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/- | ||
Copyright © 2020 Nicolò Cavalleri. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Author: Nicolò Cavalleri. | ||
-/ | ||
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import geometry.manifold.times_cont_mdiff | ||
import topology.continuous_map | ||
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/-! | ||
# Smooth bundled map | ||
In this file we define the type `times_cont_mdiff_map` of `n` times continuously differentiable | ||
bundled maps. | ||
-/ | ||
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variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] | ||
{E : Type*} [normed_group E] [normed_space 𝕜 E] | ||
{E' : Type*} [normed_group E'] [normed_space 𝕜 E'] | ||
{H : Type*} [topological_space H] | ||
{H' : Type*} [topological_space H'] | ||
(I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') | ||
(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] | ||
(M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] | ||
{E'' : Type*} [normed_group E''] [normed_space 𝕜 E''] | ||
{H'' : Type*} [topological_space H''] | ||
{I'' : model_with_corners 𝕜 E'' H''} | ||
{M'' : Type*} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M''] | ||
(n : with_top ℕ) | ||
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/- Bundled `n` times continuously differentiable maps. -/ | ||
@[protect_proj] | ||
structure times_cont_mdiff_map := | ||
(to_fun : M → M') | ||
(times_cont_mdiff_to_fun : times_cont_mdiff I I' n to_fun) | ||
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/-- Bundled smooth maps. -/ | ||
@[reducible] def smooth_map := times_cont_mdiff_map I I' M M' ⊤ | ||
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notation `C[` n `](` I `, ` M `; ` I' `, ` M' `)` := times_cont_mdiff_map I I' M M' n | ||
notation `C[` n `](` I `, ` M `; ` k `)` := times_cont_mdiff_map I (model_with_corners_self k k) M k n | ||
notation `C[` n `](` I `, ` M `)` := times_cont_mdiff_map I (model_with_corners_self ℝ ℝ) M ℝ n | ||
notation `C∞(` I `, ` M `; ` I' `, ` M' `)` := times_cont_mdiff_map I I' M M' ⊤ | ||
notation `C∞(` I `, ` M `; ` k `)` := times_cont_mdiff_map I (model_with_corners_self k k) M k ⊤ | ||
notation `C∞(` I `, ` M `)` := times_cont_mdiff_map I (model_with_corners_self ℝ ℝ) M ℝ ⊤ | ||
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namespace times_cont_mdiff_map | ||
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variables {I} {I'} {M} {M'} {n} | ||
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instance : has_coe_to_fun C[n](I, M; I', M') := ⟨_, times_cont_mdiff_map.to_fun⟩ | ||
instance : has_coe C[n](I, M; I', M') C(M, M') := | ||
⟨λ f, ⟨f.to_fun, f.times_cont_mdiff_to_fun.continuous⟩⟩ | ||
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variables {f g : C[n](I, M; I', M')} | ||
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protected lemma times_cont_mdiff (f : C[n](I, M; I', M')) : | ||
times_cont_mdiff I I' n f := f.times_cont_mdiff_to_fun | ||
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protected lemma smooth (f : C∞(I, M; I', M')) : | ||
smooth I I' f := f.times_cont_mdiff_to_fun | ||
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lemma coe_inj ⦃f g : C[n](I, M; I', M')⦄ (h : (f : M → M') = g) : f = g := | ||
by cases f; cases g; cases h; refl | ||
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@[ext] theorem ext (h : ∀ x, f x = g x) : f = g := | ||
by cases f; cases g; congr'; exact funext h | ||
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/-- The identity as a smooth map. -/ | ||
def id : C[n](I, M; I, M) := ⟨id, times_cont_mdiff_id⟩ | ||
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/-- The composition of smooth maps, as a smooth map. -/ | ||
def comp (f : C[n](I', M'; I'', M'')) (g : C[n](I, M; I', M')) : C[n](I, M; I'', M'') := | ||
{ to_fun := λ a, f (g a), | ||
times_cont_mdiff_to_fun := f.times_cont_mdiff_to_fun.comp g.times_cont_mdiff_to_fun, } | ||
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@[simp] lemma comp_apply (f : C[n](I', M'; I'', M'')) (g : C[n](I, M; I', M')) (x : M) : | ||
f.comp g x = f (g x) := rfl | ||
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instance [inhabited M'] : inhabited C[n](I, M; I', M') := | ||
⟨⟨λ _, default _, times_cont_mdiff_const⟩⟩ | ||
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/-- Constant map as a smooth map -/ | ||
def const (y : M') : C[n](I, M; I', M') := ⟨λ x, y, times_cont_mdiff_const⟩ | ||
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end times_cont_mdiff_map | ||
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open_locale manifold | ||
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instance continuous_linear_map.has_coe_to_times_cont_mdiff_map : | ||
has_coe (E →L[𝕜] E') (C[n](Isf(𝕜, E), E; Isf(𝕜, E'), E')) := | ||
⟨λ f, ⟨f.to_fun, f.times_cont_mdiff⟩⟩ |