| import Mathlib |
| import OAI.Geometry.Cannon.Energy.Square |
|
|
| namespace OAI |
|
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| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Metric Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
|
|
| namespace CannonChart |
|
|
| variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] |
|
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| def liftPath (e : X → Y) (he : IsEmbedding e) {a b : X} |
| (p : Path (e a) (e b)) (hp : ∀ t, p t ∈ range e) : Path a b where |
| toFun := fun t => he.toHomeomorph.symm ⟨p t,hp t⟩ |
| continuous_toFun := he.toHomeomorph.symm.continuous.comp (p.continuous.subtype_mk hp) |
| source' := by |
| apply he.toHomeomorph.injective |
| simp |
| target' := by |
| apply he.toHomeomorph.injective |
| simp |
|
|
| lemma liftPath_apply (e : X → Y) (he : IsEmbedding e) {a b : X} |
| (p : Path (e a) (e b)) (hp : ∀ t, p t ∈ range e) (t : unitInterval) : |
| e (liftPath e he p hp t) = p t := by |
| have hh := he.toHomeomorph.apply_symm_apply ⟨p t,hp t⟩ |
| exact congrArg Subtype.val hh |
|
|
| lemma lift_joinedIn (e : X → Y) (he : IsEmbedding e) {a b : X} {S : Set Y} |
| (hS : S ⊆ range e) (hp : JoinedIn S (e a) (e b)) : |
| JoinedIn (e ⁻¹' S) a b := by |
| obtain ⟨p,hp⟩ := hp |
| let hpr : ∀ t, p t ∈ range e := fun t => hS (hp t) |
| refine ⟨liftPath e he p hpr,fun t => ?_⟩ |
| change e (liftPath e he p hpr t) ∈ S |
| rw [liftPath_apply e he p hpr] |
| exact hp t |
|
|
| end CannonChart |
|
|
| namespace CannonLevel |
|
|
| open CannonPlanar CannonRegularization |
|
|
| instance : PathConnectedSpace Square := |
| pathConnectedSpace_iff_connectedSpace.mpr inferInstance |
|
|
| theorem regularized_values {f : Square → ℝ} {A B : Set Square} |
| (hA : A.Nonempty) (hB : B.Nonempty) |
| (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1) |
| (hAf : ∀ z ∈ A, f z = 0) (hBf : ∀ z ∈ B, f z = 1) : |
| (∀ z, drain (flood f A) B z ∈ Icc (0 : ℝ) 1) ∧ |
| (∀ z ∈ A, drain (flood f A) B z = 0) ∧ |
| (∀ z ∈ B, drain (flood f A) B z = 1) := by |
| have hF0 : ∀ z, 0 ≤ flood f A z := fun z => (hf0 z).trans (le_flood hA hf1 z) |
| have hF1 := flood_le_upper hA hf1 |
| have hH0 := drain_lower hB hF0 |
| have hHF := drain_le hB hF0 |
| refine ⟨fun z => ⟨hH0 z, (hHF z).trans (hF1 z)⟩, ?_, ?_⟩ |
| · intro z hz |
| apply le_antisymm _ (hH0 z) |
| simpa [flood_eq_on_band hA hf1 hz, hAf z hz] using hHF z |
| · intro z hz |
| rw [drain_eq_on_band hB hF0 hz] |
| apply le_antisymm (hF1 z) |
| simpa [hBf z hz] using le_flood hA hf1 z |
|
|
| theorem separating_level {f : Square → ℝ} {A B : Set Square} |
| (hA : IsPathConnected A) (hB : IsPathConnected B) |
| (hAedge : ∀ s, (s,0) ∈ A) (hBedge : ∀ s, (s,1) ∈ B) |
| (hf : Continuous f) (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1) |
