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allrules.v
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allrules.v
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(* ---------------------------------------------------------------------
This file contains definitions and proof scripts related to
(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 the paper "Formalization of the
pumping lemma for context-free languages", submitted to
LATA 2016.
Marcus Vinícius Midena Ramos
--------------------------------------------------------------------- *)
Require Import List.
Require Import Ring.
Require Import Omega.
Require Import Compare_dec.
Require Import misc_arith.
Require Import misc_list.
Require Import cfg.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import ListNotations.
Open Scope list_scope.
Section AllRules.
Variables terminal non_terminal: Type.
Notation symbol:= (non_terminal + terminal)%type.
Notation sf:= (list symbol).
Notation sentence := (list terminal).
Notation term_lift:= ((terminal_lift non_terminal) terminal).
Notation rlist:= (list (non_terminal * sf)).
Notation tlist:= (list terminal).
Notation nlist:= (list non_terminal).
Notation vlist:= (list symbol).
Notation slist:= (list sf).
Fixpoint length_max_sf (l: slist): nat:=
match l with
| [] => 0
| s :: l' => if leb (length_max_sf l') (length s) then length s else length_max_sf l'
end.
Lemma length_max_sf_cons_v1:
forall s: slist,
forall a: sf,
length_max_sf (a :: s) = length a \/
length_max_sf (a :: s) = length_max_sf s.
Proof.
destruct s.
- intros a.
simpl.
left.
reflexivity.
- intros a0.
assert (H1: s = [] \/ s <> []).
{
apply nil_not_nil.
}
destruct H1 as [H1 | H1].
+ subst.
simpl.
assert (H1: (length l) <= (length a0) \/
(length l) > (length a0)) by omega.
destruct H1 as [H1 | H1].
* apply leb_correct in H1.
rewrite H1.
left.
reflexivity.
* apply leb_correct_conv in H1.
rewrite H1.
right.
reflexivity.
+ simpl.
assert (H2: (length_max_sf s) <= (length l) \/
(length_max_sf s) > (length l)) by omega.
destruct H2 as [H2 | H2].
* apply leb_correct in H2.
rewrite H2.
assert (H3: (length l) <= (length a0) \/
(length l) > (length a0)) by omega.
{
destruct H3 as [H3 | H3].
- apply leb_correct in H3.
rewrite H3.
left.
reflexivity.
- apply leb_correct_conv in H3.
rewrite H3.
right.
reflexivity.
}
* apply leb_correct_conv in H2.
rewrite H2.
assert (H3: (length_max_sf s) <= (length a0) \/
(length_max_sf s) > (length a0)) by omega.
{
destruct H3 as [H3 | H3].
- apply leb_correct in H3.
rewrite H3.
left.
reflexivity.
- apply leb_correct_conv in H3.
rewrite H3.
right.
reflexivity.
}
Qed.
Lemma length_max_sf_cons_gt:
forall s: slist,
forall n: nat,
forall a: sf,
length_max_sf (a :: s) > n ->
length a > n \/ length_max_sf s > n.
Proof.
intros s n a H.
assert (H1: length_max_sf (a :: s) = length a \/
length_max_sf (a :: s) = length_max_sf s).
{
apply length_max_sf_cons_v1.
}
destruct H1 as [H1 | H1].
- rewrite H1 in H.
left.
exact H.
- rewrite H1 in H.
right.
exact H.
Qed.
Lemma length_max_sf_cons_le:
forall s: slist,
forall n: nat,
forall a: sf,
length_max_sf (a :: s) <= n ->
length a <= n /\ length_max_sf s <= n.
Proof.
intros s.
destruct s.
- intros n a H.
simpl in H.
split.
+ exact H.
+ simpl.
omega.
- intros n a H2.
simpl in H2.
assert (H3: (length_max_sf s) <= (length l) \/
(length_max_sf s) > (length l)) by omega.
destruct H3 as [H3 | H3].
+ apply leb_correct in H3.
rewrite H3 in H2.
assert (H4: (length l) <= (length a) \/
(length l) > (length a)) by omega.
destruct H4 as [H4 | H4].
* apply leb_correct in H4.
rewrite H4 in H2.
{
split.
- exact H2.
