| import Mathlib |
| import OAI.Geometry.BallPacking.Models.RationalSlack |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PackingSufficiencySupport.Hamiltonian |
| open scoped ContDiff Manifold Topology BigOperators |
| open Set Function Filter Manifold |
|
|
| variable {E V : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| theorem globalHorizontalCoupling_congr_germs (Ω : V →L[ℝ] V →L[ℝ] ℝ) |
| {Γ Γ' : V → ManifoldOneForm E M} {K : Set (M × V)} |
| (h : ∀ z∈K,∀ᶠ q in 𝓝 z,Γ q.2 q.1=Γ' q.2 q.1) : |
| globalHorizontalCoupling Ω Γ=ᶠ[𝓝ˢ K] globalHorizontalCoupling Ω Γ' := by |
| apply mem_nhdsSet_iff_forall.mpr |
| intro z hz |
| filter_upwards [(h z hz).eventually_nhds] with q hq |
| exact globalHorizontalCoupling_congr_joint Ω hq |
|
|
| end PackingSufficiencySupport.Hamiltonian |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped ContDiff Manifold Topology BigOperators |
| open Set Function Filter Manifold |
| open DiagonalQuadrics DiagonalQuadrics.Explicit Hamiltonian FiniteMoment MomentPolytope |
|
|
| theorem physicalParameter_fiber_lower_near {A B : ℕ} {L : ℝ} (hL : 0≤L) |
| {z : BaseCurve × PlanePhase (Fin 2)} |
| (hz : physicalParameter L (planeMoments z.2)∈Radial.lowerTrapezoid A B) : |
| ∀ᶠ q in 𝓝 z,physicalParameter L (planeMoments q.2)∈Radial.lowerTrapezoid A B := by |
| have hc : Continuous (fun q : BaseCurve × PlanePhase (Fin 2) => physicalParameter L (planeMoments q.2)) := |
| (physicalParameter_smooth L).continuous.comp (planeMoments_smooth.continuous.comp continuous_snd) |
| have h0 := (hc.fst.continuousAt.eventually_lt continuousAt_const hz.2.1) |
| have h1 := ((hc.fst.add hc.snd).continuousAt.eventually_lt continuousAt_const hz.2.2.2) |
| filter_upwards [h0,h1] with q hq0 hq1 |
| exact ⟨mul_nonneg hL (planeMoments_nonneg _ 0),hq0, |
| mul_nonneg hL (planeMoments_nonneg _ 1),hq1⟩ |
|
|
| theorem physicalPrimitive_representative_fiber_germ {A B D S q : ℕ} {c L : ℝ} (hL : 0≤L) |
| {bF : Fin q → Moments 2} {cF : Fin q → ℝ} |
| (hPS : MapsTo (physicalParameter L) (region bF cF) (Radial.lowerTrapezoid A B)) |
| {ℓ : ℕ} {Γ : Moments 2 → ManifoldOneForm RealModel BaseCurve} |
| (he : ∀ᶠ q in 𝓝ˢ (region bF cF ×ˢ cubicExhaustion ℓ), |
| physicalParameter L q.1∈Radial.lowerTrapezoid A B → |
| Γ q.1 q.2=physicalPrimitive (A := A) (B := B) D S c L q.1 q.2) : |
| ∀ z∈cubicExhaustion ℓ ×ˢ planeRegion bF cF,∀ᶠ q in 𝓝 z, |
| Γ (planeMoments q.2) q.1=physicalPrimitive (A := A) (B := B) D S c L (planeMoments q.2) q.1 := by |
| intro z hz |
| have hn := mem_nhdsSet_iff_forall.mp he (planeMoments z.2,z.1) ⟨hz.2,hz.1⟩ |
| have hc : Continuous (fun q : BaseCurve × PlanePhase (Fin 2) => (planeMoments q.2,q.1)) := |
| (planeMoments_smooth.continuous.comp continuous_snd).prodMk continuous_fst |
| have hi := physicalParameter_fiber_lower_near (A := A) (B := B) (z := z) hL (hPS hz.2) |
| filter_upwards [hc.continuousAt.eventually hn,hi] with q hq hqp |
| exact hq hqp |
|
|
| attribute [local irreducible] physicalPrimitive |
|
|
| theorem physicalCoupling_representative_near {A B D S q : ℕ} {c L : ℝ} (hL : 0≤L) |
| {bF : Fin q → Moments 2} {cF : Fin q → ℝ} |
| (hPS : MapsTo (physicalParameter L) (region bF cF) (Radial.lowerTrapezoid A B)) |
| {ℓ : ℕ} {Γ : Moments 2 → ManifoldOneForm RealModel BaseCurve} |
| (he : ∀ᶠ q in 𝓝ˢ (region bF cF ×ˢ cubicExhaustion ℓ), |
| physicalParameter L q.1∈Radial.lowerTrapezoid A B → |
| Γ q.1 q.2=physicalPrimitive (A := A) (B := B) D S c L q.1 q.2) : |
