openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /CoordinateScheme.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.GrowthProcess | |
| import OAI.Probability.CriticalPercolation.Gluing.CorridorNormalization | |
| import OAI.Probability.CriticalPercolation.Gluing.RelayScales | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.Corridor | |
| open BondGraph Boundary BernoulliProduct | |
| structure CoordinateData (V E : Type*) where | |
| graph : BondGraph V E | |
| origin : V | |
| time : V → ℝ | |
| height : V → ℝ | |
| J : ℝ | |
| L : ℕ | |
| R : ℕ | |
| m : ℕ | |
| D : ℕ | |
| k : ℕ | |
| M : ℕ | |
| VM : ℕ | |
| e : ℝ | |
| rho : ℝ | |
| kappa : ℝ | |
| eta : ℝ | |
| delta : ℝ | |
| contour : ℝ | |
| pc : unitInterval | |
| p : unitInterval | |
| namespace CoordinateData | |
| variable {V E : Type*} (C : CoordinateData V E) | |
| def T : ℝ := C.L+1/2 | |
| def scheme : Scheme V E where | |
| graph := C.graph | |
| coordinate := normalized C.time C.height C.T | |
| j := C.J/C.T | |
| h := 2520/C.T | |
| seed := C.graph.vertexBall C.origin C.m | |
| mark := Corridor.mark C.time C.height C.T C.rho | |
| parameter := fun _ => C.p | |
| delta := C.delta | |
| structure Bounds : Prop where | |
| connected : C.graph.fullGraph.Connected | |
| locallyFinite : C.graph.LocallyFinite | |
| time_origin : C.time C.origin=0 | |
| height_origin : C.height C.origin=0 | |
| time_step : ∀ ⦃u v⦄, C.graph.fullGraph.Adj u v → |C.time u-C.time v|≤C.rho | |
| height_step : ∀ ⦃u v⦄, C.graph.fullGraph.Adj u v → |C.height u-C.height v|≤C.rho | |
| J_lower : 26200<C.J | |
| L_lower : 10*(C.J+210)<C.L | |
| e_pos : 0<C.e | |
| rho_nonneg : 0≤C.rho | |
| e_small : (C.L+3)*C.e<1/100 | |
| rho_small : (C.R+C.m+1)*C.rho<C.e/10 | |
| R_gt : C.m<C.R | |
| pc_pos : 0<(C.pc : ℝ) | |
| pc_lt : (C.pc : ℝ)<1 | |
| p_lower : (C.pc : ℝ)/2≤C.p | |
| p_upper : (C.p : ℝ)≤C.pc | |
| degree : ∀ v, (C.graph.incidentEdges {v}).ncard≤C.D | |
| M_pos : 0<C.M | |
| M_bound : ∀ v, (C.graph.vertexBall v (4*C.m+2)).ncard≤C.M | |
| VM_bound : ∀ v, (C.graph.vertexBall v C.m).ncard≤C.VM | |
| kappa_pos : 0<C.kappa | |
| kappa_le : C.kappa≤1 | |
| eta_nonneg : 0≤C.eta | |
| eta_small : C.eta≤C.kappa*(C.kappa/4)/(4*(Real.log (4/C.kappa)+1)) | |
| k_large : C.M*⌈Real.log (4/C.kappa)/((C.pc : ℝ)/2)^(C.m+1+C.VM)⌉₊≤C.k | |
| R_large : 4/(C.kappa*(1-(C.pc : ℝ))^(C.D*C.k))<(C.R : ℝ)-C.m | |
| delta_pos : 0<C.delta | |
| root_error : 2*(C.eta+C.L*C.kappa)<C.delta | |
| contour_nonneg : 0≤C.contour | |
| contour_small : 8*C.contour<1 | |
| test_error : C.delta+2*((C.L+1)*C.kappa)/C.delta≤C.contour^4 | |
| lemma T_pos : 0<C.T := by unfold T; positivity | |
| lemma L_large (h : C.Bounds) : 416≤C.L := by | |
| have : (416 : ℝ)≤C.L := by linarith [h.J_lower,h.L_lower] | |
| exact_mod_cast this | |
| lemma small_rho (h : C.Bounds) : C.rho≤C.J := by | |
| have he := h.e_small | |
| have hr := h.rho_small | |
| have hle : C.rho≤C.e := (relay_coordinate_scale C.L C.R C.m C.e C.rho | |
| h.e_pos h.rho_nonneg he hr).1 | |
| have hLp : 0≤(C.L : ℝ)*C.e := mul_nonneg (Nat.cast_nonneg _) h.e_pos.le | |
| linarith [h.J_lower] | |
| lemma layout (h : C.Bounds) : | |
