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concatenation.v
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concatenation.v
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(* ---------------------------------------------------------------------
This file is part of a repository containing the definitions and
proof scripts related to the formalization of context-free language
theory in Coq. Specifically, the following results were obtained:
(i) closure operations for context-free grammars,
(ii) context-free grammars simplification
(iii) context-free grammar Chomsky normalization and
(iv) pumping lemma for context-free languages.
More information can be found in thesis "Formalization of
Context-Free Language Theory", submitted to the Informatics
Center of the Pernambuco Federal University (CIn/UFPE) in
Brazil.
The file README.md descbrides the contents of each file and
provides instructions to compile them.
Marcus Vinícius Midena Ramos
--------------------------------------------------------------------- *)
Require Import List.
Require Import misc_list.
Require Import cfg.
Require Import cfl.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import ListNotations.
(* --------------------------------------------------------------------- *)
(* CONCATENATION - DEFINITIONS *)
(* --------------------------------------------------------------------- *)
Section Concatenation.
Variables non_terminal_1 non_terminal_2 terminal: Type.
Inductive g_cat_nt: Type:=
| Start_cat
| Transf1_cat_nt: non_terminal_1 -> g_cat_nt
| Transf2_cat_nt: non_terminal_2 -> g_cat_nt.
Notation sf1:= (list (non_terminal_1 + terminal)).
Notation sf2:= (list (non_terminal_2 + terminal)).
Notation sfc:= (list (g_cat_nt + terminal)).
Notation nlist:= (list g_cat_nt).
Notation tlist:= (list terminal).
Definition g_cat_sf_lift1 (c: non_terminal_1 + terminal): g_cat_nt + terminal:=
match c with
| inl nt => inl (Transf1_cat_nt nt)
| inr t => inr t
end.
Definition g_cat_sf_lift2 (c: non_terminal_2 + terminal): g_cat_nt + terminal:=
match c with
| inl nt => inl (Transf2_cat_nt nt)
| inr t => inr t
end.
Inductive g_cat_rules (g1: cfg non_terminal_1 terminal) (g2: cfg non_terminal_2 terminal): g_cat_nt -> sfc -> Prop :=
| New_cat: g_cat_rules g1 g2 Start_cat ([inl (Transf1_cat_nt (start_symbol g1))]++[inl (Transf2_cat_nt (start_symbol g2))])
| Lift1_cat: forall nt s,
rules g1 nt s ->
g_cat_rules g1 g2 (Transf1_cat_nt nt) (map g_cat_sf_lift1 s)
| Lift2_cat: forall nt s,
rules g2 nt s ->
g_cat_rules g1 g2 (Transf2_cat_nt nt) (map g_cat_sf_lift2 s).
Lemma g_cat_finite:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
exists n: nat,
exists ntl: list g_cat_nt,
exists tl: tlist,
In Start_cat ntl /\
forall left: g_cat_nt,
forall right: sfc,
g_cat_rules g1 g2 left right ->
(length right <= n) /\
(In left ntl) /\
(forall s: g_cat_nt, In (inl s) right -> In s ntl) /\
(forall s: terminal, In (inr s) right -> In s tl).
Proof.
intros g1 g2.
destruct (rules_finite g1) as [n1 [ntl1 [tl1 H1]]].
destruct (rules_finite g2) as [n2 [ntl2 [tl2 H2]]].
exists (S (S (max n1 n2))), (Start_cat :: (map Transf1_cat_nt ntl1) ++ (map Transf2_cat_nt ntl2)), (tl1 ++ tl2).
split.
- simpl.
left.
reflexivity.
- split.
+ inversion H.
* simpl.
omega.
* destruct H1 as [_ H1].
subst.
specialize (H1 nt s H0).
destruct H1 as [H1 _].
rewrite map_length.
assert (H3: n1 >= n2 \/ n1 <= n2) by omega.
{
destruct H3 as [H3 | H3].
- apply max_l in H3.
rewrite H3.
omega.
- assert (H3':= H3).
apply max_r in H3.
rewrite H3.
omega.
}
* destruct H2 as [_ H2].
subst.
specialize (H2 nt s H0).
destruct H2 as [H2 _].
rewrite map_length.
assert (H3: n1 >= n2 \/ n1 <= n2) by omega.
