--- /dev/null
+
+
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "arithmetics/nat.ma".
+include "basics/list.ma".
+
+interpretation "iff" 'iff a b = (iff a b).
+
+record Alpha : Type[1] ≝ { carr :> Type[0];
+ eqb: carr → carr → bool;
+ 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).
+
+definition word ≝ λS:Alpha.list S.
+
+inductive re (S: Alpha) : Type[0] ≝
+ z: re S
+ | e: re S
+ | s: S → re S
+ | c: re S → re S → re S
+ | o: re S → re S → re S
+ | k: re S → re S.
+
+(* notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.*)
+notation "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
+interpretation "star" 'pk a = (k ? a).
+interpretation "or" 'plus a b = (o ? a b).
+
+notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
+interpretation "cat" 'pc a b = (c ? a b).
+
+(* to get rid of \middot
+ncoercion c : ∀S:Alpha.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
+*)
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
+interpretation "epsilon" 'epsilon = (e ?).
+
+notation "∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (z ?).
+
+let rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
+
+let 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 ].
+// qed.
+
+definition empty_lang ≝ λS.λw:word S.False.
+notation "{}" non associative with precedence 90 for @{'empty_lang}.
+interpretation "empty lang" 'empty_lang = (empty_lang ?).
+
+definition sing_lang ≝ λS.λx,w:word S.x=w.
+(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.*)
+interpretation "sing lang" 'singl x = (sing_lang ? x).
+
+definition 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).
+
+definition 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).
+
+definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
+interpretation "star lang" 'pk l = (star ? l).
+
+let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝
+match r with
+[ z ⇒ {}
+| e ⇒ { [ ] }
+| s x ⇒ { [x] }
+| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
+| o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
+| k r1 ⇒ (in_l ? r1) ^*].
+
+notation "\sem{term 19 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).
+
+notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+interpretation "orb" 'orb a b = (orb a b).
+
+definition 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).
+
+inductive pitem (S: Alpha) : Type[0] ≝
+ pz: pitem S
+ | pe: pitem S
+ | ps: S → pitem S
+ | pp: S → pitem S
+ | pc: pitem S → pitem S → pitem S
+ | po: pitem S → pitem S → pitem S
+ | pk: pitem S → pitem S.
+
+definition pre ≝ λS.pitem S × bool.
+
+interpretation "pstar" 'pk a = (pk ? a).
+interpretation "por" 'plus a b = (po ? a b).
+interpretation "pcat" 'pc a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "ppatom" 'pp a = (pp ? a).
+(* to get rid of \middot
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
+*)
+interpretation "patom" 'ps a = (ps ? a).
+interpretation "pepsilon" 'epsilon = (pe ?).
+interpretation "pempty" 'empty = (pz ?).
+
+let 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 ⇒ (forget ? E1) · (forget ? E2)
+ | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
+ | pk E ⇒ (forget ? E)^* ].
+
+(* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
+interpretation "forget" 'norm a = (forget ? a).
+
+
+let 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 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
+| po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
+| pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
+
+interpretation "in_pl" 'in_l E = (in_pl ? E).
+interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+
+definition 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).
+
+definition in_prl ≝ λS : Alpha.λp:pre S.
+ if (\snd p) then \sem{\fst p} ∪ { ([ ] : word S) } else \sem{\fst p}.
+
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_l E = (in_prl ? E).
+
+lemma sem_pre_true : ∀S.∀i:pitem S.
+ \sem{〈i,true〉} = \sem{i} ∪ { ([ ] : word S) }.
+// qed.
+
+lemma sem_pre_false : ∀S.∀i:pitem S.
+ \sem{〈i,false〉} = \sem{i}.
+// qed.
+
+lemma sem_cat: ∀S.∀i1,i2:pitem S.
+ \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
+// qed.
+
+lemma sem_plus: ∀S.∀i1,i2:pitem S.
+ \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
+// qed.
+
+lemma sem_star : ∀S.∀i:pitem S.
+ \sem{i^*} = \sem{i} · \sem{|i|}^*.
+// qed.
+
+lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+#S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
+
+lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e).
+#S #e elim e normalize /2/
+ [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
+ >(append_eq_nil …H…) /2/
+ |#r1 #r2 #n1 #n2 % * /2/
+ |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
+ ]
+qed.