- simpl.
rewrite H3.
apply leb_complete in H4.
omega.
}
* {
split.
- apply leb_correct_conv in H4.
rewrite H4 in H2.
apply leb_complete_conv in H4.
omega.
- simpl.
rewrite H3.
apply leb_correct_conv in H4.
rewrite H4 in H2.
exact H2.
}
+ apply leb_correct_conv in H3.
rewrite H3 in H2.
assert (H4: (length_max_sf s) <= (length a) \/
(length_max_sf s) > (length a)) by omega.
destruct H4 as [H4 | H4].
* apply leb_correct in H4.
rewrite H4 in H2.
{
split.
- exact H2.
- simpl.
rewrite H3.
apply leb_complete in H4.
omega.
}
* {
split.
- apply leb_correct_conv in H4.
rewrite H4 in H2.
apply leb_complete_conv in H4.
omega.
- simpl.
rewrite H3.
apply leb_correct_conv in H4.
rewrite H4 in H2.
exact H2.
}
Qed.
Lemma length_max_sf_cons_v2:
forall s: slist,
forall a: sf,
forall n: nat,
length_max_sf (a :: s) = n ->
length a = n /\ length_max_sf s <= n \/
length a <= n /\ length_max_sf s = n.
Proof.
intros s a n H1.
assert (H2: length_max_sf (a :: s) = length a \/
length_max_sf (a :: s) = length_max_sf s).
{
apply length_max_sf_cons_v1.
}
destruct H2 as [H2 | H2].
- left.
split.
+ omega.
+ assert (H3: length_max_sf (a :: s) <= n). { admit. }
apply length_max_sf_cons_le in H3.
destruct H3 as [_ H3].
exact H3.
- right.
split.
+ assert (H3: length_max_sf (a :: s) <= n). { admit. }
apply length_max_sf_cons_le in H3.
destruct H3 as [H3 _].
exact H3.
+ omega.
Qed.
Lemma length_max_sf_or:
forall l1 l2: slist,
length_max_sf (l1 ++ l2) = length_max_sf l1 \/
length_max_sf (l1 ++ l2) = length_max_sf l2.
Proof.
induction l1.
- intros l2.
right.
simpl.
reflexivity.
- intros l2.
specialize (IHl1 l2).
destruct IHl1 as [IHl1 | IHl1].
+ left.
simpl.
assert (H1: (length_max_sf (l1 ++ l2)) <= (length a) \/
(length_max_sf (l1 ++ l2)) > (length a)) by omega.
destruct H1 as [H1 | H1].
* apply leb_correct in H1.
rewrite H1.
apply leb_complete in H1.
assert (H2: length_max_sf l1 <= length a).
{
rewrite IHl1 in H1.
exact H1.
}
apply leb_correct in H2.
rewrite H2.
reflexivity.
* apply leb_correct_conv in H1.
rewrite H1.
apply leb_complete_conv in H1.
rewrite IHl1 in H1.
apply leb_correct_conv in H1.
rewrite H1.
exact IHl1.
+ right.
assert (H1: (length_max_sf l2) <= (length a) \/
(length_max_sf l2) > (length a)) by omega.
destruct H1 as [H1 | H1].
* admit. (* +++ *)
* rewrite <- IHl1 in H1.
simpl.
apply leb_correct_conv in H1.
rewrite H1.
exact IHl1.
Qed.
Lemma length_max_sf_or_cons:
forall s: slist,
forall a: sf,
forall n: nat,
length_max_sf s <= n /\ length a = n \/
length_max_sf s = n /\ length a <= n ->
length_max_sf (a :: s) = n.
Proof.
induction s.
- intros a n H.
destruct H as [H | H].
+ destruct H as [_ H].
simpl.
exact H.
+ destruct H as [H1 H2].
simpl in H1.
subst.
simpl.
omega.
- intros a0 n H.
destruct H as [H | H].
+ destruct H as [H1 H2].
apply length_max_sf_cons_le in H1.
destruct H1 as [H1 H3].
simpl.
assert (H5: (length_max_sf s) <= (length a) \/
(length_max_sf s) > (length a)) by omega.
destruct H5 as [H5 | H5].
* apply leb_correct in H5.
rewrite H5.
rewrite <- H2 in H1.
apply leb_correct in H1.
rewrite H1.
exact H2.