{
destruct H3 as [H3 | H3].
- assert (H3':= H3).
apply max_l in H3.
rewrite H3.
omega.
- apply max_r in H3.
rewrite H3.
omega.
}
+ split.
* {
inversion H.
- simpl.
left.
reflexivity.
- simpl.
right.
apply in_or_app.
left.
destruct H1 as [_ H1].
specialize (H1 nt s H0).
destruct H1 as [_ [H5 [_ _]]].
apply in_split in H5.
destruct H5 as [l1 [l2 H5]].
rewrite H5.
rewrite map_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
- simpl.
right.
apply in_or_app.
right.
destruct H2 as [_ H2].
specialize (H2 nt s H0).
destruct H2 as [_ [H5 [_ _]]].
apply in_split in H5.
destruct H5 as [l1 [l2 H5]].
rewrite H5.
rewrite map_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
}
* {
split.
- inversion H.
+ subst.
intros s H4.
simpl in H4.
destruct H4 as [H4 | H4].
* inversion H4.
simpl.
right.
apply in_or_app.
left.
destruct H1 as [H1 _].
apply in_split in H1.
destruct H1 as [l1 [l2 H1]].
rewrite H1.
rewrite map_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
* {
destruct H4 as [H4 | H4].
- inversion H4.
simpl.
right.
apply in_or_app.
right.
destruct H2 as [H2 _].
apply in_split in H2.
destruct H2 as [l1 [l2 H2]].
rewrite H2.
rewrite map_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
- contradiction.
}
+ subst.
intros s0 H4.
destruct H1 as [_ H1].
specialize (H1 nt s H0).
destruct H1 as [_ [_ [H5 _]]].
simpl.
right.
apply in_or_app.
left.
destruct s0.
* apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl Start_cat :: l2) with ([inl Start_cat] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
{
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
}
* assert (H6: In (inl n) s).
{
apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl (Transf1_cat_nt n) :: l2) with ([inl (Transf1_cat_nt n)] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
destruct s1'0.
- inversion H5''.
- inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
subst.
apply in_or_app.
right.
simpl.
left.
reflexivity.
+ simpl in H3.
inversion H3.
}
specialize (H5 n H6).
apply in_map.
exact H5.
* apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl (Transf2_cat_nt n) :: l2) with ([inl (Transf2_cat_nt n)] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
{
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
}
+ subst.
intros s0 H4.
destruct H2 as [_ H2].
specialize (H2 nt s H0).
destruct H2 as [_ [_ [H5 _]]].
simpl.
right.
apply in_or_app.
right.
destruct s0.
* apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl Start_cat :: l2) with ([inl Start_cat] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
{
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
}
* apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl (Transf1_cat_nt n) :: l2) with ([inl (Transf1_cat_nt n)] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
{
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
}
* assert (H6: In (inl n) s).
{
apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inl (Transf2_cat_nt n) :: l2) with ([inl (Transf2_cat_nt n)] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
destruct s1'0.
- inversion H5''.
- inversion H5''.
destruct s0.
+ simpl in H3.
inversion H3.
subst.
apply in_or_app.
right.
simpl.
left.
reflexivity.
+ simpl in H3.
inversion H3.
}
specialize (H5 n H6).
apply in_map.
exact H5.
- inversion H.
+ subst.
intros s H4.
simpl in H4.
destruct H4 as [H4 | H4].
* inversion H4.
* {
destruct H4 as [H4 | H4].
- inversion H4.
- contradiction.
}
+ subst.
intros s0 H4.
destruct H1 as [_ H1].
specialize (H1 nt s H0).
destruct H1 as [_ [_ [_ H5]]].
assert (H6: In (inr s0) s).
{
apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inr s0 :: l2) with ([inr s0] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s1.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
subst.
apply in_or_app.
right.
simpl.
left.
reflexivity.
}
specialize (H5 s0 H6).
apply in_or_app.
left.
exact H5.
+ subst.
intros s0 H4.
destruct H2 as [_ H2].
specialize (H2 nt s H0).
destruct H2 as [_ [_ [_ H5]]].
assert (H6: In (inr s0) s).
