X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter10.ma;h=15961e5367ff674cf433e224f7f882eaa4b721fd;hb=26d2ecb945a881c61d03f3c259996374209f5d7f;hp=08c78c63281a770230da31b5d38639e38999dfdb;hpb=102c93c0642ae14649d501332a075124ae18e5b3;p=helm.git diff --git a/weblib/tutorial/chapter10.ma b/weblib/tutorial/chapter10.ma index 08c78c632..15961e536 100644 --- a/weblib/tutorial/chapter10.ma +++ b/weblib/tutorial/chapter10.ma @@ -3,19 +3,18 @@ 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 -$b_1 = b_2$. *) +(* We say that two pres 〈i_1,b_1〉 and 〈i_1,b_1〉 are {\em cofinal} if and +only if b_1 = b_2. *) -definition cofinal ≝ λS.λp:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). +img class="anchor" src="icons/tick.png" id="cofinal"definition cofinal ≝ λS.λp:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a 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:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. - a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1} a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2} a title="iff" href="cic:/fakeuri.def(1)"↔/a ∀w.a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2〉. +img class="anchor" src="icons/tick.png" id="equiv_sem"theorem equiv_sem: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="iff" href="cic:/fakeuri.def(1)"↔/a ∀w.a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. #S #e1 #e2 % [#same_sem #w cut (∀b1,b2. a href="cic:/matita/basics/logic/iff.def(1)"iff/a (b1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) (b2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) → (b1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b2)) @@ -30,12 +29,12 @@ 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:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. - a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1|)) (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2|)). +img class="anchor" src="icons/tick.png" id="occ"definition occ ≝ λS.λe1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a)) (a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"occur/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a)). -lemma occ_enough: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. -(∀w.(a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S w (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2))→ a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2〉) - →∀w.a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2〉. +img class="anchor" src="icons/tick.png" id="occ_enough"lemma occ_enough: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. +(∀w.(a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S w (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2))→ a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) + →∀w.a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. #S #e1 #e2 #H #w cases (a href="cic:/matita/tutorial/chapter5/decidable_sublist.def(6)"decidable_sublist/a S w (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2)) [@H] -H #H >a href="cic:/matita/tutorial/chapter9/to_pit.def(10)"to_pit/a [2: @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … H) #H1 #a #memba @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"sublist_unique_append_l1/a @H1 //] @@ -46,9 +45,9 @@ qed. (* The following is a stronger version of equiv_sem, relative to characters occurring the given regular expressions. *) -lemma equiv_sem_occ: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. -(∀w.(a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S w (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2))→ a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2〉) -→ a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1}a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2}. +img class="anchor" src="icons/tick.png" id="equiv_sem_occ"lemma equiv_sem_occ: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. +(∀w.(a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a S w (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2))→ a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a ? w e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) +→ a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/aa title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a. #S #e1 #e2 #H @(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (a href="cic:/matita/tutorial/chapter10/equiv_sem.def(16)"equiv_sem/a …)) @a href="cic:/matita/tutorial/chapter10/occ_enough.def(11)"occ_enough/a #w @H qed. @@ -59,10 +58,10 @@ 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:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a S.λp:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). - a href="cic:/matita/basics/list/map.fix(0,3,1)"map/a ?? (λa.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p)),a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p))〉) l. +img class="anchor" src="icons/tick.png" id="sons"definition sons ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a S.λp:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). + a href="cic:/matita/basics/list/map.fix(0,3,1)"map/a ?? (λa.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p)),a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a S a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p))a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) l. -lemma memb_sons: ∀S,l.∀p,q:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a ? l q) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → +img class="anchor" src="icons/tick.png" id="memb_sons"lemma memb_sons: ∀S,l.∀p,q:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a ? l q) 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="exists" href="cic:/fakeuri.def(1)"∃/aa.(a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a q)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p a title="logical and" href="cic:/fakeuri.def(1)"∧/a a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"move/a ? a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a q)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p). #S #l elim l [#p #q normalize in ⊢ (%→?); #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/] @@ -70,15 +69,15 @@ lemma memb_sons: ∀S,l.