import global.relation

/-!
# Gromov's theorem

We prove the h-principle for open and ample first order differential relations.
-/

noncomputable theory

open set
open_locale topological_space manifold

variables
{E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
{H : Type*} [topological_space H] {I : model_with_corners ℝ E H}
{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E']
{H' : Type*} [topological_space H'] {I' : model_with_corners ℝ E' H'}
{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']
{F : Type*} [normed_add_comm_group F] [normed_space ℝ F]
{G : Type*} [topological_space G] (J : model_with_corners ℝ F G)
(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]
{F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F']
{G' : Type*} [topological_space G'] (J' : model_with_corners ℝ F' G')
(N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N']
{R : rel_mfld I M I' M'}

/-- The non-parametric version of Gromov's theorem -/
lemma rel_mfld.ample.satisfies_h_principle (h1 : R.ample) (h2 : is_open R) :
  R.satisfies_h_principle :=
sorry

/-- **Gromov's Theorem** -/
theorem rel_mfld.ample.satisfies_h_principle_with (h1 : R.ample) (h2 : is_open R) :
  R.satisfies_h_principle_with J N :=
sorry