proposition: Theorem g_eill_mono_Si Σ Σ' Γ u : incl Σ Σ' -> Σ; Γ ⊦ u -> Σ'; Γ ⊦ u.
proof: Proof. revert Σ Σ' Γ; intros Si Si' Ga. intros H H1; revert H1 Si' H. induction 1 as [ u | Ga De p H1 H2 IH2 | Ga a p q H1 H2 IH2 | Ga De p q r H1 H2 IH2 H3 IH3 | Ga p q r H1 H2 IH2 H3 IH3 ]; intros Si' HSi. + apply in_geill_ax. + apply in_geill_perm with (1 := H1); auto. + apply in_geill_inc with (1 := HSi _ H1); auto. + apply in_geill_dec with (1 := HSi _ H1); auto. + apply in_geill_fork with (1 := HSi _ H1); auto. Qed.
imports: List Permutation Arith Undecidability.Shared.Libs.DLW(utils pos vec) Undecidability.ILL(ILL EILL ill)