openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Partitions /FiberCardinalityConstancy.lean
| import Mathlib | |
| import OAI.NumberTheory.Jacobsthal.Estimates.LocalGraphIff | |
| import OAI.NumberTheory.Jacobsthal.Estimates.NestedDegreeBounds | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| namespace ErdosGlobalFiberBranches | |
| noncomputable def orderIsoOfNatCardEq (α β : Type*) [LinearOrder α] [LinearOrder β] | |
| [Finite α] [Finite β] (h : Nat.card α = Nat.card β) : α ≃o β := by | |
| letI := Fintype.ofFinite α | |
| letI := Fintype.ofFinite β | |
| exact (Fintype.orderIsoFinOfCardEq α rfl).symm.trans | |
| (Fintype.orderIsoFinOfCardEq β (by simpa only [Nat.card_eq_fintype_card] using h.symm)) | |
| theorem finite_orderIso_unique {α β : Type*} [LinearOrder α] [LinearOrder β] | |
| [Finite α] (f g : α ≃o β) : f = g := Subsingleton.elim f g | |
| noncomputable def orderIsoOfStrictMonoSurjective {α β : Type*} | |
| [LinearOrder α] [LinearOrder β] (f : α → β) (hf : StrictMono f) | |
| (hs : Function.Surjective f) : α ≃o β := | |
| { Equiv.ofBijective f ⟨hf.injective,hs⟩ with map_rel_iff' := hf.le_iff_le } | |
| theorem orderIsoOfStrictMonoSurjective_apply {α β : Type*} | |
| [LinearOrder α] [LinearOrder β] (f : α → β) (hf : StrictMono f) | |
| (hs : Function.Surjective f) (a : α) : | |
| orderIsoOfStrictMonoSurjective f hf hs a = f a := rfl | |
| end ErdosGlobalFiberBranches | |
| end | |
| section | |
| open Filter Set | |
| open scoped Topology ContDiff | |
| namespace ErdosLocalFiberGraphs | |
| open ErdosImplicitCurvature | |
| structure FiberCharts (Q : Bivariate) (x L U : ℝ) where | |
| a : ℝ | |
| b : ℝ | |
| left : a < x | |
| right : x < b | |
| finite_base : (boundedFiber Q x L U).Finite | |
| graph : boundedFiber Q x L U → ℝ → ℝ | |
| tube : boundedFiber Q x L U → Set ℝ | |
| tube_open : ∀ r, IsOpen (tube r) | |
| tube_base : ∀ r, (r : ℝ) ∈ tube r | |
| tube_interior : ∀ r, tube r ⊆ Ioo L U | |
| tube_disjoint : Pairwise fun r s => Disjoint (tube r) (tube s) | |
| graph_base : ∀ r, graph r x = r | |
| graph_smooth : ∀ r, ContDiffOn ℝ 2 (graph r) (Ioo a b) | |
| graph_inside : ∀ r, ∀ t ∈ Ioo a b, graph r t ∈ tube r | |
| graph_regular : ∀ r, ∀ t ∈ Ioo a b, peval (partialY Q) t (graph r t) ≠ 0 | |
| zero_iff : ∀ r, ∀ t ∈ Ioo a b, ∀ y ∈ tube r, peval Q t y = 0 ↔ y = graph r t | |
| coverage : ∀ t ∈ Ioo a b, boundedFiber Q t L U = range (fun r => graph r t) | |
| ordered : ∀ t ∈ Ioo a b, StrictMono (fun r => graph r t) | |
| theorem exists_bounded_fiber_charts (Q : Bivariate) (x L U : ℝ) | |
| (hL : peval Q x L ≠ 0) (hU : peval Q x U ≠ 0) | |
| (hregular : ∀ y ∈ boundedFiber Q x L U, peval (partialY Q) x y ≠ 0) : | |
| Nonempty (FiberCharts Q x L U) := by | |
| classical | |
| let R := boundedFiber Q x L U | |
| have hR : R.Finite := boundedFiber_finite Q x L U L hL | |
| let : Fintype R := hR.fintype | |
| obtain ⟨V,hV,hsep⟩ := hR.t2_separation | |
| have hcharts (r : R) := exists_local_graph_iff Q x r r.property.2 | |
| (hregular r r.property) (V r ∩ Ioo L U) ((hV r).2.inter isOpen_Ioo) | |