+
+(* lemma 12 *)
+lemma epsilon_to_true : ∀S.∀e:pre S. [ ] ∈ e → \snd e = true.
+#S * #i #b cases b // normalize #H @False_ind /2/
+qed.
+
+lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → [ ] ∈ e.
+#S * #i #b #btrue normalize in btrue >btrue %2 //
+qed.
+
+definition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
+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:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1||b2〉.
+// qed.
+
+definition pre_concat_r ≝ λS:Alpha.λi:pitem S.λe:pre S.
+ match e with [ pair i1 b ⇒ 〈i · i1, b〉].
+
+notation "i ▸ e" left associative with precedence 60 for @{'trir $i $e}.
+interpretation "pre_concat_r" 'trir i e = (pre_concat_r ? i e).
+
+definition 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}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
+ A = B → A =1 B.
+#S #A #B #H >H /2/ qed.
+
+lemma ext_eq_trans: ∀S.∀A,B,C:word S → Prop.
+ A =1 B → B =1 C → A =1 C.
+#S #A #B #C #eqAB #eqBC #w cases (eqAB w) cases (eqBC w) /4/
+qed.
+
+lemma union_assoc: ∀S.∀A,B,C:word S → Prop.
+ A ∪ B ∪ C =1 A ∪ (B ∪ C).
+#S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
+qed.
+
+lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
+ \sem{i ▸ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+#S #i * #i1 #b1 cases b1 /2/
+>sem_pre_true >sem_cat >sem_pre_true /2/
+qed.
+
+definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pitem S → pre S.λe1:pre S.λi2:pitem S.
+ match e1 with
+ [ pair i1 b1 ⇒ match b1 with
+ [ true ⇒ (i1 ▸ (bcast ? i2))
+ | false ⇒ 〈i1 · i2,false〉
+ ]
+ ].
+
+definition lift ≝ λf:∀S.pitem S →pre S.λS.λe:pre S.
+ match e with
+ [ pair i b ⇒ 〈\fst (f S i), \snd (f S i) || b〉].
+
+notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
+interpretation "lc" 'lc op a b = (lc ? op a b).
+notation > "a ⊙ b" left associative with precedence 60 for @{'lc eclose $a $b}.
+
+definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S.
+ match e with
+ [ pair i1 b1 ⇒
+ match b1 with
+ [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
+ |false ⇒ 〈i1^*,true〉
+ ]
+ ].
+
+(* 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}.
+
+notation > "•" non associative with precedence 60 for @{eclose ?}.
+
+let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝
+ match i with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, true 〉
+ | ps x ⇒ 〈 `.x, false〉
+ | pp x ⇒ 〈 `.x, false 〉
+ | po i1 i2 ⇒ •i1 ⊕ •i2
+ | pc i1 i2 ⇒ •i1 ⊙ i2
+ | pk i ⇒ 〈(\fst(•i))^*,true〉].
+
+
+notation "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
+
+lemma eclose_plus: ∀S:Alpha.∀i1,i2:pitem S.
+ •(i1 + i2) = •i1 ⊕ •i2.
+// qed.
+
+lemma eclose_dot: ∀S:Alpha.∀i1,i2:pitem S.
+ •(i1 · i2) = •i1 ⊙ i2.
+// qed.
+
+lemma eclose_star: ∀S:Alpha.∀i:pitem S.
+ •i^* = 〈(\fst(•i))^*,true〉.
+// qed.
+
+definition reclose ≝ lift eclose.
+interpretation "reclose" 'eclose x = (reclose ? x).
+
+lemma epsilon_or : ∀S:Alpha.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2.
+#S #b1 #b2 #w % cases b1 cases b2 normalize /2/ * /2/ * ;
+qed.
+
+(*
+lemma 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 *)
+(*
+lemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.
+ \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
+#S * #i1 #b1 * #i2 #b2 #w %
+[normalize * [* /3/ | cases b1 cases b2 normalize /3/ ]
+|normalize * * /3/ cases b1 cases b2 normalize /3/ *]
+qed. *)
+
+lemma odot_true :
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,true〉 ⊙ i2 = i1 ▸ (•i2).
+// qed.
