Section Prod2List.
Variable A : Type.
Variable f : A -> nat.
(* In the product the sum of the leaves is multiplied by the integer and added to the weight of the tree *)
Definition prod2list l1 l2 :=
fold_left plus
(map2 (fun a b => a * sum_leaves f b + weight_tree f b) l1 l2) 0.
Proof using.
intros l1 l2 l3 l4 H; unfold prod2list in |- *.
rewrite map2_app; auto.
rewrite fold_left_app.
rewrite plus_comm.
apply sym_equal.
repeat
rewrite
fold_left_eta with (f := plus) (f1 := fun a b : nat => a + (fun x => x) b);
auto.
apply sym_equal; rewrite <- fold_plus_split with (f := fun x : nat => x);
auto.
apply plus_comm.
Qed.
Proof using.
intros a b c d l1 l2 l3 l4 l5 l6 H H0 H1 H2 H3;
change
(prod2list (l1 ++ (a :: nil) ++ l2 ++ (b :: nil) ++ l3)
(l4 ++ (d :: nil) ++ l5 ++ (c :: nil) ++ l6) <=
prod2list (l1 ++ (a :: nil) ++ l2 ++ (b :: nil) ++ l3)
(l4 ++ (c :: nil) ++ l5 ++ (d :: nil) ++ l6))
in |- *.
repeat rewrite prod2list_app; auto.
apply plus_le_compat; auto with arith.
repeat rewrite plus_assoc; apply plus_le_compat; auto.
repeat rewrite (fun x y z => plus_comm (prod2list (x :: y) z)).
repeat rewrite plus_assoc_reverse; apply plus_le_compat; auto.
unfold prod2list in |- *; simpl in |- *.
rewrite le_plus_minus with (1 := H3); auto.
rewrite le_plus_minus with (1 := H2); auto.
replace
(a * (sum_leaves f c + (sum_leaves f d - sum_leaves f c)) + weight_tree f d +
((a + (b - a)) * sum_leaves f c + weight_tree f c)) with
(a * sum_leaves f c + weight_tree f c +
(a * (sum_leaves f d - sum_leaves f c) + (a + (b - a)) * sum_leaves f c +
weight_tree f d)); [ idtac | ring ].
apply plus_le_compat; auto with arith.
apply plus_le_compat; auto with arith.
repeat rewrite mult_plus_distr_l || rewrite mult_plus_distr_r;
auto with arith.
replace
(a * (sum_leaves f d - sum_leaves f c) +
(a * sum_leaves f c + (b - a) * sum_leaves f c)) with
(a * sum_leaves f c + (b - a) * sum_leaves f c +
(a * (sum_leaves f d - sum_leaves f c) + 0)); [ auto with arith | ring ].
Qed.
Proof using.
intros a b c d l1 l2 l3 l4 l5 l6 H H0 H1 H2 H3;
change
(prod2list (l1 ++ (a :: nil) ++ l2 ++ (b :: nil) ++ l3)
(l4 ++ (d :: nil) ++ l5 ++ (c :: nil) ++ l6) <=
prod2list (l1 ++ (a :: nil) ++ l2 ++ (b :: nil) ++ l3)
(l4 ++ (c :: nil) ++ l5 ++ (d :: nil) ++ l6))
in |- *.
repeat rewrite prod2list_app; auto.
apply plus_le_compat; auto with arith.
repeat rewrite plus_assoc; apply plus_le_compat; auto.
repeat rewrite (fun x y z => plus_comm (prod2list (x :: y) z)).
repeat rewrite plus_assoc_reverse; apply plus_le_compat; auto.
unfold prod2list in |- *; simpl in |- *.
rewrite le_plus_minus with (1 := H3); auto.
rewrite le_plus_minus with (1 := H2); auto.
replace
((b + (a - b)) * (sum_leaves f d + (sum_leaves f c - sum_leaves f d)) +
weight_tree f c + (b * sum_leaves f d + weight_tree f d)) with
((b + (a - b)) * sum_leaves f d + weight_tree f d +
((b + (a - b)) * (sum_leaves f c - sum_leaves f d) + b * sum_leaves f d +
weight_tree f c)); [ idtac | ring ].
apply plus_le_compat; auto with arith.
apply plus_le_compat; auto with arith.
repeat rewrite mult_plus_distr_l || rewrite mult_plus_distr_r;
auto with arith.
replace (b * sum_leaves f d + b * (sum_leaves f c - sum_leaves f d)) with
(b * (sum_leaves f c - sum_leaves f d) + 0 + b * sum_leaves f d);
[ auto with arith | ring ].
