| import OAI.Combinatorics.HadwigerCounterexample.PairedMassIdentities |
| import OAI.Combinatorics.HadwigerCounterexample.UnaryPositivityBounds |
| import OAI.Combinatorics.HadwigerCounterexample.RecursiveComparisonErrors |
| import OAI.Combinatorics.HadwigerCounterexample.MixerExistence |
|
|
| namespace OAI |
|
|
| /-! # Analytic Parameters -/ |
|
|
| noncomputable section |
| namespace HadwigerCounterexample |
| open Filter Finset Matrix |
| open scoped BigOperators Topology |
| open StatusAnalysis StatusAnalysis.Station |
|
|
| def frameBills (g K : ℕ) := |
| 2*Fintype.card (Comp g)*K*qorth K |
|
|
| structure AnalyticParameters (p : Shape) (M : ℕ) (b : PreparationParameters) (r : ℕ) : Prop where |
| enter : b.entry p M |
| thickness : 4≤ M |
| profiles : 2*profileA p b.pins < r |
| jump : 1≤p.J |
| bigJ : r≤p.J |
| gleaming : bp (-((p.J-r:ℕ):ℝ))*bp r < glintRate b.pins |
| penRoom : (b.pins+1)*qorth b.pins≤p.j+1 |
| forbid : frameBills p.g b.pins+1000 ≤ b.B |
| spare : 3*b.B < Fintype.card (Label p.b) |
| secure : ∀ᶠ n : ℕ in atTop, Station.earnsScore p n (M*n) b.pins (2*b.ζ) |
|
|
| namespace PreparationParameters |
| variable {b : PreparationParameters} {p : Shape} {N M r : ℕ} {l : Filter ℕ} {m : cycleM p} |
| {s : rhythms p M} {w : tale p M} |
|
|
| def conflicted {n : ℕ} (p : Shape) (m : Mixer p n) (N : ℕ) (w : Wgt (UnitType p n N)) := |
| pr (pairW w w) (clash p m N)<bp (-100*p.g*(N:ℝ)) |
|
|
| structure veil (p : Shape) (M : ℕ) (b : PreparationParameters) (r : ℕ) |
| (l : Filter ℕ) (m : cycleM p) (s : rhythms p M) (w : tale p M) : Prop where |
| easel : AnalyticParameters p M b r |
| timeL : l≤atTop |
| durableS : b.durable l p M m w s |
| moment : ∀ᶠ n in l,b.nascent p (M*n) (w n) (m n) ∧ unaryGood (m n) b.pins r ∧ |
| binaryGood (m n) ∧ conflicted p (m n) (M*n) (w n) |
|
|
| abbrev Stations (p : Shape) (M K : ℕ) (s : rhythms p M) := |
| ∀ n : ℕ, Station p n (M*n) K (s n) |
|
|
| theorem beam_cost (p : Shape) (N : ℕ) (h : 3≤p.g) : |
| beamPrice p N * bp (-100*p.g*(N:ℝ)) ≤ bp (-(N:ℝ)) := by |
| unfold beamPrice; rw [←bp_add]; refine bp_le ?_ |
| have hh : kmax p ≤56*p.g := by unfold kmax; omega |
| have ht : (kmax p:ℝ)≤56*p.g := by exact_mod_cast hh |
| have hg : 3 ≤ (p.g:ℝ) := by exact_mod_cast h |
| apply le_trans ?_ (le_of_eq (neg_one_mul _)) |
| rw [← add_mul] |
| exact mul_le_mul_of_nonneg_right (by linarith only [ht,hg]) (by positivity) |
|
|
| set_option maxHeartbeats 1200000 in |
| theorem veil.score_cost (h : veil p M b r l m s w) |
| (v : Stations p M b.pins s) : |
| ∃ δ : ℕ → ℝ,ExpSmall l (fun n=> ((M*n:ℕ):ℝ)) δ ∧ |
| ∀ᶠ n in l, ∀ (q : Format) (d e : Rates) (x : Typus p q d e) (k : Fin (2^deckMax p b.pins)), |
| |arena (v n) x (m n) (w n) k|≤δ n := by |
| let N:=fun n : ℕ=> M*n; let z:=fun n=>((N n):ℝ) |
| have hq := h.easel |
| have ht := hq.enter.first |
| have hz : Tendsto z l atTop := (trend_sizes l h.timeL ht.hM).2 |
| let C n:= ((s n).amount (w n))⁻¹ |
| have hc := h.durableS.mass.inv_cap |
| let δ n := bp (-1*z n)*C n*C n + bp (-0.005*z n)*2 |
| have hd : ExpSmall l z δ := |
| (((ExpSmall.bp_n zero_lt_one).mul hc).mul hc).add hz |
| ((ExpSmall.bp_n (by norm_num)).mul (expO_const hz _)) |
| refine ⟨δ,hd,?_⟩ |
| filter_upwards [h.moment,h.durableS.mass.positive,h.durableS.shape,hq.secure.filter_mono h.timeL] |
| with n hn hN hA hB q d e x k |