+
+lemma odot_true_bis :
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,true〉 ⊙ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+#S #i1 #i2 normalize cases (•i2) // qed.
+
+lemma odot_false:
+ ∀S.∀i1,i2:pitem S.
+ 〈i1,false〉 ⊙ i2 = 〈i1 · i2, false〉.
+// qed.
+
+lemma LcatE : ∀S.∀e1,e2:pitem S.
+ \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}.
+// qed.
+
+(*
+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.*)
+
+lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
+// qed.
+
+lemma erase_plus : ∀S.∀i1,i2:pitem S.
+ |i1 + i2| = |i1| + |i2|.
+// qed.
+
+lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
+// qed.
+
+(*
+definition 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. *)
+
+lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
+#S #i elim i //
+ [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
+ cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
+ | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
+ cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //
+ | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
+ ]
+qed.
+
+axiom eq_ext_sym: ∀S.∀A,B:word S →Prop.
+ A =1 B → B =1 A.
+
+axiom union_ext_l: ∀S.∀A,B,C:word S →Prop.
+ A =1 C → A ∪ B =1 C ∪ B.
+
+axiom union_ext_r: ∀S.∀A,B,C:word S →Prop.
+ B =1 C → A ∪ B =1 A ∪ C.
+
+axiom union_comm : ∀S.∀A,B:word S →Prop.
+ A ∪ B =1 B ∪ A.
+
+lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
+ (A ∪ B) · C =1 A · C ∪ B · C.
+#S #A #B #C #w %
+ [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/]
+qed.
+
+(* this kind of results are pretty bad for automation;
+ better not index them *)
+lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
+ A · { [ ] } =1 A.
+#S #A #w %
+ [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
+ |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
+ ]
+qed-.
+
+lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
+ { [ ] } · A =1 A.
+#S #A #w %
+ [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
+ |#inA @(ex_intro … [ ]) @(ex_intro … w) /3/
+ ]
+qed-.
+
+
+lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
+ (A ∪ { [ ] }) · C =1 A · C ∪ C.
+#S #A #C @ext_eq_trans [|@distr_cat_r |@union_ext_r @epsilon_cat_l]
+qed.
+
+(* axiom eplison_cut_l: ∀S.∀A:word S →Prop.
+ { [ ] } · A =1 A.
+
+ axiom eplison_cut_r: ∀S.∀A:word S →Prop.
+ A · { [ ] } =1 A. *)
+
+(*
+lemma 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 *)
+lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
+ \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
+ \sem{e1 ⊙ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+#S * #i1 #b1 #i2 cases b1
+ [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
+ |#H >odot_true >sem_pre_true @(ext_eq_trans … (sem_pre_concat_r …))
+ >erase_bull
+ @ext_eq_trans
+ [|@(union_ext_r … H)
+ |@ext_eq_trans
+ [|@union_ext_r [|@union_comm ]
+ |@ext_eq_trans (* /3 by eq_ext_sym, union_ext_l/; *)
+ [|@eq_ext_sym @union_assoc
+ |/3/
+ (*
+ by eq_ext_sym, union_ext_l; @union_ext_l /3
+ /3/ ext_eq_trans /2/
+ /3 width=5 by eq_ext_sym, union_ext_r/ *)
+ ]
+ ]
+ ]
+ ]
+qed.
+
+(* 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 ≝
+ match E with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, false 〉
+ | ps y ⇒ 〈 `y, false 〉
+ | pp y ⇒ 〈 `y, x == y 〉
+ | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
+ | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
+ | pk e ⇒ (\move x e)^⊛ ].
+notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
+notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
+interpretation "move" 'move x E = (move ? x 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.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+
+nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
+#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
+nqed.
+
+
+naxiom in_move_cat:
+ ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
+ (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
+#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
+ncases e1 in H; ncases e2;
+##[##1: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##|##2: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##| #r; *; ##[ *; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
+##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
+ #H; ninversion H; nnormalize; #; ndestruct;
+ ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
+##| #r1 r2; *; ##[ *; #defw]
+ ...
+nqed.
