Notation "'[<' Γ '|-' A '>]'" := (ll_sequent_tps s Γ A) (at level 65).Definition ll_vars := nat.Inductive eill_cmd : Set :=
  | in_ll_cmd_inc  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_dec  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_fork : ll_vars -> ll_vars -> ll_vars -> eill_cmd.Inductive eill_cmd : Set :=
  | in_ll_cmd_inc  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_dec  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_fork : ll_vars -> ll_vars -> ll_vars -> eill_cmd.Inductive eill_cmd : Set :=
  | in_ll_cmd_inc  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_dec  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_fork : ll_vars -> ll_vars -> ll_vars -> eill_cmd.Inductive eill_cmd : Set :=
  | in_ll_cmd_inc  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_dec  : ll_vars -> ll_vars -> ll_vars -> eill_cmd
  | in_ll_cmd_fork : ll_vars -> ll_vars -> ll_vars -> eill_cmd.Notation LL_INC  := in_ll_cmd_inc.Notation LL_DEC  := in_ll_cmd_dec.Notation LL_FORK := in_ll_cmd_fork.Definition eill_cmd_vars c := 
  match c with
    | LL_INC  a p q => a::p::q::nil
    | LL_DEC  a p q => a::p::q::nil
    | LL_FORK p q r => p::q::r::nil
  end.Definition EILL_SEQUENT := (list eill_cmd * list ll_vars * ll_vars)%type.Definition EILL_PROVABILITY (c : EILL_SEQUENT) := match c with (Σ,Γ,A) => Σ; Γ ⊦ A end.

Theorem g_eill_mono_Si Σ Σ' Γ u : incl Σ Σ' -> Σ; Γ ⊦ u -> Σ'; Γ ⊦ u.