| (hAf : ∀ z ∈ A, f z = 0) (hBf : ∀ z ∈ B, f z = 1) |
| {t : ℝ} (ht0 : 0 < t) (ht1 : t < 1) : |
| let H := drain (flood f A) B |
| ∃ C : Set Square, IsCompact C ∧ IsConnected C ∧ |
| (∀ z ∈ C, H z = t) ∧ (∃ z ∈ C, z.1 = 0) ∧ (∃ z ∈ C, z.1 = 1) ∧ |
| (∀ {x y : Square} (p : Path x y), H x < t → t < H y → |
| ∃ s, p s ∈ C) := by |
| let F := flood f A |
| let H := drain F B |
| have hF0 : ∀ z, 0 ≤ F z := fun z => (hf0 z).trans (le_flood hA.nonempty hf1 z) |
| have hFc : Continuous F := continuous_flood hA.nonempty hf hf1 |
| have hHc : Continuous H := continuous_drain hB.nonempty hFc hF0 |
| obtain ⟨hHr,hHA,hHB⟩ := regularized_values hA.nonempty hB.nonempty hf0 hf1 hAf hBf |
| obtain ⟨C,hCc,hCn,hCH,⟨a,ha,ha0⟩,⟨b,hb,hb1⟩⟩ := level_continuum hHc |
| (c := t) (fun s => by rw [show H (s,0) = 0 from hHA _ (hAedge s)]; exact ht0.le) |
| (fun s => by rw [show H (s,1) = 1 from hHB _ (hBedge s)]; exact ht1.le) |
| refine ⟨C,hCc,hCn,hCH,⟨a,ha,ha0⟩,⟨b,hb,hb1⟩,?_⟩ |
| intro x y p hx hy |
| obtain ⟨u,hu,lo,hlo⟩ := regularized_low_side hA.nonempty hB.nonempty hf hf0 hf1 hAf ht0 hx |
| obtain ⟨v,hv,hi,hhi⟩ := drain_strict_side hB.nonempty hF0 hy |
| obtain ⟨qa,hqa⟩ := hA.joinedIn u hu (0,0) (hAedge 0) |
| obtain ⟨qb,hqb⟩ := hB.joinedIn v hv (0,1) (hBedge 0) |
| let low : Path x (0,0) := lo.trans qa |
| let high : Path y (0,1) := hi.trans qb |
| have hlow : ∀ s, H (low s) < t := by |
| intro s |
| have hs : low s ∈ range lo ∪ range qa := by rw [← Path.trans_range]; exact mem_range_self s |
| rcases hs with ⟨r,hr⟩ | ⟨r,hr⟩ |
| · rw [← hr]; exact hlo r |
| · rw [← hr, show H (qa r) = 0 from hHA _ (hqa r)]; exact ht0 |
| have hhigh : ∀ s, t < H (high s) := by |
| intro s |
| have hs : high s ∈ range hi ∪ range qb := by rw [← Path.trans_range]; exact mem_range_self s |
| rcases hs with ⟨r,hr⟩ | ⟨r,hr⟩ |
| · rw [← hr]; exact hhi r |
| · rw [← hr, show H (qb r) = 1 from hHB _ (hqb r)]; exact ht1 |
| let q := high.symm.trans (p.symm.trans low) |
| obtain ⟨s,hs⟩ := continuum_crosses_path hCn.isPreconnected ha hb ha0 hb1 q.continuous |
| (by simp [q]) (by simp [q]) |
| have hqs : q s ∈ range high.symm ∪ (range p.symm ∪ range low) := by |
| rw [← Path.trans_range, ← Path.trans_range] |
| exact mem_range_self s |
| rcases hqs with ⟨r,hr⟩ | ⟨r,hr⟩ | ⟨r,hr⟩ |
| · have ht := hhigh (unitInterval.symm r) |
| have he := hCH _ hs |
| change H (high.symm r) > t at ht |
| rw [hr,he] at ht |
| exact False.elim (lt_irrefl _ ht) |
| · exact ⟨unitInterval.symm r, hr.symm ▸ hs⟩ |
| · have ht := hlow r |
| rw [hr,hCH _ hs] at ht |
| exact False.elim (lt_irrefl _ ht) |
|
|
| lemma slab_pathConnected {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (hb : b ≤ 1) : |
| IsPathConnected {z : Square | a ≤ (z.2 : ℝ) ∧ (z.2 : ℝ) ≤ b} := by |