+
+ntheorem move_ok:
+ ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
+#S E; ncases E; #r b; nelim r;
+##[##1,2: #a w; @;
+ ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
+ #H; ninversion H; #; ndestruct;
+ ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
+ #H; ninversion H; #; ndestruct;##]
+##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
+ *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
+##|#a c w; @; nnormalize;
+ ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
+ #H; ninversion H; #; ndestruct;
+ ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
+ #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
+##|#r1 r2 H1 H2 a w; @;
+ ##[ #H; ncases (in_move_cat … H);
+ ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
+ ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
+ nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
+ ##|
+ ...
+##|
+##|
+##]
+nqed.
+
+
+notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
+nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
+ match w with
+ [ nil ⇒ E
+ | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
+
+ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
+
+ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
+ mk_equiv:
+ ∀E1,E2: bool × (pre S).
+ \fst E1 = \fst E2 →
+ (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
+ equiv S E1 E2.
+
+ndefinition NAT: decidable.
+ @ nat eqb; /2/.
+nqed.
+
+include "hints_declaration.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
+
+ninductive unit: Type[0] ≝ I: unit.
+
+nlet corec foo_nop (b: bool):
+ equiv ?
+ 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
+ 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
+ @; //; #x; ncases x
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
+nqed.
+
+(*
+nlet corec foo (a: unit):
+ equiv NAT
+ (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
+ (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
+≝ ?.
+ @;
+ ##[ nnormalize; //
+ ##| #x; ncases x
+ [ nnormalize in ⊢ (??%%);
+ nnormalize in foo: (? → ??%%);
+ @; //; #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply foo_nop
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%);
+
+ ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
+ ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
+ ##]
+nqed.
+*)
+
+ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
+ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
+ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
+
+
+nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
+ nnormalize in match test3;
+ nnormalize;
+//;
+nqed.
+
+(**********************************************************)
+
+ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
+ der_z: der S a (z S) (z S)
+ | der_e: der S a (e S) (z S)
+ | der_s1: der S a (s S a) (e ?)
+ | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
+ | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
+ der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
+ | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
+ der S a (c ? e1 e2) (c ? e1' e2)
+ | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
+ der S a (o ? e1 e2) (o ? e1' e2').
+
+nlemma eq_rect_CProp0_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+nqed.
+
+nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
+
+naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
+(* #S; #r1; #r2; #w; nelim r1
+ [ #K; ninversion K
+ | #H1; #H2; napply (in_c ? []); //
+ | (* tutti casi assurdi *) *)
+
+ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
+ in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
+ | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
+
+ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
+ mk_eq_re: ∀E1,E2.
+ (in_l S [] E1 → in_l S [] E2) →
+ (in_l S [] E2 → in_l S [] E1) →
+ (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
+ eq_re S E1 E2.
+
+(* serve il lemma dopo? *)
+ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
+ #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
+ [ #r; #K (* ok *)
+ | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
+
+(* IL VICEVERSA NON VALE *)
+naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
+(* #S; #w; #E; #H; nelim H
+ [ //
+ | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
+ ]
+nqed. *)
+
+ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
+ #S; #a; #E; #E'; #w; #H; nelim H
+ [##1,2: #H1; ninversion H1
+ [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
+ |##2,9: #X; #Y; #K; ncases (?:False); /2/
+ |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##6,13: #x; #y; #K; ncases (?:False); /2/
+ |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
+##| #H1; ninversion H1
+ [ //
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #H1; #H2; #H3; ninversion H3
+ [ #_; #K; ncases (?:False); /2/
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;
+
+lemma
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "arithmetics/nat.ma".
+include "basics/list.ma".
+
+interpretation "iff" 'iff a b = (iff a b).
+
+record Alpha : Type[1] ≝ { carr :> Type[0];
+ eqb: carr → carr → bool;
+ 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).
+
+definition word ≝ λS:Alpha.list S.
+
+let rec eqbw S (x,y:word S) on x ≝
+ match x with
+ [ nil ⇒ match y with [nil ⇒ true | _ ⇒ false]
+ | cons a xtl ⇒ match y with
+ [nil ⇒ false | cons b ytl ⇒ a == b ∧ eqbw S xtl ytl ]
+ ]
+.
+
+inductive re (S: Alpha) : Type[0] ≝
+ z: re S
+ | e: re S
+ | s: S → re S
+ | c: re S → re S → re S
+ | o: re S → re S → re S
+ | k: re S → re S.
+
+notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
+notation > "a ^ *" non associative with precedence 90 for @{ 'pk $a}.
