+nlemma sub_dot_star :
+ ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
+#S X b; napply extP; #w; @;
+##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+ *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
+ @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+ @; //; napply (subW … sube);
+##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
+ #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
+ ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
+ @; ncases b in H1; #H1;
+ ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
+ nrewrite > (associative_append ? w' w1 w2);
+ nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
+ ##| ncases w' in Pw';
+ ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
+ ##| #x xs Px; @(x::xs); @(w1@w2);
+ nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
+ @wl'; @; //; ##] ##]
+ ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
+ nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
+ ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
+ nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
+ @[]; @; //;
+ ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //]
+ @; //; @; //; @; *;##]##]##]
+nqed.
+
+(* theorem 16: 1 *)
+alias symbol "pc" (instance 13) = "cat lang".
+alias symbol "in_pl" (instance 23) = "in_pl".
+alias symbol "in_pl" (instance 5) = "in_pl".
+alias symbol "eclose" (instance 21) = "eclose".
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S. 𝐋\p (•e) = 𝐋\p e ∪ 𝐋 |e|.
+#S e; nelim e; //;
+ ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
+ ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
+ ##| #e1 e2 IH1 IH2;
+ nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
+ nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
+ nrewrite > (IH1 …); nrewrite > (cup_dotD …);
+ nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
+ nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
+ nrewrite < (erase_dot …); nrewrite < (cupA …); //;
+ ##| #e1 e2 IH1 IH2;
+ nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
+ nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
+ nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
+ nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …);
+ nrewrite < (erase_plus …); //.
+ ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
+ nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
+ nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
+ nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+ nrewrite > (erase_bull…e);
+ nrewrite > (erase_star …);
+ nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2:
+ nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH;
+ ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH;
+ nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
+ ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
+ nrewrite > (cup_dotD…); nrewrite > (cupA…);
+ nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //;
+ nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##]
+ nqed.
+
+(* theorem 16: 3 *)
+nlemma odot_dot:
+ ∀S.∀e1,e2: pre S. 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
+#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
+
+nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} = 𝐋 e^*.
+#S e; napply extP; #w; nnormalize; @;
+##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2;
+ *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
+ nrewrite < defw; nrewrite < defw2; @; //; @;//;
+##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
+ #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
+ @; /2/; @xs; /2/;##]
+ nqed.
+
+nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
+#S e; @[]; /2/; nqed.
+
+nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
+#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
+
+nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
+#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
+
+nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
+ ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
+#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
+##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
+nqed.
+
+(* theorem 16: 4 *)
+nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
+#S p; ncases p; #e b; ncases b;
+##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
+ nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
+ nchange in ⊢ (??%?) with (?∪?);
+ nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+ nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
+ nlapply (bull_cup ? e); #bc;
+ nchange in match (𝐋\p (•e)) in bc with (?∪?);
+ nchange in match b' in bc with b';
+ ncases b' in bc; ##[##2: nrewrite > (cup0…); nrewrite > (sub0…); //]
+ nrewrite > (cup_sub…); ##[napply rcanc_sing] //;##]
+ nrewrite > (cup_dotD…); nrewrite > (cupA…);nrewrite > (erase_bull…);
+ nrewrite > (sub_dot_star…);
+ nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;
+##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (cup0…);
+ nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
+ nrewrite < (cup0 ? (𝐋\p e)); //;##]
+nqed.
+
+nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
+ match e with
+ [ z ⇒ pz ?
+ | e ⇒ pe ?
+ | s x ⇒ ps ? x
+ | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
+ | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
+ | k e1 ⇒ pk ? (pre_of_re ? e1)].
+
+nlemma notFalse : ¬False. @; //; nqed.
+
+nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
+#S A; nnormalize; napply extP; #w; @; ##[##2: *]
+*; #w1; *; #w2; *; *; //; nqed.
+
+nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
+#S e; nelim e; ##[##1,2,3: //]
+##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
+ nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
+##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
+ nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
+##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
+ nrewrite > H1; napply dot0; ##]
+nqed.
+
+nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
+#S A; nelim A; //;
+##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
+ nrewrite < H1; nrewrite < H2; //
+##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
+ nrewrite < H1; nrewrite < H2; //
+##| #e1 H1; nchange in match (𝐋 (e1^* )) with ((𝐋 e1)^* );
+ nrewrite < H1; //]
+nqed.
+
+(* corollary 17 *)
+nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
+#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
+nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
+nqed.
+
+nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
+#S f g H; nrewrite > H; //; nqed.
+
+(* corollary 18 *)
+ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
+#S e; @;
+##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
+ nrewrite > defsnde; #H;
+ nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
+
+STOP
+