2 \ 5h1
\ 6Broadcasting points
\ 5/h1
\ 6
3 Intuitively, a regular expression e must be understood as a pointed expression with a single
4 point in front of it. Since however we only allow points before symbols, we must broadcast
5 this initial point inside e traversing all nullable subexpressions, that essentially corresponds
6 to the ϵ-closure operation on automata. We use the notation •(_) to denote such an operation;
7 its definition is the expected one: let us start discussing an example.
10 Let us broadcast a point inside (a + ϵ)(b*a + b)b. We start working in parallel on the
11 first occurrence of a (where the point stops), and on ϵ that gets traversed. We have hence
12 reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Again, we work in
13 parallel on the two additive subterms b^*a and b; the first point is allowed to both enter the
14 star, and to traverse it, stopping in front of a; the second point just stops in front of b.
15 No point reached that end of b^*a + b hence no further propagation is possible. In conclusion:
16 •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉
19 include "tutorial/chapter7.ma".
21 (* Broadcasting a point inside an item generates a pre, since the point could possibly reach
22 the end of the expression.
23 Broadcasting inside a i1+i2 amounts to broadcast in parallel inside i1 and i2.
25 〈i1,b1〉 ⊕ 〈i2,b2〉 = 〈i1 + i2, b1∨ b2〉
26 then, we just have •(i1+i2) = •(i1)⊕ •(i2).
29 include "tutorial/chapter7.ma".
31 definition lo ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λa,b:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 a
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 b,
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 a
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 b〉.
32 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
33 interpretation "oplus" 'oplus a b = (lo ? a b).
35 lemma lo_def: ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.∀b1,b2.
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,b1〉
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i2,b2〉
\ 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
\ 6i1
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6i2,b1
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6b2〉.
39 Concatenation is a bit more complex. In order to broadcast a point inside i1 · i2
40 we should start broadcasting it inside i1 and then proceed into i2 if and only if a
41 point reached the end of i1. This suggests to define •(i1 · i2) as •(i1) ▹ i2, where
42 e ▹ i is a general operation of concatenation between a pre and an item, defined by
43 cases on the boolean in e:
45 〈i1,true〉 ▹ i2 = i1 ◃ •(i_2)
46 〈i1,false〉 ▹ i2 = i1 · i2
47 In turn, ◃ says how to concatenate an item with a pre, that is however extremely simple:
48 i1 ◃ 〈i1,b〉 = 〈i_1 · i2, b〉
49 Let us come to the formalized definitions:
52 definition pre_concat_r ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λi:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.λe:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
53 match e with [ mk_Prod i1 b ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i1, b〉].
55 notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
56 interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
58 lemma eq_to_ex_eq: ∀S.∀A,B:
\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"
\ 6word
\ 5/a
\ 6 S → Prop.
59 A
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 B → A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 B.
60 #S #A #B #H >H #x % // qed.
62 (* The behaviour of ◃ is summarized by the following, easy lemma: *)
64 lemma sem_pre_concat_r : ∀S,i.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
65 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i
\ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"
\ 6◃
\ 5/a
\ 6 e}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 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 e|}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e}.
66 #S #i * #i1 #b1 cases b1 [2: @
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6 //]
67 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"
\ 6sem_cat
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5/span
\ 6\ 5/span
\ 6/
70 (* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive.
71 In this situation, a viable alternative that is usually simpler to reason about,
72 is to abstract one of the two functions with respect to the other. In particular
73 we abstract pre_concat_l with respect to an input bcast function from items to
76 definition pre_concat_l ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λbcast:∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S →
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.λe1:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.λi2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
78 [ mk_Prod i1 b1 ⇒ match b1 with
79 [ true ⇒ (i1
\ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"
\ 6◃
\ 5/a
\ 6 (bcast ? i2))
80 | false ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i2,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
84 notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
85 interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
87 (* We are ready to give the formal definition of the broadcasting operation. *)
89 notation "•" non associative with precedence 60 for @{eclose ?}.
91 let rec eclose (S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6) (i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S) on i :
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S ≝
93 [ pz ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"
\ 6pz
\ 5/a
\ 6 ?,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 〉
94 | pe ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"
\ 6ϵ
\ 5/a
\ 6,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6 〉
95 | ps x ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"
\ 6`
\ 5/a
\ 6.x,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
96 | pp x ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"
\ 6`
\ 5/a
\ 6.x,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6 〉
97 | po i1 i2 ⇒ •i1
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6 •i2
98 | pc i1 i2 ⇒ •i1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2
99 | pk i ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6(
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (•i))
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉].
