Library Lambda.Russell.Thinning


Require
 Import
 Lambda.Terms.

Require
 Import
 Lambda.Reduction.

Require
 Import
 Lambda.LiftSubst.

Require
 Import
 Lambda.Env.

Require
 Import
 Lambda.Russell.Types.

Require
 Import
 Lambda.Conv.







Implicit
 Types
 i
 k
 m
 n
 p
 : nat.

Implicit
 Type
 s
 : sort.

Implicit
 Types
 A
 B
 M
 N
 T
 t
 u
 v
 : term.

Implicit
 Types
 e
 f
 g
 : env.




Lemma
 double_weak_weak : ∀ A
,

  (∀ e
 t
 T
, e
 |-- t
 : T
 →

   ∀ n
 f
, ins_in_env A
 n
 e
 f
 → wf
 f
 → 

   f
 |-- (lift_rec 1 t
 n
) : (lift_rec 1 T
 n
)) ∧

 (∀ e
 T
 U
 s
, e
 |-- T
 >> U
 : s
 →

  ∀ n
 f
, ins_in_env A
 n
 e
 f
 → wf
 f
 → 

  f
 |-- (lift_rec 1 T
 n
) >> (lift_rec 1 U
 n
) : s
).




  Theorem
 thinning :

   ∀ e
 t
 T
,

   typ e
 t
 T
 → ∀ A
, wf
 (A
 :: e
) → typ (A
 :: e
) (lift 1 t
) (lift 1 T
).




Theorem
 thinning_coerce : 

   ∀ e
 T
 U
 s
,

   e
 |-- T
 >> U
 : s
 →

   ∀ A
, wf
 (A
 :: e
) → (A
 :: e
) |-- (lift 1 T
) >> (lift 1 U
) : s
.




Lemma
 thinning_n :

   ∀ n
 e
 f
,

   trunc _
 n
 e
 f
 →

   ∀ t
 T
, typ f
 t
 T
 → wf
 e
 → typ e
 (lift n
 t
) (lift n
 T
).




Lemma
 thinning_n_coerce : ∀ n
 e
 f
, trunc _
 n
 e
 f
 →

  ∀ T
 U
 s
, f
 |-- T
 >> U
 : s
 → wf
 e
 → e
 |-- (lift n
 T
) >> (lift n
 U
) : s
.




Lemma
 wf_sort_lift :

 ∀ n
 e
 t
, wf
 e
 → item_lift t
 e
 n
 → ∃ s
 : sort, typ e
 t
 (Srt s
).




Hint
 Resolve
 thinning_n thinning_n_coerce : coc
.