| globalHorizontalCoupling (E := RealModel) (M := BaseCurve) (V := PlanePhase (Fin 2)) phaseArea (Γ ∘ planeMoments)=ᶠ[𝓝ˢ (cubicExhaustion ℓ ×ˢ planeRegion bF cF)] |
| globalHorizontalCoupling (E := RealModel) (M := BaseCurve) (V := PlanePhase (Fin 2)) phaseArea (physicalPrimitive (A := A) (B := B) D S c L ∘ planeMoments) := by |
| have hg : ∀ z∈cubicExhaustion ℓ ×ˢ planeRegion bF cF,∀ᶠ q in 𝓝 z, |
| Γ (planeMoments q.2) q.1=physicalPrimitive (A := A) (B := B) D S c L (planeMoments q.2) q.1 := |
| physicalPrimitive_representative_fiber_germ hL hPS he |
| exact globalHorizontalCoupling_congr_germs (E := RealModel) (M := BaseCurve) (V := PlanePhase (Fin 2)) phaseArea (Γ := Γ ∘ planeMoments) |
| (Γ' := physicalPrimitive (A := A) (B := B) D S c L ∘ planeMoments) hg |
|
|
| end PackingSufficiencySupport.CubicModel |
|
|
| namespace PackingSufficiencySupport |
| open scoped BigOperators |
| open MeasureTheory Set |
| open scoped Pointwise |
| open MomentPolytope |
|
|
| theorem exists_positive_common_denominator {ι : Type*} [Fintype ι] |
| (q : ι → ℚ) (hq : ∀ i,0≤q i) : |
| ∃ L : ℕ,0<L ∧ ∃ k : ι → ℕ,∀ i,(k i:ℝ)=(L:ℝ)*(q i:ℝ) := by |
| classical |
| let L := ∏ i,(q i).den |
| have hL : 0<L := Finset.prod_pos (fun i _ => (q i).den_pos) |
| have hd i : (q i).den∣L := Finset.dvd_prod_of_mem _ (Finset.mem_univ i) |
| have hn i : 0≤(q i).num := Rat.num_nonneg.mpr (hq i) |
| let k := fun i => (q i).num.toNat*(L/(q i).den) |
| refine ⟨L,hL,k,fun i => ?_⟩ |
| have hc : ((q i).num.toNat:ℝ)=((q i).num:ℝ) := by |
| exact_mod_cast Int.toNat_of_nonneg (hn i) |
| have hmul : ((L/(q i).den:ℕ):ℝ)*((q i).den:ℝ)=(L:ℝ) := by |
| exact_mod_cast Nat.div_mul_cancel (hd i) |
| have hd0 : ((q i).den:ℝ)≠0 := by exact_mod_cast (q i).den_ne_zero |
| dsimp only [k] |
| rw [Nat.cast_mul,hc,Rat.cast_def] |
| field_simp |
| nlinarith [hmul] |
|
|
| def truncatedSimplex (q : ℕ) (h b : ℝ) : Set (ℝ × (Fin q → ℝ)) := |
| {x | x.1 ∈ Icc 0 h ∧ x.2 ∈ momentSimplex q (b-x.1)} |
|
|
| theorem truncatedSimplex_isClosed (q : ℕ) (h b : ℝ) : |
| IsClosed (truncatedSimplex q h b) := by |
| have hn : IsClosed {x : ℝ × (Fin q → ℝ) | ∀ i,0≤x.2 i} := by |
| simp only [ofPred_forall] |
| exact isClosed_iInter fun i => isClosed_le continuous_const ((continuous_apply i).comp continuous_snd) |
| exact (isClosed_Icc.preimage continuous_fst).inter (hn.inter |
| (isClosed_le (continuous_finsetSum _ fun i _ => (continuous_apply i).comp continuous_snd) |
| (continuous_const.sub continuous_fst))) |
|
|
| theorem truncatedSimplex_isCompact (q : ℕ) (h b : ℝ) : |
| IsCompact (truncatedSimplex q h b) := by |
| refine (isCompact_Icc (a := (0,fun _ : Fin q => (0:ℝ))) |
| (b := (h,fun _ => b))).of_isClosed_subset (truncatedSimplex_isClosed q h b) ?_ |
| intro x hx |
| refine ⟨⟨hx.1.1,hx.2.1⟩,hx.1.2,fun i => ?_⟩ |
| have hi := (Finset.single_le_sum (fun j _ => hx.2.1 j) (Finset.mem_univ i)).trans hx.2.2 |
| linarith [hx.1.1] |
|
|
| theorem truncatedSimplex_fubini (q : ℕ) (h b : ℝ) |
| (f : ℝ×(Fin q → ℝ) → ℝ) (hf : ContinuousOn f (truncatedSimplex q h b)) : |
| (∫ x in truncatedSimplex q h b, f x) = |
| ∫ u in Icc 0 h, ∫ p in momentSimplex q (b-u), f (u,p) := by |
| let D := truncatedSimplex q h b |
| have hD := (truncatedSimplex_isClosed q h b).measurableSet |
| have hi : Integrable (D.indicator f) := (integrable_indicator_iff hD).2 |
| (hf.integrableOn_compact (truncatedSimplex_isCompact q h b)) |
| rw [←integral_indicator hD] |
| change (∫ x, D.indicator f x ∂(volume.prod volume)) = _ |
| rw [integral_prod _ hi,←integral_indicator measurableSet_Icc] |