| let e : Square → ℝ × ℝ := Prod.map Subtype.val Subtype.val |
| have he : Topology.IsInducing e := Topology.IsInducing.subtypeVal.prodMap |
| Topology.IsInducing.subtypeVal |
| rw [he.isPathConnected_iff] |
| have himage : e '' {z : Square | a ≤ (z.2 : ℝ) ∧ (z.2 : ℝ) ≤ b} = |
| Icc ((0 : ℝ),a) (1,b) := by |
| ext z |
| constructor |
| · rintro ⟨w,hw,rfl⟩ |
| exact ⟨⟨w.1.property.1,hw.1⟩,⟨w.1.property.2,hw.2⟩⟩ |
| · intro hz |
| refine ⟨(⟨z.1,hz.1.1,hz.2.1⟩,⟨z.2,ha.trans hz.1.2,hz.2.2.trans hb⟩), ?_, rfl⟩ |
| exact ⟨hz.1.2,hz.2.2⟩ |
| rw [himage] |
| exact (convex_Icc _ _).isPathConnected ⟨(0,a),⟨le_rfl,⟨zero_le_one,hab⟩⟩⟩ |
|
|
| def topBand : Set Square := {z | (z.2 : ℝ) ≤ 1 / 3} |
|
|
| def bottomBand : Set Square := {z | 2 / 3 ≤ (z.2 : ℝ)} |
|
|
| def centralRectangle : Set Square := {z | (z.2 : ℝ) ∈ Icc (1 / 3) (2 / 3)} |
|
|
| lemma topBand_pathConnected : IsPathConnected topBand := by |
| convert slab_pathConnected (a := 0) (b := 1 / 3) (by norm_num) (by norm_num) (by norm_num) using 1 |
| ext z |
| exact ⟨fun hz => ⟨z.2.property.1,hz⟩,fun hz => hz.2⟩ |
|
|
| lemma bottomBand_pathConnected : IsPathConnected bottomBand := by |
| convert slab_pathConnected (a := 2 / 3) (b := 1) (by norm_num) (by norm_num) (by norm_num) using 1 |
| ext z |
| exact ⟨fun hz => ⟨hz,z.2.property.2⟩,fun hz => hz.1⟩ |
|
|
| theorem rectangle_separating_level {f : Square → ℝ} |
| (hf : Continuous f) (hf0 : ∀ z, 0 ≤ f z) (hf1 : ∀ z, f z ≤ 1) |
| (hAf : ∀ z ∈ topBand, f z = 0) (hBf : ∀ z ∈ bottomBand, f z = 1) |
| {t : ℝ} (ht0 : 0 < t) (ht1 : t < 1) : |
| let H := drain (flood f topBand) bottomBand |
| ∃ C : Set Square, IsCompact C ∧ IsConnected C ∧ C ⊆ centralRectangle ∧ |
| (∀ z ∈ C, H z = t) ∧ (∃ z ∈ C, z.1 = 0) ∧ (∃ z ∈ C, z.1 = 1) ∧ |
| (∀ {x y : Square} (p : Path x y), H x < t → t < H y → ∃ s, p s ∈ C) := by |
| obtain ⟨C,hCc,hCn,hCH,hCl,hCr,hCs⟩ := separating_level topBand_pathConnected bottomBand_pathConnected |
| (by intro s; change (0 : ℝ) ≤ 1 / 3; norm_num) |
| (by intro s; change (2 : ℝ) / 3 ≤ 1; norm_num) |
| hf hf0 hf1 hAf hBf ht0 ht1 |
| refine ⟨C,hCc,hCn,?_,hCH,hCl,hCr,hCs⟩ |
| obtain ⟨_,hHA,hHB⟩ := regularized_values topBand_pathConnected.nonempty |
| bottomBand_pathConnected.nonempty hf0 hf1 hAf hBf |
| intro z hz |
| constructor |
| · by_contra h |
| have he := hHA z (show z ∈ topBand from (lt_of_not_ge h).le) |
| rw [hCH z hz] at he |
| exact (ne_of_gt ht0) he |
| · by_contra h |
| have he := hHB z (show z ∈ bottomBand from (lt_of_not_ge h).le) |
| rw [hCH z hz] at he |
| exact (ne_of_lt ht1) he |
|
|
| end CannonLevel |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
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| end OAI |
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