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misc_list.v
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misc_list.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 JFR.
Marcus Vinícius Midena Ramos
--------------------------------------------------------------------- *)
(* --------------------------------------------------------------------- *)
(* LIST LEMMAS *)
(* --------------------------------------------------------------------- *)
Require Import List.
Require Import Omega.
Require Import misc_arith.
Import ListNotations.
Open Scope list_scope.
Lemma map_expand:
forall (A B: Type) (f: A -> B) (l: list A) (s1 s2: list B),
s1 ++ s2 = map f l ->
exists s1' s2': list A,
l = s1' ++ s2' /\ map f s1' = s1 /\ map f s2' = s2.
Proof.
intros A B f l.
induction l.
- intros s1 s2 H.
simpl in H.
apply app_eq_nil in H.
destruct H as [H1 H2].
subst.
exists [], [].
auto.
- intros s1 s2 H.
simpl in H.
destruct s1.
+ simpl in H.
destruct s2.
* inversion H.
* inversion H.
exists [], (a::l). {
split.
- simpl.
reflexivity.
- split.
+ simpl.
reflexivity.
+ simpl.
reflexivity. }
+ destruct s2.
* inversion H.
exists (a::l), []. {
split.
- simpl.
rewrite app_nil_r.
reflexivity.
- split.
+ simpl.
rewrite <- H2.
rewrite app_nil_r.
reflexivity.
+ simpl.
reflexivity. }
* inversion H.
specialize (IHl s1 (b0::s2)).
specialize (IHl H2).
destruct IHl as [s1' [s2' [H3 [H4 H5]]]].
exists (a::s1').
exists s2'. {
split.
- rewrite H3.
simpl.
reflexivity.
- split.
+ simpl.
rewrite H4.
reflexivity.
+ rewrite H5.
reflexivity. }
Qed.
Lemma equal_app (A: Type) :
forall l1 l2 s1 s2 : list A,
l1 ++ l2 = s1 ++ s2 ->
(exists l, s1 = l1 ++ l /\ l2 = l ++ s2) \/ (exists l, l1 = s1 ++ l /\ s2 = l ++ l2).
Proof.
intros l1.
induction l1 as [|a l1 IH].
- intros l2 s1 s2 H.
simpl in H.
subst.
simpl.
left.
exists s1.
split.
reflexivity.
reflexivity.
- intros l2 s1 s2 H.
simpl in H.
destruct s1 as [|a' s1].
+ simpl in H.
subst.
simpl.
right.
exists (a::l1).
split.
* reflexivity.
* simpl.
reflexivity.
+ simpl in H.
inversion H.
subst a'.
clear H.
apply IH in H2.
destruct H2 as [(l & ? & ?) | (l & ? & ?)].
* subst.
simpl.
left.
exists l. {
split.
- reflexivity.
- reflexivity. }
* right.
exists l. {
split.
- rewrite H.
simpl.
reflexivity.
- rewrite H0.
reflexivity. }
Qed.
Lemma last_last:
forall A: Type,
forall s d: A,
forall l: list A,
last (l++[s]) d = s.
Proof.
intros A s d l.
induction l.
- simpl.
reflexivity.
- simpl.
destruct l.
+ simpl.
reflexivity.
+ rewrite <- app_comm_cons.
exact IHl.
Qed.
Lemma length_ge_0:
forall A: Type,
forall l: list A,
length l >= 0.
Proof.
intros A l.
destruct l.
- simpl.
omega.
- simpl.
omega.
Qed.
Lemma length_ge_2:
forall A: Type,
forall a1 a2: A,
forall l: list A,
length (a1::a2::l) >= 2.
Proof.
intros A a1 a2 l.
simpl.
apply Le.le_n_S.
apply Le.le_n_S.
apply Le.le_0_n.
Qed.
Lemma length_lt_1:
forall A: Type,
forall l: list A,
length l < 1 -> l = [].
Proof.
intros A l H.
destruct l.
- reflexivity.
- replace (a::l) with ([a]++l) in H.
+ rewrite app_length in H.
simpl in H.
apply lt_S_n in H.
apply lt_n_O in H.
contradiction.
+ simpl.
reflexivity.
Qed.
Lemma nth_cons:
forall A: Type,
forall a e: A,
forall l: list A,
forall i: nat,
nth (S i) (a::l) e = nth i l e.