∀p,q:(a href="cic:/matita/tutorial/chapter7/pre.def(1 [#H @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … a) >(\P H) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ |#H @Hind @H] qed. -definition is_bisim ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?.λalpha:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a S. +img class="anchor" src="icons/tick.png" id="is_bisim"definition is_bisim ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?.λalpha:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a S. ∀p:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p l 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 href="cic:/matita/tutorial/chapter10/cofinal.def(2)"cofinal/a ? p a title="logical and" href="cic:/fakeuri.def(1)"∧/a (a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a ? (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a ? alpha p) l). (* Using lemma equiv_sem_occ it is easy to prove the following result: *) -lemma bisim_to_sem: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀l:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. - a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"is_bisim/a S l (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2) → a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2〉 l 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="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1}a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2}. +img class="anchor" src="icons/tick.png" id="bisim_to_sem"lemma bisim_to_sem: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀l:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"is_bisim/a S l (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a S e1 e2) → a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a l 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="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/aa title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a. #S #l #e1 #e2 #Hbisim #Hmemb @a href="cic:/matita/tutorial/chapter10/equiv_sem_occ.def(17)"equiv_sem_occ/a -#w #Hsub @(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hbisim a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e2〉 ?)) +#w #Hsub @(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hbisim a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e1,a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a ?)) lapply Hsub @(a href="cic:/matita/basics/list/list_elim_left.def(10)"list_elim_left/a … w) [//] #a #w1 #Hind #Hsub >a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"moves_left/a >a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"moves_left/a @(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a …(Hbisim …(Hind ?))) [#x #Hx @Hsub @a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"memb_append_l1/a // @@ -112,17 +111,17 @@ 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: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?) on n ≝ +img class="anchor" src="icons/tick.png" id="bisim"let rec bisim S l n (frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?) on n ≝ match n with - [ O ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉 (* assert false *) + [ O ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a (* assert false *) | S m ⇒ match frontier with - [ nil ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visited〉 + [ nil ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a | cons hd tl ⇒ if a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a hd)) (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a hd)) then - bisim S l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (hda title="cons" href="cic:/fakeuri.def(1)":/a:visited))) - (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a S l hd)) tl) (hda title="cons" href="cic:/fakeuri.def(1)":/a:visited) - else a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉 + bisim S l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (hda title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited))) + (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a S l hd)) tl) (hda title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited) + else a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a ] ]. @@ -134,43 +133,43 @@ 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: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. +img class="anchor" src="icons/tick.png" id="unfold_bisim"lemma unfold_bisim: ∀S,l,n.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l n frontier visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a match n with - [ O ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉 (* assert false *) + [ O ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a (* assert false *) | S m ⇒ match frontier with - [ nil ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visited〉 + [ nil ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a | cons hd tl ⇒ if a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a hd)) (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a hd)) then - a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a(a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (hda title="cons" href="cic:/fakeuri.def(1)":/a:visited))) - (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a S l hd)) tl) (hda title="cons" href="cic:/fakeuri.def(1)":/a:visited) - else a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉 + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a(a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (hda title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited))) + (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a S l hd)) tl) (hda title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited) + else a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a ] ]. #S #l #n cases n // qed. -lemma bisim_never: ∀S,l.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. - a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a frontier visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉. +img class="anchor" src="icons/tick.png" id="bisim_never"lemma bisim_never: ∀S,l.