| congr 1 |
| funext u |
| by_cases hu : u∈Icc 0 h |
| · rw [indicator_of_mem hu,←integral_indicator (momentSimplex_isClosed q (b-u)).measurableSet] |
| congr 1 |
| funext p |
| by_cases hp : p∈momentSimplex q (b-u) |
| · rw [indicator_of_mem (show (u,p)∈D from ⟨hu,hp⟩),indicator_of_mem hp] |
| · rw [indicator_of_notMem (show (u,p)∉D from fun hx => hp hx.2),indicator_of_notMem hp] |
| · rw [indicator_of_notMem hu] |
| have hz (p : Fin q → ℝ) : D.indicator f (u,p)=0 := |
| indicator_of_notMem (show (u,p)∉D from fun hx => hu hx.1) f |
| simp_rw [hz] |
| exact integral_zero _ _ |
|
|
| theorem integral_truncated_simplex_power (q k : ℕ) {h b : ℝ} (hh : 0≤h) (hhb : h≤b) : |
| (∫ x in truncatedSimplex q h b, (b-x.1-∑ i,x.2 i)^k) = |
| (k.factorial : ℝ)/((q+k+1).factorial : ℝ)* |
| (b^(q+k+1)-(b-h)^(q+k+1)) := by |
| rw [truncatedSimplex_fubini q h b _ (by fun_prop)] |
| have heq : (∫ u in Icc 0 h, ∫ p in momentSimplex q (b-u), (b-u-∑ i,p i)^k) = |
| ∫ u in Icc 0 h, (k.factorial : ℝ)/((q+k).factorial : ℝ)*(b-u)^(q+k) := by |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro u hu |
| exact integral_simplex_power q k (by linarith [hu.2]) |
| rw [heq,integral_const_mul,integral_Icc_eq_integral_Ioc, |
| ←intervalIntegral.integral_of_le hh, |
| intervalIntegral.integral_comp_sub_left (fun u : ℝ => u^(q+k)) b, |
| sub_zero,integral_pow] |
| rw [Nat.factorial_succ (q+k),Nat.cast_mul] |
| simp only [Nat.cast_add,Nat.cast_one,div_eq_mul_inv,mul_inv_rev] |
| ring |
|
|
| theorem volumeReal_truncatedSimplex (q : ℕ) {h b : ℝ} (hh : 0≤h) (hhb : h≤b) : |
| volume.real (truncatedSimplex q h b) = |
| (b^(q+1)-(b-h)^(q+1))/((q+1).factorial : ℝ) := by |
| simpa [div_eq_mul_inv,mul_comm] using integral_truncated_simplex_power q 0 hh hhb |
|
|
| theorem integral_truncated_tent (q : ℕ) {h b : ℝ} (hh : 0≤h) (hhb : h≤b) : |
| (∫ x in truncatedSimplex q h b, b-x.1-∑ i,x.2 i) = |
| (b^(q+2)-(b-h)^(q+2))/((q+2).factorial : ℝ) := by |
| simpa [div_eq_mul_inv,mul_comm] using integral_truncated_simplex_power q 1 hh hhb |
|
|
| theorem integral_truncated_cap (q : ℕ) {h s b : ℝ} |
| (hh : 0 ≤ h) (hhs : h ≤ s) (hsb : s ≤ b) : |
| (∫ x in truncatedSimplex q h b, max (s-x.1-∑ i,x.2 i) 0) = |
| (s^(q+2)-(s-h)^(q+2))/((q+2).factorial : ℝ) := by |
| rw [←integral_truncated_tent q hh hhs, |
| ←integral_indicator (truncatedSimplex_isClosed q h b).measurableSet, |
| ←integral_indicator (truncatedSimplex_isClosed q h s).measurableSet] |
| congr 1 |
| funext x |
| by_cases hb : x∈truncatedSimplex q h b |
| · rw [indicator_of_mem hb] |
| by_cases hs : x∈truncatedSimplex q h s |
| · rw [indicator_of_mem hs,max_eq_left] |
| linarith [hs.2.2] |
| · rw [indicator_of_notMem hs,max_eq_right] |
| have hn : ¬ (∑ i,x.2 i) ≤ s-x.1 := fun he => hs ⟨hb.1,hb.2.1,he⟩ |
| linarith [not_le.1 hn] |
| · rw [indicator_of_notMem hb,indicator_of_notMem] |
| intro hs |
| apply hb |
| exact ⟨hs.1,hs.2.1,hs.2.2.trans (by linarith)⟩ |
|
|
| theorem integral_truncated_affine {h b d : ℝ} (hh : 0 ≤ h) (hhd : h ≤ d) : |
| (∫ x in truncatedSimplex 1 h d, b-x.1-∑ i,x.2 i) = |
| (b-d)*(d^2-(d-h)^2)/2+(d^3-(d-h)^3)/6 := by |
| have ht : IntegrableOn (fun x : ℝ×(Fin 1→ℝ) => d-x.1-∑ i,x.2 i) |
| (truncatedSimplex 1 h d) := by |
| apply ContinuousOn.integrableOn_compact (truncatedSimplex_isCompact 1 h d) |
| fun_prop |
| have hc : IntegrableOn (fun _ : ℝ×(Fin 1→ℝ) => b-d) (truncatedSimplex 1 h d) := |
| continuousOn_const.integrableOn_compact (truncatedSimplex_isCompact 1 h d) |
