openAI-math-dataset / lean /OAI /Combinatorics /PolynomialVacuum /Bridge /VisitNestingComparison.lean
| import OAI.Combinatorics.HoneycombCorridor.VisitRecursion | |
| import OAI.Combinatorics.PolynomialVacuum.Geometry.IncidentNesting | |
| namespace OAI | |
| namespace PolynomialVacuumBridge.Corridor | |
| theorem Polygon.faceCount_cases (P : Polygon) (v : Vertex) (hv : v ∈ P.vertices) : | |
| faceCount P.interiorFaces v = 1 ∨ faceCount P.interiorFaces v = 2 := by | |
| obtain ⟨k,w,hw⟩ := P.exists_rooted v hv | |
| obtain ⟨b,u,rfl⟩ := List.exists_cons_of_ne_nil hw.1 | |
| obtain ⟨p,ε,hp,hpne,hε,hvals⟩ := simple_closed_word_indicator (v,k) (b :: u) | |
| (by simp) hw.2.1 hw.2.2.1 | |
| have hs : p.support = P.interiorFaces := (P.interiorFaces_of_traversal (v,k) (b :: u) hw.2.1 hw.2.2.1 hw.2.2.2 p hp).symm | |
| have ht := contour_faceCount_turn p ε hε hvals _ hp (v,k) b | |
| ((simple_closed_word_contour _ _ hw.2.1 hw.2.2.1).1) | |
| rw [hs] at ht | |
| rcases hε with rfl | rfl <;> cases b <;> simp [turnUnit] at ht <;> omega | |
| theorem through_power_ratio (x c m : ℝ) (n : ℕ) (hx : x ≠ 0) (hm : m ≠ 0) (hn : 0 < n) : | |
| 2 * x ^ (n - 1) * c = (c / (x * m)) * (2 * m * x ^ n) := by | |
| have he : x ^ n = x ^ (n - 1) * x := by | |
| rw [← pow_succ] | |
| congr 1 | |
| omega | |
| rw [he] | |
| field_simp | |
| noncomputable def lowThroughRatio : ℝ := Real.cos (3 * Real.pi / 8) / criticalActivity | |
| noncomputable def highThroughRatio : ℝ := Real.cos (Real.pi / 8) / (2 * criticalActivity) | |
| theorem lowThroughRatio_pos : 0 < lowThroughRatio := by | |
| apply div_pos _ criticalActivity_pos | |
| apply Real.cos_pos_of_mem_Ioo | |
| have hpi := Real.pi_pos | |
| constructor <;> linarith | |
| theorem highThroughRatio_pos : 0 < highThroughRatio := | |
| div_pos critical_cos_pos (mul_pos (by norm_num) criticalActivity_pos) | |
| theorem through_coefficient_two_ratios (P : Polygon) (v : Vertex) (hv : v ∈ P.vertices) : | |
| ∃ r : ℝ, (r = lowThroughRatio ∨ r = highThroughRatio) ∧ | |
| 2 * criticalActivity ^ (P.length - 1) * | |
| (-Real.cos ((3 / 8 : ℝ) * (2 * Real.pi - P.ccwTurn v))) = | |
| r * (2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length) := by | |
| have hx := ne_of_gt criticalActivity_pos | |
| rcases P.faceCount_cases v hv with hm | hm | |
| · refine ⟨lowThroughRatio,Or.inl rfl,?_⟩ | |
| have ht : P.ccwTurn v = Real.pi / 3 := by simp only [Polygon.ccwTurn,hm,Int.cast_one]; ring | |
| rw [ht,hm] | |
| norm_num only [Int.cast_one,mul_one] | |
| rw [show (3 / 8 : ℝ) * (2 * Real.pi - Real.pi / 3) = Real.pi - 3 * Real.pi / 8 by ring, | |
| Real.cos_pi_sub,neg_neg] | |
| simpa only [mul_one,lowThroughRatio] using | |
| through_power_ratio criticalActivity (Real.cos (3 * Real.pi / 8)) 1 P.length hx one_ne_zero P.length_pos | |
| · refine ⟨highThroughRatio,Or.inr rfl,?_⟩ | |
| have ht : P.ccwTurn v = -(Real.pi / 3) := by simp only [Polygon.ccwTurn,hm,Int.cast_ofNat]; ring | |
| rw [ht,hm] | |
| norm_num only [Int.cast_ofNat] | |
| rw [show (3 / 8 : ℝ) * (2 * Real.pi - -(Real.pi / 3)) = Real.pi - Real.pi / 8 by ring, | |
| Real.cos_pi_sub,neg_neg] | |
| simpa only [highThroughRatio,mul_comm criticalActivity (2 : ℝ), show (2 : ℝ) * 2 = 4 by norm_num] using | |
| through_power_ratio criticalActivity (Real.cos (Real.pi / 8)) 2 P.length hx (by norm_num) P.length_pos | |
| noncomputable def visitLowerConstant : ℝ := min 1 (min lowThroughRatio highThroughRatio) | |
| noncomputable def visitUpperConstant : ℝ := max 1 (max lowThroughRatio highThroughRatio) | |
| theorem visit_constants_positive : 0 < visitLowerConstant ∧ 0 < visitUpperConstant := by | |
| constructor | |
| · exact lt_min (by norm_num) (lt_min lowThroughRatio_pos highThroughRatio_pos) | |
| · exact lt_of_lt_of_le (by norm_num : (0 : ℝ) < 1) (le_max_left _ _) | |
| theorem through_coefficient_bounds (P : Polygon) (v : Vertex) (hv : v ∈ P.vertices) : | |
| visitLowerConstant * (2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length) ≤ | |
| 2 * criticalActivity ^ (P.length - 1) * | |
| (-Real.cos ((3 / 8 : ℝ) * (2 * Real.pi - P.ccwTurn v))) ∧ | |
