-include "cahpter9.ma".
+(*
+\ 5h1\ 6Regular Expressions Equivalence\ 5/h1\ 6*)
-(* bisimulation *)
-definition cofinal ≝ λS.λp:(pre S)×(pre S).
- \snd (\fst p) = \snd (\snd p).
-
-theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S.
- \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+include "tutorial/chapter9.ma".
+
+(* We say that two pres \langle i_1,b_1\rangle$ and
+$\langle i_1,b_1\rangle$ are {\em cofinal} if and only if
+$b_1 = b_2$. *)
+
+definition cofinal ≝ λS.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+ \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
+
+(* As a corollary of decidable_sem, we have that two expressions
+e1 and e2 are equivalent iff for any word w the states reachable
+through w are cofinal. *)
+
+theorem equiv_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 ∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
#S #e1 #e2 %
[#same_sem #w
- cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2))
- [* * // * #H1 #H2 [@sym_eq @H1 //| @H2 //]]
- #Hcut @Hcut @iff_trans [|@decidable_sem]
- @iff_trans [|@same_sem] @iff_sym @decidable_sem
-|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
+ cut (∀b1,b2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (b1 \ 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) (b2 \ 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) → (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2))
+ [* * // * #H1 #H2 [@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @H1 //| @H2 //]]
+ #Hcut @Hcut @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter9/decidable_sem.def(15)"\ 6decidable_sem\ 5/a\ 6]
+ @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [|@same_sem] @\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter9/decidable_sem.def(15)"\ 6decidable_sem\ 5/a\ 6
+|#H #w1 @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter9/decidable_sem.def(15)"\ 6decidable_sem\ 5/a\ 6] <H @\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter9/decidable_sem.def(15)"\ 6decidable_sem\ 5/a\ 6]
qed.
-definition occ ≝ λS.λe1,e2:pre S.
- unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
+(* This does not directly imply decidability: we have no bound over the
+length of w; moreover, so far, we made no assumption over the cardinality
+of S. Instead of requiring S to be finite, we may restrict the analysis
+to characters occurring in the given pres. *)
+
+definition occ ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|)) (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)).
-lemma occ_enough: ∀S.∀e1,e2:pre S.
-(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
- →∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+lemma occ_enough: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
+ →∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
#S #e1 #e2 #H #w
-cases (decidable_sublist S w (occ S e1 e2)) [@H] -H #H
- >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l1 @H1 //]
- >to_pit [2: @(not_to_not … H) #H1 #a #memba @sublist_unique_append_l2 @H1 //]
+cases (\ 5a href="cic:/matita/tutorial/chapter5/decidable_sublist.def(6)"\ 6decidable_sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2)) [@H] -H #H
+ >\ 5a href="cic:/matita/tutorial/chapter9/to_pit.def(10)"\ 6to_pit\ 5/a\ 6 [2: @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 #a #memba @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 @H1 //]
+ >\ 5a href="cic:/matita/tutorial/chapter9/to_pit.def(10)"\ 6to_pit\ 5/a\ 6 [2: @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 #a #memba @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 @H1 //]
//
qed.
-lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
-(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
-→ \sem{e1}=1\sem{e2}.
-#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H
+(* The following is a stronger version of equiv_sem, relative to characters
+occurring the given regular expressions. *)
+
+lemma equiv_sem_occ: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
+→ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+#S #e1 #e2 #H @(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem.def(16)"\ 6equiv_sem\ 5/a\ 6 …)) @\ 5a href="cic:/matita/tutorial/chapter10/occ_enough.def(11)"\ 6occ_enough\ 5/a\ 6 #w @H
qed.
-definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S).
- map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
+(*
+\ 5h2\ 6Bisimulations\ 5/h2\ 6
-lemma memb_sons: ∀S,l.∀p,q:(pre S)×(pre S). memb ? p (sons ? l q) = true →
- ∃a.(move ? a (\fst (\fst q)) = \fst p ∧
- move ? a (\fst (\snd q)) = \snd p).
