| import OAI.Combinatorics.HadwigerCounterexample.ProjectionError |
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| namespace OAI |
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| /-! # Common Partitions -/ |
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| noncomputable section |
| namespace HadwigerCounterexample |
| open Finset Module Matrix Filter Function |
| open scoped BigOperators |
| variable {A B I J X : Type} [Fintype A] [Fintype B] [Fintype I] [Fintype J] [Fintype X] |
| variable (u : Wgt A) (w : Wgt I) (v : Wgt J) |
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| set_option maxHeartbeats 800000 in |
| theorem rollup [Nonempty B] (hw : IsLaw w) (hv : IsLaw v) (hu : IsLaw u) |
| (f : I×B→Wgt A) (g : J×B→Wgt A) (h k : I×J→ B) (R : ℝ) (hr : 0<R) |
| (hg : ∀ i x, |g i x|≤R) |
| (p : A→X) (ε : ℝ) (hx : wanes (pairW w unif) u f p ε) : |
| avg (pairW w v) (fun i=> |rebate u (f (i.1,h i)) (g (i.2,k i)) p|) ≤ |
| num B ^ 2 * (R*ε) := by |
| classical |
| let H (i : I × B) (j : J × B) := |rebate u (f i) (g j) p| |
| let T := num B |
| have ht : 0<T := by unfold T num; positivity |
| have he j : |
| avg (pairW w (unif : Wgt B)) (fun i=> H i j)≤ R*ε := by |
| let g' (i:A) := g j i/R |
| have hh (i) : |g' i|≤1 := by |
| unfold g'; rw [abs_div,abs_of_pos hr,div_le_one hr]; exact hg _ _ |
| have hz (i) : |
| H i j=R* |dip u (fun x=> f i x-banner u p (f i) x) g'| := by |
| unfold H; rw [rebate_calc _ _ _ _ hu.nz] |
| unfold g' dip; simp_rw [←mul_div_assoc,avg_div] |
| rw [abs_div,abs_of_pos hr]; field_simp |
| simp_rw [hz,avg_mul']; exact mul_le_mul_of_nonneg_left (hx g' hh) hr.le |
| have hh i j : |
| H (i,h (i,j)) (j,k (i,j))≤ ∑ b : B,∑ c : B,H (i,b) (j,c) := |
| (Finset.single_le_sum (f:=fun x=> H (i,h (i,j)) (j,x)) (fun _ _ => abs_nonneg _) (mem_univ (k (i,j)))).trans |
| (Finset.single_le_sum (f:=fun x => ∑ c:B,H (i,x) (j,c)) (fun _ _=> Finset.sum_nonneg (fun _ _=>abs_nonneg _)) (mem_univ (h (i,j)))) |
| have hi j c : |
| avg w (fun i=>∑ b : B,H (i,b) (j,c))≤ T*(R*ε) := by |
| have hg : ∀ i : I,(∑ b:B,H (i,b) (j,c))=T*avg unif (fun b=> H (i,b) (j,c)) := by |
| intro i; simp only [avg,unif_apply,← Finset.mul_sum] |
| change _ = T*(T⁻¹ * _) |
| rw [←mul_assoc,mul_inv_cancel₀ ht.ne',one_mul] |
| simp_rw [hg,avg_mul'] |
| refine mul_le_mul_of_nonneg_left ?_ ht.le; rw [← pairW_avg w unif (fun p : I×B=>H p (j,c))] |
| exact he _ |
| change avg (pairW w v) _ ≤ T^2*(R*ε) |
| rw [pairW_avg,avg_avg] |
| refine hv.avg_le fun j=> ?_ |
| have hz i : ∑ b:B,∑ c:B,H (i,b) (j,c)=∑ c:B,∑ b:B,H (i,b) (j,c) := |
| Finset.sum_comm |
| apply (avg_mono hw.nz (fun i => (hh i j).trans_eq (hz i))).trans |
| rw [avg_sum] |
| exact (Finset.sum_le_sum fun i _=>hi j i).trans_eq (by |
| simp only [sum_const,nsmul_eq_mul,Finset.card_univ,T,num]; ring) |
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| set_option maxHeartbeats 1200000 in |
| theorem global_net (R ε : ℝ) (hr : 0<R) (hε : 0<ε) : |
