| import OAI.Combinatorics.HadwigerCounterexample.UnseenCrossingFactorization |
| import OAI.Combinatorics.HadwigerCounterexample.GramConditionedLaws |
|
|
| namespace OAI |
|
|
| /-! # Conditional Four Corners -/ |
|
|
| noncomputable section |
| namespace HadwigerCounterexample |
| open Finset Matrix Module LinearMap Function Filter |
| open scoped BigOperators |
| variable {p : Shape} {n N : ℕ} {I J : Type} [Fintype I] [DecidableEq I] |
| set_option maxHeartbeats 600000 in |
| private theorem stip_mask (E : Rail I) (t : Vec E) (p : Wgt (Rack I N)) (δ : ℝ) |
| (hδ : 0≤δ) (hh : |avg unif p|≤δ) (h : ∀ f:Vec E,f≠0 → |
| |avg unif (fun i=>phasedBy (ledgerM E f) i*p i)|≤bp (-(N:ℝ)/2)) : |
| |partMean (fun x=>traced x E t) p|≤δ+bp (-(N:ℝ)/2) := by |
| classical |
| have he (f : Vec E) : |
| |vectorPhase f t*avg unif (fun i : Rack I N=>phasedBy (ledgerM E f) i*p i)|≤ δ+bp (-(N:ℝ)/2) := by |
| rw [abs_mul,vectorPhase,wsign_abs,one_mul] |
| by_cases hz : f=0 |
| · subst f; simpa only [ledger_zero,layer_zero,one_mul] using hh.trans (le_add_of_nonneg_right (bp_pos (-(N:ℝ)/2)).le) |
| exact (h f hz).trans (le_add_of_nonneg_left hδ) |
| have hf : partMean (fun x=>traced x E t) p=avg unif (fun f:Vec E => vectorPhase f t*avg unif |
| (fun i:Rack I N=>phasedBy (ledgerM E f) i*p i)) := by |
| unfold partMean; simp_rw [rail_expand,←avg_mul] |
| rw [avg_avg]; simp_rw [←avg_mul']; exact avg_congr (fun i _=>avg_congr fun j _=>by ring) |
| rw [hf] |
| exact (abs_avg_le unif_law.nz).trans (unif_law.avg_le he) |
| namespace Template |
| variable (v : Template p I) |
|
|
| theorem pack_bound (v : Template p I) (j : I) (U : Wgt (Keys (p:=p) v.t N)) |
| (hU : ∀ i, |U i|≤1) (i) : |packSecs v j U i| ≤1 := by |
| unfold packSecs; rw [Finset.abs_prod] |
| exact (Finset.prod_le_prod₀ (fun i _=>abs_nonneg _) (fun i _ => hU _)).trans_eq (by simp) |
|
|
| set_option maxHeartbeats 800000 in |
| theorem lopsided (v : Template p I) (j : I) (q : I→F) |
| (U : Wgt (Keys (p:=p) v.t N)) (δ : ℝ) (hδ : 0≤δ) |
| (hU : ∀ x,|U x|≤1) (h : chairCut v j U δ) |
| (hW : carried U (ownPart (p:=p) v.t q)) |
| (Z : GeoM v.seam) : |
| |partMean (gramPart Z) (packSecs v j U)|≤δ+bp (-(N:ℝ)/2) := by |
| classical |
| let P := packSecs v j U |
| let Q := gramPart Z (N:=N) |
| have hb := bp_pos (-(N:ℝ)/2) |
| by_cases hp : ∃ x,Q x∧P x≠0 |
| · obtain ⟨x,hx,hP⟩ := hp |
| let E := cantRail v j |
| let T (y : v.PackStack N) := traced y E (cantVec v j Z) |
| have hg i : ind (Q i)*P i=ind (T i)*P i := by |
| by_cases hh : P i=0 |
| · rw [hh,mul_zero,mul_zero] |
| rw [show Q i=T i from propext |
| (cant_bridge v j q Z x i (pack_gate v j q U hW hP) (pack_gate v j q U hW hh) hx)] |
| change |avg unif (fun x => ind (Q x)*P x)|≤_ |
| simp_rw [hg] |
| exact stip_mask E (cantVec v j Z) P δ hδ (global_quart v j U hU δ h) (cant_leak v j U hU) |
| have hh (x) : ind (Q x)*P x=0 := by |
| by_cases he : Q x <;> simp only [he,ind_true,ind_false,one_mul,zero_mul] |
| exact not_ne_iff.mp (fun hx => hp ⟨x,he,hx⟩) |
| change |avg unif (fun x=>ind (Q x)*P x)|≤_ |
| simp_rw [hh,avg_zero,abs_zero] |
| exact add_nonneg hδ hb.le |
|
|
| set_option maxHeartbeats 800000 in |
| theorem pack_hot (v : Template p I) (j : I) (q : I→F) (U : Wgt (Keys (p:=p) v.t N)) |
| (δ : ℝ) (hδ : 0≤δ) (hU : ∀ x, |U x|≤1) (h : chairCut v j U δ) |
| (hW : carried U (ownPart (p:=p) v.t q)) |
| (Z : GeoM v.seam) (k : ℝ) (hk : ∀ i,num (Port v.seam i)≤ k) |
| (hN : 2*k-(N:ℝ) ≤ -1) : |
| let W := netFact v.seam |
| |avg (rackStation Z N) (packSecs v j U)|≤ |
| 2*W*(num (Ports p)*(2*bp (k-N)))+W*(δ+bp (-(N:ℝ)/2)) := by |
| classical |
| intro W |
| let B := pr unif (gramPart Z (N:=N)) |
| have hb := wave_norm Z N (fun i => by linarith only [hN,hk i]) |
| have h0 : 0<W := netFact_pos |
| have he : 0<B := (inv_pos_of_pos h0).trans_le hb |
| let V := packSecs v j U |
| let D := avg (rackWave Z N) V |
| have hl := lopsided v j q U δ hδ hU h hW Z |
| rw [part_reset _ _ he,abs_mul,abs_of_nonneg he.le] at hl |
| have hh : |
| |D| ≤ W*(δ+bp (-(N:ℝ)/2)) := by |
| apply le_of_mul_le_mul_left _ (inv_pos_of_pos h0) |
| exact (le_trans (mul_le_mul_of_nonneg_right hb (abs_nonneg _)) hl).trans_eq (by field_simp) |
| let A := avg (rackStation Z N) V |
| have hs := compare_station Z V (pack_bound v j U hU) k hk hN |
| change |A|≤_ |
| exact le_trans (by simpa only [sub_zero] using (abs_sub_le A D 0)) (add_le_add hs hh) |
| end Template |
| end HadwigerCounterexample |
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| end |
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| end OAI |
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