(* *)
(**************************************************************************)
-(*include "logic/connectives.ma".*)
-(*include "logic/equality.ma".*)
include "datatypes/list.ma".
include "datatypes/pairs.ma".
include "arithmetics/nat.ma".
-(*include "Plogic/equality.ma".*)
+interpretation "iff" 'iff a b = (iff a b).
-nrecord Alpha : Type[1] ≝
- { carr :> Type[0];
+nrecord Alpha : Type[1] ≝ { carr :> Type[0];
eqb: carr → carr → bool;
- eqb_true: ∀x,y. (eqb x y = true) = (x=y);
- eqb_false: ∀x,y. (eqb x y = false) = (x≠y)
- }.
+ eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
+}.
notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
interpretation "eqb" 'eqb a b = (eqb ? a b).
notation "∅" non associative with precedence 90 for @{ 'empty }.
interpretation "empty" 'empty = (z ?).
-notation > "w ∈ E" non associative with precedence 45 for @{in_l ? $w $E}.
-ninductive in_l (S : Alpha) : word S → re S → Prop ≝
- | in_e: [ ] ∈ ϵ
- | in_s: ∀x:S. [x] ∈ `x
- | in_c: ∀w1,w2,e1,e2. w1 ∈ e1 → w2 ∈ e2 → w1@w2 ∈ e1 · e2
- | in_o1: ∀w,e1,e2. w ∈ e1 → w ∈ e1 + e2
- | in_o2: ∀w,e1,e2. w ∈ e2 → w ∈ e1 + e2
- | in_ke: ∀e. [ ] ∈ e^*
- | in_ki: ∀w1,w2,e. w1 ∈ e → w2 ∈ e^* → w1@w2 ∈ e^*.
-interpretation "in_l" 'mem w l = (in_l ? w l).
+nlet rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
-(*
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
-interpretation "orb" 'orb a b = (orb a b).
+nlet rec conjunct (S : Alpha) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
+match l with [ nil ⇒ ? | cons w tl ⇒ r w ∧ conjunct ? tl r ]. napply True. nqed.
-notation "a && b" left associative with precedence 40 for @{'andb $a $b}.
-interpretation "andb" 'andb a b = (andb a b).
+ndefinition empty_lang ≝ λS.λw:word S.False.
+notation "{}" non associative with precedence 90 for @{'empty_lang}.
+interpretation "empty lang" 'empty_lang = (empty_lang ?).
-nlet rec weq (S : Alpha) (l1, l2 : word S) on l1 : bool ≝
-match l1 with
-[ nil ⇒ match l2 with [ nil ⇒ true | cons _ _ ⇒ false ]
-| cons x xs ⇒ match l2 with [ nil ⇒ false | cons y ys ⇒ (x == y) && weq S xs ys]].
+ndefinition sing_lang ≝ λS.λx,w:word S.x=w.
+notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.
+interpretation "sing lang" 'sing_lang x = (sing_lang ? x).
-ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
-notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
+ndefinition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.l1 w ∨ l2 w.
+interpretation "union lang" 'union a b = (union ? a b).
-*)
+ndefinition cat : ∀S,l1,l2,w.Prop ≝
+ λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
+interpretation "cat lang" 'pc a b = (cat ? a b).
-nlemma in_l_inv_e:
- ∀S.∀w:word S. w ∈ ∅ → w = [].
- #S; #w; #H; ninversion H; #; ndestruct;
-nqed.
+ndefinition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
+interpretation "star lang" 'pk l = (star ? l).
-nlemma in_l_inv_s: ∀S.∀w:word S.∀x. w ∈ `x → w = [x].
-#S w x H; ninversion H; #; ndestruct; //.
-nqed.
+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 ⇒ {}
+| e ⇒ { [ ] }
+| s x ⇒ { [x] }
+| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
+| o r1 r2 ⇒ 𝐋 r1 ∪ 𝐋 r2
+| k r1 ⇒ (𝐋 r1) ^*].
+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).
-nlemma in_l_inv_c:
- ∀S.∀w:word S.∀E1,E2. w ∈ E1 · E2 → ∃w1.∃w2. w = w1@w2 ∧ w1 ∈ E1 ∧ w2 ∈ E2.
-#S w e1 e2 H; ninversion H; ##[##1,2,4,5,6,7: #; ndestruct; ##]
-#w1 w2 r1 r2 w1r1 w2r2; #_; #_; #defw defe; @w1; @w2; ndestruct; /3/.
-nqed.
+notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+interpretation "orb" 'orb a b = (orb a b).
+
+ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
ninductive pitem (S: Alpha) : Type[0] ≝
pz: pitem S
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 > "w .∈ E" non associative with precedence 40 for @{in_pl ? $w $E}.
-ninductive in_pl (S: Alpha): word S → pitem S → Prop ≝
- | in_pp: ∀x:S.[x] .∈ `.x
- | in_pc1: ∀w1,w2:word S.∀e1,e2:pitem S.
- w1 .∈ e1 → w2 ∈ .|e2| → (w1@w2) .∈ e1 · e2
- | in_pc2: ∀w,e1,e2. w .∈ e2 → w .∈ e1 · e2
- | in_po1: ∀w,e1,e2. w .∈ e1 → w .∈ e1 + e2
- | in_po2: ∀w,e1,e2. w .∈ e2 → w .∈ e1 + e2
- | in_pki: ∀w1,w2,e. w1 .∈ e → w2 ∈ .|e|^* → (w1@w2) .∈ e^*.
-
-interpretation "in_pl" 'in_pl w l = (in_pl ? w l).
-
-ndefinition in_prl ≝ λS : Alpha.λw:word S.λp:pre S.