| ⟨(hV r).1,boundedFiber_interior Q x L U hL hU r.property⟩ | |
| choose a b W f ha hb hWo hrW hWV hbase hC hmem hder hiff using hcharts | |
| have hWint (r : R) : W r ⊆ Ioo L U := fun y hy => (hWV r hy).2 | |
| have hWd : Pairwise fun r s : R => Disjoint (W r) (W s) := by | |
| intro r s hrs | |
| apply (hsep r.property s.property (fun he => hrs (Subtype.ext he))).mono | |
| · exact fun y hy => (hWV r hy).1 | |
| · exact fun y hy => (hWV s hy).1 | |
| let K := Icc L U \ ⋃ r : R, W r | |
| have hK : IsCompact K := isCompact_Icc.diff (isOpen_iUnion hWo) | |
| have hKzero : ∀ y ∈ K, peval Q x y ≠ 0 := by | |
| intro y hy hzero | |
| have hyR : y ∈ R := ⟨hy.1,hzero⟩ | |
| exact hy.2 (mem_iUnion.mpr ⟨⟨y,hyR⟩,hrW ⟨y,hyR⟩⟩) | |
| have hnoextra := compact_nonzero_eventually Q x K hK hKzero | |
| have hdomains : ∀ᶠ t in 𝓝 x, ∀ r : R, t ∈ Ioo (a r) (b r) := | |
| eventually_all.mpr fun r => isOpen_Ioo.mem_nhds ⟨ha r,hb r⟩ | |
| have hcont (r : R) : ContinuousAt (f r) x := | |
| (hC r).continuousOn.continuousAt (isOpen_Ioo.mem_nhds ⟨ha r,hb r⟩) | |
| have horder : ∀ᶠ t in 𝓝 x, ∀ r s : R, r < s → f r t < f s t := by | |
| apply eventually_all.mpr | |
| intro r | |
| apply eventually_all.mpr | |
| intro s | |
| by_cases hrs : r < s | |
| · have hlt : f r x < f s x := by simpa only [hbase] using (show (r : ℝ) < (s : ℝ) from hrs) | |
| exact ((hcont r).eventually_lt (hcont s) hlt).mono fun t ht _ => ht | |
| · exact Eventually.of_forall fun t ht => (hrs ht).elim | |
| have hN : {t | (∀ r : R, t ∈ Ioo (a r) (b r)) ∧ | |
| (∀ y ∈ K, peval Q t y ≠ 0) ∧ (∀ r s : R, r < s → f r t < f s t)} ∈ 𝓝 x := | |
| hdomains.and (hnoextra.and horder) | |
| obtain ⟨eps,heps,hball⟩ := Metric.mem_nhds_iff.mp hN | |
| have hs : Ioo (x-eps) (x+eps) ⊆ {t | (∀ r : R, t ∈ Ioo (a r) (b r)) ∧ | |
| (∀ y ∈ K, peval Q t y ≠ 0) ∧ (∀ r s : R, r < s → f r t < f s t)} := by | |
| simpa only [Real.ball_eq_Ioo] using hball | |
| refine ⟨{ | |
| a := x-eps | |
| b := x+eps | |
| left := by linarith | |
| right := by linarith | |
| finite_base := hR, graph := f, tube := W, tube_open := hWo, tube_base := hrW, | |
| tube_interior := hWint, tube_disjoint := hWd, graph_base := hbase, | |
| graph_smooth := fun r => (hC r).mono (fun t ht => (hs ht).1 r), | |
| graph_inside := fun r t ht => hmem r t ((hs ht).1 r), | |
| graph_regular := fun r t ht => hder r t ((hs ht).1 r), | |
| zero_iff := fun r t ht => hiff r t ((hs ht).1 r), | |
| coverage := ?_, ordered := fun t ht => (hs ht).2.2 }⟩ | |
| intro t ht | |
| ext y | |
| constructor | |
| · intro hy | |
| have hyW : y ∈ ⋃ r : R, W r := by | |
| by_contra hn | |
| exact (hs ht).2.1 y ⟨hy.1,hn⟩ hy.2 | |
| obtain ⟨r,hyr⟩ := mem_iUnion.mp hyW | |
| exact ⟨r,((hiff r t ((hs ht).1 r) y hyr).mp hy.2).symm⟩ | |
| · rintro ⟨r,rfl⟩ | |
| have hfW := hmem r t ((hs ht).1 r) | |
| exact ⟨Ioo_subset_Icc_self (hWint r hfW), | |
| (hiff r t ((hs ht).1 r) (f r t) hfW).mpr rfl⟩ | |
| end ErdosLocalFiberGraphs | |
| end | |
| section | |
| open Set | |
| namespace ErdosGlobalFiberBranches | |
| open ErdosImplicitCurvature ErdosLocalFiberGraphs | |