* apply leb_correct_conv in H5.
rewrite H5.
rewrite <- H2 in H3.
apply leb_correct in H3.
rewrite H3.
exact H2.
+ destruct H as [H1 H2].
apply length_max_sf_cons_v2 in H1.
assert (H5: (length_max_sf s) <= (length a) \/
(length_max_sf s) > (length a)) by omega.
destruct H5 as [H5 | H5].
* apply leb_correct in H5.
simpl.
rewrite H5.
{
destruct H1 as [H1 | H1].
- destruct H1 as [H3 H4].
rewrite <- H3 in H2.
apply leb_correct in H2.
admit.
- destruct H1 as [H3 H4].
admit.
}
* apply leb_correct_conv in H5.
simpl.
rewrite H5.
simpl in IHs.
apply IHs.
{
destruct H1 as [H1 | H1].
- admit.
- admit.
}
Qed.
Lemma length_max_sf_app_le:
forall l1 l2: slist,
forall n: nat,
length_max_sf l1 <= n ->
length_max_sf l2 <= n ->
length_max_sf (l1 ++ l2) <= n.
Proof.
induction l1.
- intros l2 n H1 H2.
simpl.
exact H2.
- intros l2 n H1 H2.
apply length_max_sf_cons_le in H1.
destruct H1 as [H3 H4].
assert (H5: length_max_sf (l1 ++ l2) > length a \/
length_max_sf (l1 ++ l2) <= length a) by omega.
destruct H5 as [H5 | H5].
+ simpl.
apply leb_correct_conv in H5.
rewrite H5.
apply IHl1.
* exact H4.
* exact H2.
+ simpl.
apply leb_correct in H5.
rewrite H5.
exact H3.
Qed.
Lemma length_max_sf_or_app_eq:
forall l1 l2: slist,
forall n: nat,
(length_max_sf l1 = n /\ length_max_sf l2 <= n) \/
(length_max_sf l2 = n /\ length_max_sf l1 <= n) ->
length_max_sf (l1 ++ l2) = n.
Proof.
induction l1.
- intros l2 n H.
simpl in H.
destruct H as [H | H].
+ subst.
simpl.
omega.
+ simpl.
omega.
- intros l2 n H.
specialize (IHl1 l2 n).
destruct H as [H | H].
+ destruct H as [H1 H2].
apply length_max_sf_cons_v2 in H1.
change ((a :: l1) ++ l2) with (a :: (l1 ++ l2)).
apply length_max_sf_or_cons.
destruct H1 as [H1 | H1].
* left.
{
split.
- destruct H1 as [H1 _].
admit.
- destruct H1 as [H3 H4].
exact H3.
}
* right.
{
split.
- destruct H1 as [H1 _].
admit.
- destruct H1 as [H1 _].
exact H1.
}
+ destruct H as [H1 H2].
apply length_max_sf_cons_le in H2.
change ((a :: l1) ++ l2) with (a :: (l1 ++ l2)).
apply length_max_sf_or_cons.
destruct H2 as [H3 H4].
right.
split.
* admit.
* exact H3.
Qed.
Fixpoint length_max (rl: rlist): nat:=
match rl with
| [] => 0
| (l, r) :: rl' => if leb (length_max rl') (length r) then length r else length_max rl'
end.
Lemma length_max_correct:
forall rl: rlist,
forall left: non_terminal,
forall right: sf,
In (left, right) rl ->
length right <= length_max rl.
Proof.
induction rl.
- intros left right H.
simpl in H.
contradiction.
- intros left right H1.
simpl in H1.
destruct H1.
+ subst.
simpl.
assert (H2: (length_max rl) <= (length right) \/
(length_max rl) > (length right)) by omega.
destruct H2 as [H2 | H2].
* apply leb_correct in H2.
rewrite H2.
omega.
* apply leb_correct_conv in H2.
rewrite H2.
assert (H3: leb (length_max rl) (length right) = false -> (length_max rl) > (length right)).
{
apply leb_complete_conv.
}
specialize (H3 H2).
omega.
+ destruct a.
simpl.
assert (H2: (length_max rl) <= (length l) \/
(length_max rl) > (length l)) by omega.
destruct H2 as [H2 | H2].
