(** ** Syntax *)

Notation var := nat (only parsing).Notation par := nat (only parsing).Inductive term : Type :=
  V (v : var) | P (p : par)
| t_f : bool -> term -> term
| t_e : term.Inductive term : Type :=
  V (v : var) | P (p : par)
| t_f : bool -> term -> term
| t_e : term.Inductive term : Type :=
  V (v : var) | P (p : par)
| t_f : bool -> term -> term
| t_e : term.Inductive term : Type :=
  V (v : var) | P (p : par)
| t_f : bool -> term -> term
| t_e : term.Inductive term : Type :=
  V (v : var) | P (p : par)
| t_f : bool -> term -> term
| t_e : term.Inductive logic := frag | full.Inductive logic := frag | full.Inductive logic := frag | full.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.Inductive form : logic -> Type :=
| Pr {b} : term -> term -> form b
| Q {b} : form b
| Fal : form full
| Impl {b} : form b -> form b -> form b
| All {b} : var -> form b -> form b.

Lemma form_frag_ind : forall P : form frag -> Prop, (forall (t t0 : term), P (Pr t t0)) -> (P Q) -> (forall (f3 : form frag), P f3 -> forall f4 : form frag, P f4 -> P (Impl f3 f4)) -> (forall (n : var) (f6 : form frag), P f6 -> P (All n f6)) -> forall (f8 : form frag), P f8.