| noncomputable def chartOrderIso {Q : Bivariate} {x L U : ℝ} | |
| (c : FiberCharts Q x L U) (t : ℝ) (ht : t ∈ Ioo c.a c.b) : | |
| boundedFiber Q x L U ≃o boundedFiber Q t L U := by | |
| let f : boundedFiber Q x L U → boundedFiber Q t L U := fun r => | |
| ⟨c.graph r t,by rw [c.coverage t ht]; exact mem_range_self r⟩ | |
| apply orderIsoOfStrictMonoSurjective f | |
| · exact fun r s hrs => (c.ordered t ht) hrs | |
| · intro y | |
| have hy := (c.coverage t ht).subset y.property | |
| obtain ⟨r,hr⟩ := hy | |
| exact ⟨r,Subtype.ext hr⟩ | |
| theorem chartOrderIso_val {Q : Bivariate} {x L U : ℝ} | |
| (c : FiberCharts Q x L U) (t : ℝ) (ht : t ∈ Ioo c.a c.b) | |
| (r : boundedFiber Q x L U) : (chartOrderIso c t ht r : ℝ) = c.graph r t := rfl | |
| end ErdosGlobalFiberBranches | |
| end | |
| section | |
| open Set Filter | |
| open scoped Topology | |
| namespace ErdosLocalFiberGraphs | |
| open ErdosImplicitCurvature ErdosCriticalGeometry | |
| theorem boundedFiber_card_le_degree (Q : Bivariate) (x L U z : ℝ) | |
| (hz : peval Q x z ≠ 0) : (boundedFiber Q x L U).ncard ≤ Q.totalDegree := by | |
| classical | |
| have hf := boundedFiber_finite Q x L U z hz | |
| have hp : verticalPolynomial Q x ≠ 0 := by | |
| intro he | |
| apply hz | |
| rw [← verticalPolynomial_eval,he,Polynomial.eval_zero] | |
| calc | |
| (boundedFiber Q x L U).ncard = hf.toFinset.card := ncard_eq_toFinset_card _ hf | |
| _ ≤ (verticalPolynomial Q x).natDegree := Polynomial.card_le_degree_of_subset_roots (by | |
| intro y hy | |
| apply (Polynomial.mem_roots hp).mpr | |
| change (verticalPolynomial Q x).eval y = 0 | |
| exact (verticalPolynomial_eval Q x y).trans (hf.mem_toFinset.mp hy).2) | |
| _ ≤ (nestedY ℝ Q).natDegree := Polynomial.natDegree_map_le | |
| _ ≤ Q.totalDegree := nestedY_degree_le ℝ Q | |
| namespace FiberCharts | |
| variable {Q : Bivariate} {x L U : ℝ} (c : FiberCharts Q x L U) | |
| theorem fiber_finite {t : ℝ} (ht : t ∈ Ioo c.a c.b) : | |
| (boundedFiber Q t L U).Finite := by | |
| let := c.finite_base.fintype | |
| rw [c.coverage t ht] | |
| exact finite_range _ | |
| theorem fiber_card_eq {t : ℝ} (ht : t ∈ Ioo c.a c.b) : | |
| (boundedFiber Q t L U).ncard = (boundedFiber Q x L U).ncard := by | |
| rw [c.coverage t ht,ncard_range_of_injective (c.ordered t ht).injective, | |
| Nat.card_coe_set_eq] | |
| theorem unique_root_in_tube (r : boundedFiber Q x L U) {t : ℝ} | |
| (ht : t ∈ Ioo c.a c.b) : | |
| ∃! y, y ∈ c.tube r ∧ peval Q t y = 0 := by | |
| refine ⟨c.graph r t,⟨c.graph_inside r t ht,?_⟩,?_⟩ | |
| · exact (c.zero_iff r t ht _ (c.graph_inside r t ht)).mpr rfl | |
| · exact fun y hy => (c.zero_iff r t ht y hy.1).mp hy.2 | |
| theorem all_roots_regular {t : ℝ} (ht : t ∈ Ioo c.a c.b) : | |
| ∀ y ∈ boundedFiber Q t L U, peval (partialY Q) t y ≠ 0 := by | |
| intro y hy | |
| rw [c.coverage t ht] at hy | |
| obtain ⟨r,rfl⟩ := hy | |
| exact c.graph_regular r t ht | |
| end FiberCharts | |
| theorem locally_constant_fiber_card (Q : Bivariate) (x L U : ℝ) | |
| (hL : peval Q x L ≠ 0) (hU : peval Q x U ≠ 0) | |