+interpretation "star" 'pk a = (k ? a).
+interpretation "or" 'plus a b = (o ? a b).
+
+notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
+interpretation "cat" 'pc a b = (c ? a b).
+
+(* to get rid of \middot
+ncoercion c : ∀S:Alpha.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
+*)
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
+interpretation "epsilon" 'epsilon = (e ?).
+
+notation "∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (z ?).
+
+let rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
+
+let rec conjunct (S : Alpha) (l : list (word S)) (r : word S → bool) on l: bool ≝
+match l with [ nil ⇒ true | cons w tl ⇒ r w ∧ conjunct ? tl r ].
+
+definition empty_lang ≝ λS.λw:word S.false.
+notation "{}" non associative with precedence 90 for @{'empty_lang}.
+interpretation "empty lang" 'empty_lang = (empty_lang ?).
+
+definition sing_lang ≝ λS.λx,w:word S.eqbw S x w.
+(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.*)
+interpretation "sing lang" 'singl x = (sing_lang ? x).
+
+definition 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).
+
+definition 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).
+
+definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
+interpretation "star lang" 'pk l = (star ? l).
+
+let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝
+match r with
+[ z ⇒ {}
+| e ⇒ { [ ] }
+| s x ⇒ { [x] }
+| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
+| o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
+| k r1 ⇒ (in_l ? r1) ^*].
+
+notation "\sem{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).
+
+notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+interpretation "orb" 'orb a b = (orb a b).
+
+definition 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).
+
+inductive pitem (S: Alpha) : Type[0] ≝
+ pz: pitem S
+ | pe: pitem S
+ | ps: S → pitem S
+ | pp: S → pitem S
+ | pc: pitem S → pitem S → pitem S
+ | po: pitem S → pitem S → pitem S
+ | pk: pitem S → pitem S.
+
+definition pre ≝ λS.pitem S × bool.
+
+interpretation "pstar" 'pk a = (pk ? a).
+interpretation "por" 'plus a b = (po ? a b).
+interpretation "pcat" 'pc a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "ppatom" 'pp a = (pp ? a).
+(* to get rid of \middot
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S ≝ pc on _p : pitem ? to ∀_:?.?.
+*)
+interpretation "patom" 'ps a = (ps ? a).
+interpretation "pepsilon" 'epsilon = (pe ?).
+interpretation "pempty" 'empty = (pz ?).
+
+notation > "| e |" non associative with precedence 65 for @{forget ? $e}.
+let 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 ⇒ (forget ? E1) · (forget ? E2)
+ | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
+ | pk E ⇒ (forget ? E)^* ].
+
+notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
+interpretation "forget" 'forget a = (forget ? a).
+
+
+let 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 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
+| po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
+| pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
+
+interpretation "in_pl" 'in_l E = (in_pl ? E).
+interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+
+definition 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).
+
+definition in_prl ≝ λS : Alpha.λp:pre S. \sem{\fst p} ∪ ϵ (\snd p).
+
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_l E = (in_prl ? E).
+
+lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+#S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
+
+lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e).
+#S #e elim e normalize /2/
+ [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
+ >(append_eq_nil …H…) /2/
+ |#r1 #r2 #n1 #n2 % * /2/
+ |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
+ ]
+qed.
+
+(* lemma 12 *)
+lemma epsilon_to_true : ∀S.∀e:pre S. [ ] ∈ e → \snd e = true.
+#S #r * [#H apply False_ind /2/ | cases (\snd r) normalize // * ;
+qed.
+
+lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → [ ] ∈ e.
+#S #e #H %2 >H //
+qed.
+
+definition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
+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:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1||b2〉.
+// qed.
+
+definition item_concat ≝ λS:Alpha.λi:pitem S.λe:pre S.
+ match e with [ pair i1 b ⇒ 〈i · i1, b〉].
+
+definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pre S → pre S.λe1.λe2:pre S.
+ match e1 with
+ [ pair i1 b1 ⇒ match b1 with
+ [ true ⇒ item_concat ? i1 (bcast ? e2)
+ | false ⇒ item_concat ? i1 e2
+ ]
+ ].
+
+definition lift ≝ λf:∀S.pitem S →pre S.λS.λe:pre S.