-#S #l elim l [#p #q normalize in ⊢ (%→?); #abs @False_ind /2/]
-#a #tl #Hind #p #q #H cases (orb_true_l … H) -H
- [#H @(ex_intro … a) >(\P H) /2/ |#H @Hind @H]
+We say that a list of pairs of pres is a bisimulation if it is closed
+w.r.t. moves, and all its members are cofinal.
+*)
+
+definition sons ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+ \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (λa.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)),\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))〉) l.
+
+lemma memb_sons: ∀S,l.∀p,q:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l q) \ 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 →
+ \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6a.(\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 q)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
+ \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 q)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
+#S #l elim l [#p #q normalize in ⊢ (%→?); #abs @\ 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/]
+#a #tl #Hind #p #q #H cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H) -H
+ [#H @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … a) >(\P H) /\ 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/ |#H @Hind @H]
qed.
-definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
- ∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
-
-lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S.
- is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
-#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ
-#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
-lapply Hsub @(list_elim_left … w) [//]
-#a #w1 #Hind #Hsub >moves_left >moves_left @(proj2 …(Hbisim …(Hind ?)))
- [#x #Hx @Hsub @memb_append_l1 //
- |cut (memb S a (occ S e1 e2) = true) [@Hsub @memb_append_l2 //] #occa
- @(memb_map … occa)
+definition is_bisim ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.λalpha:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.
+ ∀p:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 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 → \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? p \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? alpha p) l).
+
+(* Using lemma equiv_sem_occ it is easy to prove the following result: *)
+
+lemma bisim_to_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S l (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 l \ 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 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+#S #l #e1 #e2 #Hbisim #Hmemb @\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem_occ.def(17)"\ 6equiv_sem_occ\ 5/a\ 6
+#w #Hsub @(\ 5a href="cic:/matita/basics/logic/proj1.def(2)"\ 6proj1\ 5/a\ 6 … (Hbisim \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2〉 ?))
+lapply Hsub @(\ 5a href="cic:/matita/basics/list/list_elim_left.def(10)"\ 6list_elim_left\ 5/a\ 6 … w) [//]
+#a #w1 #Hind #Hsub >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 …(Hbisim …(Hind ?)))
+ [#x #Hx @Hsub @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 //
+ |cut (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) \ 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) [@Hsub @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 //] #occa
+ @(\ 5a href="cic:/matita/tutorial/chapter5/memb_map.def(5)"\ 6memb_map\ 5/a\ 6 … occa)
]
qed.
-(* the algorithm *)
-let rec bisim S l n (frontier,visited: list ?) on n ≝
+(* This is already an interesting result: checking if l is a bisimulation
+is decidable, hence we could generate l with some untrusted piece of code
+and then run a (boolean version of) is_bisim to check that it is actually
+a bisimulation.
+However, in order to prove that equivalence of regular expressions
+is decidable we must prove that we can always effectively build such a list
+(or find a counterexample).
+The idea is that the list we are interested in is just the set of
+all pair of pres reachable from the initial pair via some
+sequence of moves.
+
+The algorithm for computing reachable nodes in graph is a very
+traditional one. We split nodes in two disjoint lists: a list of
+visited nodes and a frontier, composed by all nodes connected
+to a node in visited but not visited already. At each step we select a node
+a from the frontier, compute its sons, add a to the set of
+visited nodes and the (not already visited) sons to the frontier.
+
+Instead of fist computing reachable nodes and then performing the
+bisimilarity test we can directly integrate it in the algorithm:
+the set of visited nodes is closed by construction w.r.t. reachability,
+so we have just to check cofinality for any node we add to visited.