* apply leb_correct in H2.
rewrite H2.
specialize (IHrl left right H).
assert (H3: leb (length_max rl) (length l) = true -> (length_max rl) <= (length l)).
{
apply leb_complete.
}
specialize (H3 H2).
omega.
* apply leb_correct_conv in H2.
rewrite H2.
apply IHrl with (left:= left).
exact H.
Qed.
Lemma length_max_or:
forall l1 l2: rlist,
forall n: nat,
(length_max l1 = n /\ length_max l1 >= length_max l2) \/
(length_max l2 = n /\ length_max l2 > length_max l1) ->
length_max (l1 ++ l2) = n.
Proof.
admit.
Qed.
Lemma length_max_max:
forall l1 l2: rlist,
length_max (l1 ++ l2) = max (length_max l1) (length_max l2).
Proof.
admit.
Qed.
Definition npair (n: non_terminal): (non_terminal + terminal):= inl n.
Definition tpair (t: terminal): (non_terminal + terminal):= inr t.
Fixpoint all_terminals_sf (l: sf): tlist:=
match l with
| [] => []
| (inr t) :: l' => t :: all_terminals_sf l'
| (inl n) :: l' => all_terminals_sf l'
end.
Lemma all_terminals_sf_app:
forall l1 l2: sf,
all_terminals_sf (l1 ++ l2) = all_terminals_sf l1 ++ all_terminals_sf l2.
Proof.
induction l1.
- simpl.
reflexivity.
- intros l2.
change (a :: l1) with ([a] ++ l1).
rewrite <- app_assoc.
destruct a.
+ simpl.
exact (IHl1 l2).
+ simpl.
apply app_eq.
apply IHl1.
Qed.
Fixpoint tin (t: terminal) (l: tlist): bool:=
match l with
| [] => false
| t' :: l' => orb false (* (t = t') ... *) (tin t l')
end.
Fixpoint remove_tduplicates (l: tlist) (p: tlist): tlist:=
match l with
| [] => []
| t :: l' => if tin t p
then remove_tduplicates l' p
else remove_tduplicates l' (t :: p)
end.
Fixpoint all_terminals (rl: rlist): tlist:=
match rl with
| [] => []
| (l, r) :: rl' => (all_terminals_sf r) ++ (all_terminals rl')
end.
Lemma all_terminals_app:
forall l1 l2: rlist,
all_terminals (l1 ++ l2) = all_terminals l1 ++ all_terminals l2.
Proof.
induction l1.
- simpl.
reflexivity.
- intros l2.
change (a :: l1) with ([a] ++ l1).
rewrite <- app_assoc.
destruct a.
simpl.
rewrite <- app_assoc.
specialize (IHl1 l2).
rewrite IHl1.
reflexivity.
Qed.
Lemma all_terminals_correct:
forall t: terminal,
forall left: non_terminal,
forall right: sf,
forall rl: rlist,
In (inr t) right -> In (left, right) rl -> In t (all_terminals rl).
Proof.
intros t left right rl.
generalize dependent right.
generalize dependent left.
generalize dependent t.
induction rl.
- intros t left right H1 H2.
simpl in H2.
contradiction.
- intros t left right H1 H2.
simpl in H2.
destruct H2 as [H2 | H2].
+ subst.
simpl.
apply in_or_app.
left.
apply in_split in H1.
destruct H1 as [l1 [l2 H1]].
rewrite H1.
rewrite all_terminals_sf_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
+ change (a :: rl) with ([a] ++ rl).
rewrite all_terminals_app.
apply in_or_app.
right.
apply IHrl with (left:= left) (right:= right).
* exact H1.
* exact H2.
Qed.
Fixpoint all_non_terminals_sf (l: sf): nlist:=
match l with
| [] => []
| (inr t) :: l' => all_non_terminals_sf l'
| (inl n) :: l' => n :: all_non_terminals_sf l'
end.
Lemma all_non_terminals_sf_app:
forall l1 l2: sf,
all_non_terminals_sf (l1 ++ l2) = all_non_terminals_sf l1 ++ all_non_terminals_sf l2.
Proof.
induction l1.
- simpl.
reflexivity.