Qed.
Proof using.
intros a b c l1 l2 l3 l4 l5 l6 H H0 H1;
change
(prod2list (l1 ++ (a :: nil) ++ l2 ++ (a :: nil) ++ l3)
(l4 ++ (b :: nil) ++ l5 ++ (c :: nil) ++ l6) =
prod2list (l1 ++ (a :: nil) ++ l2 ++ (a :: nil) ++ l3)
(l4 ++ (c :: nil) ++ l5 ++ (b :: nil) ++ l6))
in |- *.
repeat rewrite prod2list_app; auto with arith.
ring.
Qed.
Proof using.
intros a b b1 l1 l2 l3 l4 l5 H H0 H1 H2 H3 H4.
cut (In b (b1 :: l5));
[ simpl in |- *; intros [HH0| HH0]
| apply permutation_in with (1 := H3); auto with datatypes ].
exists l3; exists l4; repeat (split; auto).
pattern b1 at 2 in |- *; rewrite HH0; apply permutation_sym; auto.
rewrite HH0; auto.
cut (In b1 (l3 ++ b :: l4));
[ intros HH1
| apply permutation_in with (1 := permutation_sym _ _ _ H3);
auto with datatypes ].
case in_app_or with (1 := HH1); intros HH2.
case in_ex_app with (1 := HH2).
intros l6 (l7, HH3); exists (l6 ++ b :: l7); exists l4; repeat (split; auto).
apply trans_equal with (1 := H).
rewrite HH3; repeat rewrite length_app; simpl in |- *; auto with arith.
apply permutation_sym; apply permutation_trans with (2 := H3); auto.
rewrite HH3.
repeat rewrite app_ass.
simpl in |- *; apply permutation_transposition.
rewrite HH3; auto.
repeat rewrite app_ass.
case (same_length_ex _ _ b1 l6 l7 l1); auto.
rewrite <- HH3; auto.
intros l8 (l9, (b2, (HH4, (HH5, HH6)))).
rewrite HH6.
repeat rewrite app_ass; simpl in |- *.
apply prod2list_le_l; auto.
change (sum_order f b1 b) in |- *.
apply ordered_trans with (2 := H4); auto.
unfold sum_order in |- *; intros a0 b0 c H5 H6; apply le_trans with (1 := H5);
auto.
apply H1; rewrite HH6; auto with datatypes.
simpl in HH2; case HH2; intros HH3.
exists l3; exists l4; repeat (split; auto); try (rewrite <- HH3; auto; fail).
pattern b1 at 2 in |- *; rewrite <- HH3; apply permutation_sym; auto.
case in_ex_app with (1 := HH3).
intros l6 (l7, HH4); exists l3; exists (l6 ++ b :: l7); repeat (split; auto).
apply trans_equal with (1 := H0).
rewrite HH4; repeat rewrite length_app; simpl in |- *; auto with arith.
apply permutation_sym; apply permutation_trans with (2 := H3); auto.
rewrite HH4.
simpl in |- *; apply permutation_transposition.
rewrite HH4; auto.
case (same_length_ex _ _ b1 l6 l7 l2); auto.
rewrite <- HH4; auto.
intros l8 (l9, (b2, (HH5, (HH6, HH7)))).
rewrite HH7.
apply prod2list_le_r; auto.
change (sum_order f b1 b) in |- *.
apply ordered_trans with (2 := H4); auto.
unfold sum_order in |- *; intros a0 b0 c H5 H6; apply le_trans with (1 := H5);
auto.
apply H2; rewrite HH7; auto with datatypes.
Qed.

(* Putting the smallest tree with the smallest integer lower the product *)
Theorem prod2list_reorder2 :
forall a b c b1 c1 l1 l2 l3 l4 l5,
length l1 = length l3 ->
length l2 = length l4 ->
(forall b, In b l1 -> b <= a) ->
(forall b, In b l2 -> b <= a) ->
permutation (l3 ++ b :: c :: l4) (b1 :: c1 :: l5) ->
ordered (sum_order f) (b1 :: c1 :: l5) ->
exists l6 : _,
(exists l7 : _,
length l1 = length l6 /\
length l2 = length l7 /\
permutation (b1 :: c1 :: l5) (l6 ++ b1 :: c1 :: l7) /\
prod2list (l1 ++ a :: a :: l2) (l6 ++ b1 :: c1 :: l7) <=
prod2list (l1 ++ a :: a :: l2) (l3 ++ b :: c :: l4)).