then, we just have •(i1+i2) = •(i1)⊕ •(i2).
*)
-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〉.
+\ 5img class="anchor" src="icons/tick.png" id="lo"\ 6definition 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
interpretation "oplus" 'oplus a b = (lo ? a b).
-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〉.
+\ 5img class="anchor" src="icons/tick.png" id="lo_def"\ 6lemma 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 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="Pair construction" 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\ 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
(*
Let us come to the formalized definitions:
*)
-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.
- 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〉].
+\ 5img class="anchor" src="icons/tick.png" id="pre_concat_r"\ 6definition 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.
+ 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
-lemma eq_to_ex_eq: ∀S.∀A,B:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eq_to_ex_eq"\ 6lemma eq_to_ex_eq: ∀S.∀A,B:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
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.
#S #A #B #H >H #x % // qed.
(* The behaviour of ◃ is summarized by the following, easy lemma: *)
-lemma sem_pre_concat_r : ∀S,i.∀e:\ 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{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}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pre_concat_r"\ 6lemma sem_pre_concat_r : ∀S,i.∀e:\ 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{i \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 e\ 5a title="in_prl" 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_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#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 //]
>\ 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/
qed.
we abstract pre_concat_l with respect to an input bcast function from items to
pres. *)
-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.
+\ 5img class="anchor" src="icons/tick.png" id="pre_concat_l"\ 6definition 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.
match e1 with
[ mk_Prod i1 b1 ⇒ match b1 with
[ true ⇒ (i1 \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 (bcast ? i2))
- | 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〉
+ | 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
notation "•" non associative with precedence 60 for @{eclose ?}.
-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 ≝
+\ 5img class="anchor" src="icons/tick.png" id="eclose"\ 6let 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 ≝
match i with
- [ 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 〉
- | 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 〉
- | 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〉
- | 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 〉
+ [ 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 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | 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 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | 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\ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | 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\ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| po i1 i2 ⇒ •i1 \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 •i2
| pc i1 i2 ⇒ •i1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2
- | 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〉].
+ | 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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
notation "• x" non associative with precedence 60 for @{'eclose $x}.
interpretation "eclose" 'eclose x = (eclose ? x).
(* Here are a few simple properties of ▹ and •(-) *)
-lemma pcl_true : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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).
+\ 5img class="anchor" src="icons/tick.png" id="pcl_true"\ 6lemma pcl_true : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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).
// qed.
-lemma pcl_true_bis : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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)〉.
+\ 5img class="anchor" src="icons/tick.png" id="pcl_true_bis"\ 6lemma pcl_true_bis : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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)\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #i1 #i2 normalize cases (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) // qed.
-lemma pcl_false: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="pcl_false"\ 6lemma pcl_false: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-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.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_plus"\ 6lemma 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.
\ 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.
// qed.
-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.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_dot"\ 6lemma 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.
\ 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.
// qed.
-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.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_star"\ 6lemma 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.
+ \ 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="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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
(* The definition of •(-) (eclose) can then be lifted from items to pres
in the obvious way. *)
-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.
+\ 5img class="anchor" src="icons/tick.png" id="lift"\ 6definition 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.
match e with
- [ 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〉].
+ [ 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
-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).
+\ 5img class="anchor" src="icons/tick.png" id="preclose"\ 6definition 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).
interpretation "preclose" 'eclose x = (preclose ? x).
(* Obviously, broadcasting does not change the carrier of the item,
as it is easily proved by structural induction. *)
-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|.
+\ 5img class="anchor" src="icons/tick.png" id="erase_bull"\ 6lemma 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="forget" 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\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#S #i elim i //
[ #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
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 //
The proof of sem_oplus is straightforward. *)
-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.
- \ 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}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_oplus"\ 6lemma 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.
+ \ 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="in_prl" 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{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S * #i1 #b1 * #i2 #b2 #w %
[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/
|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/
Then, using the previous result, we prove sem_bullet by induction
on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
-lemma LcatE : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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}.
+\ 5img class="anchor" src="icons/tick.png" id="LcatE"\ 6lemma LcatE : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="in_pl" 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="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-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.
- \ 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|} →
- \ 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}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pcl_aux"\ 6lemma 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.
+ \ 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="in_prl" 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_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 →
+ \ 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="in_prl" 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{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S * #i1 #b1 #i2 cases b1
[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/
|#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 …))
]
qed.
-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.
- \ 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 ]}).
+\ 5img class="anchor" src="icons/tick.png" id="minus_eps_pre_aux"\ 6lemma 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.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" 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_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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="in_pl" 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_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6).