101 notation "• x" non associative with precedence 60 for @{'eclose $x}.
102 interpretation "eclose" 'eclose x = (eclose ? x).
104 (* Here are a few simple properties of ▹ and •(-) *)
106 lemma pcl_true : ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
107 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 i1
\ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"
\ 6◃
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2).
110 lemma pcl_true_bis : ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
111 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2
\ 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
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2),
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2)〉.
112 #S #i1 #i2 normalize cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2) // qed.
114 lemma pcl_false: ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
115 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2
\ 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
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i2,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉.
118 lemma eclose_plus: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
119 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6(i1
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6 i2)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i1
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2.
122 lemma eclose_dot: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
123 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6(i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i2)
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2.
126 lemma eclose_star: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
127 \ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6(
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6(
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i))
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉.
130 (* The definition of •(-) (eclose) can then be lifted from items to pres
131 in the obvious way. *)
133 definition lift ≝ λS.λf:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S →
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.λe:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
135 [ mk_Prod i b ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (f i),
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 (f i)
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 b〉].
137 definition preclose ≝ λS.
\ 5a href="cic:/matita/tutorial/chapter8/lift.def(2)"
\ 6lift
\ 5/a
\ 6 S (
\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"
\ 6eclose
\ 5/a
\ 6 S).
138 interpretation "preclose" 'eclose x = (preclose ? x).
140 (* Obviously, broadcasting does not change the carrier of the item,
141 as it is easily proved by structural induction. *)
143 lemma erase_bull : ∀S.∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 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 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i)|
\ 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
\ 6i|.
145 [ #i1 #i2 #IH1 #IH2 >
\ 5a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"
\ 6erase_dot
\ 5/a
\ 6 <IH1 >
\ 5a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"
\ 6eclose_dot
\ 5/a
\ 6
146 cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i1) #i11 #b1 cases b1 // <IH2 >
\ 5a href="cic:/matita/tutorial/chapter8/pcl_true_bis.def(5)"
\ 6pcl_true_bis
\ 5/a
\ 6 //
147 | #i1 #i2 #IH1 #IH2 >
\ 5a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"
\ 6eclose_plus
\ 5/a
\ 6 >(
\ 5a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"
\ 6erase_plus
\ 5/a
\ 6 … i1) <IH1 <IH2
148 cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i1) #i11 #b1 cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2) #i21 #b2 //
149 | #i #IH >
\ 5a href="cic:/matita/tutorial/chapter8/eclose_star.def(5)"
\ 6eclose_star
\ 5/a
\ 6 >(
\ 5a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"
\ 6erase_star
\ 5/a
\ 6 … i) <IH cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i) //
153 (* We are now ready to state the main semantic properties of ⊕, ◃ and •(-):
155 sem_oplus: \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}
156 sem_pcl: \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}
157 sem_bullet \sem{•i} =1 \sem{i} ∪ \sem{|i|}
159 The proof of sem_oplus is straightforward. *)
161 lemma sem_oplus: ∀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.
162 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e1
\ 5a title="oplus" href="cic:/fakeuri.def(1)"
\ 6⊕
\ 5/a
\ 6 e2}
\ 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{e1}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e2}.
163 #S * #i1 #b1 * #i2 #b2 #w %
164 [cases b1 cases b2 normalize /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5/span
\ 6\ 5/span
\ 6/ * /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"
\ 6or_introl
\ 5/a
\ 6,
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ * /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"
\ 6or_introl
\ 5/a
\ 6,
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
165 |cases b1 cases b2 normalize /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5/span
\ 6\ 5/span
\ 6/ * /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"
\ 6or_introl
\ 5/a
\ 6,
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ * /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"
\ 6or_introl
\ 5/a
\ 6,
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
169 (* For the others, we proceed as follow: we first prove the following
170 auxiliary lemma, that assumes sem_bullet:
173 \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
174 \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
176 Then, using the previous result, we prove sem_bullet by induction
177 on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
179 lemma LcatE : ∀S.∀e1,e2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
180 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 e2}
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e1}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="forget" href="cic:/fakeuri.def(1)"
\ 6|
\ 5/a
\ 6e2|}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e2}.
183 lemma sem_pcl_aux : ∀S.∀e1:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.∀i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
184 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i2}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="forget" href="cic:/fakeuri.def(1)"
\ 6|
\ 5/a
\ 6i2|} →
185 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i2}
\ 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{e1}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="forget" href="cic:/fakeuri.def(1)"
\ 6|
\ 5/a
\ 6i2|}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i2}.
