(**************************************************************************)
include "arithmetics/nat.ma".
-include "basics/list.ma".
+include "basics/lists/list.ma".
+include "basics/sets.ma".
-interpretation "iff" 'iff a b = (iff a b).
+definition word ≝ λS:DeqSet.list S.
-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] ≝
+inductive re (S: DeqSet) : Type[0] ≝
z: re S
| e: re S
| s: S → re S
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 ∀_:?.?.
+ncoercion c : ∀S:DeqSet.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?.
*)
notation < "a" non associative with precedence 90 for @{ 'ps $a}.
notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
interpretation "epsilon" 'epsilon = (e ?).
-notation "∅" non associative with precedence 90 for @{ 'empty }.
+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 ≝
+let rec flatten (S : DeqSet) (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 ≝
+let rec conjunct (S : DeqSet) (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 ?).
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).
+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.
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 ≝
+let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
match r with
-[ z ⇒ {}
+[ z ⇒ ∅
| e ⇒ { [ ] }
| s x ⇒ { [x] }
| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
// qed.
-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] ≝
+inductive pitem (S: DeqSet) : Type[0] ≝
pz: pitem S
| pe: pitem S
| ps: S → pitem S
interpretation "pepsilon" 'epsilon = (pe ?).
interpretation "pempty" 'empty = (pz ?).
-let rec forget (S: Alpha) (l : pitem S) on l: re S ≝
+let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
match l with
- [ pz ⇒ ∅
+ [ pz ⇒ `∅
| pe ⇒ ϵ
| ps x ⇒ `x
| pp x ⇒ `x
(* 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 ≝
+let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
match r with
-[ pz ⇒ {}
-| pe ⇒ {}
-| ps _ ⇒ {}
+[ 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)
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 {}.
+(*
+definition epsilon : ∀S:DeqSet.bool → word S → Prop
+≝ 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).
+interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b). *)
-definition in_prl ≝ λS : Alpha.λp:pre S.
+definition in_prl ≝ λS : DeqSet.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).
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).
+lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ ([ ] ∈ e).
#S #e elim e normalize /2/
[#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
>(append_eq_nil …H…) /2/
#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〉.
+definition lo ≝ λS:DeqSet.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〉.
+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.
+definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
match e with [ pair i1 b ⇒ 〈i · i1, b〉].
notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}.
A = B → A =1 B.
#S #A #B #H >H /2/ qed.
-lemma ext_eq_trans: ∀S.∀A,B,C:word S → Prop.
+(* lemma eqP_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.
+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.
+qed. *)
lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
\sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
>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.
+definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
match e1 with
[ pair i1 b1 ⇒ match b1 with
[ true ⇒ (i1 ◂ (bcast ? i2))
definition lift ≝ λS.λf:pitem S →pre S.λe:pre S.
match e with
- [ pair i b ⇒ 〈\fst (f i), \snd (f i) || b〉].
+ [ pair i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
interpretation "lc" 'lc op a b = (lc ? op a b).
-definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S.
+definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S.
match e with
[ pair i1 b1 ⇒
match b1 with
].
(*
-lemma oplus_tt : ∀S: Alpha.∀i1,i2:pitem S.
+lemma oplus_tt : ∀S: DeqSet.∀i1,i2:pitem S.
〈i1,true〉 ⊕ 〈i2,true〉 = 〈i1 + i2,true〉.
// qed.
-lemma oplus_tf : ∀S: Alpha.∀i1,i2:pitem S.
+lemma oplus_tf : ∀S: DeqSet.∀i1,i2:pitem S.
〈i1,true〉 ⊕ 〈i2,false〉 = 〈i1 + i2,true〉.
// qed.
-lemma oplus_ft : ∀S: Alpha.∀i1,i2:pitem S.
+lemma oplus_ft : ∀S: DeqSet.∀i1,i2:pitem S.
〈i1,false〉 ⊕ 〈i2,true〉 = 〈i1 + i2,true〉.
// qed.
