| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Amenability.BarrierCore |
| |
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section BarrierSignalSystem |
| open Classical Set |
|
|
| structure BarrierSignalSystem (a : ℕ) (ha : 0 < a) (Q B : ℝ) where |
| R₀ : ℝ |
| R₁ : ℝ |
| R₀_pos : 0 < R₀ |
| gap : R₀+1 < R₁ |
| first : FieldSignalData |
| second : Fin 4 → FieldSignalData |
| first_q : first.q=-3 |
| second_q : ∀j,(second j).q=-3 |
| first_pos : ∀x,‖x‖ ≤ R₀ → first.signal x=3 |
| first_neg : ∀x,R₁ ≤ ‖x‖ → first.signal x=-3 |
| second_pos : ∀j x,‖x‖ ≤ R₁ → |conjugateLineForm a j x| ≤ 1 → (second j).signal x=3 |
| second_neg : ∀j x,3 ≤ |conjugateLineForm a j x| → (second j).signal x=-3 |
| intersection_core : ∀N : ℝ,1 ≤ N → ∀i j : Fin 4,i ≠ j → ∀c d : CutRing, |
| |conjugate c| ≤ 4*N+Q → |conjugate d| ≤ B → ∀p : ℝ×ℝ, |
| conjugateLineForm a i p=conjugate c → conjugateLineForm a j p=conjugate d → ‖p‖ ≤ R₀*N |
| incoming_core : ∀m : ℕ,∀i : Fin m,∀N : ℝ,1 ≤ N → ∀j : Fin 4,∀c : CutRing, |
| ∀hc : c ∈ barrierCandidate a j N,‖flagSiteConjugate (barrierIncomingSite ha i hc)‖ ≤ R₀*N |
|
|
| theorem exists_barrierSignalSystem {a : ℕ} (ha : 0 < a) (Q B : ℝ) : |
| Nonempty (BarrierSignalSystem a ha Q B) := by |
| obtain ⟨R,hR,hcore⟩ := exists_barrierIntersection_core ha Q B |
| obtain ⟨T,hT,hin⟩ := exists_barrierIncoming_core ha |
| let R₀ := R+T+1 |
| let R₁ := R₀+2 |
| have hR₀ : 0 < R₀ := by dsimp [R₀] |
| have hR₁ : 0 < R₁ := by dsimp [R₁] |
| obtain ⟨first,hq,hp,hn⟩ := exists_firstBarrierSignal hR₀ (show R₀ < R₁ by dsimp [R₁] |
| choose second hsq hsp hsn using fun j : Fin 4 => exists_secondBarrierSignal (conjugateLineForm a j) hR₁ |
| refine ⟨⟨R₀,R₁,hR₀,by dsimp [R₁] |
| · intro N hN i j hij c d hc hd p hi hj |
| exact (hcore N hN i j hij c d hc hd p hi hj).trans |
| (mul_le_mul_of_nonneg_right (by dsimp [R₀] |
| · intro m i N hN j c hc |
| exact (hin m i N hN j c hc).trans |
| (mul_le_mul_of_nonneg_right (by dsimp [R₀] |
|
|
| namespace BarrierSignalSystem |
| variable {a : ℕ} {ha : 0 < a} {Q B : ℝ} (S : BarrierSignalSystem a ha Q B) |
| |
| theorem R₁_pos : 0 < S.R₁ := by linarith [S.R₀_pos,S.gap] |
|
|
| theorem incoming_first_true {m : ℕ} (i : Fin m) {N : ℝ} (hN : 1 ≤ N) {j : Fin 4} |
| {c : CutRing} (hc : c ∈ barrierCandidate a j N) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (hs : signalSuccess (flagMeanSignal S.first.signal N) b) : b (barrierIncomingSite ha i hc)=true := by |
| apply (hs _).1 |
| rw [flagMeanSignal,S.first_pos] |
| · norm_num |
| · rw [norm_scaledSiteConjugate (by linarith : 0 < N),div_le_iff₀ (by linarith : 0 < N)] |
| exact S.incoming_core m i N hN j c hc |
|
|
| noncomputable def marks {m : ℕ} {N : ℝ} (hN : 0 < N) (j : Fin 4) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) := |
| finiteFirstMarks (commonVertexDenominator_pos ha) hN S.R₁_pos b |
| |