+ match e with
+ [ pair i b ⇒ 〈\fst (f S i), \snd (f S i) || b〉].
+
+notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
+interpretation "lc" 'lc op a b = (lc ? op a b).
+notation > "a ⊙ b" left associative with precedence 60 for @{'lc (lift eclose) $a $b}.
+
+definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S.
+ match e with
+ [ pair i1 b1 ⇒
+ match b1 with
+ [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
+ |false ⇒ 〈i1^*,true〉
+ ]
+ ].
+
+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}.
+
+
+notation > "•" non associative with precedence 60 for @{eclose ?}.
+let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝
+ match i with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, true 〉
+ | ps x ⇒ 〈 `.x, false〉
+ | pp x ⇒ 〈 `.x, false 〉
+ | po E1 E2 ⇒ •E1 ⊕ •E2
+ | pc E1 E2 ⇒ •E1 ⊙ 〈E2,false〉
+ | 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}.
+
+definition reclose ≝ lift eclose.
+interpretation "reclose" 'eclose x = (reclose ? x).
+
+definition 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}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+(*
+lemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) =1 ϵ b1 ∪ ϵ b2.
+##[##2: napply S]
+#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
+nqed.
+
+lemma 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 *)
+lemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.\sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
+#S * #i1 #b1 * #i2 #b2 >lo_def normalize in ⊢ (?%?);
+
+#w cases b1 cases b2 normalize % #w 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 ≝
+ match E with
+ [ pz ⇒ 〈 ∅, false 〉
+ | pe ⇒ 〈 ϵ, false 〉
+ | ps y ⇒ 〈 `y, false 〉
+ | pp y ⇒ 〈 `y, x == y 〉
+ | po e1 e2 ⇒ \move x e1 ⊕ \move x e2
+ | pc e1 e2 ⇒ \move x e1 ⊙ \move x e2
+ | pk e ⇒ (\move x e)^⊛ ].
+notation < "\move\shy x\shy E" non associative with precedence 60 for @{'move $x $E}.
+notation > "\move term 90 x term 90 E" non associative with precedence 60 for @{'move $x $E}.
+interpretation "move" 'move x E = (move ? x 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.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+
+nlemma XXe : ∀S:Alpha.∀w:word S. w .∈ ϵ → False.
+#S w abs; ninversion abs; #; ndestruct;
+nqed.
+
+nlemma XXze : ∀S:Alpha.∀w:word S. w .∈ (∅ · ϵ) → False.
+#S w abs; ninversion abs; #; ndestruct; /2/ by XXz,XXe;
+nqed.
+
+
+naxiom in_move_cat:
+ ∀S.∀w:word S.∀x.∀E1,E2:pitem S. w .∈ \move x (E1 · E2) →
+ (∃w1.∃w2. w = w1@w2 ∧ w1 .∈ \move x E1 ∧ w2 ∈ .|E2|) ∨ w .∈ \move x E2.
+#S w x e1 e2 H; nchange in H with (w .∈ \move x e1 ⊙ \move x e2);
+ncases e1 in H; ncases e2;
+##[##1: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##|##2: *; ##[*; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz,XXze;
+##| #r; *; ##[ *; nnormalize; #; ndestruct]
+ #H; ninversion H; ##[##1,4,5,6: nnormalize; #; ndestruct]
+ ##[##2: nnormalize; #; ndestruct; @2; @2; //.##]
+ nnormalize; #; ndestruct; ncases (?:False); /2/ by XXz;
+##| #y; *; ##[ *; nnormalize; #defw defx; ndestruct; @2; @1; /2/ by conj;##]
+ #H; ninversion H; nnormalize; #; ndestruct;
+ ##[ncases (?:False); /2/ by XXz] /3/ by or_intror;
+##| #r1 r2; *; ##[ *; #defw]
+ ...
+nqed.
+
+ntheorem move_ok:
+ ∀S:Alpha.∀E:pre S.∀a,w.w .∈ \move a E ↔ (a :: w) .∈ E.