{
apply in_split in H4.
destruct H4 as [l1 [l2 H4]].
symmetry in H4.
apply map_expand in H4.
destruct H4 as [s1' [s2' [H4' [H4'' H4''']]]].
change (inr s0 :: l2) with ([inr s0] ++ l2) in H4'''.
symmetry in H4'''.
apply map_expand in H4'''.
destruct H4''' as [s1'0 [s2'0 [H5' [H5'' H5''']]]].
destruct s1'0.
- inversion H5''.
- simpl in H5''.
inversion H5''.
destruct s1.
+ simpl in H3.
inversion H3.
+ simpl in H3.
inversion H3.
subst.
apply in_or_app.
right.
simpl.
left.
reflexivity.
}
specialize (H5 s0 H6).
apply in_or_app.
right.
exact H5.
}
Qed.
Definition g_cat (g1: cfg non_terminal_1 terminal) (g2: cfg non_terminal_2 terminal): (cfg g_cat_nt terminal):= {|
start_symbol:= Start_cat;
rules:= g_cat_rules g1 g2;
rules_finite:= g_cat_finite g1 g2
|}.
(* --------------------------------------------------------------------- *)
(* CONCATENATION - LEMMAS *)
(* --------------------------------------------------------------------- *)
Lemma derives_add_cat_left:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
forall s s': sf1,
derives g1 s s' -> derives (g_cat g1 g2) (map g_cat_sf_lift1 s) (map g_cat_sf_lift1 s').
Proof.
intros g1 g2 s s' H.
induction H as [| x y z left right H1 H2 H3].
- apply derives_refl.
- rewrite map_app.
rewrite map_app.
rewrite map_app in H2.
simpl in H2.
apply derives_step with (left:=(Transf1_cat_nt left)).
exact H2.
simpl.
apply Lift1_cat.
apply H3.
Qed.
Lemma derives_add_cat_right:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
forall s s': sf2,
derives g2 s s' -> derives (g_cat g1 g2) (map g_cat_sf_lift2 s) (map g_cat_sf_lift2 s').
Proof.
intros g1 g2 s s' H.
induction H as [| x y z left right H1 H2 H3].
- apply derives_refl.
- rewrite map_app.
rewrite map_app.
rewrite map_app in H2.
simpl in H2.
apply derives_step with (left:=(Transf2_cat_nt left)).
exact H2.
simpl.
apply Lift2_cat.
apply H3.
Qed.
Lemma g_cat_correct_aux:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
forall s11 s12: sf1,
forall s21 s22: sf2,
derives g1 s11 s12 /\ derives g2 s21 s22 ->
derives (g_cat g1 g2)
((map g_cat_sf_lift1 s11)++(map g_cat_sf_lift2 s21))
((map g_cat_sf_lift1 s12)++(map g_cat_sf_lift2 s22)).
Proof.
intros g1 g2 s11 s12 s21 s22 [H1 H2].
induction H1, H2.
- apply derives_refl.
- rewrite map_app.
rewrite map_app.
apply derives_trans with (s2:=
(map g_cat_sf_lift1 s ++
map g_cat_sf_lift2 (s2 ++ [inl left] ++ s3))).
apply derives_context_free_add_left.
apply derives_add_cat_right.
simpl.
exact H2.
apply derives_context_free_add_left.
rewrite map_app.
rewrite map_app.
simpl.
apply derives_step with (left:= Transf2_cat_nt left).
+ apply derives_refl.
+ simpl.
apply Lift2_cat.
exact H.
- apply derives_context_free_add_right.
apply derives_add_cat_left.
apply derives_step with (left:=left).
+ exact H1.
+ exact H.
- rewrite map_app.
rewrite map_app.
rewrite <- app_assoc.
rewrite <- app_assoc.
rewrite map_app.
rewrite map_app.
rewrite map_app in IHderives.
rewrite map_app in IHderives.
rewrite <- app_assoc in IHderives.
simpl in IHderives.
rewrite map_app in IHderives.
remember ((map g_cat_sf_lift1 s1 ++
map g_cat_sf_lift2 s0)) as w1.
remember (map g_cat_sf_lift1 s3 ++
map g_cat_sf_lift2 s4 ++
map g_cat_sf_lift2 right0 ++
map g_cat_sf_lift2 s5) as w2.
apply derives_step with (left:= Transf1_cat_nt left).
exact IHderives.
simpl.
apply Lift1_cat.
exact H.
Qed.