186 #S * #i1 #b1 #i2 cases b1
187 [2:#th >
\ 5a href="cic:/matita/tutorial/chapter8/pcl_false.def(5)"
\ 6pcl_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"
\ 6sem_pre_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"
\ 6sem_pre_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"
\ 6sem_cat
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
188 |#H >
\ 5a href="cic:/matita/tutorial/chapter8/pcl_true.def(5)"
\ 6pcl_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 @(
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 … (
\ 5a href="cic:/matita/tutorial/chapter8/sem_pre_concat_r.def(10)"
\ 6sem_pre_concat_r
\ 5/a
\ 6 …))
189 >
\ 5a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"
\ 6erase_bull
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@(
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 … H)]
190 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6[|@
\ 5a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"
\ 6union_comm
\ 5/a
\ 6 ]]
191 @
\ 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/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6 ] /
\ 5span class="autotactic"
\ 63
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6,
\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"
\ 6eqP_sym
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
195 lemma minus_eps_pre_aux: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.∀A.
196 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 A →
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 e}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 (A
\ 5a title="substraction" href="cic:/fakeuri.def(1)"
\ 6-
\ 5/a
\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"
\ 6{
\ 5/a
\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6 ]}).
198 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter7/minus_eps_pre.def(10)"
\ 6minus_eps_pre
\ 5/a
\ 6]
199 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"
\ 6eqP_sym
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter7/minus_eps_item.def(9)"
\ 6minus_eps_item
\ 5/a
\ 6]]
200 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/distribute_substract.def(3)"
\ 6distribute_substract
\ 5/a
\ 6]
201 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_substract_r.def(3)"
\ 6eqP_substract_r
\ 5/a
\ 6 //
204 theorem sem_bull: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6. ∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
\ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{i}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a title="forget" href="cic:/fakeuri.def(1)"
\ 6|
\ 5/a
\ 6i|}.
206 [#w normalize % [/
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ | * //]
207 |/
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
208 |#x normalize #w % [ /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ | * [@
\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"
\ 6False_ind
\ 5/a
\ 6 | //]]
209 |#x normalize #w % [ /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"
\ 6or_intror
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/ | * // ]
210 |#i1 #i2 #IH1 #IH2 >
\ 5a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"
\ 6eclose_dot
\ 5/a
\ 6
211 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"
\ 6sem_pcl_aux
\ 5/a
\ 6 //] >
\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"
\ 6sem_cat
\ 5/a
\ 6
212 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6
213 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6
214 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@(
\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"
\ 6cat_ext_l
\ 5/a
\ 6 … IH1)] @
\ 5a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"
\ 6distr_cat_r
\ 5/a
\ 6]]
215 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6]
216 @
\ 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/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6]
217 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 //
218 |#i1 #i2 #IH1 #IH2 >
\ 5a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"
\ 6eclose_plus
\ 5/a
\ 6
219 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/sem_oplus.def(9)"
\ 6sem_oplus
\ 5/a
\ 6] >
\ 5a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"
\ 6sem_plus
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"
\ 6erase_plus
\ 5/a
\ 6
220 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@(
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 … IH2)]
221 @
\ 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/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6]
222 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6
223 @
\ 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/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6]
224 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"
\ 6union_comm
\ 5/a
\ 6]]
225 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6] /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
226 |#i #H >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"
\ 6sem_star
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"
\ 6erase_bull
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"
\ 6sem_star
\ 5/a
\ 6
227 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"
\ 6cat_ext_l
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"
\ 6minus_eps_pre_aux
\ 5/a
\ 6 //]]]
228 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"
\ 6distr_cat_r
\ 5/a
\ 6]]
229 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"
\ 6erase_star
\ 5/a
\ 6
230 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"
\ 6eqP_sym
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"
\ 6star_fix_eps
\ 5/a
\ 6
235 \ 5h2
\ 6Blank item
\ 5/h2
\ 6
237 As a corollary of theorem sem_bullet, given a regular expression e, we can easily
238 find an item with the same semantics of $e$: it is enough to get an item (blank e)
239 having e as carrier and no point, and then broadcast a point in it. The semantics of
240 (blank e) is obviously the empty language: from the point of view of the automaton,
241 it corresponds with the pit state. *)
243 let rec blank (S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6) (i:
\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"
\ 6re
\ 5/a
\ 6 S) on i :
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S ≝
245 [ z ⇒
\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"
\ 6pz
\ 5/a
\ 6 ?