| have heq (x : ℝ×(Fin 1→ℝ)) : b-x.1-∑ i,x.2 i = (b-d)+(d-x.1-∑ i,x.2 i) := by ring |
| simp_rw [heq] |
| rw [integral_add hc ht,setIntegral_const,integral_truncated_tent 1 hh hhd] |
| simp only [smul_eq_mul,volumeReal_truncatedSimplex 1 hh hhd] |
| norm_num |
| ring |
|
|
| def sixModelMean (s b h₁ h₂ : ℝ) : ℝ := |
| (3/2)*(b^3-(b-h₁)^3) -(9/2)*h₁*(b-h₂)^2 -(1/3)*(s^3-(s-h₁)^3) |
|
|
| theorem integral_six_model {s b h₁ h₂ : ℝ} |
| (hh₁ : 0 ≤ h₁) (hhs : h₁ ≤ s) (hsh : s ≤ h₂) : |
| (∫ x in truncatedSimplex 1 h₁ h₂, |
| 9*(b-x.1-∑ i,x.2 i)-2*max (s-x.1-∑ i,x.2 i) 0) = |
| sixModelMean s b h₁ h₂ := by |
| have hi (f : ℝ×(Fin 1→ℝ) → ℝ) (hf : Continuous f) : |
| IntegrableOn f (truncatedSimplex 1 h₁ h₂) := |
| hf.continuousOn.integrableOn_compact (truncatedSimplex_isCompact 1 h₁ h₂) |
| rw [integral_sub (hi _ (by fun_prop)) (hi _ (by fun_prop)), |
| integral_const_mul,integral_const_mul, |
| integral_truncated_affine hh₁ (hhs.trans hsh), |
| integral_truncated_cap 1 hh₁ hhs hsh] |
| norm_num [sixModelMean] |
| ring |
|
|
| theorem continuous_sixModelMean (s b : ℝ) : |
| Continuous (fun h : ℝ×ℝ => sixModelMean s b h.1 h.2) := by |
| unfold sixModelMean |
| fun_prop |
|
|
| theorem sixModelMean_limit_value (a : ℝ) : |
| sixModelMean (1-a) ((2-a)/3) (1-a) ((2-a)/3) = (1-a^3)/6 := by |
| unfold sixModelMean |
| ring |
|
|
| def momentHeadTail (q : ℕ) : (Fin (q+1) → ℝ) ≃ᵐ (ℝ×(Fin q → ℝ)) := |
| MeasurableEquiv.piFinSuccAbove (fun _ : Fin (q+1) => ℝ) 0 |
|
|
| @[simp] theorem momentHeadTail_apply (q : ℕ) (p : Fin (q+1) → ℝ) : |
| momentHeadTail q p=(p 0,fun i => p i.succ) := by |
| rfl |
|
|
| @[simp] theorem momentHeadTail_symm_apply (q : ℕ) (x : ℝ×(Fin q → ℝ)) : |
| (momentHeadTail q).symm x=Fin.cons x.1 x.2 := by |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i <;> |
| simp [momentHeadTail,MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv] |
|
|
| theorem continuous_momentHeadTail (q : ℕ) : Continuous (momentHeadTail q) := by |
| change Continuous (fun p : Fin (q+1) → ℝ => (p 0,fun i : Fin q => p i.succ)) |
| fun_prop |
|
|
| theorem continuous_momentHeadTail_symm (q : ℕ) : Continuous (momentHeadTail q).symm := by |
| change Continuous (fun x : ℝ×(Fin q → ℝ) => (momentHeadTail q).symm x) |
| simp_rw [momentHeadTail_symm_apply] |
| apply continuous_pi |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i <;> simp only [Fin.cons_zero,Fin.cons_succ] <;> fun_prop |
|
|
| theorem isCompact_preimage_momentHeadTail (q : ℕ) {D : Set (ℝ×(Fin q → ℝ))} |
| (hD : IsCompact D) : IsCompact ((momentHeadTail q) ⁻¹' D) := by |
| have he : (momentHeadTail q) ⁻¹' D = (momentHeadTail q).symm '' D := by |
| ext p |
| constructor |
| · intro hp |
| exact ⟨momentHeadTail q p,hp,(momentHeadTail q).symm_apply_apply p⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| simpa using hx |
| rw [he] |
| exact hD.image (continuous_momentHeadTail_symm q) |
|
|
| theorem integral_momentHeadTail (q : ℕ) {D : Set (ℝ×(Fin q → ℝ))} |
| (hD : MeasurableSet D) (f : ℝ×(Fin q → ℝ) → ℝ) : |
| (∫ p in (momentHeadTail q) ⁻¹' D, f (momentHeadTail q p)) = ∫ x in D,f x := by |
| let e := momentHeadTail q |
| have he : MeasurePreserving e := volume_preserving_piFinSuccAbove (fun _ : Fin (q+1) => ℝ) 0 |
| rw [←integral_indicator (hD.preimage e.measurable),←integral_indicator hD] |
| have hf : (e ⁻¹' D).indicator (fun p => f (e p)) = (D.indicator f) ∘ e := by |
| funext p |
| by_cases hp : e p∈D <;> simp [hp] |
| rw [hf] |
| exact he.integral_comp e.measurableEmbedding _ |