- intros l2.
change (a :: l1) with ([a] ++ l1).
rewrite <- app_assoc.
destruct a.
+ simpl.
specialize (IHl1 l2).
rewrite IHl1.
reflexivity.
+ simpl.
specialize (IHl1 l2).
rewrite IHl1.
reflexivity.
Qed.
Fixpoint nin (n: non_terminal) (l: nlist): bool:=
match l with
| [] => false
| n' :: l' => orb false (* (n = n') ... *) (nin n l')
end.
Fixpoint remove_nduplicates (l: nlist) (p: nlist): nlist:=
match l with
| [] => []
| n :: l' => if nin n p
then remove_nduplicates l' p
else remove_nduplicates l' (n :: p)
end.
Fixpoint all_non_terminals (rl: rlist): nlist:=
match rl with
| [] => []
| (l, r) :: rl' => l :: (all_non_terminals_sf r) ++ (all_non_terminals rl')
end.
Lemma all_non_terminals_app:
forall l1 l2: rlist,
all_non_terminals (l1 ++ l2) = all_non_terminals l1 ++ all_non_terminals l2.
Proof.
induction l1.
- simpl.
reflexivity.
- intros l2.
change (a :: l1) with ([a] ++ l1).
rewrite <- app_assoc.
destruct a.
simpl.
rewrite <- app_assoc.
specialize (IHl1 l2).
rewrite IHl1.
reflexivity.
Qed.
Lemma all_non_terminals_correct:
forall n: non_terminal,
forall left: non_terminal,
forall right: sf,
forall rl: rlist,
(n = left) \/ In (inl n) right ->
In (left, right) rl -> In n (all_non_terminals rl).
Proof.
intros n left right rl.
generalize dependent right.
generalize dependent left.
generalize dependent n.
induction rl.
- intros n left right H1 H2.
simpl in H2.
contradiction.
- intros n left right H1 H2.
destruct H1 as [H1 | H1].
+ subst.
destruct H2 as [H2 | H2].
* subst.
simpl.
left.
reflexivity.
* destruct a.
simpl.
right.
apply in_or_app.
right.
specialize (IHrl left left right).
{
apply IHrl.
- left.
reflexivity.
- exact H2.
}
+ simpl in H2.
destruct H2 as [H2 | H2].
* apply in_split in H1.
destruct H1 as [l1 [l2 H1]].
subst.
simpl.
right.
apply in_or_app.
left.
rewrite all_non_terminals_sf_app.
apply in_or_app.
right.
simpl.
left.
reflexivity.
* change (a :: rl) with ([a] ++ rl).
rewrite all_non_terminals_app.
apply in_or_app.
right.
{
apply IHrl with (left:= left) (right:= right).
- right.
exact H1.
- exact H2.
}
Qed.
Lemma all_non_terminals_not_nil:
forall rl: rlist,
rl <> [] ->
all_non_terminals rl <> [].
Proof.
destruct rl.
- intros H.
destruct H.
reflexivity.
- intros H.
destruct p.
simpl.
change (n :: all_non_terminals_sf l ++ all_non_terminals rl) with ((n :: all_non_terminals_sf l) ++ all_non_terminals rl).
apply app_not_nil_inv.
left.
apply not_eq_sym.
apply nil_cons.
Qed.
Definition all_symbols (rl: rlist): vlist:=
let t: tlist:= all_terminals rl in
let l: nlist:= all_non_terminals rl in
(map tpair t) ++ (map npair l).
Lemma all_symbols_correct:
forall n: non_terminal,
forall t: terminal,
forall rl: rlist,
(In (inl n) (map npair (all_non_terminals rl)) -> In (inl n) (all_symbols rl)) /\
(In (inr t) (map tpair (all_terminals rl)) -> In (inr t) (all_symbols rl)).
Proof.
intros n t rl.
split.
- intros H.
unfold all_symbols.
apply in_or_app.
right.
exact H.
- intros H.
unfold all_symbols.
apply in_or_app.
left.
exact H.
Qed.
Lemma all_symbols_not_nil:
forall rl: rlist,
rl <> [] ->
all_symbols rl <> [].
Proof.
destruct rl.
- intros H.
destruct H.
reflexivity.
