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From Coq Require Import Sorting.Permutation.
From Huffman Require Export Cover WeightTree.

Set Default Proof Using "Type".

Section SameSumLeaves.
Variable A : Type.
Variable f : A -> nat.

Definition same_sum_leaves (l1 l2 : list (btree A)) : Prop :=
  exists l3 : list (btree A),
    (exists l4 : list (btree A),
       Permutation l1 l3 /\
       Permutation l2 l4 /\
       map (sum_leaves f) l3 = map (sum_leaves f) l4).

Theorem same_sum_leaves_length :
 forall l1 l2 : list (btree A),
 same_sum_leaves l1 l2 -> length l1 = length l2.
Proof.
intros l1 l2 (l3, (l4, (H0, (H1, H2)))).
rewrite (Permutation_length H0).
rewrite (Permutation_length H1).
repeat rewrite <- (map_length (sum_leaves f)); auto.
apply f_equal with (f := length (A:=nat)); auto.
Qed.

End SameSumLeaves.

Arguments same_sum_leaves [A].