+
+Here is the extremely simple algorithm: *)
+
+let rec bisim S l n (frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?) on n ≝
match n with
- [ O ⇒ 〈false,visited〉 (* assert false *)
+ [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
| S m ⇒
match frontier with
- [ nil ⇒ 〈true,visited〉
+ [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
| cons hd tl ⇒
- if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
- bisim S l m (unique_append ? (filter ? (λx.notb (memb ? x (hd::visited)))
- (sons S l hd)) tl) (hd::visited)
- else 〈false,visited〉
+ if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
+ bisim S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
+ else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
]
].
-
-lemma unfold_bisim: ∀S,l,n.∀frontier,visited: list ?.
- bisim S l n frontier visited =
+
+(* The integer n is an upper bound to the number of recursive call,
+equal to the dimension of the graph. It returns a pair composed
+by a boolean and a the set of visited nodes; the boolean is true
+if and only if all visited nodes are cofinal.
+
+The following results explicitly state the behaviour of bisim is the general
+case and in some relevant instances *)
+
+lemma unfold_bisim: ∀S,l,n.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
match n with
- [ O ⇒ 〈false,visited〉 (* assert false *)
+ [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
| S m ⇒
match frontier with
- [ nil ⇒ 〈true,visited〉
+ [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
| cons hd tl ⇒
- if beqb (\snd (\fst hd)) (\snd (\snd hd)) then
- bisim S l m (unique_append ? (filter ? (λx.notb(memb ? x (hd::visited)))
- (sons S l hd)) tl) (hd::visited)
- else 〈false,visited〉
+ if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
+ else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
]
].
#S #l #n cases n // qed.
-
-lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
- bisim S l O frontier visited = 〈false,visited〉.
-#frontier #visited >unfold_bisim //
+
+lemma bisim_never: ∀S,l.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+#frontier #visited >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 //
qed.
-lemma bisim_end: ∀Sig,l,m.∀visited: list ?.
- bisim Sig l (S m) [] visited = 〈true,visited〉.
-#n #visisted >unfold_bisim //
+lemma bisim_end: ∀Sig,l,m.∀visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉.
+#n #visisted >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 //
qed.
-lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
-beqb (\snd (\fst p)) (\snd (\snd p)) = true →
- bisim Sig l (S m) (p::frontier) visited =
- bisim Sig l m (unique_append ? (filter ? (λx.notb(memb ? x (p::visited)))
- (sons Sig l p)) frontier) (p::visited).
-#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
+lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 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 →
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 Sig l p)) frontier) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited).
+#Sig #l #m #p #frontier #visited #test >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 whd in ⊢ (??%?); >test //
qed.
-lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: list ?.
-beqb (\snd (\fst p)) (\snd (\snd p)) = false →
- bisim Sig l (S m) (p::frontier) visited = 〈false,visited〉.
-#Sig #l #m #p #frontier #visited #test >unfold_bisim normalize nodelta >test //
+lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 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 href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+#Sig #l #m #p #frontier #visited #test >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 whd in ⊢ (??%?); >test //
qed.
-lemma notb_eq_true_l: ∀b. notb b = true → b = false.
+(* In order to prove termination of bisim we must be able to effectively
+enumerate all possible pres: *)
+
+(* lemma notb_eq_true_l: ∀b. notb b = true → b = false.
#b cases b normalize //
-qed.
+qed. *)
-let rec pitem_enum S (i:re S) on i ≝
+let rec pitem_enum S (i:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
match i with
- [ z ⇒ [pz S]
- | e ⇒ [pe S]
- | s y ⇒ [ps S y; pp S y]
- | o i1 i2 ⇒ compose ??? (po S) (pitem_enum S i1) (pitem_enum S i2)
- | c i1 i2 ⇒ compose ??? (pc S) (pitem_enum S i1) (pitem_enum S i2)
- | k i ⇒ map ?? (pk S) (pitem_enum S i)
+ [ z ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+ | e ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"\ 6pe\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+ | s y ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"\ 6ps\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:(\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"\ 6pp\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+ | o i1 i2 ⇒ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,6,1)"\ 6po\ 5/a\ 6 S) (pitem_enum S i1) (pitem_enum S i2)
+ | c i1 i2 ⇒ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,5,1)"\ 6pc\ 5/a\ 6 S) (pitem_enum S i1) (pitem_enum S i2)
+ | k i ⇒ \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,7,1)"\ 6pk\ 5/a\ 6 S) (pitem_enum S i)
].