- (w = [ ] ∧ \snd p = true) ∨ w .∈ (\fst p).
+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
+| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
+| 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).
+
+ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
+
+interpretation "epsilon" 'epsilon = (epsilon ?).
+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).
-notation > "w .∈ E" non associative with precedence 40 for @{'in_pl $w $E}.
-notation < "w\shy .∈\shy E" non associative with precedence 40 for @{'in_pl $w $E}.
-interpretation "in_prl" 'in_pl w l = (in_prl ? w l).
-
-interpretation "iff" 'iff a b = (iff a b).
-
-nlemma append_eq_nil :
- ∀S.∀w1,w2:word S. [ ] = w1 @ w2 → w1 = [ ].
-#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct;
-nqed.
-
-nlemma append_eq_nil_r :
- ∀S.∀w1,w2:word S. [ ] = w1 @ w2 → w2 = [ ].
-#S w1; nelim w1; ##[ #w2 H; nrewrite > H; // ]
-#x tl IH w2; nnormalize; #abs; ndestruct;
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_pl E = (in_prl ? E).
+
+nlemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct; nqed.
+
+(* lemma 12 *)
+nlemma epsilon_in_true : ∀S.∀e:pre S. [ ] ∈ e ↔ \snd e = true.
+#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//;
+nnormalize; *; ##[##2:*] nelim e;
+##[ ##1,2: *; ##| #c; *; ##| #c; nnormalize; #; ndestruct; ##| ##7: #p H;
+##| #r1 r2 H G; *; ##[##2: /3/ by or_intror]
+##| #r1 r2 H1 H2; *; /3/ by or_intror, or_introl; ##]
+*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
nqed.
-nlemma lemma16 :
- ∀S.∀e:pre S. [ ] .∈ e ↔ \snd e = true.
-#S p; ncases p; #e b; @; ##[##2: #H; nrewrite > H; @; @; //. ##]
-ncases b; //; *; ##[*; //] nelim e;
-##[##1,2: #abs; ninversion abs; #; ndestruct;
-##|##3,4: #x abs; ninversion abs; #; ndestruct;
-##|#p1 p2 H1 H2 H; ninversion H; ##[##1,3,4,5,6: #; ndestruct; /2/. ##]
- #w1 w2 r1 r2 w1r1 w2fr2 H3 H4 Ep1p2; ndestruct;
- nrewrite > (append_eq_nil … H4) in w1r1; /2/ by {};
-##|#r1 r2 H1 H2 H; ninversion H; #; ndestruct; /2/ by {};
-##|#r H1 H2; ninversion H2; ##[##1,2,3,4,5: #; ndestruct; ##]
- #w1 w2 r1 w1r1 w1er1 H11 H21 H31;
- nrewrite > (append_eq_nil … H21) in w1r1 H1;
- nrewrite > (?: r = r1); /2/ by {};
- ndestruct; //. ##]
+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;
+ nrewrite > (append_eq_nil …H…); /2/;
+##| #r1 r2 n1 n2; @; *; /2/;
+##| #r n; @; *; #w1; *; #w2; *; *; #H;
+ nrewrite > (append_eq_nil …H…); /2/;##]
nqed.
ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
match a with [ mk_pair e1 b1 ⇒
- match b with [ mk_pair e2 b2 ⇒
match b1 with
- [ false ⇒ 〈e1 · e2, b2〉
- | true ⇒ match bcast ? e2 with [ mk_pair e2' b2' ⇒ 〈e1 · e2', b2 || b2'〉 ]]]].
+ [ false ⇒ 〈e1 · \fst b, \snd b〉
+ | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
interpretation "lc" 'lc op a b = (lc ? op a b).
match a with [ mk_pair e1 b1 ⇒
match b1 with
[ false ⇒ 〈e1^*, false〉
- | true ⇒ match bcast ? e1 with [ mk_pair e1' b1' ⇒ 〈e1'^*, true〉 ]]].
+ | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
-notation < "a \sup ⊛" non associative with precedence 90 for @{'lk ? $a}.
+notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
interpretation "lk" 'lk op a = (lk ? op a).
notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
| pp x ⇒ 〈 `.x, false 〉
| po E1 E2 ⇒ •E1 ⊕ •E2
| pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉
- | pk E ⇒ 〈E,true〉^⊛].
+ | pk E ⇒ 〈(\fst (•E))^*,true〉].
notation < "• x" non associative with precedence 60 for @{'eclose $x}.
interpretation "eclose" 'eclose x = (eclose ? x).
notation > "• x" non associative with precedence 60 for @{'eclose $x}.
ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
interpretation "reclose" 'eclose x = (reclose ? x).
-nlemma lemma19_2 :
- ∀S:Alpha.∀e1,e2:pre S.∀w. w .∈ e1 ⊕ e2 → w .∈ e1 ∨ w .∈ e2.
-#S e1 e2 w H; nnormalize in H; ncases H;
-##[ *; #defw; ncases e1; #p b; ncases b; nnormalize;
- ##[ #_; @1; @1; /2/ by conj;
- ##| #H1; @2; @1; /2/ by conj; ##]
-##| #H1; ninversion H1; #; ndestruct; /4/ by or_introl, or_intror; ##]
+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.
+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
+
notation > "\move term 90 x term 90 E"
non associative with precedence 60 for @{move ? $x $E}.
nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
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.
#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
nqed.
-nlemma eqb_t : ∀S:Alpha.∀a,b:S.∀p:a == b = true. a = b.
-#S a b H; nrewrite < (eqb_true ? a b); //.
-nqed.
naxiom in_move_cat:
∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
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