Theorem g_cat_correct:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
forall s1: sf1,
forall s2: sf2,
generates g1 s1 /\ generates g2 s2 ->
generates (g_cat g1 g2)
((map g_cat_sf_lift1 s1)++(map g_cat_sf_lift2 s2)).
Proof.
unfold generates.
intros g1 g2 s1 s2 H.
destruct H as [H1 H2].
apply derives_trans with (s2:= (map g_cat_sf_lift1 [(inl (start_symbol g1))] ++ map g_cat_sf_lift2 [(inl (start_symbol g2))])).
- match goal with
| |- derives _ ?s1 ?s2 =>
change s1 with ([] ++ s1);
change s2 with ([] ++ s2 ++ [])
end.
apply derives_step with (left:= (start_symbol (g_cat g1 g2))).
+ apply derives_refl.
+ simpl.
apply New_cat.
- apply g_cat_correct_aux.
split.
+ exact H1.
+ exact H2.
Qed.
Theorem g_cat_correct_inv:
forall g1: cfg non_terminal_1 terminal,
forall g2: cfg non_terminal_2 terminal,
forall s: sfc,
generates (g_cat g1 g2) s ->
s = [inl (start_symbol (g_cat g1 g2))] \/
exists s1: sf1,
exists s2: sf2,
s =(map g_cat_sf_lift1 s1)++(map g_cat_sf_lift2 s2) /\
generates g1 s1 /\
generates g2 s2.
Proof.
unfold generates.
intros g1 g2 s.
remember ([inl (start_symbol (g_cat g1 g2))]) as init.
intros H.
induction H.
- left. reflexivity.
- subst.
specialize (IHderives eq_refl).
destruct IHderives.
+ (* IHderives, first case *)
(* A primeira cadeia gerada é a raiz da gramática *)
destruct s2.
* (* Se s2 é vazio, a primeira regra está sendo aplicada *)
simpl in H1. inversion H1. subst. clear H1.
simpl in H0. inversion H0. clear H0 H2.
simpl in *.
right.
exists [inl (start_symbol g1)], [inl (start_symbol g2)].
{
split.
- simpl. reflexivity.
- split. apply derives_refl. apply derives_refl.
}
* (* Se s2 não é vazio, temos uma contradição *)
simpl in H1.
{
destruct s2.
- simpl in *. inversion H1.
- simpl in *. inversion H1.
}
+ (* IHderives, second case *)
(* A primeira cadeia gerada é a concatenação de map x com map x0 *)
destruct H1 as (? & ? & ? & ? & ?).
(* Análise da regra usada *)
inversion H0.
subst.
clear H0.
* (* Case 1: rule New_cat *)
(* Contradiction: H1 cannot contain Start_cat *)
assert (IN : In (inl Start_cat) (map g_cat_sf_lift1 x ++ map g_cat_sf_lift2 x0)).
{
rewrite <- H1.
rewrite in_app_iff.
right.
simpl.
left.
reflexivity.
}
rewrite in_app_iff in IN.
(* Start_cat is in x or x0 *)
{
destruct IN.
- (* Start_cat is in x, contradiction *)
rewrite in_map_iff in H0.
destruct H0 as (? & ? & ?).
(* But Start_cat = lift x1 and x cannot contain x1 *)
destruct x1.
+ simpl in H0.
inversion H0.
+ simpl in H0.
inversion H0.
- (* Start_cat is in x0, contradiction *)
rewrite in_map_iff in H0.
destruct H0 as (? & ? & ?).
destruct x1.
+ simpl in H0.
inversion H0.
+ simpl in H0.
inversion H0.
}
* (* Case 2: rule Lift1_cat *)
(* A rule of g1 has been used *)
subst.
(* Consider both situations in H1 *)
assert (IN : In (inl (Transf1_cat_nt nt)) (map g_cat_sf_lift1 x ++ map g_cat_sf_lift2 x0)).
{
rewrite <- H1.
rewrite in_app_iff.
right.
simpl.
left.
reflexivity.
}
rewrite in_app_iff in IN.
(* Transf1_cat_nt is in x or x0 *)
{
destruct IN.
- (* Transf1_cat_nt is in x, ok *)
apply equal_app in H1.
destruct H1.
+ destruct H1 as [l [H1 H6]].
right.
destruct l.
