| import Mathlib.Topology.Instances.AddCircle.Real |
| import Mathlib.Topology.Algebra.Order.Field |
|
|
| namespace OAI |
|
|
| /-!+# Compact interval-band graphs for bounded real coordinates |
|
|
| Take the closure of the physical graph and fill the interval between |
| cluster values in each fiber. This constructs a compact band graph; |
| on the open physical domain it is exactly the original graph. Its |
| projection modulo one consequently gives compact level sets with the |
| correct physical fibers. |
| -/ |
|
|
| open Set |
|
|
| namespace CircleDomainRigidity |
|
|
| def clusterGraph {X : Type*} [TopologicalSpace X] (G : Set X) (f : X → ℝ) : Set (X × ℝ) := |
| closure ((fun x => (x, f x)) '' G) |
|
|
| def intervalFiberHull {X : Type*} (K : Set (X × ℝ)) : Set (X × ℝ) := |
| (fun p : (X × ℝ) × (X × ℝ) × ℝ => |
| (p.1.1, (1 - p.2.2) * p.1.2 + p.2.2 * p.2.1.2)) '' |
| ((K ×ˢ (K ×ˢ Icc 0 1)) ∩ {p | p.1.1 = p.2.1.1}) |
|
|
| theorem isCompact_clusterGraph {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| {G : Set X} {f : X → ℝ} {M : ℝ} (hM : ∀ x ∈ G, |f x| ≤ M) : |
| IsCompact (clusterGraph G f) := by |
| apply (isCompact_univ.prod (isCompact_Icc : IsCompact (Icc (-M) M))).of_isClosed_subset |
| isClosed_closure |
| apply closure_minimal _ (isClosed_univ.prod isClosed_Icc) |
| rintro _ ⟨x, hx, rfl⟩ |
| exact ⟨mem_univ x, abs_le.mp (hM x hx)⟩ |
|
|
| theorem isCompact_intervalFiberHull {X : Type*} [TopologicalSpace X] [T2Space X] |
| {K : Set (X × ℝ)} (hK : IsCompact K) : IsCompact (intervalFiberHull K) := by |
| apply ((hK.prod (hK.prod isCompact_Icc)).inter_right |
| (isClosed_eq (continuous_fst.fst) (continuous_snd.fst.fst))).image |
| exact continuous_fst.fst.prodMk |
| (((continuous_const.sub continuous_snd.snd).mul continuous_fst.snd).add |
| (continuous_snd.snd.mul continuous_snd.fst.snd)) |
|
|
| theorem mem_intervalFiberHull_iff {X : Type*} {K : Set (X × ℝ)} {x : X} {t : ℝ} : |
| (x, t) ∈ intervalFiberHull K ↔ |
| ∃ a b s : ℝ, (x, a) ∈ K ∧ (x, b) ∈ K ∧ s ∈ Icc 0 1 ∧ |
| t = (1 - s) * a + s * b := by |
| constructor |
| · rintro ⟨⟨⟨y, a⟩, ⟨z, b⟩, s⟩, ⟨⟨ha, hb, hs⟩, heq⟩, hout⟩ |
| obtain ⟨rfl, ht⟩ := Prod.mk.inj hout |
| change y = z at heq |
| subst z |
| exact ⟨a, b, s, ha, hb, hs, ht.symm⟩ |
| · rintro ⟨a, b, s, ha, hb, hs, rfl⟩ |
| exact ⟨((x, a), (x, b), s), ⟨⟨ha, hb, hs⟩, rfl⟩, rfl⟩ |
|
|
| theorem subset_intervalFiberHull {X : Type*} (K : Set (X × ℝ)) : |
| K ⊆ intervalFiberHull K := by |
| rintro ⟨x, t⟩ ht |
| apply mem_intervalFiberHull_iff.mpr |
| exact ⟨t, t, 0, ht, ht, by norm_num, by ring⟩ |
|
|
| theorem isCompact_real_fiber {X : Type*} [TopologicalSpace X] [T2Space X] |
| {K : Set (X × ℝ)} (hK : IsCompact K) (x : X) : |
| IsCompact {t : ℝ | (x, t) ∈ K} := by |
