Library Lambda.JRussell.Coercion


Require
 Import
 Lambda.Terms.

Require
 Import
 Lambda.Reduction.

Require
 Import
 Lambda.Conv.

Require
 Import
 Lambda.LiftSubst.

Require
 Import
 Lambda.Env.

Require
 Import
 Lambda.JRussell.Types.

Require
 Import
 Lambda.JRussell.Basic.







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.




Inductive
 coerce_in_env : env → env → Prop
 :=

  | coerce_env_hd : ∀ e
 t
 u
 s
, e
 |-= t
 >> u
 : s
 → e
 |-= u
 : Srt s
 → 

        coerce_in_env (t
 :: e
) (u
 :: e
)

  | coerce_env_tl :

        ∀ e
 f
 t
, coerce_in_env e
 f
 → coerce_in_env (t
 :: e
) (t
 :: f
).




Hint
 Resolve
 coerce_env_hd coerce_env_tl: coc
.




Lemma
 ind_coerce_env :

  (∀ e
 t
 T
, e
 |-= t
 : T
 → 

  ∀ f
, coerce_in_env e
 f
 → f
 |-= t
 : T
) ∧

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

  ∀ f
, coerce_in_env e
 f
 → f
 |-= T
 >> U
 : s
) ∧

  (∀ e
 U
 V
 T
, e
 |-= U
 = V
 : T
 →

  ∀ f
, coerce_in_env e
 f
 → f
 |-= U
 = V
 : T
).




Lemma
 typ_coerce_env : ∀ e
 t
 T
, e
 |-= t
 : T
 → ∀ f
, coerce_in_env e
 f
 → 

  f
 |-= t
 : T
.




Lemma
 coerce_coerce_env : ∀ e
 T
 U
 s
, e
 |-= T
 >> U
 : s
 → 

  ∀ f
, coerce_in_env e
 f
 → 

  f
 |-= T
 >> U
 : s
.




Lemma
 jeq_coerce_env : ∀ e
 U
 V
 T
, e
 |-= U
 = V
 : T
 → 

  ∀ f
, coerce_in_env e
 f
 → 

  f
 |-= U
 = V
 : T
.