Library Lambda.JRussell.PreFunctionality


Require
 Import
 Lambda.Terms.

Require
 Import
 Lambda.Reduction.

Require
 Import
 Lambda.LiftSubst.

Require
 Import
 Lambda.Env.

Require
 Import
 Lambda.JRussell.Types.

Require
 Import
 Lambda.JRussell.Coercion.

Require
 Import
 Lambda.JRussell.Substitution.




Require
 Import
 Coq.Arith.Arith.







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
 pre_functionality_rec : ∀ g
 (d
 : term) t
, g
 |-= d
 : t
 → ∀ d'
, g
 |-= d
 = d'
 : t
 →

  ∀ e
 U
 T
, e
 |-= U
 : T
 →

  ∀ f
 n
, sub_in_env d
 t
 n
 e
 f
 → trunc _
 n
 f
 g
 → 

  f
 |-= (subst_rec d
 U
 n
) = (subst_rec d'
 U
 n
) : (subst_rec d
 T
 n
).




Lemma
 pre_functionality : ∀ e
 (d
 : term) t
, e
 |-= d
 : t
 → 

  ∀ d'
, e
 |-= d
 = d'
 : t
 →

  ∀ U
 T
, t
 :: e
 |-= U
 : T
 →

  e
 |-= (subst d
 U
) = (subst d'
 U
) : (subst d
 T
).