|
|
| def sixModelBounds : Fin 2 → Moments 2 := ![![1,0],![1,1]] |
|
|
| theorem sixModelBounds_nonneg : ∀ ν j,0 ≤ sixModelBounds ν j := by |
| intro ν j |
| fin_cases ν <;> fin_cases j <;> norm_num [sixModelBounds] |
|
|
| theorem sixModelBounds_le_one : ∀ ν j,sixModelBounds ν j ≤ 1 := by |
| intro ν j |
| fin_cases ν <;> fin_cases j <;> norm_num [sixModelBounds] |
|
|
| theorem sixModelRegion_eq (h₁ h₂ : ℝ) : |
| region sixModelBounds ![h₁,h₂] = (momentHeadTail 1) ⁻¹' truncatedSimplex 1 h₁ h₂ := by |
| ext p |
| change ((∀ j,0≤p j) ∧ ∀ ν,∑ j,sixModelBounds ν j*p j ≤ ![h₁,h₂] ν) ↔ |
| (p 0∈Icc 0 h₁ ∧ (fun j : Fin 1 => p j.succ)∈momentSimplex 1 (h₂-p 0)) |
| simp only [Fin.forall_fin_two,Fin.sum_univ_two,Fin.sum_univ_one, |
| sixModelBounds,Matrix.cons_val_zero,Matrix.cons_val_one,momentSimplex, |
| mem_ofPred_eq,Fin.forall_fin_one,mem_Icc,one_mul,zero_mul,add_zero] |
| norm_num only [Fin.succ_zero_eq_one] |
| constructor |
| · rintro ⟨⟨h0,h1⟩,hh₁,hh₂⟩ |
| exact ⟨⟨h0,hh₁⟩,h1,by linarith⟩ |
| · rintro ⟨⟨h0,hh₁⟩,h1,hh₂⟩ |
| exact ⟨⟨h0,h1⟩,hh₁,by linarith⟩ |
|
|
| theorem sixModelRegion_isCompact (h₁ h₂ : ℝ) : |
| IsCompact (region sixModelBounds ![h₁,h₂]) := by |
| rw [sixModelRegion_eq] |
| exact isCompact_preimage_momentHeadTail 1 (truncatedSimplex_isCompact 1 h₁ h₂) |
|
|
| theorem sixModelRegion_contains_cap {r h₁ h₂ : ℝ} (hr₁ : r ≤ h₁) (hr₂ : r ≤ h₂) : |
| momentSimplex 2 r ⊆ region sixModelBounds ![h₁,h₂] := by |
| intro p hp |
| refine ⟨hp.1,fun ν => ?_⟩ |
| fin_cases ν |
| · change (∑ j,(![1,0] : Fin 2→ℝ) j*p j) ≤ h₁ |
| simp only [Fin.sum_univ_two,Matrix.cons_val_zero,Matrix.cons_val_one,one_mul,zero_mul,add_zero] |
| have hs : p 0+p 1 ≤ r := by simpa only [Fin.sum_univ_two] using hp.2 |
| linarith [hp.1 1] |
| · change (∑ j,(![1,1] : Fin 2→ℝ) j*p j) ≤ h₂ |
| simpa only [Fin.sum_univ_two,Matrix.cons_val_zero,Matrix.cons_val_one,one_mul] using hp.2.trans hr₂ |
|
|
| def sixModelDensity (s b : ℝ) (p : Moments 2) : ℝ := |
| 9*(b-∑ j,p j)-2*max (s-∑ j,p j) 0 |
|
|
| def sixModelGap {ι : Type*} [Fintype ι] (s b : ℝ) (r : ι → ℝ) (p : Moments 2) : ℝ := |
| sixModelDensity s b p-∑ i,max (r i-∑ j,p j) 0 |
|
|
| theorem continuous_sixModelGap {ι : Type*} [Fintype ι] (s b : ℝ) (r : ι → ℝ) : |
| Continuous (sixModelGap s b r) := by |
| unfold sixModelGap sixModelDensity |
| fun_prop |
|
|
| theorem sixModelGap_eq_cappedAffine {ι : Type*} [Fintype ι] (s b : ℝ) (r : ι → ℝ) : |
| sixModelGap s b r = cappedAffine (9*b) (fun _ => 9) |
| (Sum.elim (fun _ : Unit => s) r) (Sum.elim (fun _ : Unit => 2) (fun _ => 1)) := by |
| funext p |
| simp only [sixModelGap,sixModelDensity,cappedAffine,Fintype.sum_sum_type, |
| Sum.elim_inl,Sum.elim_inr,Fintype.sum_unique,one_mul,←Finset.mul_sum] |
| ring |
|
|
| theorem integral_sixModelDensity {s b h₁ h₂ : ℝ} |
| (hh₁ : 0 ≤ h₁) (hhs : h₁ ≤ s) (hsh : s ≤ h₂) : |
| (∫ p in region sixModelBounds ![h₁,h₂],sixModelDensity s b p) = sixModelMean s b h₁ h₂ := by |
| rw [sixModelRegion_eq] |
| have he : sixModelDensity s b = (fun x : ℝ×(Fin 1→ℝ) => |
| 9*(b-x.1-∑ j,x.2 j)-2*max (s-x.1-∑ j,x.2 j) 0) ∘ momentHeadTail 1 := by |
| funext p |
| simp only [sixModelDensity,Fin.sum_univ_succ,Function.comp_apply,momentHeadTail_apply] |
| congr 2 <;> ring_nf |
| rw [he] |
| simp only [Function.comp_apply] |
| rw [integral_momentHeadTail 1 (truncatedSimplex_isClosed 1 h₁ h₂).measurableSet |
| (fun x : ℝ×(Fin 1→ℝ) => 9*(b-x.1-∑ j,x.2 j)-2*max (s-x.1-∑ j,x.2 j) 0)] |
| exact integral_six_model hh₁ hhs hsh |
|
|
