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(* generated by Ott 0.10.14 from: ../tests/test10st.ott *)

Require Import Arith.
Require Import Bool.
Require Import List.

(** syntax *)
Definition termvar := nat.
Lemma eq_termvar: forall (x y : termvar), {x = y} + {x <> y}.
Proof.
  decide equality; auto with ott_coq_equality arith.
Defined.
Hint Resolve eq_termvar : ott_coq_equality.

Definition typvar := nat.
Lemma eq_typvar: forall (x y : typvar), {x = y} + {x <> y}.
Proof.
  decide equality; auto with ott_coq_equality arith.
Defined.
Hint Resolve eq_typvar : ott_coq_equality.


Inductive 
T : Set := 
   T_var : typvar -> T
 | T_arrow : T -> T -> T
.

Inductive 
t : Set := 
   t_Var : termvar -> t
 | t_Lam : termvar -> t -> t
 | t_App : t -> t -> t
.

Definition
G : Set := list (termvar*T)
.


(** library functions *)
Fixpoint list_mem (A:Set) (eq:forall a b:A,{a=b}+{a<>b}) (x:A) (l:list A) {struct l} : bool :=
  match l with
  | nil => false
  | cons h t => if eq h x then true else list_mem A eq x t
end.
Implicit Arguments list_mem.

Fixpoint list_minus (A:Set) (eq:forall a b:A,{a=b}+{a<>b}) (l1:list A) (l2:list A) {struct l1} : list A :=
  match l1 with
  | nil => nil
  | cons h t => if (list_mem (A:=A) eq h l2) then list_minus A eq t l2 else cons h (list_minus A eq t l2)
end.
Implicit Arguments list_minus.


(** subrules *)
Definition is_v_of_t (t_6:t) : Prop :=
  match t_6 with
  | (t_Var x) => False
  | (t_Lam x t5) => (True)
  | (t_App t5 t') => False
end.


(** free variables *)
Fixpoint fv_t (t_6:t) : list termvar :=
  match t_6 with
  | (t_Var x) => (cons x nil)
  | (t_Lam x t5) => ((list_minus eq_termvar (fv_t t5) (cons x nil)))
  | (t_App t5 t') => (app (fv_t t5) (fv_t t'))
end.


(** substitutions *)
Fixpoint tsubst_t (t_6:t) (x5:termvar) (t__7:t) {struct t__7} : t :=
  match t__7 with
  | (t_Var x) => (if eq_termvar x x5 then t_6 else (t_Var x))
  | (t_Lam x t5) => t_Lam x (if list_mem eq_termvar x5 (cons x nil) then t5 else (tsubst_t t_6 x5 t5))
  | (t_App t5 t') => t_App (tsubst_t t_6 x5 t5) (tsubst_t t_6 x5 t')
end.

Notation G_nil := (@nil (termvar*T)).
Definition bound x T0 G :=
  exists G1, exists G2,
    (G = List.app G1 (List.cons (x,T0) G2)) /\
    ~In x (List.map (@fst termvar T) G1).


(** definitions *)
(* defns Jtype *)

Inductive
(* defn GtT *)

GtT : G -> t -> T -> Prop :=
 | GtT_value_name : forall (G5:G) (x:termvar) (T5:T),
      (bound  x   T5   G5 )  ->
     GtT G5 (t_Var x) T5
 | GtT_apply : forall (G5:G) (t5:t) (t':t) (T2:T) (T1:T),
     GtT G5 t5 (T_arrow T1 T2) ->
     GtT G5 t' T1 ->
     GtT G5 (t_App t5 t') T2
 | GtT_lambda : forall (G5:G) (x1:termvar) (t5:t) (T1:T) (T_5:T),
     GtT  (cons ( x1 , T1 )  G5 )  t5 T_5 ->
     GtT G5 (t_Lam x1 t5) (T_arrow T1 T_5)

.
(* defns Jop *)

Inductive
(* defn reduce *)

reduce : t -> t -> Prop :=
 | ax_app : forall (x:termvar) (t12:t) (v2:t),
     is_v_of_t v2 ->
     reduce (t_App  (t_Lam x t12)  v2)  ( tsubst_t  v2   x   t12  ) 
 | ctx_app_fun : forall (t1:t) (t_5:t) (t1':t),
     reduce t1 t1' ->
     reduce (t_App t1 t_5) (t_App t1' t_5)
 | ctx_app_arg : forall (v5:t) (t1:t) (t1':t),
     is_v_of_t v5 ->
     reduce t1 t1' ->
     reduce (t_App v5 t1) (t_App v5 t1')

.
Hint Constructors reduce GtT : rules.