| import Mathlib | |
| import OAI.Probability.JammingConcavity.RowWeightedStep | |
| namespace OAI | |
| /-! Row Quadratic Terminal Limit. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped NNReal ENNReal | |
| open Set Filter | |
| open scoped Topology | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal NNReal BigOperators | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped ENNReal NNReal Topology | |
| open scoped NNReal ENNReal Topology | |
| open scoped NNReal Topology | |
| open Set | |
| open Set Filter MeasureTheory | |
| open scoped BigOperators | |
| open scoped Topology NNReal | |
| open scoped Topology BigOperators | |
| open scoped ENNReal NNReal | |
| open MeasureTheory Set | |
| open MeasureTheory ProbabilityTheory | |
| open scoped ENNReal NNReal BigOperators Classical | |
| open Classical | |
| open scoped ENNReal NNReal Topology BigOperators MatrixOrder | |
| open scoped NNReal BigOperators | |
| open MeasureTheory Metric Set | |
| open Metric | |
| open scoped RealInnerProductSpace | |
| open Filter | |
| open Finset Set | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter Metric | |
| open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction | |
| open scoped BigOperators Classical | |
| open scoped ENNReal NNReal Topology BigOperators Matrix MatrixOrder | |
| open scoped BigOperators RealInnerProductSpace | |
| open scoped NNReal Topology BigOperators | |
| open scoped NNReal BigOperators RealInnerProductSpace | |
| open scoped ENNReal NNReal BigOperators MatrixOrder | |
| open scoped MatrixOrder | |
| open scoped NNReal | |
| open scoped BigOperators NNReal | |
| open scoped NNReal ENNReal BigOperators Topology | |
| open scoped Topology ENNReal NNReal | |
| open scoped Matrix.Norms.L2Operator MatrixOrder Topology NNReal ENNReal BigOperators | |
| open scoped Topology ENNReal NNReal BigOperators MatrixOrder Matrix.Norms.L2Operator | |
| open scoped Topology NNReal ENNReal BigOperators | |
| open scoped BigOperators NNReal Topology | |
| open scoped Topology NNReal ENNReal BigOperators MatrixOrder | |
| open scoped Topology NNReal ENNReal BigOperators MatrixOrder Matrix.Norms.L2Operator | |
| open scoped Topology NNReal ENNReal | |
| open Set Filter MeasureTheory ProbabilityTheory | |
| open scoped Topology BigOperators NNReal | |
| open Set Filter MeasureTheory ProbabilityTheory | |
| open scoped Topology | |
| namespace MicroscopicJamming | |
| lemma row_quadratic_operator_limit {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {K a T : ℝ} | |
| (hK : 0 ≤ K) (ha : 0 ≤ a) (hu : Measurable u) (hv : ∀ n,Measurable (v n)) | |
| (hb : ∀ n x,|v n x| ≤ K*(1+x^2)) (hup : ∀ n x,v n x ≤ 0) | |
| (hlim : ∀ x,Tendsto (fun n => v n x) atTop (𝓝 (u x))) (x : ℝ) : | |
| Tendsto (fun n => gaussianRowOperator a T (v n) x) atTop (𝓝 (gaussianRowOperator a T u x)) := by | |
| have hup' (y : ℝ) : u y ≤ 0 := le_of_tendsto (hlim y) (Eventually.of_forall (fun n => hup n y)) | |
| have hi : Integrable (fun z => K*(1+(x+Real.sqrt T*z)^2)) (gaussianReal 0 1) := | |
| heat_affine_quadratic_integrable (v:=fun y => K*(1+y^2)) (by fun_prop) ⟨K,hK,fun y => by | |
| rw [abs_of_nonneg (by positivity)] | |