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"O/a frontier visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. #frontier #visited >a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"unfold_bisim/a // qed. -lemma bisim_end: ∀Sig,l,m.∀visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. - a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) a title="nil" href="cic:/fakeuri.def(1)"[/a] visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visited〉. +img class="anchor" src="icons/tick.png" id="bisim_end"lemma bisim_end: ∀Sig,l,m.∀visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) a title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. #n #visisted >a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"unfold_bisim/a // qed. -lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. +img class="anchor" src="icons/tick.png" id="bisim_step_true"lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p)) (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p)) 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 href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) (pa title="cons" href="cic:/fakeuri.def(1)":/a:frontier) visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a - a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a(a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (pa title="cons" href="cic:/fakeuri.def(1)":/a:visited))) - (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a Sig l p)) frontier) (pa title="cons" href="cic:/fakeuri.def(1)":/a:visited). + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) (pa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/afrontier) visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l m (a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"unique_append/a ? (a href="cic:/matita/basics/list/filter.def(2)"filter/a ? (λx.a href="cic:/matita/basics/bool/notb.def(1)"notb/a(a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x (pa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited))) + (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a Sig l p)) frontier) (pa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/avisited). #Sig #l #m #p #frontier #visited #test >a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"unfold_bisim/a whd in ⊢ (??%?); >test // qed. -lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. +img class="anchor" src="icons/tick.png" id="bisim_step_false"lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p)) (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p)) 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 href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) (pa title="cons" href="cic:/fakeuri.def(1)":/a:frontier) visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visited〉. + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a Sig l (a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a m) (pa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/afrontier) visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,visiteda title="Pair construction" href="cic:/fakeuri.def(1)"〉/a. #Sig #l #m #p #frontier #visited #test >a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"unfold_bisim/a whd in ⊢ (??%?); >test // qed. @@ -181,18 +180,18 @@ enumerate all possible pres: *) #b cases b normalize // qed. *) -let rec pitem_enum S (i:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S) on i ≝ +img class="anchor" src="icons/tick.png" id="pitem_enum"let rec pitem_enum S (i:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S) on i ≝ match i with - [ z ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/aspan class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"/spanspan class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"/span S)a title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] - | e ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"pe/a S)a title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] - | s y ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"ps/a S y)a title="cons" href="cic:/fakeuri.def(1)":/a:(a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"pp/a S y)a title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" href="cic:/fakeuri.def(1)"[/a] + [ z ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/aspan class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"/spanspan class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"/span S)a title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a + | e ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"pe/a S)a title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a + | s y ⇒ (a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"ps/a S y)a title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/a(a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"pp/a S y)a title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a | o i1 i2 ⇒ a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (a href="cic:/matita/tutorial/chapter7/pitem.con(0,6,1)"po/a S) (pitem_enum S i1) (pitem_enum S i2) | c i1 i2 ⇒ a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (a href="cic:/matita/tutorial/chapter7/pitem.con(0,5,1)"pc/a S) (pitem_enum S i1) (pitem_enum S i2) | k i ⇒ a href="cic:/matita/basics/list/map.fix(0,3,1)"map/a ?? (a href="cic:/matita/tutorial/chapter7/pitem.con(0,7,1)"pk/a S) (pitem_enum S i) ]. -lemma pitem_enum_complete : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. - a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a (a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"DeqItem/a S) i (a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"pitem_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/ai|)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. +img class="anchor" src="icons/tick.png" id="pitem_enum_complete"lemma pitem_enum_complete : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. + a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a (a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"DeqItem/a S) i (a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"pitem_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/a)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. #S #i elim i [1,2:// |3,4:#c normalize >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … c)) // @@ -201,47 +200,47 @@ lemma pitem_enum_complete : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pit ] qed. -definition pre_enum ≝ λS.