| (hregular : ∀ y ∈ boundedFiber Q x L U, peval (partialY Q) x y ≠ 0) : | |
| ∃ a b : ℝ, a < x ∧ x < b ∧ ∀ t ∈ Ioo a b, | |
| (boundedFiber Q t L U).Finite ∧ | |
| (boundedFiber Q t L U).ncard = (boundedFiber Q x L U).ncard := by | |
| obtain ⟨c⟩ := exists_bounded_fiber_charts Q x L U hL hU hregular | |
| exact ⟨c.a,c.b,c.left,c.right,fun t ht => ⟨c.fiber_finite ht,c.fiber_card_eq ht⟩⟩ | |
| end ErdosLocalFiberGraphs | |
| end | |
| section | |
| open Set Filter | |
| open scoped Topology | |
| namespace ErdosLocalFiberGraphs | |
| open ErdosImplicitCurvature ErdosDivisibleInflection | |
| def regularParameters (Q : Bivariate) (L U : ℝ) : Set ℝ := | |
| {x | peval Q x L ≠ 0 ∧ peval Q x U ≠ 0 ∧ | |
| ∀ y ∈ boundedFiber Q x L U, peval (partialY Q) x y ≠ 0} | |
| theorem continuous_horizontal_eval (Q : Bivariate) (y : ℝ) : | |
| Continuous (fun t => peval Q t y) := by | |
| change Continuous (planeEval Q ∘ (fun t : ℝ => (t,y))) | |
| exact (contDiff_planeEval Q 0).continuous.comp (continuous_id.prodMk continuous_const) | |
| theorem regularParameters_isOpen (Q : Bivariate) (L U : ℝ) : | |
| IsOpen (regularParameters Q L U) := by | |
| apply isOpen_iff_mem_nhds.mpr | |
| intro x hx | |
| obtain ⟨c⟩ := exists_bounded_fiber_charts Q x L U hx.1 hx.2.1 hx.2.2 | |
| have hleft := (continuous_horizontal_eval Q L).continuousAt.eventually_ne hx.1 | |
| have hright := (continuous_horizontal_eval Q U).continuousAt.eventually_ne hx.2.1 | |
| filter_upwards [hleft,hright,isOpen_Ioo.mem_nhds ⟨c.left,c.right⟩] with t htL htU ht | |
| exact ⟨htL,htU,c.all_roots_regular ht⟩ | |
| theorem fiber_card_eventually_eq (Q : Bivariate) (L U : ℝ) {x : ℝ} | |
| (hx : x ∈ regularParameters Q L U) : | |
| ∀ᶠ t in 𝓝 x, (boundedFiber Q t L U).ncard = (boundedFiber Q x L U).ncard := by | |
| obtain ⟨a,b,ha,hb,h⟩ := locally_constant_fiber_card Q x L U hx.1 hx.2.1 hx.2.2 | |
| filter_upwards [isOpen_Ioo.mem_nhds ⟨ha,hb⟩] with t ht | |
| exact (h t ht).2 | |
| end ErdosLocalFiberGraphs | |
| end | |
| section | |
| open Set Filter | |
| open scoped Topology | |
| namespace ErdosGlobalFiberBranches | |
| open ErdosImplicitCurvature ErdosLocalFiberGraphs | |
| theorem continuousOn_fiber_card (Q : Bivariate) (L U : ℝ) (I : Set ℝ) | |
| (hI : I ⊆ regularParameters Q L U) : | |
| ContinuousOn (fun x => (boundedFiber Q x L U).ncard) I := by | |
| intro x hx | |
| have he : (fun t => (boundedFiber Q t L U).ncard) =ᶠ[𝓝 x] | |
| (fun _ => (boundedFiber Q x L U).ncard) := fiber_card_eventually_eq Q L U (hI hx) | |
| have hc : ContinuousAt (fun t => (boundedFiber Q t L U).ncard) x := | |
| continuousAt_const.congr he.symm | |
| exact hc.continuousWithinAt | |
| theorem fiber_card_eq_on_preconnected (Q : Bivariate) (L U : ℝ) (I : Set ℝ) | |
| (hconn : IsPreconnected I) (hI : I ⊆ regularParameters Q L U) | |
| {x t : ℝ} (hx : x ∈ I) (ht : t ∈ I) : | |
| (boundedFiber Q x L U).ncard = (boundedFiber Q t L U).ncard := | |
| hconn.constant (continuousOn_fiber_card Q L U I hI) hx ht | |
| end ErdosGlobalFiberBranches | |
| end | |
| end Erdos970 | |
| end OAI | |