| have hh : 1+y^2 ≤ (1+|y|)^2 := by nlinarith [abs_nonneg y,sq_abs y] | |
| exact mul_le_mul_of_nonneg_left hh hK⟩ _ _ | |
| by_cases hz : a=0 | |
| · simp only [gaussianRowOperator,hz,ite_true] | |
| exact tendsto_integral_of_dominated_convergence (fun z => K*(1+(x+Real.sqrt T*z)^2)) | |
| (fun n => ((hv n).comp (by fun_prop)).aestronglyMeasurable) hi | |
| (fun n => Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs] using hb n (x+Real.sqrt T*z))) | |
| (Eventually.of_forall (fun z => hlim _)) | |
| · have heb (n : ℕ) (y : ℝ) : ‖Real.exp (a*v n y)‖ ≤ 1 := by | |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),Real.exp_le_one_iff] | |
| exact mul_nonpos_of_nonneg_of_nonpos ha (hup n y) | |
| have he : Tendsto (fun n => gaussianHeat (fun y => Real.exp (a*v n y)) T x) atTop | |
| (𝓝 (gaussianHeat (fun y => Real.exp (a*u y)) T x)) := by | |
| exact tendsto_integral_of_dominated_convergence (fun _ => (1:ℝ)) | |
| (fun n => (Real.measurable_exp.comp (measurable_const.mul ((hv n).comp (by fun_prop)))).aestronglyMeasurable) (integrable_const 1) (fun n => Eventually.of_forall (fun z => heb n _)) | |
| (Eventually.of_forall (fun z => (Real.continuous_exp.tendsto _).comp ((hlim (x+Real.sqrt T*z)).const_mul a))) | |
| have hui : Integrable (fun z => Real.exp (a*u (x+Real.sqrt T*z))) (gaussianReal 0 1) := by | |
| apply (integrable_const (1:ℝ)).mono' (Real.measurable_exp.comp (measurable_const.mul (hu.comp (by fun_prop)))).aestronglyMeasurable | |
| filter_upwards [] with z | |
| change ‖Real.exp (a*u (x+Real.sqrt T*z))‖ ≤ 1 | |
| rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _),Real.exp_le_one_iff] | |
| exact mul_nonpos_of_nonneg_of_nonpos ha (hup' _) | |
| have hp : 0<gaussianHeat (fun y => Real.exp (a*u y)) T x := integral_exp_pos hui | |
| simpa only [gaussianRowOperator,hz,ite_false] using (he.log hp.ne').const_mul (1/a) | |
| lemma row_quadratic_composition_limit {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {A Q : ℝ} {C : ℕ → ℝ} | |
| (hQ : 0<Q) (hv : ∀ n,RowAnalyticTerminal (v n) A 0 (C n) 0 Q) | |
| (hlim : ∀ x,Tendsto (fun n => v n x) atTop (𝓝 (u x))) | |
| (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) | |
| (hT : gaussianStepTime rs ≤ Q) : | |
| ∀ x,Tendsto (fun n => gaussianRowComposition rs (v n) x) atTop (𝓝 (gaussianRowComposition rs u x)) := by | |
| induction rs with | |
| | nil => exact hlim | |
| | cons r rs ih => | |
| have hr := hrs r (by simp) | |
| have htail : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2 := fun r hr => hrs r (List.mem_cons_of_mem _ hr) | |
| have ht : gaussianStepTime rs ≤ Q := by | |
| have hh : r.2+gaussianStepTime rs ≤ Q := by simpa [gaussianStepTime] using hT | |
| linarith [hr.2.2] | |
| have hlimit := ih htail ht | |
| have hbound (n : ℕ) := gaussian_step_bounds (v n) A 0 (C n) 0 Q hQ (hv n) rs htail ht | |
| have hm (n : ℕ) : Measurable (gaussianRowComposition rs (v n)) := (hbound n).1.continuous.measurable | |
| have hmu : Measurable (gaussianRowComposition rs u) := measurable_of_tendsto_metrizable hm (tendsto_pi_nhds.mpr hlimit) | |
| have hA : 0 ≤ A := (hv 0).2.1 | |
| have hb (n : ℕ) (x : ℝ) : |gaussianRowComposition rs (v n) x| ≤ (A*(1+Q))*(1+x^2) := by | |
| have hh := (hbound n).2.2 x | |
