Module Type cut_mod (B: base_mod) (S: sound_mod B) (C: complete_mod B S) (G: sequent_mod B S C).
(** * Cut Elimination  We first give some definitions*)
Reserved Notation "Γ ⊃c A" (at level 80).
Inductive Gcf : list PropF->list PropF->Prop :=
| Gcax  : forall v Γ Δ      , In #v Γ            -> In #v Δ          -> Γ ⊃c Δ
| GcBot : forall Γ Δ        , In ⊥ Γ                                 -> Γ ⊃c Δ
| cAndL : forall A B Γ1 Γ2 Δ, Γ1++A::B::Γ2 ⊃c Δ                      -> Γ1++A∧B::Γ2 ⊃c Δ
| cAndR : forall A B Γ Δ1 Δ2, Γ ⊃c Δ1++A::Δ2    -> Γ ⊃c Δ1++B::Δ2   -> Γ ⊃c Δ1++A∧B::Δ2
| cOrL  : forall A B Γ1 Γ2 Δ, Γ1++A::Γ2 ⊃c Δ    -> Γ1++B::Γ2 ⊃c Δ   -> Γ1++A∨B::Γ2 ⊃c Δ
| cOrR  : forall A B Γ Δ1 Δ2, Γ ⊃c Δ1++A::B::Δ2                      -> Γ ⊃c Δ1++A∨B::Δ2
| cImpL : forall A B Γ1 Γ2 Δ, Γ1++B::Γ2 ⊃c Δ    -> Γ1++Γ2 ⊃c A::Δ   -> Γ1++A→B::Γ2 ⊃c Δ
| cImpR : forall A B Γ Δ1 Δ2, A::Γ ⊃c Δ1++B::Δ2                     -> Γ ⊃c Δ1++A→B::Δ2
where "Γ ⊃c Δ" := (Gcf Γ Δ) : My_scope.
Notation "Γ =⊃ Δ" := (forall v,Satisfies v Γ->Validates v Δ) (at level 80).
Inductive Atomic : Set :=
| AVar : PropVars -> Atomic
| ABot : Atomic
.
Fixpoint AtomicF (P:Atomic) : PropF := match P with
| AVar P => #P
| ABot   => ⊥
end.
Fixpoint size A : nat := match A with
| # P   => 0
| ⊥     => 0
| B ∨ C => S (size B + size C)
| B ∧ C => S (size B + size C)
| B → C => S (size B + size C)
end.
Definition sizel := map_fold_right size plus 0.
Definition sizes Γ Δ:= sizel Γ + sizel Δ.
(** * Theorems  reverse implication*)

Theorem G_to_Gcf : forall Γ Δ, Γ ⊃c Δ -> Γ ⊃ Δ.