| theorem integral_sixModelGap {ι : Type*} [Fintype ι] (r : ι → ℝ) (hr : ∀ i,0 ≤ r i) |
| {s b h₁ h₂ : ℝ} (hh₁ : 0 ≤ h₁) (hhs : h₁ ≤ s) (hsh : s ≤ h₂) |
| (hr₁ : ∀ i,r i ≤ h₁) (hr₂ : ∀ i,r i ≤ h₂) : |
| (∫ p in region sixModelBounds ![h₁,h₂],sixModelGap s b r p) = |
| sixModelMean s b h₁ h₂-(∑ i,r i^3)/6 := by |
| have hC := sixModelRegion_isCompact h₁ h₂ |
| have hcap (i : ι) := sixModelRegion_contains_cap (hr₁ i) (hr₂ i) |
| unfold sixModelGap |
| rw [integral_sub] |
| · rw [integral_sixModelDensity hh₁ hhs hsh, |
| integral_sum_tents r hr hC.measurableSet (fun _ hp => hp.1) hcap] |
| norm_num |
| · apply ContinuousOn.integrableOn_compact hC |
| unfold sixModelDensity |
| fun_prop |
| · apply ContinuousOn.integrableOn_compact hC |
| fun_prop |
|
|
| theorem sixModelDensity_lower {s b h₁ h₂ : ℝ} (hsh : s ≤ h₂) |
| {p : Moments 2} (hp : p ∈ region sixModelBounds ![h₁,h₂]) : |
| 9*(b-h₂) ≤ sixModelDensity s b p := by |
| have hs : (∑ j,p j) ≤ h₂ := by |
| have ht := hp.2 1 |
| simpa [sixModelBounds,Fin.sum_univ_two] using ht |
| have hm : max (s-∑ j,p j) 0 ≤ h₂-∑ j,p j := max_le (by linarith) (by linarith) |
| unfold sixModelDensity |
| linarith |
|
|
| end PackingSufficiencySupport |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped BigOperators |
| open Set Function |
| open MomentPolytope FiniteMoment |
|
|
| theorem physicalArea_eq_sixModelDensity {L D S : ℕ} (hL : 0<L) {a : ℝ} |
| (hD : (D:ℝ)=(L:ℝ)*(2-a)) (hS : (S:ℝ)=(L:ℝ)*(1-a)) : |
| physicalArea D S (1/((L:ℝ)*Real.pi)) L=sixModelDensity (1-a) ((2-a)/3) := by |
| have hLr : 0<(L:ℝ) := Nat.cast_pos.mpr hL |
| funext p |
| unfold physicalArea sixModelDensity |
| rw [hD,hS,Fin.sum_univ_two] |
| have he : (L:ℝ)*(1-a)-((L:ℝ)*p 0+(L:ℝ)*p 1)=(L:ℝ)*((1-a)-(p 0+p 1)) := by ring |
| rw [he] |
| have hm : max ((L:ℝ)*((1-a)-(p 0+p 1))) 0 = (L:ℝ)*max ((1-a)-(p 0+p 1)) 0 := by |
| simpa only [mul_zero] using (mul_max_of_nonneg ((1-a)-(p 0+p 1)) 0 hLr.le).symm |
| rw [hm] |
| field_simp [hLr.ne',Real.pi_ne_zero] |
| ring |
|
|
| theorem exists_six_cubic_weights {a d₁ d₂ : ℚ} (ha : (1:ℚ)/2<a) (ha1 : a<1) |
| (hd₁ : 1-a<d₁) (hdd : d₁<d₂) (hd₂ : d₂<(2-a)/3) : |
| ∃ L A B D S : ℕ,0<L ∧ 0<A ∧ 0<S ∧ A≤B ∧ S≤B ∧ 3*B<D ∧ 3*S<D ∧ |
| (A:ℝ)=(L:ℝ)*(d₁:ℝ) ∧ (B:ℝ)=(L:ℝ)*(d₂:ℝ) ∧ |
| (D:ℝ)=(L:ℝ)*(2-(a:ℝ)) ∧ (S:ℝ)=(L:ℝ)*(1-(a:ℝ)) := by |
| let q : Fin 4 → ℚ := ![1-a,2-a,d₁,d₂] |
| have hq : ∀ i,0≤q i := by |
| intro i |
| fin_cases i <;> dsimp [q] <;> linarith |
| obtain ⟨L,hL,k,hk⟩ := exists_positive_common_denominator q hq |
| have hS : (k 0:ℝ)=(L:ℝ)*(1-(a:ℝ)) := by simpa [q] using hk 0 |
| have hD : (k 1:ℝ)=(L:ℝ)*(2-(a:ℝ)) := by simpa [q] using hk 1 |
| have hA : (k 2:ℝ)=(L:ℝ)*(d₁:ℝ) := by simpa [q] using hk 2 |
| have hB : (k 3:ℝ)=(L:ℝ)*(d₂:ℝ) := by simpa [q] using hk 3 |
| have hLr : 0<(L:ℝ) := Nat.cast_pos.mpr hL |
| have ha' : (1:ℝ)/2<(a:ℝ) := by |
| have hh : (((1:ℚ)/2:ℚ):ℝ)<(a:ℝ) := Rat.cast_lt.mpr ha |
| norm_num at hh ⊢ |
| exact hh |
| have ha1' : (a:ℝ)<1 := by exact_mod_cast ha1 |
| have hd₁' : 1-(a:ℝ)<d₁ := by exact_mod_cast hd₁ |
| have hdd' : (d₁:ℝ)<d₂ := by exact_mod_cast hdd |
| have hd₂' : (d₂:ℝ)<(2-(a:ℝ))/3 := by exact_mod_cast hd₂ |
| refine ⟨L,k 2,k 3,k 1,k 0,hL,?_,?_,?_,?_,?_,?_,hA,hB,hD,hS⟩ |
| · apply (Nat.cast_pos (α := ℝ)).mp |
| rw [hA] |
| exact mul_pos hLr (by linarith) |
| · apply (Nat.cast_pos (α := ℝ)).mp |
| rw [hS] |
| exact mul_pos hLr (by linarith) |
| · apply (Nat.cast_le (α := ℝ)).mp |
| rw [hA,hB] |
| exact mul_le_mul_of_nonneg_left hdd'.le hLr.le |