-lemma pitem_enum_complete : ∀S.∀i:pitem S.
- memb (DeqItem S) i (pitem_enum S (|i|)) = true.
+lemma pitem_enum_complete : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) i (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|)) \ 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.
#S #i elim i
[1,2://
- |3,4:#c normalize >(\b (refl … c)) //
- |5,6:#i1 #i2 #Hind1 #Hind2 @(memb_compose (DeqItem S) (DeqItem S)) //
- |#i #Hind @(memb_map (DeqItem S)) //
+ |3,4:#c normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … c)) //
+ |5,6:#i1 #i2 #Hind1 #Hind2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S)) //
+ |#i #Hind @(\ 5a href="cic:/matita/tutorial/chapter5/memb_map.def(5)"\ 6memb_map\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S)) //
]
qed.
-definition pre_enum ≝ λS.λi:re S.
- compose ??? (λi,b.〈i,b〉) (pitem_enum S i) [true;false].
+definition pre_enum ≝ λS.λi:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉) (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S i) (\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="cons" 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="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
-lemma pre_enum_complete : ∀S.∀e:pre S.
- memb ? e (pre_enum S (|\fst e|)) = true.
-#S * #i #b @(memb_compose (DeqItem S) DeqBool ? (λi,b.〈i,b〉))
+lemma pre_enum_complete : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? e (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|)) \ 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.
+#S * #i #b @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) \ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6 ? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
// cases b normalize //
qed.
-definition space_enum ≝ λS.λi1,i2:re S.
- compose ??? (λe1,e2.〈e1,e2〉) (pre_enum S i1) (pre_enum S i2).
+definition space_enum ≝ λS.λi1,i2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λe1,e2.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i1) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i2).
-lemma space_enum_complete : ∀S.∀e1,e2: pre S.
- memb ? 〈e1,e2〉 (space_enum S (|\fst e1|) (|\fst e2|)) = true.
-#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
+lemma space_enum_complete : ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)) \ 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.
+#S #e1 #e2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 … (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
// qed.
-definition all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
-uniqueb ? l = true ∧
- ∀p. memb ? p l = true →
- ∃w.(moves S w e1 = \fst p) ∧ (moves S w e2 = \snd p).
+definition all_reachable ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 ? l \ 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 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
+ ∀p. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 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 →
+ \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w.(\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
-definition disjoint ≝ λS:DeqSet.λl1,l2.
- ∀p:S. memb S p l1 = true → memb S p l2 = false.
+definition disjoint ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl1,l2.
+ ∀p:S. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S p l1 \ 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 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S p l2 \ 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.
-lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
- ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
- |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
- all_reachable S e1 e2 visited →
- all_reachable S e1 e2 frontier →
- disjoint ? frontier visited →
- \fst (bisim S l n frontier visited) = true.
+(* We are ready to prove that bisim is correct; we use the invariant
+that at each call of bisim the two lists visited and frontier only contain
+nodes reachable from \langle e_1,e_2\rangle, hence it is absurd to suppose
+to meet a pair which is not cofinal. *)
+
+lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} →
+ ∀l,n.∀frontier,visited:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ((\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)).
+ \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)| \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6visited|→
+ \ 5a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"\ 6all_reachable\ 5/a\ 6 S e1 e2 visited →
+ \ 5a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"\ 6all_reachable\ 5/a\ 6 S e1 e2 frontier →
+ \ 5a href="cic:/matita/tutorial/chapter10/disjoint.def(5)"\ 6disjoint\ 5/a\ 6 ? frontier visited →
+ \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited) \ 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.