Proof.
intros A a e l i.
simpl.
destruct i.
- simpl.
reflexivity.
- reflexivity.
Qed.
Lemma length_not_zero:
forall A: Type,
forall l: list A,
length l > 0 -> [] <> l.
Proof.
intros A l H.
destruct l.
- simpl in H.
apply gt_irrefl in H.
contradiction.
- apply nil_cons.
Qed.
Lemma length_not_zero_inv:
forall A: Type,
forall l: list A,
l <> [] -> length l > 0.
Proof.
intros A l H.
destruct l.
- destruct H.
reflexivity.
- simpl.
omega.
Qed.
Lemma not_nil:
forall A: Type,
forall l: list A,
l <> [] -> length l <> 0.
Proof.
destruct l.
- intro H.
auto.
- intro H.
replace (a::l) with ([a]++l).
+ rewrite app_length.
simpl.
apply not_eq_sym.
apply O_S.
+ simpl.
reflexivity.
Qed.
Lemma not_nil_ge_1:
forall A: Type,
forall l: list A,
l <> [] -> length l >= 1.
Proof.
intros A l H.
apply not_nil in H.
destruct l.
- simpl in H.
destruct H.
reflexivity.
- simpl.
apply le_n_S.
apply le_0_n.
Qed.
Lemma list_last:
forall A: Type,
forall d: A,
forall l: list A,
removelast l <> [] -> nth (length (removelast l) - 1) (removelast l ++ [last l d]) d = last (removelast l) d.
Proof.
intros A d l H.
rewrite app_nth1 with (l:=removelast l).
- rewrite app_removelast_last with (l:=removelast l)(d:=d).
+ rewrite app_nth2.
* rewrite app_length.
assert (H1: Datatypes.length (removelast (removelast l)) + 1 - 1 -
Datatypes.length (removelast (removelast l)) = 0).
{
omega.
}
simpl.
rewrite H1.
rewrite last_last.
reflexivity.
* rewrite app_length.
simpl.
omega.
+ exact H.
- apply n_minus_1.
apply not_nil.
exact H.
Qed.
Lemma length_zero:
forall A: Type,
forall l: list A,
length l = 0 -> l = [].
Proof.
intros A l H.
destruct l.
- reflexivity.
- simpl in H.
symmetry in H.
apply O_S in H.
contradiction.
Qed.
Lemma list_single:
forall A: Type,
forall l: list A,
forall d e: A,
nth 0 l d = e -> length l = 1 -> l = [e].
Proof.
intros A l d e H1 H2.
destruct l.
- simpl in H2.
apply O_S in H2.
contradiction.
- simpl in H1.
rewrite H1.
simpl in H2.
inversion H2.
apply length_zero in H0.
rewrite H0.
reflexivity.
Qed.
Lemma hd_nth0:
forall A: Type,
forall d: A,
forall l: list A,
hd d l = nth 0 l d.
Proof.
intros A d l.
destruct l.
- simpl.
reflexivity.
- simpl.
reflexivity.
Qed.
Lemma last_nth1:
forall A: Type,
forall d: A,
forall l: list A,
length l = 2 -> last l d = nth 1 l d.
Proof.
intros A d l H.
destruct l.
- simpl.
reflexivity.
- simpl.
destruct l.
+ simpl in H.
inversion H.
+ replace (a :: a0 :: l) with ([a] ++ [a0] ++ l) in H.
* repeat rewrite app_length in H.
simpl in H.
inversion H.
apply length_zero in H1.
subst.
simpl.
reflexivity.
* simpl.
reflexivity.
Qed.
Lemma hd_tl_nth1:
forall A: Type,
forall d: A,
forall l: list A,
length l > 2 -> hd d (tl l) = nth 1 l d.
Proof.
intros A d l H.
destruct l.
- simpl.
reflexivity.
- simpl.
apply hd_nth0.
Qed.
Lemma last_tl_last:
forall A: Type,
forall d: A,
forall l: list A,
length l > 2 -> last (tl l) d = last l d.
Proof.
intros A d l H.
destruct l.
- simpl.
reflexivity.
- simpl.
destruct l.
+ simpl in H.
apply gt_S_n in H.
apply lt_n_0 in H.
contradiction.