246 | e ⇒
\ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"
\ 6ϵ
\ 5/a
\ 6
247 | s y ⇒
\ 5a title="pitem ps" href="cic:/fakeuri.def(1)"
\ 6`
\ 5/a
\ 6y
248 | o e1 e2 ⇒ (blank S e1)
\ 5a title="pitem or" href="cic:/fakeuri.def(1)"
\ 6+
\ 5/a
\ 6 (blank S e2)
249 | c e1 e2 ⇒ (blank S e1)
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 (blank S e2)
250 | k e ⇒ (blank S e)
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6* ].
252 lemma forget_blank: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"
\ 6re
\ 5/a
\ 6 S.
\ 5a title="forget" href="cic:/fakeuri.def(1)"
\ 6|
\ 5/a
\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"
\ 6blank
\ 5/a
\ 6 S e|
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 e.
253 #S #e elim e normalize //
256 lemma sem_blank: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"
\ 6re
\ 5/a
\ 6 S.
\ 5a title="in_pl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{
\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"
\ 6blank
\ 5/a
\ 6 S e}
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61
\ 5a title="empty set" href="cic:/fakeuri.def(1)"
\ 6∅
\ 5/a
\ 6.
258 [1,2:@
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6 //
259 |#s @
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6 //
260 |#e1 #e2 #Hind1 #Hind2 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"
\ 6sem_cat
\ 5/a
\ 6
261 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@(
\ 5a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"
\ 6union_empty_r
\ 5/a
\ 6 …
\ 5a title="empty set" href="cic:/fakeuri.def(1)"
\ 6∅
\ 5/a
\ 6)]
262 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6[|@Hind2]] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6
263 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@(
\ 5a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"
\ 6cat_empty_l
\ 5/a
\ 6 … ?)] @
\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"
\ 6cat_ext_l
\ 5/a
\ 6 @Hind1
264 |#e1 #e2 #Hind1 #Hind2 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"
\ 6sem_plus
\ 5/a
\ 6
265 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@(
\ 5a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"
\ 6union_empty_r
\ 5/a
\ 6 …
\ 5a title="empty set" href="cic:/fakeuri.def(1)"
\ 6∅
\ 5/a
\ 6)]
266 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6[|@Hind2]] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 @Hind1
267 |#e #Hind >
\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"
\ 6sem_star
\ 5/a
\ 6
268 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@(
\ 5a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"
\ 6cat_empty_l
\ 5/a
\ 6 … ?)] @
\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"
\ 6cat_ext_l
\ 5/a
\ 6 @Hind
272 theorem re_embedding: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"
\ 6re
\ 5/a
\ 6 S.
273 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 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 S e)}
\ 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{e}.
274 #S #e @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/sem_bull.def(12)"
\ 6sem_bull
\ 5/a
\ 6] >
\ 5a href="cic:/matita/tutorial/chapter8/forget_blank.def(4)"
\ 6forget_blank
\ 5/a
\ 6
275 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/sem_blank.def(9)"
\ 6sem_blank
\ 5/a
\ 6]]
276 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"
\ 6union_comm
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"
\ 6union_empty_r
\ 5/a
\ 6.
280 \ 5h2
\ 6Lifted Operators
\ 5/h2
\ 6
282 Plus and bullet have been already lifted from items to pres. We can now
283 do a similar job for concatenation ⊙ and Kleene's star ⊛.*)
285 definition lifted_cat ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λe:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
286 \ 5a href="cic:/matita/tutorial/chapter8/lift.def(2)"
\ 6lift
\ 5/a
\ 6 S (
\ 5a href="cic:/matita/tutorial/chapter8/pre_concat_l.def(3)"
\ 6pre_concat_l
\ 5/a
\ 6 S
\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"
\ 6eclose
\ 5/a
\ 6 e).
288 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
290 interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
292 lemma odot_true_b : ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.∀b.
293 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i2,b〉
\ 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
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 (
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2)),
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2)
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 b〉.
294 #S #i1 #i2 #b normalize in ⊢ (??%?); cases (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i2) //
297 lemma odot_false_b : ∀S.∀i1,i2:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.∀b.
298 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i2,b〉
\ 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
\ 6i1
\ 5a title="pitem cat" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 i2 ,b〉.
302 lemma erase_odot:∀S.∀e1,e2:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
303 \ 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="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 e2)|
\ 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="re cat" 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|).
304 #S * #i1 * * #i2 #b2 // >
\ 5a href="cic:/matita/tutorial/chapter8/odot_true_b.def(6)"
\ 6odot_true_b
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"
\ 6erase_dot
\ 5/a
\ 6 //
307 (* Let us come to the star operation: *)
309 definition lk ≝ λS:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.λe:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
313 [true ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6(
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (
\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"
\ 6eclose
\ 5/a
\ 6 ? i1))
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
314 |false ⇒
\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i1
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
318 (* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
319 interpretation "lk" 'lk a = (lk ? a).