- intros H.
destruct p.
unfold all_symbols.
apply app_not_nil_inv.
right.
simpl.
apply not_eq_sym.
apply nil_cons.
Qed.
Fixpoint concat_symbol (s: non_terminal + terminal) (l: slist): slist:=
match l with
| [] => []
| e :: l' => (s :: e) :: concat_symbol s l'
end.
Lemma concat_symbol_correct:
forall e: symbol,
forall s: sf,
forall l: slist,
In s l ->
In (e :: s) (concat_symbol e l).
Proof.
intros e s l.
generalize dependent e.
generalize dependent s.
induction l.
- intros s e H1.
simpl in H1.
contradiction.
- intros s e H1.
simpl in H1.
simpl.
destruct H1 as [H1 | H1].
+ left.
rewrite H1.
reflexivity.
+ right.
specialize (IHl s e H1).
exact IHl.
Qed.
Lemma concat_symbol_length_correct:
forall l: slist,
forall e: symbol,
forall d: sf,
length l > 0 ->
forall i: nat,
i < length l ->
length (nth i (concat_symbol e l) d) = length (nth i l d) + 1.
Proof.
induction l.
- intros e d H.
simpl in H.
apply lt_irrefl in H.
contradiction.
- intros e d H1 i H2.
simpl concat_symbol.
assert (H3: i = 0 \/ i > 0) by omega.
destruct H3 as [H3 | H3].
+ subst.
simpl.
omega.
+ apply gt_zero_exists in H3.
destruct H3 as [j H3].
subst.
simpl.
apply IHl.
* simpl in H2.
omega.
* simpl in H2.
apply le_S_n.
exact H2.
Qed.
Lemma concat_symbol_length_max_sf_correct_aux:
forall l: slist,
forall s: symbol,
forall n: nat,
length_max_sf l <= n ->
length_max_sf (concat_symbol s l) <= S n.
Proof.
induction l.
- intros s n H.
simpl.
omega.
- intros s n H1.
apply length_max_sf_cons_le in H1.
simpl concat_symbol.
change ((s :: a) :: concat_symbol s l) with ([(s :: a)] ++ concat_symbol s l).
apply length_max_sf_app_le.
+ simpl.
omega.
+ apply IHl.
destruct H1 as [_ H1].
exact H1.
Qed.
Lemma concat_symbol_length_max_sf_correct:
forall l: slist,
forall s: symbol,
forall n: nat,
l <> [] ->
length_max_sf l = n ->
length_max_sf (concat_symbol s l) = S n.
Proof.
induction l.
- intros s n H.
destruct H.
reflexivity.
- intros s n H1 H2.
apply length_max_sf_cons_v2 in H2.
simpl concat_symbol.
change ((s :: a) :: concat_symbol s l) with ([(s :: a)] ++ concat_symbol s l).
apply length_max_sf_or_app_eq.
assert (H: l = [] \/ l <> []).
{
apply nil_not_nil.
}
destruct H as [H | H].
+ subst.
simpl.
destruct H2 as [H2 | H2].
* left.
{
split.
- omega.
- omega.
}
* left.
{
split.
- simpl in H2.
destruct H2 as [H3 H4].
subst.
omega.
- omega.
}
+ destruct H2 as [H2 | H2].
* left.
simpl.
{
split.
- apply eq_S.
destruct H2 as [H2 _].
exact H2.
- apply concat_symbol_length_max_sf_correct_aux.
destruct H2 as [_ H2].
exact H2.
}
* right.
simpl.
{
split.
- apply IHl.
+ exact H.
+ destruct H2 as [_ H2].
exact H2.
- omega.
}
Qed.
Lemma concat_symbol_nil:
forall s: symbol,
concat_symbol s [] = [].
Proof.
intros s.
simpl.
reflexivity.
Qed.
Lemma concat_symbol_not_nil:
forall l: slist,
forall s: symbol,
l <> [] ->
concat_symbol s l <> [].
Proof.
intros l s H.
destruct l.
- destruct H.
reflexivity.
- simpl.
apply not_eq_sym.
apply nil_cons.
Qed.
Lemma concat_symbol_not_nil_inv:
forall l: slist,
forall s: symbol,
concat_symbol s l <> [] ->
l <> [].
Proof.
destruct l.
- intros s H.