-lemma oplus_ff : ∀S: Alpha.∀i1,i2:pitem S.
+lemma oplus_ff : ∀S: DeqSet.∀i1,i2:pitem S.
〈i1,false〉 ⊕ 〈i2,false〉 = 〈i1 + i2,false〉.
// qed. *)
notation "•" non associative with precedence 60 for @{eclose ?}.
-let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝
+let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
match i with
- [ pz ⇒ 〈 ∅, false 〉
+ [ pz ⇒ 〈 `∅, false 〉
| pe ⇒ 〈 ϵ, true 〉
| ps x ⇒ 〈 `.x, false〉
| pp x ⇒ 〈 `.x, false 〉
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.
+lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
•(i1 + i2) = •i1 ⊕ •i2.
// qed.
-lemma eclose_dot: ∀S:Alpha.∀i1,i2:pitem S.
+lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
•(i1 · i2) = •i1 ▸ i2.
// qed.
-lemma eclose_star: ∀S:Alpha.∀i:pitem S.
+lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
•i^* = 〈(\fst(•i))^*,true〉.
// qed.
definition reclose ≝ λS. lift S (eclose S).
interpretation "reclose" 'eclose x = (reclose ? x).
-lemma epsilon_or : ∀S:Alpha.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2.
+(*
+lemma epsilon_or : ∀S:DeqSet.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2.
#S #b1 #b2 #w % cases b1 cases b2 normalize /2/ * /2/ * ;
-qed.
+qed. *)
(*
lemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
#S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.*)
(* theorem 16: 2 *)
-lemma sem_oplus: ∀S:Alpha.∀e1,e2:pre S.
+lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
\sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
#S * #i1 #b1 * #i2 #b2 #w %
[cases b1 cases b2 normalize /2/ * /3/ * /3/
]
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.
-
-axiom union_idemp: ∀S.∀A:word S →Prop.
- A ∪ A =1 A.
-
axiom cat_ext_l: ∀S.∀A,B,C:word S →Prop.
A =1 C → A · B =1 C · B.
axiom fix_star: ∀S.∀A:word S → Prop.
A^* =1 A · A^* ∪ { [ ] }.
-axiom star_epsilon: ∀S:Alpha.∀A:word S → Prop.
+axiom star_epsilon: ∀S:DeqSet.∀A:word S → Prop.
A^* ∪ { [ ] } =1 A^*.
-lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S.
+lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
\sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ { [ ] }.
/2/ qed.
(*
-lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S.
+lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
\sem{〈i^*,true〉} =1 \sem{〈i,true〉}·\sem{|i|}^* ∪ { [ ] }.
/2/ 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]
+#S #A #C @eqP_trans [|@distr_cat_r |@eqP_union_l @epsilon_cat_l]
qed.
(* axiom eplison_cut_l: ∀S.∀A:word S →Prop.
\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 [|@eq_ext_sym @union_assoc ] /3/
+ |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
+ >erase_bull @eqP_trans [|@(eqP_union_l … H)]
+ @eqP_trans [|@eqP_union_l[|@union_comm ]]
+ @eqP_trans [|@eqP_sym @union_assoc ] /3/
]
qed.
\sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
(* theorem 16: 1 *)
-theorem sem_bull: ∀S:Alpha. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
+theorem sem_bull: ∀S:DeqSet. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
#S #e elim e
[#w normalize % [/2/ | * //]
|/2/
|#x normalize #w % [ /2/ | * [@False_ind | //]]
|#x normalize #w % [ /2/ | * // ]
|#i1 #i2 #IH1 #IH2 >eclose_dot
- @ext_eq_trans [|@odot_dot_aux //] >sem_cat
- @ext_eq_trans
- [|@union_ext_l
- [|@ext_eq_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
- @ext_eq_trans [|@union_assoc]
- @ext_eq_trans [||@eq_ext_sym @union_assoc]
- @union_ext_r //
+ @eqP_trans [|@odot_dot_aux //] >sem_cat
+ @eqP_trans
+ [|@eqP_union_r
+ [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
+ @eqP_trans [|@union_assoc]
+ @eqP_trans [||@eqP_sym @union_assoc]
+ @eqP_union_l //
|#i1 #i2 #IH1 #IH2 >eclose_plus
- @ext_eq_trans [|@sem_oplus] >sem_plus >erase_plus
- @ext_eq_trans [|@(union_ext_r … IH2)]
- @ext_eq_trans [|@eq_ext_sym @union_assoc]
- @ext_eq_trans [||@union_assoc] @union_ext_l
- @ext_eq_trans [||@eq_ext_sym @union_assoc]
- @ext_eq_trans [||@union_ext_r [|@union_comm]]
- @ext_eq_trans [||@union_assoc] /3/
+ @eqP_trans [|@sem_oplus] >sem_plus >erase_plus
+ @eqP_trans [|@(eqP_union_l … IH2)]
+ @eqP_trans [|@eqP_sym @union_assoc]
+ @eqP_trans [||@union_assoc] @eqP_union_r
+ @eqP_trans [||@eqP_sym @union_assoc]
+ @eqP_trans [||@eqP_union_l [|@union_comm]]
+ @eqP_trans [||@union_assoc] /3/
|#i #H >sem_pre_true >sem_star >erase_bull >sem_star
- @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]]
- @ext_eq_trans [|@union_ext_l [|@distr_cat_r]]
- @ext_eq_trans [|@union_assoc] @union_ext_r >erase_star @star_fix
+ @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@sem_fst_aux //]]]
+ @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+ @eqP_trans [|@union_assoc] @eqP_union_l >erase_star @star_fix
]
qed.
-definition lifted_cat ≝ λS:Alpha.λe:pre S.
+definition lifted_cat ≝ λS:DeqSet.λe:pre S.
lift S (lc S eclose e).
notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-lemma sem_odot_true: ∀S:Alpha.∀e1:pre S.∀i.
+lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
\sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }.
#S #e1 #i
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || true〉) [//]
+cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//]
#H >H cases (e1 ▸ i) #i1 #b1 cases b1
- [>sem_pre_true @ext_eq_trans [||@eq_ext_sym @union_assoc]
- @union_ext_r /2/
+ [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
+ @eqP_union_l /2/
|/2/
]
qed.
-lemma eq_odot_false: ∀S:Alpha.∀e1:pre S.∀i.
+lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i.
e1 ⊙ 〈i,false〉 = e1 ▸ i.
#S #e1 #i
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || false〉) [//]
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ false〉) [//]
cases (e1 ▸ i) #i1 #b1 cases b1 #H @H
qed.
∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
#S #e1 * #i2 *
[>sem_pre_true
- @ext_eq_trans [|@sem_odot_true]
- @ext_eq_trans [||@union_assoc] @union_ext_l @odot_dot_aux //
+ @eqP_trans [|@sem_odot_true]
+ @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
|>sem_pre_false >eq_odot_false @odot_dot_aux //
]
qed.
\sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
#S * #i #b cases b
[>sem_pre_true >sem_pre_true >sem_star >erase_bull
- @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]]
- @ext_eq_trans [|@union_ext_l [|@distr_cat_r]]
- @ext_eq_trans [||@eq_ext_sym @distr_cat_r]
- @ext_eq_trans [|@union_assoc] @union_ext_r
- @ext_eq_trans [||@eq_ext_sym @epsilon_cat_l] @star_fix
+ @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]]
+ @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+ @eqP_trans [||@eqP_sym @distr_cat_r]
+ @eqP_trans [|@union_assoc] @eqP_union_l
+ @eqP_trans [||@eqP_sym @epsilon_cat_l] @star_fix
|>sem_pre_false >sem_pre_false >sem_star /2/
]
qed.
(*
-nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝
+nlet rec pre_of_re (S : DeqSet) (e : re S) on e : pitem S ≝
match e with
[ z ⇒ pz ?
| e ⇒ pe ?
projects / helm.git / commitdiff