| · apply (Nat.cast_le (α := ℝ)).mp |
| rw [hS,hB] |
| exact mul_le_mul_of_nonneg_left (by linarith) hLr.le |
| · have hm := mul_lt_mul_of_pos_left (show 3*(d₂:ℝ)<2-(a:ℝ) by linarith) hLr |
| have hp : 3*(k 3:ℝ)<(k 1:ℝ) := by rw [hB,hD]; nlinarith |
| exact_mod_cast hp |
| · have hm := mul_lt_mul_of_pos_left (show 3*(1-(a:ℝ))<2-(a:ℝ) by linarith) hLr |
| have hp : 3*(k 0:ℝ)<(k 1:ℝ) := by rw [hS,hD]; nlinarith |
| exact_mod_cast hp |
|
|
| theorem physicalParameter_maps_six_region {L A B : ℕ} (hL : 0<L) |
| {h₁ h₂ d₁ d₂ : ℝ} (hhd₁ : h₁<d₁) (hhd₂ : h₂<d₂) |
| (hA : (A:ℝ)=(L:ℝ)*d₁) (hB : (B:ℝ)=(L:ℝ)*d₂) : |
| MapsTo (physicalParameter L) (region sixModelBounds ![h₁,h₂]) (Radial.lowerTrapezoid A B) := by |
| intro p hp |
| have hn := hp.1 |
| have hp0 : p 0≤h₁ := by |
| simpa [sixModelBounds,Fin.sum_univ_two] using hp.2 0 |
| have hpt : p 0+p 1≤h₂ := by |
| simpa [sixModelBounds,Fin.sum_univ_two] using hp.2 1 |
| have hLr : 0<(L:ℝ) := Nat.cast_pos.mpr hL |
| refine ⟨mul_nonneg hLr.le (hn 0),?_,mul_nonneg hLr.le (hn 1),?_⟩ |
| · change (L:ℝ)*p 0<(A:ℝ) |
| rw [hA] |
| exact mul_lt_mul_of_pos_left (hp0.trans_lt hhd₁) hLr |
| · change (L:ℝ)*p 0+(L:ℝ)*p 1<(B:ℝ) |
| rw [hB,← mul_add] |
| exact mul_lt_mul_of_pos_left (hpt.trans_lt hhd₂) hLr |
|
|
| theorem exists_six_cubic_inner_weights {a d₁ d₂ : ℚ} (ha : (1:ℚ)/2<a) (ha1 : a<1) |
| (hd₁0 : 0<d₁) (hd₁ : d₁<1-a) (hd₂s : 1-a<d₂) (hd₂ : d₂<(2-a)/3) : |
| ∃ L A B D S : ℕ,0<L ∧ 0<A ∧ 0<S ∧ A<S ∧ A≤B ∧ S≤B ∧ 3*B<D ∧ 3*S<D ∧ |
| (A:ℝ)=(L:ℝ)*(d₁:ℝ) ∧ (B:ℝ)=(L:ℝ)*(d₂:ℝ) ∧ |
| (D:ℝ)=(L:ℝ)*(2-(a:ℝ)) ∧ (S:ℝ)=(L:ℝ)*(1-(a:ℝ)) := by |
| let q : Fin 4 → ℚ := ![1-a,2-a,d₁,d₂] |
| have hq : ∀ i,0≤q i := by |
| intro i |
| fin_cases i <;> dsimp [q] <;> linarith |
| obtain ⟨L,hL,k,hk⟩ := exists_positive_common_denominator q hq |
| have hS : (k 0:ℝ)=(L:ℝ)*(1-(a:ℝ)) := by simpa [q] using hk 0 |
| have hD : (k 1:ℝ)=(L:ℝ)*(2-(a:ℝ)) := by simpa [q] using hk 1 |
| have hA : (k 2:ℝ)=(L:ℝ)*(d₁:ℝ) := by simpa [q] using hk 2 |
| have hB : (k 3:ℝ)=(L:ℝ)*(d₂:ℝ) := by simpa [q] using hk 3 |
| have hLr : 0<(L:ℝ) := Nat.cast_pos.mpr hL |
| have ha' : (1:ℝ)/2<(a:ℝ) := by |
| have hh : (((1:ℚ)/2:ℚ):ℝ)<(a:ℝ) := Rat.cast_lt.mpr ha |
| norm_num at hh ⊢ |
| exact hh |
| have ha1' : (a:ℝ)<1 := by exact_mod_cast ha1 |
| have hd₁0' : 0<(d₁:ℝ) := by exact_mod_cast hd₁0 |
| have hd₁' : (d₁:ℝ)<1-(a:ℝ) := by exact_mod_cast hd₁ |
| have hd₂s' : 1-(a:ℝ)<d₂ := by exact_mod_cast hd₂s |
| have hdd' : (d₁:ℝ)<d₂ := hd₁'.trans hd₂s' |
| have hd₂' : (d₂:ℝ)<(2-(a:ℝ))/3 := by exact_mod_cast hd₂ |
| refine ⟨L,k 2,k 3,k 1,k 0,hL,?_,?_,?_,?_,?_,?_,?_,hA,hB,hD,hS⟩ |
| · apply (Nat.cast_pos (α := ℝ)).mp |
| rw [hA] |
| exact mul_pos hLr (by linarith) |
| · apply (Nat.cast_pos (α := ℝ)).mp |
| rw [hS] |
| exact mul_pos hLr (by linarith) |
| · apply (Nat.cast_lt (α := ℝ)).mp |
| rw [hA,hS] |
| exact mul_lt_mul_of_pos_left hd₁' hLr |
| · apply (Nat.cast_le (α := ℝ)).mp |
| rw [hA,hB] |
| exact mul_le_mul_of_nonneg_left hdd'.le hLr.le |
| · apply (Nat.cast_le (α := ℝ)).mp |
| rw [hS,hB] |
| exact mul_le_mul_of_nonneg_left (by linarith) hLr.le |
| · have hm := mul_lt_mul_of_pos_left (show 3*(d₂:ℝ)<2-(a:ℝ) by linarith) hLr |
| have hp : 3*(k 3:ℝ)<(k 1:ℝ) := by rw [hB,hD]; nlinarith |
| exact_mod_cast hp |
| · have hm := mul_lt_mul_of_pos_left (show 3*(1-(a:ℝ))<2-(a:ℝ) by linarith) hLr |
| have hp : 3*(k 0:ℝ)<(k 1:ℝ) := by rw [hS,hD]; nlinarith |
| exact_mod_cast hp |
|
|
| theorem exists_six_cubic_inner_weights_large (Q : ℕ) {a d₁ d₂ : ℚ} |
| (ha : (1:ℚ)/2<a) (ha1 : a<1) |