#S #e #i #A #seme
@\ 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]
@\ 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]]
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_substract_r.def(3)"\ 6eqP_substract_r\ 5/a\ 6 //
qed.
-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|}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_bull"\ 6theorem 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="in_prl" 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_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e elim e
[#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/ | * //]
|/\ 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/
(blank e) is obviously the empty language: from the point of view of the automaton,
it corresponds with the pit state. *)
-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 ≝
+\ 5img class="anchor" src="icons/tick.png" id="blank"\ 6let 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 ≝
match i with
[ z ⇒ \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 ?
| e ⇒ \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6
| s y ⇒ \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y
| o e1 e2 ⇒ (blank S e1) \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (blank S e2)
| c e1 e2 ⇒ (blank S e1) \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (blank S e2)
- | k e ⇒ (blank S e)\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ].
+ | k e ⇒ (blank S e)\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ].
-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.
+\ 5img class="anchor" src="icons/tick.png" id="forget_blank"\ 6lemma 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
#S #e elim e normalize //
qed.
-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.
+\ 5img class="anchor" src="icons/tick.png" id="sem_blank"\ 6lemma 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="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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.
#S #e elim e
[1,2:@\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6 //
|#s @\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6 //
]
qed.
-theorem re_embedding: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 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\ 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}.
+\ 5img class="anchor" src="icons/tick.png" id="re_embedding"\ 6theorem re_embedding: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 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\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e)\ 5a title="in_prl" 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_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#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
@\ 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]]
@\ 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.
Plus and bullet have been already lifted from items to pres. We can now
do a similar job for concatenation ⊙ and Kleene's star ⊛.*)
-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.
+\ 5img class="anchor" src="icons/tick.png" id="lifted_cat"\ 6definition 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.
\ 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).
notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-lemma odot_true_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="odot_true_b"\ 6lemma odot_true_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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="Pair construction" 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\ 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #i1 #i2 #b normalize in ⊢ (??%?); cases (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) //
qed.
-lemma odot_false_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="odot_false_b"\ 6lemma odot_false_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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="Pair construction" 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\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2 ,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
//
qed.
-lemma erase_odot:∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="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|).
+\ 5img class="anchor" src="icons/tick.png" id="erase_odot"\ 6lemma erase_odot:∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 e2)\ 5a title="forget" 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 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 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\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6).
#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 //
qed.
(* Let us come to the star operation: *)
-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.
+\ 5img class="anchor" src="icons/tick.png" id="lk"\ 6definition 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.
match e with
[ mk_Prod i1 b1 ⇒
match b1 with
- [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〉
- |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〉
+ [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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ |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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
interpretation "lk" 'lk a = (lk ? a).
notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
-lemma ostar_true: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="ostar_true"\ 6lemma ostar_true: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-lemma ostar_false: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 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〉.
+\ 5img class="anchor" src="icons/tick.png" id="ostar_false"\ 6lemma ostar_false: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 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 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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-lemma erase_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e\ 5a title="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*.
+\ 5img class="anchor" src="icons/tick.png" id="erase_ostar"\ 6lemma erase_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6)\ 5a title="forget" 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
#S * #i * // qed.
-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.
- \ 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 ] }.
+\ 5img class="anchor" src="icons/tick.png" id="sem_odot_true"\ 6lemma 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.
+ \ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="in_prl" 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{e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e1 #i
-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〉) [//]
+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="Pair construction" 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 (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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
#H >H cases (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) #i1 #b1 cases b1
[>\ 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]
@\ 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/
]
qed.
-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.
- 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.
+\ 5img class="anchor" src="icons/tick.png" id="eq_odot_false"\ 6lemma 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.
+ 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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 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.
#S #e1 #i
-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〉) [//]
+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="Pair construction" 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 (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\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
cases (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) #i1 #b1 cases b1 #H @H
qed.
(* We conclude this section with the proof of the main semantic properties
of ⊙ and ⊛. *)
-lemma sem_odot:
- ∀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}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_odot"\ 6lemma sem_odot:
+ ∀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="in_prl" 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{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e1 * #i2 *
[>\ 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_odot_true.def(10)"\ 6sem_odot_true\ 5/a\ 6]
]
qed.
-theorem sem_ostar: ∀S.∀e:\ 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{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*.
+\ 5img class="anchor" src="icons/tick.png" id="sem_ostar"\ 6theorem sem_ostar: ∀S.∀e:\ 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{e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6\ 5a title="in_prl" 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="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 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="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
#S * #i #b cases b
[>\ 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
@\ 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 //]]]
projects / helm.git / blobdiff