* {
destruct x0.
- simpl in H6.
inversion H6.
- simpl in H6.
inversion H6.
destruct s0.
+ inversion H8.
+ inversion H8.
}
* inversion H6.
subst.
clear H6.
{
apply derives_step with (right:=(map g_cat_sf_lift1 s)) in H.
- symmetry in H1.
apply map_expand in H1.
destruct H1 as [s1' [s2' [H6 [H7 H8]]]].
replace (inl (Transf1_cat_nt nt) :: l) with ([inl (Transf1_cat_nt nt)]++l) in H8.
+ symmetry in H8.
apply map_expand in H8.
destruct H8 as [s1'' [s2'' [H9 [H10 H11]]]].
exists (s1'++s++s2''), x0.
split.
* repeat rewrite map_app.
subst.
repeat rewrite <- app_assoc.
reflexivity.
* {
split.
- subst.
destruct s1''.
+ simpl in H10.
inversion H10.
+ inversion H10.
destruct s0.
* inversion H6.
apply map_eq_nil in H7.
subst.
{
apply derives_step with (right:= s) in H2.
- exact H2.
- exact H4.
}
* inversion H6.
- exact H3.
}
+ simpl.
reflexivity.
- exact H0.
}
+ destruct H1 as [l [H1 H6]].
assert (IN: In (inl (Transf1_cat_nt nt)) (map g_cat_sf_lift2 x0)).
{
rewrite H6.
apply in_app_iff.
right.
simpl.
left.
reflexivity.
}
apply in_map_iff in IN.
destruct IN.
destruct H7.
destruct x1.
inversion H7.
inversion H7.
- (* Transf1_cat_nt is in x0, contradiction *)
rewrite in_map_iff in H5.
destruct H5.
destruct H5.
destruct x1.
+ inversion H5.
+ inversion H5.
}
* (* Case 3: rule Lift2_cat *)
(* A rule of g2 has been used *)
subst.
(* Consider both situations in H1 *)
assert (IN : In (inl (Transf2_cat_nt nt)) (map g_cat_sf_lift1 x ++ map g_cat_sf_lift2 x0)).
{
rewrite <- H1.
rewrite in_app_iff.
right.
simpl.
left.
reflexivity.
}
rewrite in_app_iff in IN.
(* Transf2_cat_nt is in x or x0 *)
{
destruct IN.
- (* Transf2_cat_nt is in x, contradiction *)
rewrite in_map_iff in H5.
destruct H5.
destruct H5.
destruct x1.
+ inversion H5.
+ inversion H5.
- (* Transf2_cat_nt is in x0, ok *)
apply equal_app in H1.
destruct H1.
+ destruct H1 as [l [H1 H6]].
right.
destruct l.
* {
destruct x0.
- simpl in H6.
inversion H6.
- simpl in H6.
inversion H6.
rewrite app_nil_r in H1.
subst.
exists x.
exists (s++x0).
split.
rewrite map_app.
reflexivity.
split.
exact H2.
inversion H6.
destruct s0.
+ simpl in H7.
inversion H7.
subst.
rewrite <- app_nil_l in H3.
apply derives_step with (right:=s) in H3.
* exact H3.
* exact H4.
+ inversion H7.
}
* inversion H6.
subst.
clear H6.
assert (IN : In (inl (Transf2_cat_nt nt)) (map g_cat_sf_lift1 x)).
{
rewrite H1.
rewrite in_app_iff.
right.
simpl.
left.
reflexivity.
}
rewrite in_map_iff in IN.
destruct IN as [x0' [H6 H7]].
{
destruct x0'.
- simpl in H6.
inversion H6.
- inversion H6.
}
+ destruct H1 as [l [H7 H8]].
subst.
symmetry in H8.
apply map_expand in H8.
destruct H8 as [ s1' [s2' [H10 [H11 H12]]]].
subst.
destruct s2'.
* inversion H12.
* inversion H12.
subst.
right.
exists x.
exists (s1'++s++s2').
split.
rewrite <- app_assoc.
repeat rewrite map_app.
reflexivity.
split.
exact H2.
{
destruct s0.
- inversion H6.
subst.
apply derives_step with (right:=s) in H3.
exact H3.
exact H4.
- inversion H6.
}
}
Qed.
End Concatenation.