+ reflexivity.
Qed.
Lemma nth_S:
forall A: Type,
forall i: nat,
forall s s0 s1: A,
forall l: list A,
nth (S i) (s::s0::s1::l) = nth i (s0::s1::l).
Proof.
intros A i s s0 s1 l.
destruct i.
- simpl.
reflexivity.
- destruct i.
+ simpl.
reflexivity.
+ destruct i.
* simpl.
reflexivity.
* simpl.
reflexivity.
Qed.
Lemma skipn_S:
forall i: nat,
forall A: Type,
forall a: A,
forall l: list A,
skipn (S i) (a :: l) = skipn i l.
Proof.
intros i A a l.
destruct i.
- simpl.
reflexivity.
- simpl.
destruct l.
+ reflexivity.
+ reflexivity.
Qed.
Lemma hd_skip:
forall i: nat,
forall A: Type,
forall d: A,
forall l: list A,
hd d (skipn i l) = nth i l d.
Proof.
intros i A d.
induction i as [| i IH].
- intros l.
destruct l.
+ trivial.
+ trivial.
- intros l.
destruct l.
+ trivial.
+ simpl.
rewrite IH.
reflexivity.
Qed.
Lemma last_skip:
forall i: nat,
forall A: Type,
forall d: A,
forall l: list A,
i < length l -> last (skipn i l) d = last l d.
Proof.
intros i A d.
induction i as [|i IH].
- intros l.
destruct l.
+ trivial.
+ trivial.
- intros l.
destruct l.
+ trivial.
+ simpl.
intros Hl.
assert (Hl' : i < length l) by omega.
rewrite IH.
* {
destruct l.
- simpl in Hl'.
omega.
- trivial.
}
* trivial.
Qed.
Lemma hd_hd:
forall A: Type,
forall d: A,
forall l l': list A,
length l >= 1 -> hd d l = hd d (l ++ l').
Proof.
intros A d l l' H.
destruct l.
- simpl in H.
apply le_Sn_0 in H.
contradiction.
- simpl.
reflexivity.
Qed.
Lemma hd_first:
forall A: Type,
forall d: A,
forall l: list A,
forall i: nat,
i >= 1 -> hd d (firstn i l) = hd d l.
Proof.
intros A d l i H.
destruct l.
- destruct i.
+ simpl.
reflexivity.
+ simpl.
reflexivity.
- destruct i.
+ apply le_Sn_0 in H.
contradiction.
+ simpl.
reflexivity.
Qed.
Lemma firstn_empty:
forall A: Type,
forall a: A,
forall i: nat,
a :: firstn i [] = [a].
Proof.
intros A a i.
destruct i.
- simpl.
reflexivity.
- simpl.
reflexivity.
Qed.
Lemma firstn_single:
forall A: Type,
forall a: A,
forall i: nat,
firstn (S i) [a] = [a].
intros A a i.
destruct i.
- simpl.
reflexivity.
- simpl.
reflexivity.
Qed.
Lemma nth_S':
forall i: nat,
forall A: Type,
forall a d: A,
forall l: list A,
nth (S i) (a :: l) d = nth i l d.
Proof.
intros i A a d l.
destruct i.
- simpl. reflexivity.
- simpl. reflexivity.
Qed.
Lemma firstn_S:
forall A: Type,
forall a: A,
forall l: list A,
forall i: nat,
firstn (S i) (a :: l) = a :: firstn i l.
Proof.
intros A a l.
induction l.
- intros i. simpl. reflexivity.
- intros i. destruct i.
+ simpl. reflexivity.
+ simpl. reflexivity.
Qed.
Lemma last_cons:
forall A: Type,
forall a d: A,
forall l: list A,
l <> [] -> last (a :: l) d = last l d.
Proof.
intros A a d l H.
induction l.
- destruct H. reflexivity.
- trivial.
Qed.
Lemma not_nil_firstn:
forall A: Type,
forall l: list A,
forall i: nat,
l <> [] -> firstn (S i) l <> [].
Proof.
intros A l i H.
destruct l.
- simpl. exact H.
- simpl. apply not_eq_sym. apply nil_cons.
Qed.
Lemma nil_not_nil:
forall A: Type,
forall l: list A,
l = [] \/ l <> [].
