(* Definition des composantes du calcul *)

  Inductive calc : Set :=
    | Pos : calc
    | Neg : calc.(* Definition des composantes du calcul *)

  Inductive calc : Set :=
    | Pos : calc
    | Neg : calc.(* Definition des composantes du calcul *)

  Inductive calc : Set :=
    | Pos : calc
    | Neg : calc.Inductive srt_v6 : Set :=
    | Sprop : calc -> srt_v6
    | Stype : nat -> srt_v6.Inductive srt_v6 : Set :=
    | Sprop : calc -> srt_v6
    | Stype : nat -> srt_v6.Inductive srt_v6 : Set :=
    | Sprop : calc -> srt_v6
    | Stype : nat -> srt_v6.Inductive axiom_v6 : srt_v6 -> srt_v6 -> Prop :=
    | ax_prop : forall (c : calc) (n : nat), axiom_v6 (Sprop c) (Stype n)
    | ax_type : forall n m : nat, n < m -> axiom_v6 (Stype n) (Stype m).Inductive axiom_v6 : srt_v6 -> srt_v6 -> Prop :=
    | ax_prop : forall (c : calc) (n : nat), axiom_v6 (Sprop c) (Stype n)
    | ax_type : forall n m : nat, n < m -> axiom_v6 (Stype n) (Stype m).Inductive axiom_v6 : srt_v6 -> srt_v6 -> Prop :=
    | ax_prop : forall (c : calc) (n : nat), axiom_v6 (Sprop c) (Stype n)
    | ax_type : forall n m : nat, n < m -> axiom_v6 (Stype n) (Stype m).Inductive rules_v6 : srt_v6 -> srt_v6 -> srt_v6 -> Prop :=
    | rule_prop_l : forall (c : calc) (s : srt_v6), rules_v6 (Sprop c) s s
    | rule_prop_r :
        forall (c : calc) (s : srt_v6), rules_v6 s (Sprop c) (Sprop c)
    | rule_type :
        forall n m p : nat,
        n <= p -> m <= p -> rules_v6 (Stype n) (Stype m) (Stype p).Inductive rules_v6 : srt_v6 -> srt_v6 -> srt_v6 -> Prop :=
    | rule_prop_l : forall (c : calc) (s : srt_v6), rules_v6 (Sprop c) s s
    | rule_prop_r :
        forall (c : calc) (s : srt_v6), rules_v6 s (Sprop c) (Sprop c)
    | rule_type :
        forall n m p : nat,
        n <= p -> m <= p -> rules_v6 (Stype n) (Stype m) (Stype p).Inductive rules_v6 : srt_v6 -> srt_v6 -> srt_v6 -> Prop :=
    | rule_prop_l : forall (c : calc) (s : srt_v6), rules_v6 (Sprop c) s s
    | rule_prop_r :
        forall (c : calc) (s : srt_v6), rules_v6 s (Sprop c) (Sprop c)
    | rule_type :
        forall n m p : nat,
        n <= p -> m <= p -> rules_v6 (Stype n) (Stype m) (Stype p).Inductive rules_v6 : srt_v6 -> srt_v6 -> srt_v6 -> Prop :=
    | rule_prop_l : forall (c : calc) (s : srt_v6), rules_v6 (Sprop c) s s
    | rule_prop_r :
        forall (c : calc) (s : srt_v6), rules_v6 s (Sprop c) (Sprop c)
    | rule_type :
        forall n m p : nat,
        n <= p -> m <= p -> rules_v6 (Stype n) (Stype m) (Stype p).Inductive univ_v6 : srt_v6 -> srt_v6 -> Prop :=
    | univ_prop : forall (c : calc) (n : nat), univ_v6 (Sprop c) (Stype n)
    | univ_type : forall n m : nat, n <= m -> univ_v6 (Stype n) (Stype m).Inductive univ_v6 : srt_v6 -> srt_v6 -> Prop :=
    | univ_prop : forall (c : calc) (n : nat), univ_v6 (Sprop c) (Stype n)
    | univ_type : forall n m : nat, n <= m -> univ_v6 (Stype n) (Stype m).Inductive univ_v6 : srt_v6 -> srt_v6 -> Prop :=
    | univ_prop : forall (c : calc) (n : nat), univ_v6 (Sprop c) (Stype n)
    | univ_type : forall n m : nat, n <= m -> univ_v6 (Stype n) (Stype m).Definition univ : srt_v6 -> srt_v6 -> Prop := clos_refl_trans _ univ_v6.Inductive inv_univ : srt_v6 -> srt_v6 -> Prop :=
    | iu_prop : forall c : calc, inv_univ (Sprop c) (Sprop c)
    | iu_pr_ty : forall (c : calc) (n : nat), inv_univ (Sprop c) (Stype n)
    | iu_type : forall n m : nat, n <= m -> inv_univ (Stype n) (Stype m).Inductive inv_univ : srt_v6 -> srt_v6 -> Prop :=
    | iu_prop : forall c : calc, inv_univ (Sprop c) (Sprop c)
    | iu_pr_ty : forall (c : calc) (n : nat), inv_univ (Sprop c) (Stype n)
    | iu_type : forall n m : nat, n <= m -> inv_univ (Stype n) (Stype m).Inductive inv_univ : srt_v6 -> srt_v6 -> Prop :=
    | iu_prop : forall c : calc, inv_univ (Sprop c) (Sprop c)
    | iu_pr_ty : forall (c : calc) (n : nat), inv_univ (Sprop c) (Stype n)
    | iu_type : forall n m : nat, n <= m -> inv_univ (Stype n) (Stype m).Inductive inv_univ : srt_v6 -> srt_v6 -> Prop :=
    | iu_prop : forall c : calc, inv_univ (Sprop c) (Sprop c)
    | iu_pr_ty : forall (c : calc) (n : nat), inv_univ (Sprop c) (Stype n)
    | iu_type : forall n m : nat, n <= m -> inv_univ (Stype n) (Stype m).

(* Specifications: *)
Lemma inv_univ_trans : forall x y z : srt_v6, inv_univ x y -> inv_univ y z -> inv_univ x z.
Lemma univ_inv : forall s s' : srt_v6, univ s s' -> forall P : Prop, (inv_univ s s' -> P) -> P.
Lemma calc_dec : forall c c' : calc, decide (c = c').
Lemma v6_sort_dec : forall s s' : srt_v6, decide (s = s').
Lemma univ_v6_dec : forall s s' : srt_v6, decide (univ s s').
Lemma v6_inf_axiom : forall s : srt_v6, {sp : srt_v6 | ppal (axiom_v6 s) univ sp}.
Lemma v6_inf_rule : forall x1 x2 : srt_v6, {x3 : srt_v6 | rules_v6 x1 x2 x3 & forall s1 s2 s3 : srt_v6, rules_v6 s1 s2 s3 -> univ x1 s1 -> univ x2 s2 -> univ x3 s3}.