Section Evaluation.
Variable (M : model).
Fixpoint eval (ρ : nat -> tpred) (s : poly_type) : tpred :=
match s with
| poly_var n => Var M (ρ n)
| poly_arr s t => Arr M (eval ρ s) (eval ρ t)
| poly_abs s => All M (fun d => eval (d .: ρ) s)
end.
Instance eval_ext :
Proper (pointwise_relation _ pw_iff ==> eq ==> pw_iff) eval.
Definition D : model.
(* ** Logical Relation  *)
(* ** Logical relation on contexts  *)
Definition C (Γ : nat -> poly_type) (ρ : nat -> tpred) : (nat -> term) -> Prop :=
fun σ => forall n, E (Γ n) ρ (σ n).

Lemma fundamental {Γ s P} :
typing Γ P s ->
forall σ τ ρ,
C (fun n => match List.nth_error Γ n with Some x => x | _ => poly_abs (poly_var 0) end) ρ τ ->
E s ρ (subst_term σ τ P).