Proof.
destruct l.
- left.
reflexivity.
- right.
apply not_eq_sym.
apply nil_cons.
Qed.
Lemma last_first_nth:
forall A: Type,
forall d: A,
forall l: list A,
forall i: nat,
length l >= S i -> last (firstn (S i) l) d = nth i l d.
Proof.
intros A d l.
induction l.
- destruct i.
+ trivial.
+ trivial.
- destruct i.
+ trivial.
+ intros H.
rewrite nth_S'.
rewrite <- IHl.
* rewrite firstn_S.
{
rewrite last_cons.
- reflexivity.
- apply not_nil_firstn.
specialize (IHl i).
replace (a::l) with ([a]++l) in H.
+ rewrite app_length in H.
simpl in H.
assert (H1: length l > 0) by omega.
destruct l.
* simpl in H1.
omega.
* apply not_eq_sym.
apply nil_cons.
+ simpl.
reflexivity.
}
* {
replace (a::l) with ([a]++l) in H.
- rewrite app_length in H.
simpl in H.
omega.
- simpl.
reflexivity.
}
Qed.
Lemma nth_last:
forall A: Type,
forall l: list A,
forall d: A,
nth (length l - 1) l d = last l d.
Proof.
intros A l d.
destruct l.
- simpl.
reflexivity.
- remember (a::l) as l0.
rewrite app_removelast_last with (l:=l0) (d:=d).
+ rewrite app_length.
simpl.
rewrite last_last.
rewrite app_nth2.
* assert (H: (Datatypes.length (removelast l0) + 1 - 1 - Datatypes.length (removelast l0) = 0)) by omega.
rewrite H.
simpl.
reflexivity.
* omega.
+ rewrite Heql0.
apply not_eq_sym.
replace (a::l) with ([]++a::l).
* apply app_cons_not_nil.
* simpl.
reflexivity.
Qed.
Lemma hd_app:
forall A: Type,
forall l l': list A,
forall a d: A,
l <> [] ->
l' <> [] ->
hd d l = hd d l' ->
hd d (l ++ [a]) = hd d (l' ++ [a]).
Proof.
intros A l l' a d H1 H2 H3.
destruct l.
- destruct l'.
+ simpl.
reflexivity.
+ destruct H1.
reflexivity.
- destruct l'.
+ destruct H2.
reflexivity.
+ simpl.
simpl in H3.
exact H3.
Qed.
Lemma last_2_1:
forall A: Type,
forall l: list A,
forall a1 a2 d: A,
last (a1 :: a2 :: l) d = last (a2 :: l) d.
Proof.
intros A l a1 a2 d.
simpl.
reflexivity.
Qed.
Lemma last_app:
forall A: Type,
forall l l': list A,
forall a d: A,
l <> [] ->
l' <> [] ->
last l d = last l' d ->
last ([a] ++ l) d = last ([a] ++ l') d.
Proof.
intros A l l' a d H1 H2 H3.
induction l, l'.
- simpl.
reflexivity.
- destruct H1.
reflexivity.
- destruct H2.
reflexivity.
- replace ([a] ++ a0 :: l) with (a :: a0 :: l).
+ replace ([a] ++ a1 :: l') with (a :: a1 :: l').
* repeat rewrite last_2_1.
exact H3.
* simpl.
reflexivity.
+ simpl.
reflexivity.
Qed.
Lemma hd_not_nil:
forall A: Type,
forall l: list A,
forall a d: A,
d <> a ->
hd d l = a ->
l <> [].
Proof.
intros A l a d H1 H2.
destruct l.
- simpl in H2.
contradiction.
- apply not_eq_sym.
apply nil_cons.
Qed.
Lemma hd_not_nil_v2:
forall A: Type,
forall l: list A,
forall d: A,
hd d l <> d -> l <> [].
Proof.
intros A l d H.
destruct l.
- simpl in H.
destruct H.
reflexivity.
- apply not_eq_sym.
apply nil_cons.
Qed.
Lemma hd_not_zero:
forall A: Type,
forall l: list A,
forall a d: A,
d <> a ->
hd d l = a ->
length l > 0.
Proof.
intros A l a d H1 H2.
destruct l.
- simpl in H2.
contradiction.
- simpl.
apply gt_Sn_O.
Qed.