320 notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
322 lemma ostar_true: ∀S.∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
323 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
\ 5a title="lk" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6⊛
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6(
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (
\ 5a title="eclose" href="cic:/fakeuri.def(1)"
\ 6•
\ 5/a
\ 6i))
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉.
326 lemma ostar_false: ∀S.∀i:
\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"
\ 6pitem
\ 5/a
\ 6 S.
327 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
\ 5a title="lk" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6⊛
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i
\ 5a title="pitem star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉.
330 lemma erase_ostar: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
331 \ 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="lk" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6⊛)|
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 \ 5a title="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="re star" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*.
334 lemma sem_odot_true: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀e1:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.∀i.
335 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e1
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉}
\ 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{e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"
\ 6{
\ 5/a
\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"
\ 6[
\ 5/a
\ 6 ] }.
337 cut (e1
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉
\ 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 title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i),
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6(e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i)
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"
\ 6true
\ 5/a
\ 6〉) [//]
338 #H >H cases (e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i) #i1 #b1 cases b1
339 [>
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 @
\ 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/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6]
340 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"
\ 6eqP_sym
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
341 |/
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
345 lemma eq_odot_false: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀e1:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.∀i.
346 e1
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i.
348 cut (e1
\ 5a title="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"
\ 6〈
\ 5/a
\ 6i,
\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉
\ 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 title="pair pi1" href="cic:/fakeuri.def(1)"
\ 6\fst
\ 5/a
\ 6 (e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i),
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"
\ 6\snd
\ 5/a
\ 6(e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i)
\ 5a title="boolean or" href="cic:/fakeuri.def(1)"
\ 6∨
\ 5/a
\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"
\ 6false
\ 5/a
\ 6〉) [//]
349 cases (e1
\ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"
\ 6▹
\ 5/a
\ 6 i) #i1 #b1 cases b1 #H @H
352 (* We conclude this section with the proof of the main semantic properties
356 ∀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="lifted cat" href="cic:/fakeuri.def(1)"
\ 6⊙
\ 5/a
\ 6 e2}
\ 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{e1}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 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|}
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e2}.
358 [>
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6
359 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/sem_odot_true.def(10)"
\ 6sem_odot_true
\ 5/a
\ 6]
360 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [||@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"
\ 6sem_pcl_aux
\ 5/a
\ 6 //
361 |>
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"
\ 6sem_pre_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter8/eq_odot_false.def(6)"
\ 6eq_odot_false
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"
\ 6sem_pcl_aux
\ 5/a
\ 6 //
365 theorem sem_ostar: ∀S.∀e:
\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"
\ 6pre
\ 5/a
\ 6 S.
366 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 5/a
\ 6{e
\ 5a title="lk" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6⊛}
\ 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{e}
\ 5a title="cat lang" href="cic:/fakeuri.def(1)"
\ 6·
\ 5/a
\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"
\ 6\sem
\ 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 e|}
\ 5a title="star lang" href="cic:/fakeuri.def(1)"
\ 6^
\ 5/a
\ 6*.
368 [>
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"
\ 6sem_pre_true
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"
\ 6sem_star
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"
\ 6erase_bull
\ 5/a
\ 6
369 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6[|@
\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"
\ 6cat_ext_l
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"
\ 6minus_eps_pre_aux
\ 5/a
\ 6 //]]]
370 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"
\ 6eqP_union_r
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"
\ 6distr_cat_r
\ 5/a
\ 6]]
371 @
\ 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/chapter6/distr_cat_r.def(5)"
\ 6distr_cat_r
\ 5/a
\ 6]
372 @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"
\ 6eqP_trans
\ 5/a
\ 6 [|@
\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"
\ 6union_assoc
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"
\ 6eqP_union_l
\ 5/a
\ 6
373 @
\ 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/chapter6/epsilon_cat_l.def(5)"
\ 6epsilon_cat_l
\ 5/a
\ 6] @
\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"
\ 6eqP_sym
\ 5/a
\ 6 @
\ 5a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"
\ 6star_fix_eps
\ 5/a
\ 6
374 |>
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"
\ 6sem_pre_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"
\ 6sem_pre_false
\ 5/a
\ 6 >
\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"
\ 6sem_star
\ 5/a
\ 6 /
\ 5span class="autotactic"
\ 62
\ 5span class="autotrace"
\ 6 trace
\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"
\ 6eq_to_ex_eq
\ 5/a
\ 6\ 5/span
\ 6\ 5/span
\ 6/
projects / helm.git / blob