Update

src/subsequences2.v

@@ -538,17 +538,14 @@ Proof.
 Qed.
 
 
-
+Theorem Subsequence_split_2 {X: Type} :
-
-
-
-
-Theorem subsequence_split_2 {X: Type} :
   forall (u v w: list X),
-    subsequence u (v++w) 
+    Subsequence u (v++w) 
-    -> (exists a b, u = a++b /\ (subsequence a v) /\ subsequence b w).
+    -> (exists a b, u = a++b /\ (Subsequence a v) /\ Subsequence b w).
 Proof.
-  intros u v w. intro H. destruct H. destruct H. destruct H.
+  intros u v w. intro H.
+  apply Subsequence_flat_map in H.
+  destruct H. destruct H. destruct H.
   exists (x ++
      flat_map (fun e : X * list X => fst e :: snd e) (combine v x0)).
   exists (
@@ -565,70 +562,71 @@ Proof.
    simpl. rewrite IHw0. reflexivity. inversion I. reflexivity.
    rewrite L. reflexivity. rewrite app_length in H. rewrite H.
    reflexivity. split.
-   exists x. exists (firstn (length v) x0). split.
+
-   rewrite firstn_length. rewrite <- H. rewrite app_length.
+   apply Subsequence_flat_map. exists x. exists (firstn (length v) x0).
+   split. rewrite firstn_length. rewrite <- H. rewrite app_length.
    rewrite min_l. reflexivity. apply PeanoNat.Nat.le_add_r.
    rewrite combine_firstn_l. reflexivity.
 
-   exists nil. exists (skipn (length v) x0). split.
+   apply Subsequence_flat_map. exists nil. exists (skipn (length v) x0).
-   rewrite app_length in H. rewrite skipn_length. rewrite <- H.
+   split. rewrite app_length in H. rewrite skipn_length. rewrite <- H.
    rewrite PeanoNat.Nat.add_sub_swap. rewrite PeanoNat.Nat.sub_diag.
    reflexivity. reflexivity. reflexivity.
  Qed.
 
 
-Theorem subsequence_filter_2 {X: Type} :
+Theorem Subsequence_filter_2 {X: Type} :
   forall (u v: list X) f,
-    subsequence u v -> subsequence (filter f u) (filter f v).
+    Subsequence u v -> Subsequence (filter f u) (filter f v).
 Proof.
   intros u v. generalize u. induction v; intros u0 f; intro H.
-  apply subsequence_nil_r.
+  apply Subsequence_nil_r.
 
   assert (f a = true \/ f a = false).
   destruct (f a); [ left | right ]; reflexivity. destruct H0.
   simpl. rewrite H0.
-  replace (a::v) with ([a] ++ v) in H. apply subsequence_split_2 in H.
+  replace (a::v) with ([a] ++ v) in H. apply Subsequence_split_2 in H.
   destruct H. destruct H. destruct H. destruct H1.
 
-  assert (subsequence ((filter f x) ++ (filter f x0))
+  assert (Subsequence ((filter f x) ++ (filter f x0))
                       ((filter f [a]) ++ (filter f v))).
-  apply subsequence_app. simpl. rewrite H0.
+  apply Subsequence_app. simpl. rewrite H0.
-  apply subsequence_incl in H1. apply incl_cons_inv in H1. destruct H1.
+  apply Subsequence_incl in H1. apply incl_cons_inv in H1. destruct H1.
   assert (In a (filter f x)). apply filter_In. split; assumption.
-  apply subsequence_in. assumption. apply IHv. assumption.
+  apply Subsequence_in. assumption. apply IHv. assumption.
   rewrite <- filter_app in H3. rewrite <- H in H3. simpl in H3.
   rewrite H0 in H3. assumption. reflexivity.
-  simpl. rewrite H0. apply IHv. apply subsequence_cons_r in H.
+  simpl. rewrite H0. apply IHv. apply Subsequence_cons_r in H.
   assumption.
 Qed.
 
 
-Theorem subsequence_add {X: Type} :
+Theorem Subsequence_add {X: Type} :
-  forall (u v w: list X) x, subsequence u v -> Add x u w -> subsequence w v.
+  forall (u v w: list X) x, Subsequence u v -> Add x u w -> Subsequence w v.
 Proof.
   intros u v w. generalize u v.
   induction w; intros u0 v0 x; intros H I; inversion I.
-  rewrite <- H2. apply subsequence_cons_l. rewrite <- H3. assumption.
+  rewrite <- H2. apply Subsequence_cons_l. rewrite <- H3. assumption.
 
