Section obvious_links_between_fragments.
Notation "P ⇒ Q" := (forall Γ A, P Γ A -> Q Γ A) (at level 70).
Fact S_ill_restr_restr_wc : S_ill_restr ⇒ S_ill_restr_wc.
Fact S_ill_full_wc : S_ill ⇒ S_ill_wc.
Fact S_ill_restr_full : S_ill_restr ⇒ S_ill.
Fact S_ill_restr_full_wc : S_ill_restr_wc ⇒ S_ill_wc.
End obvious_links_between_fragments.
Local Notation "Γ '⊢r' A" := (S_ill_restr Γ A) (at level 70, no associativity).
Local Notation "Γ '⊢' A" := (S_ill_wc Γ A) (at level 70, no associativity).
Fact S_ill_restr_weak Γ Δ B : Δ ⊢r B -> ‼Γ++Δ ⊢r B.
Fact S_ill_wc_cntr Γ Δ B : ‼Γ++‼Γ++Δ ⊢ B -> ‼Γ++ Δ ⊢ B.

Theorem S_ill_restr_weak_cntr Σ Γ A B : In A Σ -> ‼Σ++Γ ⊢r B <-> !A::‼Σ++Γ ⊢r B.