ndefinition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
interpretation "star lang" 'pk l = (star ? l).
-notation > "𝐋 term 90 E" non associative with precedence 90 for @{in_l ? $E}.
+notation > "𝐋 term 70 E" non associative with precedence 75 for @{in_l ? $E}.
nlet rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝
match r with
[ z ⇒ {}
| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
| o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2
| k r1 ⇒ (𝐋 r1) ^*].
-notation "𝐋 term 90 E" non associative with precedence 90 for @{'in_l $E}.
+notation "𝐋 term 70 E" non associative with precedence 75 for @{'in_l $E}.
interpretation "in_l" 'in_l E = (in_l ? E).
interpretation "in_l mem" 'mem w l = (in_l ? l w).
interpretation "pepsilon" 'epsilon = (pe ?).
interpretation "pempty" 'empty = (pz ?).
-notation > ".|term 19 e|" non associative with precedence 90 for @{forget ? $e}.
+notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
match l with
[ pz ⇒ ∅
| pe ⇒ ϵ
| ps x ⇒ `x
| pp x ⇒ `x
- | pc E1 E2 ⇒ (.|E1| · .|E2|)
- | po E1 E2 ⇒ (.|E1| + .|E2|)
- | pk E ⇒ .|E|^* ].
-notation < ".|term 19 e|" non associative with precedence 90 for @{'forget $e}.
+ | pc E1 E2 ⇒ (|E1| · |E2|)
+ | po E1 E2 ⇒ (|E1| + |E2|)
+ | pk E ⇒ |E|^* ].
+notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
interpretation "forget" 'forget a = (forget ? a).
notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
interpretation "snd" 'snd x = (snd ? ? x).
-notation > "𝐋\p\ term 90 E" non associative with precedence 90 for @{in_pl ? $E}.
+notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝
match r with
[ pz ⇒ {}
| pe ⇒ {}
| ps _ ⇒ {}
| pp x ⇒ { [x] }
-| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 .|r2| ∪ 𝐋\p\ r2
+| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 |r2| ∪ 𝐋\p\ r2
| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
-| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (.|r1|^* ) ].
-notation > "𝐋\p term 90 E" non associative with precedence 90 for @{'in_pl $E}.
-notation "𝐋\sub(\p) term 90 E" non associative with precedence 90 for @{'in_pl $E}.
+| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
+notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'in_pl $E}.
+notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'in_pl $E}.
interpretation "in_pl" 'in_pl E = (in_pl ? E).
interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
-ndefinition in_prl ≝ λS : Alpha.λp:pre S. 𝐋\p\ (\fst p) ∪ ϵ (\snd p).
+ndefinition in_prl ≝ λS : Alpha.λp:pre S. 𝐋\p (\fst p) ∪ ϵ (\snd p).
interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
interpretation "in_prl" 'in_pl E = (in_prl ? E).
*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
nqed.
-nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
+nlemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ((𝐋\p e) [ ]).
#S e; nelim e; nnormalize; /2/ by nmk;
##[ #; @; #; ndestruct;
##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
∀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 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; @;
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|.
+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|.
+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.
+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 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|.
+nlemma erase_bull : ∀S.∀a:pitem S. |\fst (•a)| = |a|.
#S a; nelim a; // by {};
-##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
+##[ #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|);
+##| #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;
+##| #e IH; nchange in ⊢ (???%) with (|e|^* ); nrewrite < IH;
nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
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.
+ 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 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'|);
+ 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);
nrewrite > (cupA…); //;##]
nqed.
+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|.
+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; *;
##| #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'|^* );
+ 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:
+ 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.
+ 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: 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^*.
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/;##]
+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 e; ncases e; #e' b'; ncases b';
-##[ nchange in match (〈e',true〉^⊛) with 〈?,?〉;
- nletin e'' ≝ (\fst (•e'));
+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.|e''|^* ∪ {[]} = (𝐋\p e' ∪ 𝐋.|e'|) · 𝐋.|e''|^* ∪ {[]}); ##[##2:
- nrewrite < (bull_cup…); nchange in ⊢ (???(??(??%?)?)) with (?∪?);
- nchange in match e'' with e'';
- ncases (\snd (•e')); ##[##2: nrewrite > (cup0…); //]
- nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…);
- nrewrite > (cupA…); nrewrite > (cup_star_nil…);
- napply key_id;##]
- nrewrite > (cup_dotD…); nrewrite > (cupA…);
- nrewrite > (?: ?·? ∪ {[]} = 𝐋.|e'|^* ); ##[##2:
- nrewrite > (erase_bull…); nrewrite > (dot_star_epsilon…); //]
- nrewrite > (erase_bull…);
+ 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 > (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')); //;##]
+ 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
notation > "\move term 90 x term 90 E"
ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
interpretation "rmove" 'move x E = (rmove ? x E).
-nlemma XXz : ∀S:Alpha.∀w:word S. w .∈ ∅ → False.
+nlemma XXz : ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
#S w abs; ninversion abs; #; ndestruct;
nqed.
projects / helm.git / blobdiff