openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Gluing /CoordinateProspective.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.RelayProbability | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CoordinateScheme | |
| import OAI.Probability.CriticalPercolation.Criticality.GrowthEstimate | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.SourceMissLaw | |
| import OAI.Probability.CriticalPercolation.Gluing.ProspectiveCorridor | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.Corridor.CoordinateData | |
| open BondGraph Boundary BernoulliProduct | |
| variable {V E : Type*} [Countable V] [Countable E] (C : CoordinateData V E) | |
| lemma active_slot_positive (n : ℕ) (S : GrowthState) (hv : S.Valid n) | |
| (ha : (incoming n S.good).isSome) : 0<n := by | |
| obtain ⟨e,he⟩ := Option.isSome_iff_exists.mp ha | |
| have ht := parent_slot_lt e (hv.2.1 _ (incoming_source he)).1 | |
| rw [incoming_target he,gridSlot_slotSite] at ht | |
| omega | |
| theorem prospective_estimate (h : C.Bounds) | |
| (hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove C.time C.height C.m C.e y s)) | |
| (n : ℕ) (S : GrowthState) (hv : S.Valid n) (_hc : S.Covered) | |
| (ha : (incoming n S.good).isSome) (ω : Set E) (b : Bool) : | |
| (blockProbability C.scheme.parameter (C.scheme.block S) | |
| (C.scheme.incomingHit n S\C.scheme.nextHit n S b) ω).toReal≤(C.L+1)*C.kappa := by | |
| let v := slotSite n | |
| let F := C.scheme.enlarged n S | |
| let U := F∪C.scheme.edgeDomain (v,b) | |
| have hdom := C.domain_eq v b | |
| have hDU : C.graph.inducedEdges (relayDomain (localTime C.time C.T v) | |
| (localHeight C.height C.T v) C.J C.T (sign b))⊆U := by | |
| rw [← hdom] | |
| exact Set.subset_union_right | |
| have hfresh : Disjoint (C.scheme.block S) | |
| (C.graph.inducedEdges (relayDomain (localTime C.time C.T v) | |
| (localHeight C.height C.T v) C.J C.T (sign b))) := by | |
| rw [← hdom] | |
| exact outgoing_unrevealed C.graph (normalized C.time C.height C.T) (C.J/C.T) (2520/C.T) | |
| (C.layout h).1 (C.layout h).2.1 (C.layout h).2.2 n S hv (active_slot_positive n S hv ha) rfl | |
| have hmoves : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove (localTime C.time C.T v) (localHeight C.height C.T v) | |
| C.m C.e y s) := by | |
| intro y s hs | |
| rw [coordinateMove_translate] | |
| exact hmove y s hs | |
| have hP : 1≤C.T := by | |
| have hh : (416 : ℝ)≤C.L := by exact_mod_cast C.L_large h | |
| dsimp [T]; linarith | |
| have hh := C.graph.coordinate_pinned_corridor_failure h.connected h.locallyFinite | |
| C.scheme.seed (C.graph.vertexBall_finite h.locallyFinite C.origin C.m) C.scheme.parameter | |
| (C.scheme.block S) F U ω Set.subset_union_left | |
| (localTime C.time C.T v) (localHeight C.height C.T v) (sign b) C.e C.rho C.J C.T | |
| C.L C.R C.m (sign_abs b) h.e_pos h.rho_nonneg h.J_lower (C.L_large h) h.e_small h.rho_small | |
| hP (C.source_behind h n (active_slot_positive n S hv ha)) | |
| (fun {_ _} hh => C.local_time_step h v hh) (fun {_ _} hh => C.local_height_step h v hh) h.R_gt C.pc C.p h.pc_pos h.pc_lt | |
| h.p_lower h.p_upper (fun _ _ => rfl) hDU hfresh C.D C.k C.M C.VM h.degree h.M_pos | |
| h.M_bound h.VM_bound C.kappa C.eta h.kappa_pos h.kappa_le h.eta_nonneg h.eta_small | |
| h.k_large h.R_large hmoves | |
| change (blockProbability C.scheme.parameter (C.scheme.block S) | |
| (C.graph.restrictedSeed C.scheme.seed (relayMark (localTime C.time C.T v) (localHeight C.height C.T v) 0 (sign b) C.rho) F\ | |
| C.graph.restrictedSeed C.scheme.seed (relayMark (localTime C.time C.T v) (localHeight C.height C.T v) C.T (sign b) C.rho) U) ω).toReal≤_ at hh | |
| rw [mark_initial,mark_target] at hh | |
| exact hh | |
| lemma root_corridor_estimate (h : C.Bounds) | |
| (hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove C.time C.height C.m C.e y s)) (b : Bool) : | |
| (BernoulliProduct.law C.scheme.parameter).real | |
| (C.scheme.hit (target ((0,0),b)) | |