λi:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. - a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,b〉) (a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"pitem_enum/a S i) a title="cons" href="cic:/fakeuri.def(1)"[/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a;a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]. +img class="anchor" src="icons/tick.png" id="pre_enum"definition pre_enum ≝ λS.λi:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. + a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) (a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"pitem_enum/a S i) (a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a). -lemma pre_enum_complete : ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. - a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? e (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e|)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. -#S * #i #b @(a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"memb_compose/a (a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"DeqItem/a S) a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"DeqBool/a ? (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,b〉)) +img class="anchor" src="icons/tick.png" id="pre_enum_complete"lemma pre_enum_complete : ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? e (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="forget" href="cic:/fakeuri.def(1)"|/a)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. +#S * #i #b @(a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"memb_compose/a (a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"DeqItem/a S) a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"DeqBool/a ? (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a)) // cases b normalize // qed. -definition space_enum ≝ λS.λi1,i2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. - a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (λe1,e2.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2〉) (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S i1) (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S i2). +img class="anchor" src="icons/tick.png" id="space_enum"definition space_enum ≝ λS.λi1,i2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. + a href="cic:/matita/basics/list/compose.def(2)"compose/a ??? (λe1,e2.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S i1) (a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"pre_enum/a S i2). -lemma space_enum_complete : ∀S.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. - a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2〉 (a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1|) (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2|)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. -#S #e1 #e2 @(a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"memb_compose/a … (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,b〉)) +img class="anchor" src="icons/tick.png" id="space_enum_complete"lemma space_enum_complete : ∀S.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. + a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a (a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a) (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. +#S #e1 #e2 @(a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"memb_compose/a … (λi,b.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a)) // qed. -definition all_reachable ≝ λS.λe1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λl: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. +img class="anchor" src="icons/tick.png" id="all_reachable"definition all_reachable ≝ λS.λe1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λl: a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"uniqueb/a ? l 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="logical and" href="cic:/fakeuri.def(1)"∧/a ∀p. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p l 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="exists" href="cic:/fakeuri.def(1)"∃/aw.(a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p) a title="logical and" href="cic:/fakeuri.def(1)"∧/a (a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"moves/a S w e2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p). -definition disjoint ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl1,l2. +img class="anchor" src="icons/tick.png" id="disjoint"definition disjoint ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λl1,l2. ∀p:S. a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S p l1 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 href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a S p l2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a. (* 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. *) +nodes reachable from 〈e_1,e_2〉, hence it is absurd to suppose to meet a pair +which is not cofinal. *) -lemma bisim_correct: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1}a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2} → +img class="anchor" src="icons/tick.png" id="bisim_correct"lemma bisim_correct: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/aa title="extensional equality" href="cic:/fakeuri.def(1)"=/a1a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a → ∀l,n.