| have hc : IsCompact (Prod.snd '' (K ∩ {p | p.1 = x})) := |
| (hK.inter_right (isClosed_eq continuous_fst continuous_const)).image continuous_snd |
| convert hc using 1 |
| ext t |
| constructor |
| · intro ht |
| exact ⟨(x, t), ⟨ht, rfl⟩, rfl⟩ |
| · rintro ⟨⟨y, a⟩, ⟨ha, heq⟩, rfl⟩ |
| change y = x at heq |
| change (x, a) ∈ K |
| simpa only [heq] using ha |
|
|
| / |
| and largest cluster values, with the endpoints attained in the compact |
| original fiber. -/ |
| theorem intervalFiberHull_fiber_eq_Icc |
| {X : Type*} [TopologicalSpace X] [T2Space X] {K : Set (X × ℝ)} |
| (hK : IsCompact K) (x : X) (hne : {t : ℝ | (x, t) ∈ K}.Nonempty) : |
| {t : ℝ | (x, t) ∈ intervalFiberHull K} = |
| Icc (sInf {t : ℝ | (x, t) ∈ K}) (sSup {t : ℝ | (x, t) ∈ K}) := by |
| let F : Set ℝ := {t | (x, t) ∈ K} |
| have hlo := (isCompact_real_fiber hK x).isLeast_sInf hne |
| have hhi := (isCompact_real_fiber hK x).isGreatest_sSup hne |
| ext t |
| constructor |
| · intro ht |
| obtain ⟨a, b, s, ha, hb, hs, rfl⟩ := mem_intervalFiberHull_iff.mp ht |
| constructor |
| · nlinarith [mul_nonneg (sub_nonneg.mpr hs.2) (sub_nonneg.mpr (hlo.2 ha)), |
| mul_nonneg hs.1 (sub_nonneg.mpr (hlo.2 hb))] |
| · nlinarith [mul_nonneg (sub_nonneg.mpr hs.2) (sub_nonneg.mpr (hhi.2 ha)), |
| mul_nonneg hs.1 (sub_nonneg.mpr (hhi.2 hb))] |
| · intro ht |
| apply mem_intervalFiberHull_iff.mpr |
| by_cases heq : sInf F = sSup F |
| · have htEq : t = sInf F := le_antisymm (ht.2.trans_eq heq.symm) ht.1 |
| refine ⟨sInf F, sSup F, 0, hlo.1, hhi.1, by norm_num, ?_⟩ |
| rw [htEq] |
| ring |
| · have hlt : sInf F < sSup F := lt_of_le_of_ne (hlo.2 hhi.1) heq |
| have hd : 0 < sSup F - sInf F := sub_pos.mpr hlt |
| refine ⟨sInf F, sSup F, (t - sInf F) / (sSup F - sInf F), hlo.1, hhi.1, |
| ⟨div_nonneg (sub_nonneg.mpr ht.1) hd.le, (div_le_one hd).mpr (by linarith [ht.2])⟩, ?_⟩ |
| field_simp |
| ring |
|
|
| theorem clusterGraph_fiber_nonempty_of_mem_closure |
| {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X] |
| {G : Set X} {f : X → ℝ} {M : ℝ} (hM : ∀ x ∈ G, |f x| ≤ M) |
| {x : X} (hx : x ∈ closure G) : {t : ℝ | (x, t) ∈ clusterGraph G f}.Nonempty := by |
| have hsub : closure G ⊆ Prod.fst '' clusterGraph G f := by |
| apply closure_minimal _ ((isCompact_clusterGraph hM).image continuous_fst).isClosed |
| intro y hy |
| exact ⟨(y, f y), subset_closure ⟨y, hy, rfl⟩, rfl⟩ |
| obtain ⟨⟨y, t⟩, ht, heq⟩ := hsub hx |
| change y = x at heq |
| exact ⟨t, heq ▸ ht⟩ |
|
|
| theorem clusterGraph_fiber_eq_of_continuousAt |
| {X : Type*} [TopologicalSpace X] {G : Set X} {f : X → ℝ} {x : X} |
| (hf : ContinuousAt f x) {t : ℝ} (ht : (x, t) ∈ clusterGraph G f) : t = f x := by |
| have hc : ContinuousAt (fun p : X × ℝ => p.2 - f p.1) (x, t) := |