+#S E; ncases E; #r b; nelim r;
+##[##1,2: #a w; @;
+ ##[##1,3: nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #abs; ncases (XXz … abs); ##]
+ #H; ninversion H; #; ndestruct;
+ ##|##*:nnormalize; *; ##[##1,3: *; #; ndestruct; ##| #H1; ncases (XXz … H1); ##]
+ #H; ninversion H; #; ndestruct;##]
+##|#a c w; @; nnormalize; ##[*; ##[*; #; ndestruct; ##] #abs; ninversion abs; #; ndestruct;##]
+ *; ##[##2: #abs; ninversion abs; #; ndestruct; ##] *; #; ndestruct;
+##|#a c w; @; nnormalize;
+ ##[ *; ##[ *; #defw; nrewrite > defw; #ca; @2; nrewrite > (eqb_t … ca); @; ##]
+ #H; ninversion H; #; ndestruct;
+ ##| *; ##[ *; #; ndestruct; ##] #H; ninversion H; ##[##2,3,4,5,6: #; ndestruct]
+ #d defw defa; ndestruct; @1; @; //; nrewrite > (eqb_true S d d); //. ##]
+##|#r1 r2 H1 H2 a w; @;
+ ##[ #H; ncases (in_move_cat … H);
+ ##[ *; #w1; *; #w2; *; *; #defw w1m w2m;
+ ncases (H1 a w1); #H1w1; #_; nlapply (H1w1 w1m); #good;
+ nrewrite > defw; @2; @2 (a::w1); //; ncases good; ##[ *; #; ndestruct] //.
+ ##|
+ ...
+##|
+##|
+##]
+nqed.
+
+
+notation > "x ↦* E" non associative with precedence 60 for @{move_star ? $x $E}.
+nlet rec move_star (S : decidable) w E on w : bool × (pre S) ≝
+ match w with
+ [ nil ⇒ E
+ | cons x w' ⇒ w' ↦* (x ↦ \snd E)].
+
+ndefinition in_moves ≝ λS:decidable.λw.λE:bool × (pre S). \fst(w ↦* E).
+
+ncoinductive equiv (S:decidable) : bool × (pre S) → bool × (pre S) → Prop ≝
+ mk_equiv:
+ ∀E1,E2: bool × (pre S).
+ \fst E1 = \fst E2 →
+ (∀x. equiv S (x ↦ \snd E1) (x ↦ \snd E2)) →
+ equiv S E1 E2.
+
+ndefinition NAT: decidable.
+ @ nat eqb; /2/.
+nqed.
+
+include "hints_declaration.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ ; X ≟ NAT ⊢ carr X ≡ nat.
+
+ninductive unit: Type[0] ≝ I: unit.
+
+nlet corec foo_nop (b: bool):
+ equiv ?
+ 〈 b, pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0))) 〉
+ 〈 b, pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0) 〉 ≝ ?.
+ @; //; #x; ncases x
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply (foo_nop false)
+ | #w; nnormalize in ⊢ (??%%); napply (foo_nop false) ]##]
+nqed.
+
+(*
+nlet corec foo (a: unit):
+ equiv NAT
+ (eclose NAT (pc ? (ps ? 0) (pk ? (pc ? (ps ? 1) (ps ? 0)))))
+ (eclose NAT (pc ? (pk ? (pc ? (ps ? 0) (ps ? 1))) (ps ? 0)))
+≝ ?.
+ @;
+ ##[ nnormalize; //
+ ##| #x; ncases x
+ [ nnormalize in ⊢ (??%%);
+ nnormalize in foo: (? → ??%%);
+ @; //; #y; ncases y
+ [ nnormalize in ⊢ (??%%); napply foo_nop
+ | #y; ncases y
+ [ nnormalize in ⊢ (??%%);
+
+ ##| #z; nnormalize in ⊢ (??%%); napply foo_nop ]##]
+ ##| #y; nnormalize in ⊢ (??%%); napply foo_nop
+ ##]
+nqed.
+*)
+
+ndefinition test1 : pre ? ≝ ❨ `0 | `1 ❩^* `0.
+ndefinition test2 : pre ? ≝ ❨ (`0`1)^* `0 | (`0`1)^* `1 ❩.
+ndefinition test3 : pre ? ≝ (`0 (`0`1)^* `1)^*.
+
+
+nlemma foo: in_moves ? [0;0;1;0;1;1] (ɛ test3) = true.
+ nnormalize in match test3;
+ nnormalize;
+//;
+nqed.