∀frontier,visited:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ((a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)a title="Product" href="cic:/fakeuri.def(1)"×/a(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S)). - a title="norm" href="cic:/fakeuri.def(1)"|/aa href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1|) (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2|)| a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n a title="natural plus" href="cic:/fakeuri.def(1)"+/a a title="norm" href="cic:/fakeuri.def(1)"|/avisited|→ + a title="norm" href="cic:/fakeuri.def(1)"|/aa href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a S (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a) (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a)a title="norm" href="cic:/fakeuri.def(1)"|/a a title="natural 'less than'" href="cic:/fakeuri.def(1)"</a n a title="natural plus" href="cic:/fakeuri.def(1)"+/a a title="norm" href="cic:/fakeuri.def(1)"|/avisiteda title="norm" href="cic:/fakeuri.def(1)"|/a→ a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"all_reachable/a S e1 e2 visited → a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"all_reachable/a S e1 e2 frontier → a href="cic:/matita/tutorial/chapter10/disjoint.def(5)"disjoint/a ? frontier visited → a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l n frontier visited) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a. #Sig #e1 #e2 #same #l #n elim n [#frontier #visited #abs * #unique #H @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @(a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a … abs) - @a href="cic:/matita/arithmetics/nat/le_to_not_lt.def(8)"le_to_not_lt/a @a href="cic:/matita/tutorial/chapter5/sublist_length.def(9)"sublist_length/a // * #e11 #e21 #membp - cut ((a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e11| a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1|) a title="logical and" href="cic:/fakeuri.def(1)"∧/a (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e21| a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2|)) + @a href="cic:/matita/arithmetics/nat/le_to_not_lt.def(8)"le_to_not_lt/a @a href="cic:/matita/tutorial/chapter5/sublist_length.def(6)"sublist_length/a // * #e11 #e21 #membp + cut ((|a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e11a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a |a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a) a title="logical and" href="cic:/fakeuri.def(1)"∧/a (|a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e21a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a |a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a)) [|* #H1 #H2

a href="cic:/matita/tutorial/chapter9/same_kernel_moves.def(9)"same_kernel_moves/a // |#m #HI * [#visited #vinv #finv >a href="cic:/matita/tutorial/chapter10/bisim_end.def(10)"bisim_end/a //] @@ -258,15 +257,15 @@ lemma bisim_correct: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.de |#p1 #H (cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … H)) [#eqp >(\P eqp) // |@r_visited] ] |whd % [@a href="cic:/matita/tutorial/chapter5/unique_append_unique.def(6)"unique_append_unique/a @(a href="cic:/matita/basics/bool/andb_true_r.def(4)"andb_true_r/a … u_frontier)] - @a href="cic:/matita/tutorial/chapter5/unique_append_elim.dec"unique_append_elim/a #q #H + @a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"unique_append_elim/a #q #H [cases (a href="cic:/matita/tutorial/chapter10/memb_sons.def(8)"memb_sons/a … (a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"memb_filter_memb/a … H)) -H - #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … (w1a title="append" href="cic:/fakeuri.def(1)"@/aa title="cons" href="cic:/fakeuri.def(1)"[/aa])) + #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(a href="cic:/matita/basics/logic/ex.con(0,1,2)"ex_intro/a … (w1a title="append" href="cic:/fakeuri.def(1)"@/a(a:a title="cons" href="cic:/fakeuri.def(1)":/a[a title="nil" href="cic:/fakeuri.def(1)"]/a))) >a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"moves_left/a >a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"moves_left/a >mw1 >mw2 >m1 >m2 % // |@r_frontier @a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"memb_cons/a // ] - |@a href="cic:/matita/tutorial/chapter5/unique_append_elim.dec"unique_append_elim/a #q #H - [@a href="cic:/matita/basics/bool/injective_notb.def(4)"injective_notb/a @(a href="cic:/matita/tutorial/chapter5/filter_true.def(5)"filter_true/a … H) - |cut ((qa title="eqb" href="cic:/fakeuri.def(1)"=/a=p) 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 href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"unique_append_elim/a #q #H + [@a href="cic:/matita/basics/bool/injective_notb.def(4)"injective_notb/a @(a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"memb_filter_true/a … H) + |cut ((q=a title="eqb" href="cic:/fakeuri.def(1)"=/ap) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) [|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"memb_cons/a //] cases (a href="cic:/matita/basics/bool/andb_true.def(5)"andb_true/a … u_frontier) #notp #_ @(\bf ?) @(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) #eqqp H // @@ -274,22 +273,22 @@ lemma bisim_correct: ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.de ] ] 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:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S) a title="Product" href="cic:/fakeuri.def(1)"×/a (a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p l a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → +img class="anchor" src="icons/tick.png" id="all_true"definition all_true ≝ λS.λl.∀p:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S) a title="Product" href="cic:/fakeuri.def(1)"×/a (a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? p l 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 href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p)) (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p)) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a). -definition sub_sons ≝ λS,l,l1,l2.∀x:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S) a title="Product" href="cic:/fakeuri.def(1)"×/a (a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). +img class="anchor" src="icons/tick.png" id="sub_sons"definition sub_sons ≝ λS,l,l1,l2.