| continuousAt_snd.sub (hf.comp continuousAt_fst) |
| have hz : t - f x ∈ closure ({0} : Set ℝ) := hc.continuousWithinAt.mem_closure ht (by |
| rintro _ ⟨y, hy, rfl⟩ |
| simp) |
| simpa only [closure_singleton, mem_singleton_iff, sub_eq_zero] using hz |
|
|
| theorem intervalFiberHull_clusterGraph_on_open |
| {X : Type*} [TopologicalSpace X] {G : Set X} (hG : IsOpen G) {f : X → ℝ} |
| (hf : ContinuousOn f G) {x : X} (hx : x ∈ G) {t : ℝ} : |
| (x, t) ∈ intervalFiberHull (clusterGraph G f) ↔ t = f x := by |
| constructor |
| · intro ht |
| obtain ⟨a, b, s, ha, hb, _, ht⟩ := mem_intervalFiberHull_iff.mp ht |
| have hfx := hf.continuousAt (hG.mem_nhds hx) |
| rw [clusterGraph_fiber_eq_of_continuousAt hfx ha, |
| clusterGraph_fiber_eq_of_continuousAt hfx hb] at ht |
| nlinarith [ht] |
| · rintro rfl |
| exact subset_intervalFiberHull _ (subset_closure ⟨x, hx, rfl⟩) |
|
|
| def circleBandGraph {X : Type*} [TopologicalSpace X] (G : Set X) (f : X → ℝ) : |
| Set (X × AddCircle (1 : ℝ)) := |
| (fun p : X × ℝ => (p.1, (p.2 : AddCircle (1 : ℝ)))) '' |
| intervalFiberHull (clusterGraph G f) |
|
|
| theorem isCompact_circleBandGraph {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| [T2Space X] {G : Set X} {f : X → ℝ} {M : ℝ} (hM : ∀ x ∈ G, |f x| ≤ M) : |
| IsCompact (circleBandGraph G f) := by |
| exact (isCompact_intervalFiberHull (isCompact_clusterGraph hM)).image |
| (continuous_fst.prodMk ((AddCircle.continuous_mk' (1 : ℝ)).comp continuous_snd)) |
| |
| theorem circleBandGraph_on_open {X : Type*} [TopologicalSpace X] |
| {G : Set X} (hG : IsOpen G) {f : X → ℝ} (hf : ContinuousOn f G) |
| {x : X} (hx : x ∈ G) {t : AddCircle (1 : ℝ)} : |
| (x, t) ∈ circleBandGraph G f ↔ (f x : AddCircle (1 : ℝ)) = t := by |
| constructor |
| · rintro ⟨⟨y, a⟩, ha, hout⟩ |
| obtain ⟨rfl, ht⟩ := Prod.mk.inj hout |
| rwa [(intervalFiberHull_clusterGraph_on_open hG hf hx).mp ha] at ht |
| · intro ht |
| exact ⟨(x, f x), (intervalFiberHull_clusterGraph_on_open hG hf hx).mpr rfl, |
| Prod.ext rfl ht⟩ |
| |
| theorem isCompact_circleBandLevel {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| [T2Space X] {G : Set X} {f : X → ℝ} {M : ℝ} (hM : ∀ x ∈ G, |f x| ≤ M) |
| (t : AddCircle (1 : ℝ)) : IsCompact {x | (x, t) ∈ circleBandGraph G f} := |
| ((isCompact_circleBandGraph hM).isClosed.preimage (continuous_id.prodMk continuous_const)).isCompact |
| |
| theorem isCompact_circleBandIncidence {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| [T2Space X] {G : Set X} {f : X → ℝ} {M : ℝ} (hM : ∀ x ∈ G, |f x| ≤ M) |
| (x : X) : IsCompact {t : AddCircle (1 : ℝ) | (x, t) ∈ circleBandGraph G f} := |
| ((isCompact_circleBandGraph hM).isClosed.preimage (continuous_const.prodMk continuous_id)).isCompact |
| |
| end CircleDomainRigidity |
| |
| end OAI |
| |