| 2 * criticalActivity ^ (P.length - 1) * | |
| (-Real.cos ((3 / 8 : ℝ) * (2 * Real.pi - P.ccwTurn v))) ≤ | |
| visitUpperConstant * (2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length) := by | |
| obtain ⟨r,hr,he⟩ := through_coefficient_two_ratios P v hv | |
| have hnon : 0 ≤ 2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length := by | |
| have hm : 0 < (faceCount P.interiorFaces v : ℝ) := by | |
| rcases P.faceCount_cases v hv with h | h <;> norm_num [h] | |
| exact (mul_pos (mul_pos (by norm_num) hm) (pow_pos criticalActivity_pos _)).le | |
| have hc : visitLowerConstant ≤ r ∧ r ≤ visitUpperConstant := by | |
| rcases hr with rfl | rfl | |
| · exact ⟨(min_le_right _ _).trans (min_le_left _ _),(le_max_left _ _).trans (le_max_right _ _)⟩ | |
| · exact ⟨(min_le_right _ _).trans (min_le_right _ _),(le_max_right _ _).trans (le_max_right _ _)⟩ | |
| rw [he] | |
| exact ⟨mul_le_mul_of_nonneg_right hc.1 hnon,mul_le_mul_of_nonneg_right hc.2 hnon⟩ | |
| theorem visitMass_incident_comparison (D : Finset Vertex) (v : Vertex) | |
| (hD : Filled D) (hv : v ∈ D) : | |
| visitLowerConstant * incidentNestingMass D v ≤ visitMass D v ∧ | |
| visitMass D v ≤ visitUpperConstant * incidentNestingMass D v := by | |
| classical | |
| induction hn : D.card using Nat.strong_induction_on generalizing D with | |
| | h n ih => | |
| rw [finite_visit_recursion D v hD hv,incidentNestingMass_recursion D v hD] | |
| simp only [mul_add,Finset.mul_sum] | |
| have hbase : visitLowerConstant * 3 ≤ 3 ∧ 3 ≤ visitUpperConstant * 3 := by | |
| have hl : visitLowerConstant ≤ 1 := min_le_left _ _ | |
| have hu : 1 ≤ visitUpperConstant := le_max_left _ _ | |
| constructor <;> linarith | |
| have hb : | |
| (∑ P ∈ polygons D, visitLowerConstant * (if v ∈ P.vertices then | |
| 2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length else 0)) ≤ | |
| (∑ P ∈ polygons D, if v ∈ P.vertices then | |
| 2 * criticalActivity ^ (P.length - 1) * | |
| (-Real.cos ((3 / 8 : ℝ) * (2 * Real.pi - P.ccwTurn v))) else 0) ∧ | |
| (∑ P ∈ polygons D, if v ∈ P.vertices then | |
| 2 * criticalActivity ^ (P.length - 1) * | |
| (-Real.cos ((3 / 8 : ℝ) * (2 * Real.pi - P.ccwTurn v))) else 0) ≤ | |
| (∑ P ∈ polygons D, visitUpperConstant * (if v ∈ P.vertices then | |
| 2 * (faceCount P.interiorFaces v : ℝ) * criticalActivity ^ P.length else 0)) := by | |
| constructor <;> apply Finset.sum_le_sum <;> intro P hP | |
| · split_ifs with hvP | |
| · exact (through_coefficient_bounds P v hvP).1 | |
| · simp | |
| · split_ifs with hvP | |
| · exact (through_coefficient_bounds P v hvP).2 | |
| · simp | |
| have hint (P : Polygon) (hP : P ∈ polygons D) (hvP : v ∈ P.interior) : | |
| visitLowerConstant * incidentNestingMass P.interior v ≤ visitMass P.interior v ∧ | |
| visitMass P.interior v ≤ visitUpperConstant * incidentNestingMass P.interior v := | |
| ih P.interior.card (hn ▸ P.interior_card_lt D hD hP) P.interior P.interior_filled hvP rfl | |
| have hi : | |
| (∑ P ∈ polygons D, visitLowerConstant * (if v ∈ P.interior then | |
| 2 * criticalActivity ^ P.length * incidentNestingMass P.interior v else 0)) ≤ | |
| (∑ P ∈ polygons D, if v ∈ P.interior then | |
| 2 * criticalActivity ^ P.length * visitMass P.interior v else 0) ∧ | |
| (∑ P ∈ polygons D, if v ∈ P.interior then | |
| 2 * criticalActivity ^ P.length * visitMass P.interior v else 0) ≤ | |
| (∑ P ∈ polygons D, visitUpperConstant * (if v ∈ P.interior then | |
| 2 * criticalActivity ^ P.length * incidentNestingMass P.interior v else 0)) := by | |
| constructor <;> apply Finset.sum_le_sum <;> intro P hP | |
| · split_ifs with hvP | |
| · have h := mul_le_mul_of_nonneg_left (hint P hP hvP).1 (recursion_coefficients_positive P).1.le | |
| nlinarith only [h] | |
| · simp | |
| · split_ifs with hvP | |
| · have h := mul_le_mul_of_nonneg_left (hint P hP hvP).2 (recursion_coefficients_positive P).1.le | |
| nlinarith only [h] | |
| · simp | |
| exact ⟨add_le_add (add_le_add hbase.1 hb.1) hi.1, | |
| add_le_add (add_le_add hbase.2 hb.2) hi.2⟩ | |
| end PolynomialVacuumBridge.Corridor | |
| end OAI | |