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed. - -nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*. -#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed. - -nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S . - ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }. -#S A C b nbA defC; nrewrite < defC; napply extP; #w; @; -##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *] -nqed. - -(* theorem 16: 4 *) -nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*. -#S p; ncases p; #e b; ncases b; -##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉; - nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); - nchange in ⊢ (??%?) with (?∪?); - nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* ); - nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2: - nlapply (bull_cup ? e); #bc; - nchange in match (𝐋\p (•e)) in bc with (?∪?); - nchange in match b' in bc with b'; - ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //] - nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##] - nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…); - nrewrite > (sub_dot_star…); - nchange in match (𝐋\p 〈?,?〉) with (?∪?); - nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //; -##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?); - nrewrite > (cup0…); - nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* ); - nrewrite < (cup0 ? (𝐋\p e)); //;##] -nqed. - -nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝ - match e with - [ z ⇒ pz ? - | e ⇒ pe ? - | s x ⇒ ps ? x - | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2) - | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2) - | k e1 ⇒ pk ? (pre_of_re ? e1)]. - -nlemma notFalse : ¬False. @; //; nqed. - -nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}. -#S A; nnormalize; napply extP; #w; @; ##[##2: *] -*; #w1; *; #w2; *; *; //; nqed. - -nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}. -#S e; nelim e; ##[##1,2,3: //] -##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?); - nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);// -##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?); - nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); // -##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?); - nrewrite > H1; napply dot0; ##] -nqed. - -nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e. -#S A; nelim A; //; -##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?); - nrewrite < H1; nrewrite < H2; // -##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?); - nrewrite < H1; nrewrite < H2; // -##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* ); - nrewrite < H1; //] -nqed. - -(* corollary 17 *) -nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e). -#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…); -nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //; -nqed. - -nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w. -#S f g H; nrewrite > H; //; nqed. +(* 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 *) -(* corollary 18 *) -ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|. -#S e; @; -##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?); - nrewrite > defsnde; #H; - nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //; - -STOP - -notation > "\move term 90 x term 90 E" -non associative with precedence 60 for @{move ? $x $E}. -nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝ - match E with - [ pz ⇒ 〈 ∅, false 〉 - | pe ⇒ 〈 ϵ, false 〉 - | ps y ⇒ 〈 `y, false 〉 - | pp y ⇒ 〈 `y, x == y 〉 - | po e1 e2 ⇒ \move x e1 ⊕ \move x e2 - | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2 - | pk e ⇒ (\move x e)^⊛ ]. -notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}. -notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}. -interpretation "move" 'move x E = (move ? x E). - -ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e). -interpretation "rmove" 'move x E = (rmove ? x E). - -nlemma XXz : ∀S:Alpha.∀w:word S. w ∈ ∅ → False. -#S w abs; ninversion abs; #; ndestruct; -nqed. - - -nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False. -#S w abs; ninversion abs; #; ndestruct; -nqed. - -nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False. -#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe; -nqed. - - -naxiom in_move_cat: - ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) → - (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2. -#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2); -ncases e1 in H; ncases e2; -##[##1: *; ##[*; nnormalize; #; ndestruct] - #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct] - nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze; -##|##2: *; ##[*; nnormalize; #; ndestruct] - #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct] - nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze; -##| #r; *; ##[ *; nnormalize; #; ndestruct] - #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct] - ##[##2: nnormalize; #; ndestruct; @2; @2; //.##] - nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz; -##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##] - #H; ninversion H; nnormalize; #; ndestruct; - ##[ncases (?:False); /2/ by XXz] /3/ by or_intror; -##| #r1 r2; *; ##[ *; #defw] - ... -nqed. - -ntheorem move_ok: - ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E. -#S E; ncases E; #r b; nelim r; -##[##1,2: #a w; @; - ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##] - #H; ninversion H; #; ndestruct; - ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##] - #H; ninversion H; #; ndestruct;##] -##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##] - *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct; -##|#a c w; @; nnormalize; - ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##] - #H; ninversion H; #; ndestruct; - ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct] - #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##] -##|#r1 r2 H1 H2 a w; @; - ##[ #H; ncases (in_move_cat … H); - ##[ *; #w1; *; #w2; *; *; #defw w1m w2m; - ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good; - nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //. - ##| - ... -##| -##| -##] -nqed. - - -notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}. -nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝ - match w with - [ nil ⇒ E - | cons x w' ⇒ w' ↦* (x ↦ \snd E)]. - -ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E). - -ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝ - mk_equiv: - ∀E1,E2: bool × (pre S). - \fst E1 = \fst E2 → - (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) → - equiv S E1 E2. - -ndefinition NAT: decidable. - @ nat eqb; /2/. -nqed. - -include "hints_declaration.ma". - -alias symbol "hint_decl" (instance 1) = "hint_decl_Type1". -unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat. - -ninductive unit: Type[0] ≝ I: unit. - -nlet corec foo_nop (b: bool): - equiv ? - 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉 - 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?. - @; //; #x; ncases x - [ nnormalize in ⊢ (??%%); napply (foo_nop false) - | #y; ncases y - [ nnormalize in ⊢ (??%%); napply (foo_nop false) - | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##] -nqed. +#S #a #b % #H -(* -nlet corec foo (a: unit): - equiv NAT - (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))))) - (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0))) -≝ ?. - @; - ##[ nnormalize; // - ##| #x; ncases x - [ nnormalize in ⊢ (??%%); - nnormalize in foo: (? → ??%%); - @; //; #y; ncases y - [ nnormalize in ⊢ (??%%); napply foo_nop - | #y; ncases y - [ nnormalize in ⊢ (??%%); - - ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##] - ##| #y; nnormalize in ⊢ (??%%); napply foo_nop - ##] -nqed. -*) +(* 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 *) + + (\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)) // ] -nqed. *) - -ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e. - #S; #a; #E; #E'; #w; #H; nelim H - [##1,2: #H1; ninversion H1 - [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/ - |##2,9: #X; #Y; #K; ncases (?:False); /2/ - |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/ - |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - |##6,13: #x; #y; #K; ncases (?:False); /2/ - |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/] -##| #H1; ninversion H1 - [ // - | #X; #Y; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - | #x; #y; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ] -##| #H1; #H2; #H3; ninversion H3 - [ #_; #K; ncases (?:False); /2/ - | #X; #Y; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/ - | #x; #y; #K; ncases (?:False); /2/ - | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ] -##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6; \ No newline at end of file +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