| (C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b)))ᶜ≤C.eta+C.L*C.kappa := by | |
| let U := C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b) | |
| have hDU : C.graph.inducedEdges (relayDomain (localTime C.time C.T (0,0)) | |
| (localHeight C.height C.T (0,0)) C.J C.T (sign b))⊆U := by | |
| rw [← C.domain_eq] | |
| exact Set.subset_union_right | |
| have hmoves : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove (localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0)) | |
| C.m C.e y s) := by | |
| intro y s hs | |
| rw [coordinateMove_translate] | |
| exact hmove y s hs | |
| have hh := C.graph.original_root_corridor_miss h.connected h.locallyFinite C.origin | |
| (localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0)) (sign b) C.e C.rho C.J | |
| C.L C.R C.m U hDU (sign_abs b) h.e_pos h.rho_nonneg h.J_lower (C.L_large h) h.e_small h.rho_small | |
| (by simp [localTime,timeCenter,h.time_origin]) (by simp [localHeight,heightCenter,h.height_origin]) | |
| (fun {_ _} hh => C.local_time_step h (0,0) hh) (fun {_ _} hh => C.local_height_step h (0,0) hh) h.R_gt C.pc C.p h.pc_pos h.pc_lt | |
| h.p_lower h.p_upper C.D C.k C.M C.VM h.degree h.M_pos h.M_bound h.VM_bound | |
| C.kappa C.eta h.kappa_pos h.kappa_le h.eta_nonneg h.eta_small h.k_large h.R_large hmoves | |
| change (BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.restrictedSeed (C.graph.vertexBall C.origin C.m) | |
| (relayMark (localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0)) C.T (sign b) C.rho) U)ᶜ≤_ at hh | |
| rw [mark_target] at hh | |
| exact hh | |
| lemma root_positive (h : C.Bounds) | |
| (hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove C.time C.height C.m C.e y s)) : | |
| 0<(BernoulliProduct.law C.scheme.parameter).real C.scheme.rootEvent := by | |
| have he : C.scheme.rootEvent=twoPredictions C.scheme.parameter (C.scheme.block GrowthState.initial) | |
| (C.scheme.hit (target ((0,0),true)) | |
| (C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),true))) | |
| (C.scheme.hit (target ((0,0),false)) | |
| (C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),false))) C.delta := by | |
| ext ω | |
| change (∀ b : Bool, 1-C.delta≤(blockProbability C.scheme.parameter | |
| (C.scheme.block GrowthState.initial) (C.scheme.hit (target ((0,0),b)) | |
| (C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b))) ω).toReal) ↔ (_ ∧ _) | |
| constructor | |
| · intro hh; exact ⟨hh true,hh false⟩ | |
| · rintro ⟨ht,hf⟩ b; cases b <;> assumption | |
| rw [he] | |
| apply twoPredictions_positive _ _ _ _ | |
| (C.graph.restrictedSeed_measurable _ _ _) (C.graph.restrictedSeed_measurable _ _ _) | |
| C.delta (C.eta+C.L*C.kappa) h.delta_pos h.root_error | |
| (C.root_corridor_estimate h hmove true) (C.root_corridor_estimate h hmove false) | |
| theorem moves_force_percolation (h : C.Bounds) | |
| (hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove C.time C.height C.m C.e y s)) : | |
| 0<C.graph.law C.p C.graph.percolates := by | |
| let hf := C.graph.vertexBall_finite h.locallyFinite C.origin C.m | |
| exact C.scheme.exploration_percolates (C.layout h).1 (C.layout h).2.1 (C.layout h).2.2 | |
| hf.toFinset (by simp only [Set.Finite.coe_toFinset]; rfl) (C.root_seed h) (C.marks_disjoint h) | |
| C.p rfl (C.root_positive h hmove) h.delta_pos ((C.L+1)*C.kappa) C.contour | |
| h.contour_nonneg h.contour_small (mul_nonneg (by positivity) h.kappa_pos.le) h.test_error (C.prospective_estimate h hmove) | |
| theorem moves_impossible_below_criticality (h : C.Bounds) | |
| (hp : C.p<C.graph.criticalProbability) | |
| (hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real | |
| (C.graph.coordinateMove C.time C.height C.m C.e y s)) : False := | |
| (not_le_of_gt hp) (C.graph.criticalProbability_le_of_pos (C.moves_force_percolation h hmove)) | |
| end CriticalPercolation.Corridor.CoordinateData | |
| end | |
| end | |
| end | |
| end OAI | |