+
+(**********************************************************)
+
+ninductive der (S: Type[0]) (a: S) : re S → re S → CProp[0] ≝
+ der_z: der S a (z S) (z S)
+ | der_e: der S a (e S) (z S)
+ | der_s1: der S a (s S a) (e ?)
+ | der_s2: ∀b. a ≠ b → der S a (s S b) (z S)
+ | der_c1: ∀e1,e2,e1',e2'. in_l S [] e1 → der S a e1 e1' → der S a e2 e2' →
+ der S a (c ? e1 e2) (o ? (c ? e1' e2) e2')
+ | der_c2: ∀e1,e2,e1'. Not (in_l S [] e1) → der S a e1 e1' →
+ der S a (c ? e1 e2) (c ? e1' e2)
+ | der_o: ∀e1,e2,e1',e2'. der S a e1 e1' → der S a e2 e2' →
+ der S a (o ? e1 e2) (o ? e1' e2').
+
+nlemma eq_rect_CProp0_r:
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; #P; #H; nassumption.
+nqed.
+
+nlemma append1: ∀A.∀a:A.∀l. [a] @ l = a::l. //. nqed.
+
+naxiom in_l1: ∀S,r1,r2,w. in_l S [ ] r1 → in_l S w r2 → in_l S w (c S r1 r2).
+(* #S; #r1; #r2; #w; nelim r1
+ [ #K; ninversion K
+ | #H1; #H2; napply (in_c ? []); //
+ | (* tutti casi assurdi *) *)
+
+ninductive in_l' (S: Type[0]) : word S → re S → CProp[0] ≝
+ in_l_empty1: ∀E.in_l S [] E → in_l' S [] E
+ | in_l_cons: ∀a,w,e,e'. in_l' S w e' → der S a e e' → in_l' S (a::w) e.
+
+ncoinductive eq_re (S: Type[0]) : re S → re S → CProp[0] ≝
+ mk_eq_re: ∀E1,E2.
+ (in_l S [] E1 → in_l S [] E2) →
+ (in_l S [] E2 → in_l S [] E1) →
+ (∀a,E1',E2'. der S a E1 E1' → der S a E2 E2' → eq_re S E1' E2') →
+ eq_re S E1 E2.
+
+(* serve il lemma dopo? *)
+ntheorem eq_re_is_eq: ∀S.∀E1,E2. eq_re S E1 E2 → ∀w. in_l ? w E1 → in_l ? w E2.
+ #S; #E1; #E2; #H1; #w; #H2; nelim H2 in E2 H1 ⊢ %
+ [ #r; #K (* ok *)
+ | #a; #w; #R1; #R2; #K1; #K2; #K3; #R3; #K4; @2 R2; //; ncases K4;
+
+(* IL VICEVERSA NON VALE *)
+naxiom in_l_to_in_l: ∀S,w,E. in_l' S w E → in_l S w E.
+(* #S; #w; #E; #H; nelim H
+ [ //
+ | #a; #w'; #r; #r'; #H1; (* e si cade qua sotto! *)
+ ]
+nqed. *)
+
+ntheorem der1: ∀S,a,e,e',w. der S a e e' → in_l S w e' → in_l S (a::w) e.
+ #S; #a; #E; #E'; #w; #H; nelim H
+ [##1,2: #H1; ninversion H1
+ [##1,8: #_; #K; (* non va ndestruct K; *) ncases (?:False); (* perche' due goal?*) /2/
+ |##2,9: #X; #Y; #K; ncases (?:False); /2/
+ |##3,10: #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ |##4,11: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##5,12: #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ |##6,13: #x; #y; #K; ncases (?:False); /2/
+ |##7,14: #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/]
+##| #H1; ninversion H1
+ [ //
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #H1; #H2; #H3; ninversion H3
+ [ #_; #K; ncases (?:False); /2/
+ | #X; #Y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #e; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #K; ncases (?:False); /2/
+ | #x; #y; #K; ncases (?:False); /2/
+ | #x; #y; #z; #w; #a; #b; #c; #d; #K; ncases (?:False); /2/ ]
+##| #r1; #r2; #r1'; #r2'; #H1; #H2; #H3; #H4; #H5; #H6;
+
projects / helm.git / commitdiff