Qed.

Lemma sizes_decr :
(forall A B Γ1 Γ2 Δ, sizes (Γ1++A::B::Γ2) Δ < sizes (Γ1++A∧B::Γ2) Δ)/\
(forall A B Γ Δ1 Δ2, sizes Γ (Δ1++A::Δ2)    < sizes Γ (Δ1++A∧B::Δ2))/\
(forall A B Γ Δ1 Δ2, sizes Γ (Δ1++B::Δ2)    < sizes Γ (Δ1++A∧B::Δ2))/\
(forall A B Γ1 Γ2 Δ, sizes (Γ1++A::Γ2) Δ    < sizes (Γ1++A∨B::Γ2) Δ)/\
(forall A B Γ1 Γ2 Δ, sizes (Γ1++B::Γ2) Δ    < sizes (Γ1++A∨B::Γ2) Δ)/\
(forall A B Γ Δ1 Δ2, sizes Γ (Δ1++A::B::Δ2) < sizes Γ (Δ1++A∨B::Δ2))/\
(forall A B Γ1 Γ2 Δ, sizes (Γ1++B::Γ2)  Δ   < sizes (Γ1++A→B::Γ2) Δ)/\
(forall A B Γ1 Γ2 Δ, sizes (Γ1++Γ2) (A::Δ)  < sizes (Γ1++A→B::Γ2) Δ)/\
(forall A B Γ Δ1 Δ2, sizes (A::Γ)(Δ1++B::Δ2)< sizes Γ (Δ1++A→B::Δ2)).