| have hm := hn.1.law |
| rw [abs_of_nonneg (arena_pos _ _ _ _ _ hN hm)] |
| let u:=(s n).sap (w n) |
| refine (arena_bound (v n) x _ _ k hA.1 hm hN hB).trans ?_ |
| have hμ i : 0≤u i := (LeafMixture.sapLaw hN hm).nz _ |
| have hk i : u i ≤ C n*w n i := LeafMixture.upper_sap hm hN i |
| have hv (x:UnitType p n (N n) × UnitType p n (N n)) : |
| pairW u u x ≤ (C n*C n)*pairW (w n) (w n) x := |
| (mul_le_mul (hk _) (hk _) (hμ _) ((hμ _).trans (hk _))).trans_eq (by unfold pairW; ring) |
| have hf : pr (pairW u u) (clash p (m n) (N n)) ≤ |
| (C n*C n)*bp (-100*p.g * (z n)) := (pr_cap hv _).trans |
| (mul_le_mul_of_nonneg_left hn.2.2.2.le (mul_self_nonneg _)) |
| unfold δ |
| rw [mul_comm _ 2] |
| refine add_le_add_left ((mul_le_mul_of_nonneg_left hf (bp_pos _).le).trans ?_) _ |
| calc _ = (beamPrice p (N n) * bp (-100 * p.g * z n))*(C n*C n) := by unfold beamPrice N; ring |
| _ ≤ _ := (mul_le_mul_of_nonneg_right (beam_cost p (N n) ht.hG) (mul_self_nonneg _)).trans_eq |
| (by unfold z; rw [neg_one_mul,mul_assoc]) |
|
|
| structure Align (b : PreparationParameters) (r : ℕ) (p : Shape) (M n : ℕ) (m : Mixer p n) |
| (u : Wgt (UnitType p n (M*n))) : Prop where |
| rum : binaryGood m |
| hymn : unaryGood m b.pins r |
| age : 2*p.r ≤ n |
| cols : 2*num (Full p n)<(M*n:ℕ) |
| drain : Template.drained m b.spJoint u |
|
|
| set_option maxHeartbeats 1200000 in |
| theorem veil.upright (h : veil p M b r l m s w) : |
| ∀ᶠ n in l,Align b r p M n (m n) ((s n).sap (w n)) := by |
| let z n:= ((M*n:ℕ):ℝ) |
| have ht := h.easel.enter.first |
| have hM : 1 ≤ (M:ℝ) := by exact_mod_cast ht.hM |
| let e : ℝ := 1/(128*M) |
| have he : 0< e:=by unfold e; positivity |
| have hz := (trend_sizes l h.timeL ht.hM).2 |
| let D n:=((s n).amount (w n))⁻¹ |
| filter_upwards [h.moment,h.durableS.mass.inv_cap e he, |
| h.durableS.mass.positive,expO_const hz b.L e he, ht.wrap.filter_mono h.timeL, |
| (eventually_ge_atTop (2*p.r)).filter_mono h.timeL] with n hn he' ht' hL hG hH |
| have hw := hn.1.law |
| have h1 (i) := LeafMixture.upper_sap hw ht' i |
| have h2 : 0≤D n := (inv_pos_of_pos ht').le |
| have hB : D n * b.L ≤ bp (1/32*(n:ℝ)) := by |
| have hi : e*z n=(1/128)*(n:ℝ) := by unfold e z; push_cast; field_simp |
| calc _ ≤ bp (e*z n)*bp (e*z n) := |
| mul_le_mul ((le_abs_self _).trans he') ((le_abs_self _).trans hL) ht.early.hL |
| (bp_pos _).le |
| _ ≤ _ := by rw [←bp_add,hi]; apply bp_le; linarith [(Nat.cast_nonneg n :0≤(n:ℝ))] |
| have hv := LeafMixture.upperSide hn.1.border h1 h2 hB |
| refine ⟨hn.2.2.1,hn.2.1,hH,hG.1, LeafMixture.sapLaw ht' hw, hv true,hv false,fun i=>?_⟩ |
| have hh := hn.1.total i |
| apply ((h1 i).trans (mul_le_mul_of_nonneg_left hh h2)).trans |
| rw [←mul_assoc] |
| refine mul_le_mul_of_nonneg_right ?_ (mul_nonneg (raw_nonneg ..) (raw_nonneg ..)) |
| have hx : e ≤1 := by unfold e; exact (div_le_iff₀ (by positivity)).mpr (by linarith) |
| refine (mul_le_mul_of_nonneg_right ((le_abs_self _).trans he') (bp_pos _).le).trans ?_ |
| rw [←bp_add] |
| unfold spJoint; refine bp_le ?_; rw [←add_mul] |
| exact mul_le_mul_of_nonneg_right (by linarith only [hx]) (by positivity) |
| end PreparationParameters |
| end HadwigerCounterexample |
|
|
| end |
|
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| end OAI |
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