Fact nat_joins_binary_le_left n f m : msum nat_join 0 n f ≲ m <-> forall i, i < n -> f i ≲ m.Fact nat_joins_binary_le_right m n f : (exists i, i < n /\ m ≲ f i) -> m ≲ msum nat_join 0 n f.Fact nat_joins_binary_le n m f g :
           (forall i, i < n -> exists j, j < m /\ f i ≲ g j)
        -> msum nat_join 0 n f ≲ msum nat_join 0 m g.Fact nat_double_joins_binary_le n m f g : 
           (forall i j, j < i < n -> exists k, k < m /\ f i j ≲ g k)
        -> msum nat_join 0 n (fun i => msum nat_join 0 i (f i)) ≲ msum nat_join 0 m g.Fact binary_le_joins_inv m n f : m ≲ msum nat_join 0 n f
                              -> { k : nat & { g : nat -> nat & { h | m = msum nat_join 0 k g
                                                                   /\ k <= n 
                                                                   /\ (forall i, i < k -> g i <> 0 /\ g i ≲ f (h i))  
                                                                   /\ (forall i, i < k -> h i < n)
                                                                   /\ (forall i j, i < j < k -> h i < h j) } } }.Fact binary_le_joins_inv' m n f : m ≲ msum nat_join 0 n f
                              -> { g | m = msum nat_join 0 n g /\ forall i, i < n -> g i ≲ f i }.

Lemma binary_le_mult_power2_inv m x q : m ≲ x * power q 2 -> m <> 0 -> { y | m = y * power q 2 /\ y <> 0 /\ y ≲ x }.