| (hd₁0 : 0<d₁) (hd₁ : d₁<1-a) (hd₂s : 1-a<d₂) (hd₂ : d₂<(2-a)/3) : |
| ∃ L A B D S : ℕ,Q<L ∧ 0<L ∧ 0<A ∧ 0<S ∧ A<S ∧ A≤B ∧ S≤B ∧ 3*B<D ∧ 3*S<D ∧ |
| (A:ℝ)=(L:ℝ)*(d₁:ℝ) ∧ (B:ℝ)=(L:ℝ)*(d₂:ℝ) ∧ |
| (D:ℝ)=(L:ℝ)*(2-(a:ℝ)) ∧ (S:ℝ)=(L:ℝ)*(1-(a:ℝ)) := by |
| obtain ⟨L,A,B,D,S,hL,hA,hS,hAS,hAB,hSB,hBD,hSD,heA,heB,heD,heS⟩ := |
| exists_six_cubic_inner_weights ha ha1 hd₁0 hd₁ hd₂s hd₂ |
| let M := Q+1 |
| have hM : 0<M := by dsimp [M]; omega |
| refine ⟨M*L,M*A,M*B,M*D,M*S,?_,Nat.mul_pos hM hL,Nat.mul_pos hM hA, |
| Nat.mul_pos hM hS,Nat.mul_lt_mul_of_pos_left hAS hM, |
| Nat.mul_le_mul_left M hAB,Nat.mul_le_mul_left M hSB,?_,?_,?_,?_,?_,?_⟩ |
| · have hh := Nat.mul_le_mul_left M (show 1≤L by omega) |
| dsimp [M] at hh ⊢ |
| nlinarith |
| · nlinarith [Nat.mul_lt_mul_of_pos_left hBD hM] |
| · nlinarith [Nat.mul_lt_mul_of_pos_left hSD hM] |
| · push_cast |
| rw [heA] |
| ring |
| · push_cast |
| rw [heB] |
| ring |
| · push_cast |
| rw [heD] |
| ring |
| · push_cast |
| rw [heS] |
| ring |
| end PackingSufficiencySupport.CubicModel |
|
|
| namespace PackingSufficiencySupport |
| open Set |
| open scoped BigOperators Pointwise |
| open MomentPolytope |
|
|
| theorem rational_truncation_pair (F : ℝ×ℝ → ℝ) (hF : Continuous F) |
| {s₁ s₂ l₁ l₂ c : ℝ} (hl₁ : l₁<s₁) (hl₂ : l₂<s₂) (hc : c<F (s₁,s₂)) : |
| ∃ h₁ h₂ d₁ d₂ : ℚ, |
| l₁<h₁ ∧ h₁<d₁ ∧ (d₁:ℝ)<s₁ ∧ |
| l₂<h₂ ∧ h₂<d₂ ∧ (d₂:ℝ)<s₂ ∧ c<F (h₁,h₂) := by |
| let U : Set (Fin 2 → ℝ) := {p | l₁<p 0 ∧ l₂<p 1 ∧ c<F (p 0,p 1)} |
| have hU : IsOpen U := (isOpen_lt continuous_const (continuous_apply 0)).inter |
| ((isOpen_lt continuous_const (continuous_apply 1)).inter |
| (isOpen_lt continuous_const (hF.comp ((continuous_apply 0).prodMk (continuous_apply 1))))) |
| let r : Fin 2 → ℝ := ![s₁,s₂] |
| have hr : r∈U := ⟨hl₁,hl₂,hc⟩ |
| obtain ⟨h,hh,hU⟩ := exists_rational_below_in_open hU hr |
| obtain ⟨d₁,hd₁,hd₁s⟩ := exists_rat_btwn (hh 0) |
| obtain ⟨d₂,hd₂,hd₂s⟩ := exists_rat_btwn (hh 1) |
| exact ⟨h 0,h 1,d₁,d₂,hU.1,by exact_mod_cast hd₁,hd₁s, |
| hU.2.1,by exact_mod_cast hd₂,hd₂s,hU.2.2⟩ |
|
|
| theorem cap_into_scaled_region {m N : ℕ} (b : Fin N → Moments m) (c : Fin N → ℝ) |
| (hb : ∀ ν j,b ν j ≤ 1) {τ r : ℝ} (hτ : 0 < τ) |
| (hr : ∀ ν,r ≤ τ*c ν) : momentSimplex m r ⊆ τ • region b c := by |
| intro p hp |
| let q := τ⁻¹ • p |
| have hq (j : Fin m) : 0 ≤ q j := mul_nonneg (inv_nonneg.2 hτ.le) (hp.1 j) |
| have hsum : (∑ j,q j) ≤ τ⁻¹*r := by |
| simpa only [q,Pi.smul_apply,smul_eq_mul,←Finset.mul_sum] using |
| mul_le_mul_of_nonneg_left hp.2 (inv_nonneg.2 hτ.le) |
| refine ⟨q,⟨hq,fun ν => ?_⟩,?_⟩ |
| · have hweighted : (∑ j,b ν j*q j) ≤ ∑ j,q j := by |
| apply Finset.sum_le_sum |
| intro j _ |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right (hb ν j) (hq j) |
| have hc : τ⁻¹*r ≤ c ν := by |
| have ht := mul_le_mul_of_nonneg_left (hr ν) (inv_nonneg.2 hτ.le) |
| simpa only [←mul_assoc,inv_mul_cancel₀ hτ.ne',one_mul] using ht |
| exact hweighted.trans (hsum.trans hc) |
| · simp [q,smul_smul,hτ.ne'] |
|
|
| theorem sum_caps_zero_outside {m : ℕ} {ι : Type*} [Fintype ι] |
| (r : ι → ℝ) {D A : Set (Fin m → ℝ)} |
| (hD : ∀ p ∈ D,∀ j,0 ≤ p j) (hcap : ∀ i,momentSimplex m (r i) ⊆ A) |
| {p : Fin m → ℝ} (hp : p ∈ D) (hout : p ∉ A) : |
| (∑ i,max (r i-∑ j,p j) 0)=0 := by |
| apply Finset.sum_eq_zero |
| intro i _ |
| apply max_eq_right |
| have hn : ¬ (∑ j,p j) ≤ r i := fun he => hout (hcap i ⟨hD p hp,he⟩) |
| linarith [not_le.1 hn] |
|
|
| end PackingSufficiencySupport |
| end |
|
|
| end OAI |
|
|