(* --------------------------------------------------------------------- *)
(* AS LANGUAGES *)
(* --------------------------------------------------------------------- *)
Section Concatenation_2.
Variable non_terminal_1 non_terminal_2 terminal: Type.
Notation sentence:= (list terminal).
Inductive l_cat (l1 l2: lang terminal): lang terminal:=
| l_cat_app: forall s1 s2: sentence, l1 s1 -> l2 s2 -> l_cat l1 l2 (s1 ++ s2).
Lemma map_cat_1:
forall s: sentence,
map (@g_cat_sf_lift1 non_terminal_1 non_terminal_2 terminal) (map (@terminal_lift non_terminal_1 terminal) s) =
map (@terminal_lift (@g_cat_nt non_terminal_1 non_terminal_2) terminal) s.
Proof.
induction s.
- simpl.
reflexivity.
- simpl.
change (inr a) with (terminal_lift (g_cat_nt non_terminal_1 non_terminal_2) a).
rewrite IHs.
reflexivity.
Qed.
Lemma map_cat_2:
forall s: sentence,
map (@g_cat_sf_lift2 non_terminal_1 non_terminal_2 terminal) (map (@terminal_lift non_terminal_2 terminal) s) =
map (@terminal_lift (@g_cat_nt non_terminal_1 non_terminal_2) terminal) s.
Proof.
induction s.
- simpl.
reflexivity.
- simpl.
change (inr a) with (terminal_lift (g_cat_nt non_terminal_1 non_terminal_2) a).
rewrite IHs.
reflexivity.
Qed.
Lemma map_cat_3:
forall s: sentence,
forall l: list (non_terminal_1 + terminal),
map (@terminal_lift (@g_cat_nt non_terminal_1 non_terminal_2) terminal) s =
map (@g_cat_sf_lift1 non_terminal_1 non_terminal_2 terminal) l ->
l = map (@terminal_lift non_terminal_1 terminal) s.
Proof.
induction s.
- intros l H.
simpl in H.
symmetry in H.
apply map_eq_nil in H.
subst.
simpl.
reflexivity.
- intros l H.
remember (terminal_lift (g_cat_nt non_terminal_1 non_terminal_2) (terminal:=terminal)) as m1.
remember (g_cat_sf_lift1 non_terminal_2 (terminal:=terminal)) as m2.
remember (terminal_lift non_terminal_1 (terminal:=terminal)) as m3.
simpl in H.
destruct l.
+ simpl in H.
inversion H.
+ simpl in H.
inversion H.
specialize (IHs l H2).
rewrite IHs.
simpl.
destruct s0.
* {
replace (m2 (inl n)) with (inl terminal (Transf1_cat_nt non_terminal_2 n)) in H1.
- replace (m1 a) with (inr (g_cat_nt non_terminal_1 non_terminal_2) a) in H1.
+ inversion H1.
+ rewrite Heqm1.
change (inr a) with (terminal_lift (g_cat_nt non_terminal_1 non_terminal_2) a).
reflexivity.
- rewrite Heqm2.
simpl.
reflexivity.
}
* {
replace (m2 (inr t)) with (inr (g_cat_nt non_terminal_1 non_terminal_2) t) in H1.
- replace (m1 a) with (inr (g_cat_nt non_terminal_1 non_terminal_2) a) in H1.
+ inversion H1.
replace (m3 t) with (inr non_terminal_1 t).
* reflexivity.
* rewrite Heqm3.
change (inr t) with (terminal_lift non_terminal_1 t).
reflexivity.
+ rewrite Heqm1.
change (inr a) with (terminal_lift (g_cat_nt non_terminal_1 non_terminal_2) a).
reflexivity.
- rewrite Heqm2.
simpl.
reflexivity.
}
Qed.
Lemma map_cat_4:
forall s: sentence,
forall l: list (non_terminal_2 + terminal),
map (@terminal_lift (@g_cat_nt non_terminal_1 non_terminal_2) terminal) s =
map (@g_cat_sf_lift2 non_terminal_1 non_terminal_2 terminal) l ->
l = map (@terminal_lift non_terminal_2 terminal) s.
Proof.
induction s.
- intros l H.
simpl in H.
symmetry in H.
apply map_eq_nil in H.
subst.
simpl.
reflexivity.
- intros l H.