Lemma hd_removelast:
forall A: Type,
forall s d: A,
forall l l': list A,
l <> [] -> hd d (removelast (s :: l) ++ l') = s.
Proof.
intros A s d l l' H.
destruct l.
- destruct H.
reflexivity.
- simpl.
reflexivity.
Qed.
Lemma length_cons_gt_1:
forall A: Type,
forall s: A,
forall l: list A,
length (s :: l) > 1 -> l <> [].
Proof.
intros A s l H.
replace (s :: l) with ([s] ++ l) in H.
- rewrite app_length in H.
simpl in H.
apply gt_S_n in H.
apply not_eq_sym.
apply length_not_zero.
exact H.
- simpl.
reflexivity.
Qed.
Lemma length_cons_eq_1:
forall A: Type,
forall s: A,
forall l: list A,
length (s :: l) = 1 -> l = [].
Proof.
intros A s l H.
destruct l.
- reflexivity.
- simpl in H.
apply eq_add_S in H.
symmetry in H.
apply O_S in H.
contradiction.
Qed.
Lemma not_exists_forall_not:
forall A B: Type,
forall l: list (A+B),
~ (exists a': A, l = [inl a']) ->
forall a': A, (l <> [inl a']).
Proof.
intros A B l H a'.
unfold not at 1 in H.
unfold not.
destruct l.
- intro H1.
inversion H1.
- intro H1.
apply H.
exists a'.
exact H1.
Qed.
Lemma in_in:
forall A: Type,
forall s: A,
forall s1 s2: list A,
In s (s1 ++ [s] ++ s2).
Proof.
intros A s s1 s2.
induction s1.
- simpl.
left.
reflexivity.
- apply in_app_iff.
right.
simpl.
left.
reflexivity.
Qed.
Lemma in_out:
forall A: Type,
forall s: A,
forall l s1 s2: list A,
l = s1 ++ [s] ++ s2 ->
In s l.
Proof.
intros A s l s1 s2 H.
rewrite H.
apply in_in.
Qed.
Lemma exists_length_eq_2:
forall A: Type,
forall l: list A,
length l = 2 ->
exists s1 s2: A,
l = [s1] ++ [s2].
Proof.
intros A l H.
destruct l.
- simpl in H.
apply O_S in H.
contradiction.
- destruct l.
+ simpl in H.
apply eq_add_S in H.
apply O_S in H.
contradiction.
+ destruct l.
* exists a, a0.
reflexivity.
* simpl in H.
repeat apply eq_add_S in H.
symmetry in H.
apply O_S in H.
contradiction.
Qed.
Lemma exists_length_gt_2:
forall A: Type,
forall d: A,
forall l: list A,
length l > 2 ->
exists s1 s2: A,
exists l': list A,
l = [s1] ++ l' ++ [s2].
Proof.
intros A d l H.
destruct l.
- simpl in H.
apply lt_n_0 in H.
contradiction.
- destruct l.
+ simpl in H.
apply gt_S_n in H.
apply lt_n_0 in H.
contradiction.
+ destruct l.
* simpl in H.
repeat apply gt_S_n in H.
apply lt_n_0 in H.
contradiction.
* assert (H1: l = [] \/ l <> []).
{
apply nil_not_nil.
}
{
destruct H1 as [H1 | H1].
- subst.
exists a, a1, [a0].
reflexivity.
- change (a :: a0 :: a1 :: l) with (([a] ++ [a0] ++ [a1]) ++ l) in H.
exists a.
rewrite app_removelast_last with (A:=A) (l:=l) (d:=d).
+ exists (last l d).
exists (a0 :: a1 :: removelast l).
reflexivity.
+ exact H1.
}
Qed.
Lemma exists_length_ge_2:
forall A: Type,
forall def: A,
forall l: list A,
length l >= 2 ->
exists s1 s2: A,
exists l': list A,
l = [s1] ++ l' ++ [s2].
Proof.
intros A def l H1.
assert (H2: length l > 2 \/ length l = 2) by omega.
destruct H2 as [H2 | H2].
- apply (exists_length_gt_2 A def l) in H2.
exact H2.
- apply exists_length_eq_2 in H2.
destruct H2 as [s1 [s2 H3]].
exists s1, s2, [].
simpl.
exact H3.
Qed.