| theorem mem_marks_iff {m : ℕ} {N : ℝ} (hN : 0 < N) (j : Fin 4) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (hs : signalSuccess (flagMeanSignal S.first.signal N) b) (z) : |
| z ∈ S.marks hN j b ↔ b z=true := |
| mem_finiteFirstMarks_iff (commonVertexDenominator_pos ha) hN S.R₁_pos S.first.signal b S.first_neg hs z |
|
|
| theorem intersection_first_true {m : ℕ} {N : ℝ} (hN : 1 ≤ N) {i j : Fin 4} (hij : i ≠ j) |
| {c d : CutRing} (hc : |conjugate c| ≤ 4*N+Q) (hd : |conjugate d| ≤ B) |
| (z : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha i)) |
| (hz : cutForm a i (barrierSitePoint z)=ordinary c) |
| (hzd : cutForm a j (barrierSitePoint z)=ordinary d) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha i) → Bool) |
| (hs : signalSuccess (flagMeanSignal S.first.signal N) b) : b z=true := by |
| apply (hs z).1 |
| rw [flagMeanSignal,S.first_pos] |
| · norm_num |
| · rw [norm_scaledSiteConjugate (by linarith : 0 < N),div_le_iff₀ (by linarith : 0 < N)] |
| exact S.intersection_core N hN i j hij c d hc hd _ |
| (conjugateLineForm_site (commonVertexDenominator_pos ha) z i c hz) |
| (conjugateLineForm_site (commonVertexDenominator_pos ha) z j d hzd) |
|
|
| theorem central_line_barrier {m : ℕ} (i : Fin m) {N : ℝ} (hN : 1 ≤ N) |
| (b₁ b₂ : ∀j,FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) |
| (hs₁ : ∀j,signalSuccess (flagMeanSignal S.first.signal N) (b₁ j)) |
| (hs₂ : ∀j,signalSuccess (flagMeanSignal (S.second j).signal N) (b₂ j)) |
| {j : Fin 4} {c : CutRing} (hc : c ∈ barrierCandidate a j N) (hband : |conjugate c| ≤ N) : |
| ∀p ∈ squareInterior,cutForm a j p=ordinary c → |
| p ∈ sampledBarriers ha i N (fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂ := by |
| have hN₀ : 0 < N := by linarith |
| apply sampledBarriers_contains_line ha i _ b₂ hc |
| intro z hzt hzline hzstart |
| have hzfirst : b₁ j z=true := by |
| rcases hzstart with hz | hz |
| · have he : z=barrierIncomingSite ha i hc := |
| flagSite_eq_of_track_point hzt hz |
| rw [he] |
| exact S.incoming_first_true i hN hc (b₁ j) (hs₁ j) |
| · exact (S.mem_marks_iff hN₀ j (b₁ j) (hs₁ j) z).mp hz |
| apply (hs₂ j z).1 |
| rw [flagMeanSignal,S.second_pos] |
| · norm_num |
| · exact (firstSignal_true_norm S.first.signal (b₁ j) S.first_neg (hs₁ j) z hzfirst).le |
| · rw [conjugateLineForm_scaledSite (commonVertexDenominator_pos ha) z j c N hzline, |
| abs_div,abs_of_pos hN₀,div_le_one hN₀] |
| exact hband |
| |
| end BarrierSignalSystem |
| end BarrierSignalSystem |
| |
| section SubdivisionCovariance |
| open Classical Set |
| |
| def subdivisionBarrier (E : Finset ℝ) (b : ℝ → Bool) : Set ℝ := |
| {x | ∃s t,ConsecutiveSubdivision E s t ∧ b s=true ∧ x ∈ Icc s t} |
|
|
| theorem consecutiveSubdivision_local {E : Finset ℝ} {L U x s t : ℝ} |
| (hL : L ∈ E) (hU : U ∈ E) (hx : x ∈ Ioo L U) |
| (hst : ConsecutiveSubdivision E s t) (hxi : x ∈ Icc s t) : L ≤ s ∧ t ≤ U := by |
| constructor |
| · by_contra h |