#Sig #e1 #e2 #same #l #n elim n
- [#frontier #visited #abs * #unique #H @False_ind @(absurd … abs)
- @le_to_not_lt @sublist_length // * #e11 #e21 #membp
- cut ((|\fst e11| = |\fst e1|) ∧ (|\fst e21| = |\fst e2|))
- [|* #H1 #H2 <H1 <H2 @space_enum_complete]
- cases (H … membp) #w * #we1 #we2 <we1 <we2 % >same_kernel_moves //
- |#m #HI * [#visited #vinv #finv >bisim_end //]
+ [#frontier #visited #abs * #unique #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … abs)
+ @\ 5a href="cic:/matita/arithmetics/nat/le_to_not_lt.def(8)"\ 6le_to_not_lt\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_length.def(9)"\ 6sublist_length\ 5/a\ 6 // * #e11 #e21 #membp
+ cut ((\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e11| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e21| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|))
+ [|* #H1 #H2 <H1 <H2 @\ 5a href="cic:/matita/tutorial/chapter10/space_enum_complete.def(9)"\ 6space_enum_complete\ 5/a\ 6]
+ cases (H … membp) #w * #we1 #we2 <we1 <we2 % >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel_moves.def(9)"\ 6same_kernel_moves\ 5/a\ 6 //
+ |#m #HI * [#visited #vinv #finv >\ 5a href="cic:/matita/tutorial/chapter10/bisim_end.def(10)"\ 6bisim_end\ 5/a\ 6 //]
#p #front_tl #visited #Hn * #u_visited #r_visited * #u_frontier #r_frontier
#disjoint
- cut (∃w.(moves ? w e1 = \fst p) ∧ (moves ? w e2 = \snd p))
- [@(r_frontier … (memb_hd … ))] #rp
- cut (beqb (\snd (\fst p)) (\snd (\snd p)) = true)
+ cut (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w.(\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))
+ [@(r_frontier … (\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6 … ))] #rp
+ cut (\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 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)
[cases rp #w * #fstp #sndp <fstp <sndp @(\b ?)
- @(proj1 … (equiv_sem … )) @same] #ptest
- >(bisim_step_true … ptest) @HI -HI
- [<plus_n_Sm //
- |% [whd in ⊢ (??%?); >(disjoint … (memb_hd …)) whd in ⊢ (??%?); //
- |#p1 #H (cases (orb_true_l … H)) [#eqp >(\P eqp) // |@r_visited]
+ @(\ 5a href="cic:/matita/basics/logic/proj1.def(2)"\ 6proj1\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem.def(16)"\ 6equiv_sem\ 5/a\ 6 … )) @same] #ptest
+ >(\ 5a href="cic:/matita/tutorial/chapter10/bisim_step_true.def(10)"\ 6bisim_step_true\ 5/a\ 6 … ptest) @HI -HI
+ [<\ 5a href="cic:/matita/arithmetics/nat/plus_n_Sm.def(4)"\ 6plus_n_Sm\ 5/a\ 6 //
+ |% [whd in ⊢ (??%?); >(disjoint … (\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6 …)) whd in ⊢ (??%?); //
+ |#p1 #H (cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H)) [#eqp >(\P eqp) // |@r_visited]
]
- |whd % [@unique_append_unique @(andb_true_r … u_frontier)]
- @unique_append_elim #q #H
- [cases (memb_sons … (memb_filter_memb … H)) -H
- #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(ex_intro … (w1@[a]))
- >moves_left >moves_left >mw1 >mw2 >m1 >m2 % //
- |@r_frontier @memb_cons //
+ |whd % [@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_unique.def(6)"\ 6unique_append_unique\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6 … u_frontier)]
+ @\ 5a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"\ 6unique_append_elim\ 5/a\ 6 #q #H
+ [cases (\ 5a href="cic:/matita/tutorial/chapter10/memb_sons.def(8)"\ 6memb_sons\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"\ 6memb_filter_memb\ 5/a\ 6 … H)) -H
+ #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (w1\ 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:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])))
+ >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >mw1 >mw2 >m1 >m2 % //
+ |@r_frontier @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //
]
- |@unique_append_elim #q #H
- [@injective_notb @(filter_true … H)
- |cut ((q==p) = false)
- [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @memb_cons //]
- cases (andb_true … u_frontier) #notp #_ @(\bf ?)