| 0≤C.scheme.h ∧ C.scheme.h≤C.scheme.j ∧ 2*C.scheme.j+C.scheme.h<1 := by | |
| change 0≤2520/C.T ∧ 2520/C.T≤C.J/C.T ∧ 2*(C.J/C.T)+2520/C.T<1 | |
| refine ⟨div_nonneg (by norm_num) C.T_pos.le, (div_le_div_iff_of_pos_right C.T_pos).mpr (by linarith [h.J_lower]),?_⟩ | |
| rw [show 2*(C.J/C.T)+2520/C.T=(2*C.J+2520)/C.T by ring,div_lt_one C.T_pos] | |
| dsimp [T] | |
| linarith [h.J_lower,h.L_lower] | |
| lemma root_seed (h : C.Bounds) : (C.scheme.seed∩C.scheme.mark (0,0)).Nonempty := by | |
| refine ⟨C.origin,C.graph.mem_vertexBall_self C.origin C.m,?_⟩ | |
| change 0≤localTime C.time C.T (0,0) C.origin ∧ | |
| localTime C.time C.T (0,0) C.origin≤C.rho ∧ |localHeight C.height C.T (0,0) C.origin|≤210 | |
| simp [localTime,localHeight,timeCenter,heightCenter,h.time_origin,h.height_origin,h.rho_nonneg] | |
| lemma marks_disjoint (h : C.Bounds) : Pairwise (Disjoint on C.scheme.mark) := by | |
| intro v w hvw | |
| exact (graph_boxes_disjoint (normalized C.time C.height C.T) (C.J/C.T) (2520/C.T) | |
| (C.layout h).2.2 hvw).mono | |
| (mark_subset_box C.time C.height C.T C.J C.rho C.T_pos (by linarith [h.J_lower]) (C.small_rho h) v) | |
| (mark_subset_box C.time C.height C.T C.J C.rho C.T_pos (by linarith [h.J_lower]) (C.small_rho h) w) | |
| lemma domain_eq (v : Site) (b : Bool) : C.scheme.edgeDomain (v,b)= | |
| C.graph.inducedEdges (relayDomain (localTime C.time C.T v) (localHeight C.height C.T v) | |
| C.J C.T (sign b)) := by | |
| change C.graph.inducedEdges (graphDomain (normalized C.time C.height C.T) (C.J/C.T) (2520/C.T) (v,b))=_ | |
| rw [normalized_domain C.time C.height C.T C.J C.T_pos] | |
| lemma local_time_step (h : C.Bounds) (v : Site) {u w : V} (huw : C.graph.fullGraph.Adj u w) : | |
| |localTime C.time C.T v u-localTime C.time C.T v w|≤C.rho := by | |
| simpa only [localTime,sub_sub_sub_cancel_right] using h.time_step huw | |
| lemma local_height_step (h : C.Bounds) (v : Site) {u w : V} (huw : C.graph.fullGraph.Adj u w) : | |
| |localHeight C.height C.T v u-localHeight C.height C.T v w|≤C.rho := by | |
| simpa only [localHeight,sub_sub_sub_cancel_right] using h.height_step huw | |
| lemma source_behind (h : C.Bounds) (n : ℕ) (hn : 0<n) : | |
| ∀ a∈C.scheme.seed, localTime C.time C.T (slotSite n) a≤C.m*C.rho-C.T := by | |
| intro a ha | |
| have ht := (abs_le.mp (C.graph.coordinate_ball_bound C.time C.rho h.rho_nonneg h.time_step ha)).2 | |
| rw [h.time_origin,sub_zero] at ht | |
| have hv := slotSite_quadrant n | |
| have hnz : slotSite n≠(0,0) := by | |
| intro he | |
| have he' := congrArg gridSlot he | |
| rw [gridSlot_slotSite] at he' | |
| change n=0 at he' | |
| omega | |
| have hsumInt : 1≤(slotSite n).1+(slotSite n).2 := by | |
| by_contra hs | |
| have hx : (slotSite n).1=0 := by dsimp [quadrant] at hv; omega | |
| have hy : (slotSite n).2=0 := by dsimp [quadrant] at hv; omega | |
| exact hnz (Prod.ext hx hy) | |
| have hsum : (1 : ℝ)≤((slotSite n).1 : ℝ)+(slotSite n).2 := by exact_mod_cast hsumInt | |
| have hmul := mul_le_mul_of_nonneg_right hsum C.T_pos.le | |
| change C.time a-(((slotSite n).1 : ℝ)+(slotSite n).2)*C.T≤_ | |
| linarith | |
| end CoordinateData | |
| end CriticalPercolation.Corridor | |
| end | |
| end | |
| end | |
| end OAI | |