| ∃ δ : ℝ,0<δ ∧ ∀ r : ℕ,∃ l : ℕ,∀ (I J A : Type) [Fintype I] [Fintype J] [Fintype A] |
| (w : Wgt I) (v : Wgt J) (u : Wgt A), IsLaw w→IsLaw v →IsLaw u → |
| ∀ (P : I→Wgt A→Prop) (Q : J→Wgt A→Prop), (∀ i,roster (P i) r δ) → |
| (∀ i,roster (Q i) r δ) → |
| ∃ k : ℕ,k ≤ l∧ ∃ p:A→Fin k, |
| ∀ f g:I × J→Wgt A,(∀ i,P i.1 (f i)) → (∀ i,Q i.2 (g i)) → |
| avg (pairW w v) (fun i => |rebate u (mcut R u (f i)) (mcut R u (g i)) p|)≤ ε := by |
| classical |
| let δ := ε/(8*R) |
| have hd : 0<δ := by positivity |
| refine ⟨δ,hd,fun r=>?_⟩ |
| let B:=Fin (r+1); let t:=num B |
| have ht : 0<t := by unfold t num; positivity |
| let c:=ε/(2*(t^2)*R) |
| have hc : 0<c := by positivity |
| obtain ⟨l,hl⟩:= erosive R c hr.le hc |
| refine ⟨l,fun I J A _ _ _ w v u hw hv hu P Q hp hq=>?_⟩ |
| choose s hs h1 using fun i=> roster_clip R u hu hr.le (P i) r δ (hp i) |
| choose y hy h2 using fun i=> roster_clip R u hu hr.le (Q i) r δ (hq i) |
| let F : I×B→Wgt A := uncurry s |
| let G : J×B→Wgt A := uncurry y |
| obtain ⟨m,hm,p,hP⟩:= hl A (I×B) (pairW w unif) u F (pairW_law hw unif_law) hu (fun i j=> hs _ _ _) |
| refine ⟨m,hm,p,fun f g hf hg=>?_⟩ |
| let f' (i : I×J):= mcut R u (f i) |
| let g' (i : I×J):= mcut R u (g i) |
| choose a hA using fun i:I×J => h1 i.1 (f i) (hf i) |
| choose b hB using fun i:I×J => h2 i.2 (g i) (hg i) |
| let H (i:I×J) := |rebate u (F (i.1,a i)) (G (i.2,b i)) p| |
| let D:=pairW w v; have hD:= pairW_law hw hv |
| have he := rollup u w v hw hv hu F G a b R hr (fun i j=> hy _ _ _) p c hP |
| have hb : 0≤2*R := by linarith only [hr] |
| have hi (i:I×J) : |rebate u (f' i) (g' i) p|≤ H i+(δ*(4*R)) := by |
| let A' := F (i.1,a i); let B' := G (i.2,b i) |
| have h1 (j) : |g' i j|≤R := by |
| rw [abs_of_nonneg ((mcut_range _ _ _ hr.le j).1)] |
| exact (mcut_range _ _ _ hr.le j).2 |
| have h3 : |rebate u (f' i) (g' i) p - rebate u A' (g' i) p|≤ δ*(2*R) := by |
| exact (rebate_cost u _ _ A' p hu.nz R hr.le h1).trans (mul_le_mul_of_nonneg_right (hA i) hb) |
| have h4 : |rebate u A' (g' i) p - rebate u A' B' p|≤δ*(2*R) := by |
| rw [rebate_sym u A',rebate_sym u A'] |
| exact (rebate_cost u _ _ B' p hu.nz R hr.le (fun i=>hs _ _ _)).trans |
| (mul_le_mul_of_nonneg_right (hB i) hb) |
| let U:=rebate u A' B' p |
| let X:=rebate u (f' i) (g' i) p |
| have hz:= (abs_sub_le X (rebate u A' (g' i) p) U).trans (add_le_add h3 h4) |
| calc |X| = |(X-U)+U| := by rw [sub_add_cancel] |
| _ ≤ _ := ((abs_add_le _ _).trans (add_le_add_left hz _)).trans_eq (by |
| change _+ |U| = |U|+δ*(4*R) |
| ring) |
| refine (avg_mono hD.nz hi).trans ?_ |
| rw [avg_add,hD.avg_const] |
| exact (add_le_add_left he _).trans_eq (by |
| change t^2*(R*c)+δ*(4*R)=_ |
| unfold δ c; field_simp; ring) |
| end HadwigerCounterexample |
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| end |
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| end OAI |
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