- @(not_to_not … not_eq_true_false) #eqqp <notp <eqqp >H //
+ |@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"\ 6unique_append_elim\ 5/a\ 6 #q #H
+ [@\ 5a href="cic:/matita/basics/bool/injective_notb.def(4)"\ 6injective_notb\ 5/a\ 6 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"\ 6memb_filter_true\ 5/a\ 6 … H\ 5span class="error" title="error location"\ 6\ 5/span\ 6)
+ |cut ((q\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=p) \ 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)
+ [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //]
+ cases (\ 5a href="cic:/matita/basics/bool/andb_true.def(5)"\ 6andb_true\ 5/a\ 6 … u_frontier) #notp #_ @(\bf ?)
+ @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … \ 5a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"\ 6not_eq_true_false\ 5/a\ 6) #eqqp <notp <eqqp >H //
]
]
]
-qed.
+qed.
+
+(* For completeness, we use the invariant that all the nodes in visited are cofinal,
+and the sons of visited are either in visited or in the frontier; since
+at the end frontier is empty, visited is hence a bisimulation. *)
-definition all_true ≝ λS.λl.∀p:(pre S) × (pre S). memb ? p l = true →
- (beqb (\snd (\fst p)) (\snd (\snd p)) = true).
+definition all_true ≝ λS.λl.∀p:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 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 →
+ (\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 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).
-definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).
-memb ? x l1 = true → sublist ? (sons ? l x) l2.
+definition sub_sons ≝ λS,l,l1,l2.∀x:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x l1 \ 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 → \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l x) l2.
lemma bisim_complete:
- ∀S,l,n.∀frontier,visited,visited_res:list ?.
- all_true S visited →
- sub_sons S l visited (frontier@visited) →
- bisim S l n frontier visited = 〈true,visited_res〉 →
- is_bisim S visited_res l ∧ sublist ? (frontier@visited) visited_res.
+ ∀S,l,n.∀frontier,visited,visited_res:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S visited →
+ \ 5a href="cic:/matita/tutorial/chapter10/sub_sons.def(8)"\ 6sub_sons\ 5/a\ 6 S l visited (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) →
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited_res〉 →
+ \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S visited_res l \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) visited_res.
#S #l #n elim n
- [#fron #vis #vis_res #_ #_ >bisim_never #H destruct
+ [#fron #vis #vis_res #_ #_ >\ 5a href="cic:/matita/tutorial/chapter10/bisim_never.def(10)"\ 6bisim_never\ 5/a\ 6 #H destruct
|#m #Hind *
[(* case empty frontier *)
-Hind #vis #vis_res #allv #H normalize in ⊢ (%→?);
#H1 destruct % #p
[#membp % [@(\P ?) @allv //| @H //]|#H1 @H1]
- |#hd cases (true_or_false (beqb (\snd (\fst hd)) (\snd (\snd hd))))
+ |#hd cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd))))
[|(* case head of the frontier is non ok (absurd) *)
- #H #tl #vis #vis_res #allv >(bisim_step_false … H) #_ #H1 destruct]
+ #H #tl #vis #vis_res #allv >(\ 5a href="cic:/matita/tutorial/chapter10/bisim_step_false.def(10)"\ 6bisim_step_false\ 5/a\ 6 … H) #_ #H1 destruct]