-  assert (J := H). apply subsequence_eq_def_1 in H. rewrite <- H1 in H.
+  assert (J := H). apply Subsequence_bools in H. rewrite <- H1 in H.
   rewrite H0 in H. destruct H. destruct H.
   destruct x1. apply O_S in H. contradiction. destruct b.
-  rewrite H4. simpl. apply subsequence_double_cons.
+  rewrite H4. simpl. apply Subsequence_double_cons.
   apply IHw with (u := l) (x := x).
-  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
+  apply Subsequence_bools. exists x1.
   inversion H. split; reflexivity. assumption.
-  apply subsequence_cons_r with (a:=a).
+  apply Subsequence_cons_r with (a:=a).
-  rewrite H4. simpl. apply subsequence_double_cons.
+  rewrite H4. simpl. apply Subsequence_double_cons.
   apply IHw with (u := l) (x := x).
-  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
+  apply Subsequence_bools. exists x1.
   inversion H. split; reflexivity. assumption.
 Qed.
 
 
-Theorem subsequence_nodup {X: Type} :
+Theorem Subsequence_nodup {X: Type} :
-  forall (u v: list X), subsequence u v -> NoDup u -> NoDup v.
+  forall (u v: list X), Subsequence u v -> NoDup u -> NoDup v.
 Proof.
-  intros u v. intros H I. apply subsequence_eq_def_1 in H.
+  intros u v. intros H I. apply Subsequence_bools in H.
   destruct H. destruct H. rewrite H0.
 
   assert (J: forall z (q: list X),
@@ -654,23 +652,22 @@ Proof.
 Qed.
 
 
-Theorem subsequence_in_2 {X: Type} :
+Theorem Subsequence_in_2 {X: Type} :
-  forall (u v: list X) a, subsequence u v -> In a v -> In a u.
+  forall (u v: list X) a, Subsequence u v -> In a v -> In a u.
 Proof.
   intros u v. generalize u. induction v; intros u0 a0; intros H I.
   contradiction I. destruct I.
-  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
   destruct H. destruct H. destruct H. rewrite H. rewrite H0.
-  apply in_elt. apply IHv. apply subsequence_cons_r in H.
+  apply in_elt. apply IHv. apply Subsequence_cons_r in H.
   assumption. assumption.
 Qed.
 
 
-Theorem subsequence_not_in {X: Type} :
+Theorem Subsequence_not_in {X: Type} :
-  forall (u v: list X) a, subsequence u v -> ~ In a u -> ~ In a v.
+  forall (u v: list X) a, Subsequence u v -> ~ In a u -> ~ In a v.
 Proof.
   intros u v. generalize u. induction v; intros u0 a0; intros H I.
-  easy. apply not_in_cons. split. apply subsequence_incl in H.
+  easy. apply not_in_cons. split. apply Subsequence_incl in H.
   apply incl_cons_inv in H. destruct H.
 
   assert (forall z (x y: X), In x z -> ~ In y z -> x <> y).
@@ -680,27 +677,27 @@ Proof.
   apply IHz. assumption.
   apply not_in_cons in K. destruct K. assumption. apply not_eq_sym.
   generalize I. generalize H. apply H1. generalize I. apply IHv.
-  apply subsequence_cons_r in H. assumption.
+  apply Subsequence_cons_r in H. assumption.
 Qed.
 
 
-Theorem subsequence_as_filter {X: Type} :
+Theorem Subsequence_as_filter {X: Type} :
-  forall (u: list X) f, subsequence u (filter f u).
+  forall (u: list X) f, Subsequence u (filter f u).
 Proof.
-  intros u f. apply subsequence_eq_def_3. apply subsequence_eq_def_2.
+  intros u f. apply Subsequence_bools.
   exists (map f u). split. apply map_length.
   induction u. reflexivity. simpl. destruct (f a).
   simpl. rewrite IHu. reflexivity. assumption.
 Qed.
 
 
-Theorem subsequence_as_partition {X: Type} :
+Theorem Subsequence_as_partition {X: Type} :
   forall (u v w: list X) f,
-    (v, w) = partition f u -> (subsequence u v) /\ (subsequence u w).
+    (v, w) = partition f u -> (Subsequence u v) /\ (Subsequence u w).
 Proof.
   intros u v w f. intro H. rewrite partition_as_filter in H. split.
   replace v with (fst (v, w)). rewrite H. simpl.
-  apply subsequence_as_filter. reflexivity.
+  apply Subsequence_as_filter. reflexivity.
   replace w with (snd (v, w)). rewrite H. simpl.
-  apply subsequence_as_filter. reflexivity.
+  apply Subsequence_as_filter. reflexivity.
 Qed.