∀x:(a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S) a title="Product" href="cic:/fakeuri.def(1)"×/a (a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S). a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? x l1 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 href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a ? (a href="cic:/matita/tutorial/chapter10/sons.def(7)"sons/a ? l x) l2. -lemma bisim_complete: +img class="anchor" src="icons/tick.png" id="bisim_complete"lemma bisim_complete: ∀S,l,n.∀frontier,visited,visited_res:a href="cic:/matita/basics/list/list.ind(1,0,1)"list/a ?. a href="cic:/matita/tutorial/chapter10/all_true.def(8)"all_true/a S visited → a href="cic:/matita/tutorial/chapter10/sub_sons.def(8)"sub_sons/a S l visited (frontiera title="append" href="cic:/fakeuri.def(1)"@/avisited) → - a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l n frontier visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visited_res〉 → + a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a S l n frontier visited a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a 〈a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,visited_resa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a → a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"is_bisim/a S visited_res l a title="logical and" href="cic:/fakeuri.def(1)"∧/a a href="cic:/matita/tutorial/chapter5/sublist.def(5)"sublist/a ? (frontiera title="append" href="cic:/fakeuri.def(1)"@/avisited) visited_res. #S #l #n elim n [#fron #vis #vis_res #_ #_ >a href="cic:/matita/tutorial/chapter10/bisim_never.def(10)"bisim_never/a #H destruct @@ -304,7 +303,7 @@ lemma bisim_complete: (* frontier = hd:: tl and hd is ok *) #H #tl #visited #visited_res #allv >(a href="cic:/matita/tutorial/chapter10/bisim_step_true.def(10)"bisim_step_true/a … H) (* new_visited = hd::visited are all ok *) - cut (a href="cic:/matita/tutorial/chapter10/all_true.def(8)"all_true/a S (hda title="cons" href="cic:/fakeuri.def(1)":/a:visited)) + cut (a href="cic:/matita/tutorial/chapter10/all_true.def(8)"all_true/a S (hd:a title="cons" href="cic:/fakeuri.def(1)":/avisited)) [#p #H1 cases (a href="cic:/matita/basics/bool/orb_true_l.def(2)"orb_true_l/a … H1) [#eqp >(\P eqp) @H |@allv]] (* we now exploit the induction hypothesis *) #allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind @@ -321,7 +320,7 @@ lemma bisim_complete: #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 (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? xa (xa title="cons" href="cic:/fakeuri.def(1)":/a:visited))) + cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a … (a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"memb/a ? xa (x:a title="cons" href="cic:/fakeuri.def(1)":/avisited))) [(* xa visited - trivial *) #membxa @a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"memb_append_l2/a // |(* xa new *) #membxa @a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"memb_append_l1/a @a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"sublist_unique_append_l1/a @a href="cic:/matita/tutorial/chapter5/memb_filter_l.def(5)"memb_filter_l/a [>membxa //|//] @@ -343,18 +342,18 @@ qed. prove that two expressions are equivalente if and only if they define the same language. *) -definition equiv ≝ λSig.λre1,re2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a Sig. +img class="anchor" src="icons/tick.png" id="equiv"definition equiv ≝ λSig.λre1,re2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a Sig. let e1 ≝ a title="eclose" href="cic:/fakeuri.def(1)"•/a(a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a ? re1) in let e2 ≝ a title="eclose" href="cic:/fakeuri.def(1)"•/a(a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a ? re2) in - let n ≝ a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/list/length.fix(0,1,1)"length/a ? (a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a Sig (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1|) (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2|))) in + let n ≝ a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/list/length.fix(0,1,1)"length/a ? (a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"space_enum/a Sig (|a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a) (|a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a))) in let sig ≝ (a href="cic:/matita/tutorial/chapter10/occ.def(7)"occ/a Sig e1 e2) in - (a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a ? sig n a title="cons" href="cic:/fakeuri.def(1)"[/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/ae1,e2〉] a title="nil" href="cic:/fakeuri.def(1)"[/a]). + (a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"bisim/a ? sig n (〈e1,e2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a:a title="cons" href="cic:/fakeuri.def(1)":/a[a title="nil" href="cic:/fakeuri.def(1)"]/a) [a title="nil" href="cic:/fakeuri.def(1)"]/a). -theorem euqiv_sem : ∀Sig.∀e1,e2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a Sig. - a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? e1 e2) 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 a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{e1} a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{e2}. +img class="anchor" src="icons/tick.png" id="euqiv_sem"theorem euqiv_sem : ∀Sig.∀e1,e2:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a Sig. + a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? e1 e2) 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 \sem{e1a title="in_l" href="cic:/fakeuri.def(1)"}/a =1 \sem{e2a title="in_l" href="cic:/fakeuri.def(1)"}/a. #Sig #re1 #re2 % [#H @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"re_embedding/a] @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"re_embedding/a] - cut (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? re1 re2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a,a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? re1 re2)〉) + cut (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? re1 re2 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="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a href="cic:/matita/tutorial/chapter10/equiv.def(9)"equiv/a ? re1 re2)a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) [