(* frontier = hd:: tl and hd is ok *)
- #H #tl #visited #visited_res #allv >(bisim_step_true … H)
+ #H #tl #visited #visited_res #allv >(\ 5a href="cic:/matita/tutorial/chapter10/bisim_step_true.def(10)"\ 6bisim_step_true\ 5/a\ 6 … H)
(* new_visited = hd::visited are all ok *)
- cut (all_true S (hd::visited))
- [#p #H1 cases (orb_true_l … H1) [#eqp >(\P eqp) @H |@allv]]
+ cut (\ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited))
+ [#p #H1 cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H1) [#eqp >(\P eqp) @H |@allv]]
(* we now exploit the induction hypothesis *)
#allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind
- [#H1 #H2 % // #p #membp @H2 -H2 cases (memb_append … membp) -membp #membp
- [cases (orb_true_l … membp) -membp #membp
- [@memb_append_l2 >(\P membp) @memb_hd
- |@memb_append_l1 @sublist_unique_append_l2 //
+ [#H1 #H2 % // #p #membp @H2 -H2 cases (\ 5a href="cic:/matita/tutorial/chapter5/memb_append.def(5)"\ 6memb_append\ 5/a\ 6 … membp) -membp #membp
+ [cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … membp) -membp #membp
+ [@\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 >(\P membp) @\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6
+ |@\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 //
]
- |@memb_append_l2 @memb_cons //
+ |@\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //
]
|(* the only thing left to prove is the sub_sons invariant *)
- #x #membx cases (orb_true_l … membx)
+ #x #membx cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … membx)
[(* case x = hd *)
#eqhdx <(\P eqhdx) #xa #membxa
(* xa is a son of x; we must distinguish the case xa
was already visited form the case xa is new *)
- cases (true_or_false … (memb ? xa (x::visited)))
- [(* xa visited - trivial *) #membxa @memb_append_l2 //
- |(* xa new *) #membxa @memb_append_l1 @sublist_unique_append_l1 @memb_filter_l
+ cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? xa (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
+ [(* xa visited - trivial *) #membxa @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 //
+ |(* xa new *) #membxa @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_l.def(5)"\ 6memb_filter_l\ 5/a\ 6
[>membxa //|//]
]
|(* case x in visited *)
- #H1 #xa #membxa cases (memb_append … (subH x … H1 … membxa))
- [#H2 (cases (orb_true_l … H2))
- [#H3 @memb_append_l2 <(\P H3) @memb_hd
- |#H3 @memb_append_l1 @sublist_unique_append_l2 @H3
+ #H1 #xa #membxa cases (\ 5a href="cic:/matita/tutorial/chapter5/memb_append.def(5)"\ 6memb_append\ 5/a\ 6 … (subH x … H1 … membxa))
+ [#H2 (cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H2))
+ [#H3 @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 <(\P H3) @\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6
+ |#H3 @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 @H3
]
- |#H2 @memb_append_l2 @memb_cons @H2
+ |#H2 @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 @H2
]
]
]
]
qed.
-definition equiv ≝ λSig.λre1,re2:re Sig.
- let e1 ≝ •(blank ? re1) in
- let e2 ≝ •(blank ? re2) in
- let n ≝ S (length ? (space_enum Sig (|\fst e1|) (|\fst e2|))) in
- let sig ≝ (occ Sig e1 e2) in
- (bisim ? sig n [〈e1,e2〉] []).
