+ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
+notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
+
+nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
+#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
+nqed.
+
+nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
+#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
+
+nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
+#S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.
+
+(* theorem 16: 2 *)
+nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
+nwhd in ⊢ (??(??%)?);
+nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
+nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
+nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
+nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
+nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
+nqed.
+
+nlemma odotEt :
+ ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
+#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
+
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) = 𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
+
+nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r).
+#S p q r; napply extP; #w; nnormalize; @;
+##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
+##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
+nqed.
+
+nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
+#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
+
+nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| = 𝐋 .|e1| · 𝐋 .|e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
+nqed.
+
+nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| = 𝐋 .|e1| ∪ 𝐋 .|e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
+
+nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
+
+ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
+interpretation "substract" 'minus a b = (substract ? a b).
+
+nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
+#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
+
+nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
+#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
+
+nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
+#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
+
+nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
+#S a b w; nnormalize; *; //; nqed.
+
+nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
+#S a; nelim a; // by {};
+##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
+ nrewrite < IH1; nrewrite < IH2;
+ nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
+ ncases (•e1); #e3 b; ncases b; nnormalize;
+ ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
+##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
+ nrewrite < IH2; nrewrite < IH1;
+ nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
+ ncases (•e1); ncases (•e2); //;
+##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
+ nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
+nqed.
+
+nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+#S p; ncases p; //; nqed.
+
+nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p.
+#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
+*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
+napply Hw2; nqed.
+
+(* theorem 16: 1 → 3 *)
+nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
+ 𝐋\p (•(\fst e2)) = 𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| →
+ 𝐋\p (e1 ⊙ e2) = 𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
+##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2);
+ nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
+ nchange in ⊢ (??%?) with (?∪?);
+ nchange in ⊢ (??(??%?)?) with (?∪?);
+ nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+ nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
+ nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
+ nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
+ nchange with (𝐋\p 〈e2'',b2''〉 = 𝐋\p e2' ∪ 𝐋 .|e2'|);
+ ngeneralize in match th1;
+ nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
+ nrewrite > (eta_lp ? e2);
+ nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
+ nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);
+ nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
+ nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
+##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
+ nchange in match (𝐋\p ?) with (?∪?);
+ nchange in match (𝐋\p (e1'·?)) with (?∪?);
+ nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
+ nrewrite > (cup0…);
+ nrewrite > (cupA…); //;##]
+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 ⊢ (???%) with ((𝐋. |e|)^* ); 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: 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.
+
+naxiom Admit : False.
+
+nlemma key_id : ∀S.∀e:pitem S. 𝐋\p e · 𝐋 .|e|^* ∪ {[]} = 𝐋\p e · 𝐋 .|e|^* ∪ 𝐋 .|e|^*.
+#S e; napply extP; #w; @;##[##2:*]
+##[*; #w1; *; #w2; *; *; #defw Hw1 Hw2;@; @w1; @w2; /3/;
+##|*; #wl; *; #H; nrewrite < H;
+(*
+ ngeneralize in match e;
+ nelim wl;##[#e;#_;@2;//] #x xs IH e;*; #Hx Hxs; ncases (IH Hxs);
+ ##[##2: #H; nnormalize; nrewrite < H; nrewrite > (append_nil…);
+
+ ncases wl; ##[#_;@2; //] #x xs; *; #Hx Hxs; @; @x; @(flatten ? xs); @;
+ ##[@;//;##|@xs; @; //]
+ ngeneralize in match Hx; ngeneralize in match x; nelim e; nnormalize; //;
+ ##[#e1 e2 IH1 IH2 x; *; #w1; *; #w2; *; *; #defx Hw1 Hw2;
+ @; @w1; @w2; /4/ by conj;
+ ##|#e1 e2 IH1 IH2 y;*; #; ##[@|@2] /2/;
+ ##|#e IH y; *; #wl; *; #delwl Hw2; nrewrite < delwl;
+ nelim wl in Hw2; ##[#_;@[];@[];@;//;
+*)
+ ncases Admit;
+##|*;##[##2: #H; nrewrite < H; @2; //] *; #w1; *; #w2; *; *; #defw Hw1 Hw2;
+ @; @w1; @w2; /3/;##]