| have hh := hst.2.2.2 L hL (lt_of_not_ge h) |
| have hlx := hx.1 |
| have hxt := hxi.2 |
| linarith |
| · exact hst.2.2.2 U hU (lt_of_le_of_lt hxi.1 hx.2) |
|
|
| theorem consecutiveSubdivision_translate {E F : Finset ℝ} {L U s t d : ℝ} |
| (hE : ∀r ∈ Icc L U,r ∈ E ↔ r+d ∈ F) (hs : L ≤ s) (ht : t ≤ U) |
| (hst : ConsecutiveSubdivision E s t) : ConsecutiveSubdivision F (s+d) (t+d) := by |
| have hsU : s ≤ U := hst.2.2.1.le.trans ht |
| have hLt : L ≤ t := hs.trans hst.2.2.1.le |
| refine ⟨(hE s ⟨hs,hsU⟩).mp hst.1,(hE t ⟨hLt,ht⟩).mp hst.2.1, |
| by linarith [hst.2.2.1],?_⟩ |
| intro r hr hsr |
| by_contra htr |
| have hrU : r-d ≤ U := by linarith |
| have hLr : L ≤ r-d := by linarith |
| have hrE : r-d ∈ E := (hE (r-d) ⟨hLr,hrU⟩).mpr (by simpa using hr) |
| have hh := hst.2.2.2 (r-d) hrE (by linarith) |
| linarith |
|
|
| theorem subdivisionBarrier_translate_iff {E F : Finset ℝ} {b c : ℝ → Bool} |
| {L U d : ℝ} (hL : L ∈ E) (hU : U ∈ E) |
| (hFL : L+d ∈ F) (hFU : U+d ∈ F) |
| (hE : ∀r ∈ Icc L U,r ∈ E ↔ r+d ∈ F) |
| (hb : ∀r ∈ E,r ∈ Ico L U → b r=c (r+d)) {x : ℝ} (hx : x ∈ Ioo L U) : |
| x ∈ subdivisionBarrier E b ↔ x+d ∈ subdivisionBarrier F c := by |
| constructor |
| · rintro ⟨s,t,hst,hbs,hxi⟩ |
| obtain ⟨hs,ht⟩ := consecutiveSubdivision_local hL hU hx hst hxi |
| refine ⟨s+d,t+d,consecutiveSubdivision_translate hE hs ht hst,?_,?_⟩ |
| · rw [←hb s hst.1 ⟨hs,lt_of_lt_of_le hst.2.2.1 ht⟩] |
| exact hbs |
| · exact ⟨by linarith [hxi.1],by linarith [hxi.2]⟩ |
| · rintro ⟨s,t,hst,hcs,hxi⟩ |
| have hx' : x+d ∈ Ioo (L+d) (U+d) := ⟨by linarith [hx.1],by linarith [hx.2]⟩ |
| obtain ⟨hs,ht⟩ := consecutiveSubdivision_local hFL hFU hx' hst hxi |
| have hFinv : ∀r ∈ Icc (L+d) (U+d),r ∈ F ↔ r+(-d) ∈ E := by |
| intro r hr |
| have hr' : r-d ∈ Icc L U := ⟨by linarith [hr.1],by linarith [hr.2]⟩ |
| simpa [sub_eq_add_neg,add_assoc] using (hE (r-d) hr').symm |
| have hst' := consecutiveSubdivision_translate hFinv hs ht hst |
| refine ⟨s+(-d),t+(-d),hst',?_,?_⟩ |
| · rw [hb _ hst'.1 ⟨by linarith,by linarith [hst.2.2.1]⟩] |
| simpa using hcs |
| · exact ⟨by linarith [hxi.1],by linarith [hxi.2]⟩ |
| |
| theorem mem_barrierActiveIntervals_image_iff {a m : ℕ} (ha : 0 < a) (i : Fin m) |
| (j : Fin 4) (c : CutRing) |
| (M : Finset (FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j))) |
| (b : FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool) (x : ℝ) : |
| (∃st ∈ barrierActiveIntervals ha i j c M b,x ∈ Icc st.1 st.2) ↔ |
| x ∈ subdivisionBarrier (barrierSubdivision ha i j c M) (barrierActivation ha i j c b) := by |
| constructor |
| · rintro ⟨⟨s,t⟩,hst,hx⟩ |
| have hh := (Finset.mem_filter.mp hst).2 |
| exact ⟨s,t,hh.1,hh.2,hx⟩ |
| · rintro ⟨s,t,hst,hb,hx⟩ |
| exact ⟨(s,t),Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨hst.1,hst.2.1⟩,hst,hb⟩,hx⟩ |
| |
| end SubdivisionCovariance |
| |
| end SimpleAmenable |
| end |
| end |
| |
| end OAI |
| |
