-interpretation "re epsilon" 'epsilon = (e ?).
-interpretation "re or" 'plus a b = (o ? a b).
-interpretation "re cat" 'middot a b = (c ? a b).
-interpretation "re star" 'star a = (k ? a).
-
-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 "`∅" non associative with precedence 90 for @{ 'empty }.
-interpretation "empty" 'empty = (z ?).
-
-let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
-match r with
-[ z ⇒ ∅
-| e ⇒ {ϵ}
-| 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).
-
-lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
-// qed.
-
-
-(* pointed items *)
-inductive pitem (S: DeqSet) : 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 "pitem star" 'star a = (pk ? a).
-interpretation "pitem or" 'plus a b = (po ? a b).
-interpretation "pitem cat" 'middot 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 "pitem pp" 'pp a = (pp ? a).
-interpretation "pitem ps" 'ps a = (ps ? a).
-interpretation "pitem epsilon" 'epsilon = (pe ?).
-interpretation "pitem empty" 'empty = (pz ?).
-
-let rec forget (S: DeqSet) (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 : DeqSet) (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 in_prl ≝ λS : DeqSet.λp:pre S.
- if (\snd p) then \sem{\fst p} ∪ {ϵ} 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} ∪ {ϵ}.
-// 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_cat_w: ∀S.∀i1,i2:pitem S.∀w.
- \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
-// qed.
-
-lemma sem_plus: ∀S.∀i1,i2:pitem S.
- \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
-// qed.
-
-lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
- \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
-// qed.
-
-lemma sem_star : ∀S.∀i:pitem S.
- \sem{i^*} = \sem{i} · \sem{|i|}^*.
-// qed.
-
-lemma sem_star_w : ∀S.∀i:pitem S.∀w.
- \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
-// qed.