Section Schellinx_observation.
(* This is an observation purely about cut-free CLL  One cannot get a cut-free CLL proof of Γ ⊢ ∅ unless ⟘ or 𝟘 or a negation occurs in Γ *)
Let schellinx_rec Γ Δ :
Γ ⊢c Δ
-> Δ = ∅
-> exists f, In f Γ /\ cll_has_bot_zero_neg f.
End cll_ill_soundness.
(* If the ILL sequent Γ ⊢ A does not contain any occurences of ⟘   then it is provable in ILL iff it is provable in CLL  Which gives a direct reduction for CLL undecidability *)

Theorem ill_cll_equiv Γ A  :
(forall f, In f (A::Γ) -> ~ ill_has_bot f)
-> Γ ⊢i A <-> ⟦Γ⟧ ⊢c [A]::∅.