+(* We can now give the definition of the equivalence algorithm, and
+prove that two expressions are equivalente if and only if they define
+the same language. *)
+
+definition equiv ≝ λSig.λre1,re2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
+ let e1 ≝ \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re1) in
+ let e2 ≝ \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re2) in
+ let n ≝ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 Sig (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|))) in
+ let sig ≝ (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 Sig e1 e2) in
+ (\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 ? sig n (\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉\ 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="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
-theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
- \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
+theorem euqiv_sem : ∀Sig.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
+ \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? e1 e2) \ 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 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
#Sig #re1 #re2 %
- [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
- cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
+ [#H @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6] @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6]
+ cut (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2)〉)
[<H //] #Hcut
- cases (bisim_complete … Hcut)
- [2,3: #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/]
- #Hbisim #Hsub @(bisim_to_sem … Hbisim)
- @Hsub @memb_hd
- |#H @(bisim_correct ? (•(blank ? re1)) (•(blank ? re2)))
- [@eqP_trans [|@re_embedding] @eqP_trans [|@H] @eqP_sym @re_embedding
+ cases (\ 5a href="cic:/matita/tutorial/chapter10/bisim_complete.def(11)"\ 6bisim_complete\ 5/a\ 6 … Hcut)
+ [2,3: #p whd in ⊢ ((??%?)→?); #abs @\ 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/]
+ #Hbisim #Hsub @(\ 5a href="cic:/matita/tutorial/chapter10/bisim_to_sem.def(18)"\ 6bisim_to_sem\ 5/a\ 6 … Hbisim)
+ @Hsub @\ 5a href="cic:/matita/tutorial/chapter5/memb_hd.def(5)"\ 6memb_hd\ 5/a\ 6
+ |#H @(\ 5a href="cic:/matita/tutorial/chapter10/bisim_correct.def(17)"\ 6bisim_correct\ 5/a\ 6 ? (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re1)) (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re2)))
+ [@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6] @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@H] @\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6
|//
- |% // #p whd in ⊢ ((??%?)→?); #abs @False_ind /2/
- |% // #p #H >(memb_single … H) @(ex_intro … ϵ) /2/
+ |% // #p whd in ⊢ ((??%?)→?); #abs @\ 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/
+ |% // #p #H >(\ 5a href="cic:/matita/tutorial/chapter5/memb_single.def(5)"\ 6memb_single\ 5/a\ 6 … H) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6) /\ 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/
|#p #_ normalize //
]
]
qed.
-lemma eqbnat_true : ∀n,m. eqbnat n m = true ↔ n = m.
-#n #m % [@eqbnat_true_to_eq | @eq_to_eqbnat_true]
-qed.
-
-definition DeqNat ≝ mk_DeqSet nat eqbnat eqbnat_true.
-
-definition a ≝ s DeqNat O.
-definition b ≝ s DeqNat (S O).
-definition c ≝ s DeqNat (S (S O)).
-
-definition exp1 ≝ ((a·b)^*·a).
-definition exp2 ≝ a·(b·a)^*.
-definition exp4 ≝ (b·a)^*.
-
-definition exp6 ≝ a·(a ·a ·b^* + b^* ).
-definition exp7 ≝ a · a^* · b^*.
-
-definition exp8 ≝ a·a·a·a·a·a·a·a·(a^* ).
-definition exp9 ≝ (a·a·a + a·a·a·a·a)^*.
-
-example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
-normalize // qed.
+definition eqbnat ≝ λn,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m.
+lemma eqbnat_true : ∀n,m. \ 5a href="cic:/matita/tutorial/chapter10/eqbnat.def(2)"\ 6eqbnat\ 5/a\ 6 n m \ 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 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+#n #m % [@\ 5a href="cic:/matita/arithmetics/nat/eqb_true_to_eq.def(6)"\ 6eqb_true_to_eq\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/eq_to_eqb_true.def(5)"\ 6eq_to_eqb_true\ 5/a\ 6]
+qed.
+definition DeqNat ≝ \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/eqbnat_true.def(7)"\ 6eqbnat_true\ 5/a\ 6.
+definition a ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+definition b ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6).
+definition c ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)).
+definition exp1 ≝ ((\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6).
+definition exp2 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+definition exp4 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+definition exp6 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ).
+definition exp7 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+definition exp8 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ).
+definition exp9 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+example ex1 : \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/exp8.def(10)"\ 6exp8\ 5/a\ 6\ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 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.
+normalize // qed.
\ No newline at end of file
projects / helm.git / blobdiff