(* Autor(s): Andrej Dudenhefner (1) Affiliation(s): (1) Saarland University, Saarbrücken, Germany  Description: Theorem "pure_typable_intro_prenex" can be used to introduce/eliminate closed assumption ∀..∀.s where s is simple. Theorem "pure_typable_intro_poly_arr_0_0" can be used to introduce/eliminate the assumption 0 -> 0 [1].  Related Work: [1] Wells, Joe B. "Typability and Type-Checking in the Second-Order lambda-Calculus are Equivalent and Undecidable." Proceedings Ninth Annual IEEE Symposium on Logic In Computer Science. IEEE, 1994. *)
From Undecidability.SystemF.Util Require Import Facts poly_type_facts pure_term_facts term_facts typing_facts iipc2_facts pure_typing_facts.
(* (\_.M) N *)
Definition K' M N := pure_app (pure_abs (ren_pure_term S M)) N.

(* self application is typed only by left-extensible types *)
Lemma pure_typable_self_application {Gamma x n t} :
not (is_poly_abs t) ->
nth_error Gamma x = Some (many_poly_abs n t) ->
pure_typable Gamma (pure_app (pure_var x) (pure_var x)) ->
exists y, y < n /\ leftmost_poly_var t = Some y.