| rw [abs_of_nonpos hh.1.2] | |
| have hh' : A*(1+x^2+gaussianStepTime rs) ≤ (A*(1+Q))*(1+x^2) := by | |
| nlinarith [mul_nonneg (mul_nonneg hA hQ.le) (sq_nonneg x),mul_le_mul_of_nonneg_left ht hA] | |
| linarith [hh.1.1] | |
| intro x | |
| exact row_quadratic_operator_limit (by positivity) hr.1 hmu hm hb (fun n y => ((hbound n).2.2 y).1.2) hlimit x | |
| end MicroscopicJamming | |
| open Set Filter | |
| open scoped Topology | |
| namespace MicroscopicJamming | |
| lemma row_hinge_profile_conclusions {β Q : ℝ} (hβ : 0<β) (hQ : 0 ≤ Q) : | |
| RowProfileConclusions (fun z => -β*rowQuadraticHinge z) Q := by | |
| rcases eq_or_lt_of_le hQ with h0 | hQ | |
| · subst Q; exact rowProfile_zero _ | |
| let R (n : ℕ) := (n:ℝ)+1 | |
| let d (n : ℕ) := 1/R n | |
| have hR (n : ℕ) : 0<R n := by dsimp [R]; linarith [Nat.cast_nonneg (α:=ℝ) n] | |
| have hd (n : ℕ) : 0<d n := by dsimp [d]; positivity | |
| have hd1 (n : ℕ) : d n ≤ 1 := by | |
| dsimp [d] | |
| apply (div_le_one (hR n)).mpr | |
| dsimp [R]; linarith [Nat.cast_nonneg (α:=ℝ) n] | |
| have hdlim : Tendsto d atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ) | |
| choose v C H hv hvb hvc herr using fun n => row_hinge_analytic_approx_one (Q:=Q) hβ (hR n) | |
| (sq_pos_of_pos (hd n)) (show (d n)^2 ≤ 1 by nlinarith [hd n,hd1 n]) | |
| let e (n : ℕ) := β*d n+2*β*(d n)^2 | |
| have hen (n : ℕ) : 0 ≤ e n := by dsimp [e]; positivity | |
| have helim : Tendsto e atTop (𝓝 0) := by | |
| simpa only [zero_pow (by omega : (2:ℕ)≠0),mul_zero,add_zero] using (hdlim.const_mul β).add ((hdlim.pow 2).const_mul (2*β)) | |
| have heq (n : ℕ) : β*R n*(d n)^2+2*β/(R n)^2=e n := by | |
| dsimp [d,e] | |
| field_simp | |
| have herror (n : ℕ) (x : ℝ) : |v n x-(-β*rowQuadraticHinge x)| ≤ e n*(1+x^2)^2 := by | |
| simpa only [heq n] using herr n x | |
| have hlim (x : ℝ) : Tendsto (fun n => v n x) atTop (𝓝 (-β*rowQuadraticHinge x)) := by | |
| rw [tendsto_iff_dist_tendsto_zero] | |
| apply squeeze_zero (g:=fun n => e n*(1+x^2)^2) (fun _ => dist_nonneg) (fun n => by rw [Real.dist_eq]; exact herror n x) | |
| simpa only [zero_mul] using helim.mul_const ((1+x^2)^2) | |
| let L := 8*((3*β)*(1+Q)) | |
| have hL : 0 ≤ L := by dsimp [L]; positivity | |
| let K := Real.exp ((10+8*L)*Q) | |
| have hbound (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) | |
| (ht : gaussianStepTime rs=Q) (n m : ℕ) : | |
| |gaussianRowComposition rs (v n) 0-gaussianRowComposition rs (v m) 0| ≤ (e n+e m)*K := by | |
| apply row_step_weighted_terminal hQ (hv n) (hv m) hL (add_nonneg (hen n) (hen m)) hrs ht | |
| · intro t ht' x; simpa only [abs_zero,zero_add] using row_step_concave_slope hQ (hv n) hrs ht ht' x | |
| · intro t ht' x; simpa only [abs_zero,zero_add] using row_step_concave_slope hQ (hv m) hrs ht ht' x | |
| · intro x | |
| have hh := abs_sub_le (v n x) (-β*rowQuadraticHinge x) (v m x) | |
| rw [abs_sub_comm (-β*rowQuadraticHinge x)] at hh | |
| nlinarith [herror n x,herror m x] | |
| have hfinite (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) (ht : gaussianStepTime rs=Q) := | |
| row_quadratic_composition_limit hQ hv hlim rs hrs ht.le 0 | |
| apply row_terminal_profile_passage hQ hv hvb hvc | |
| (show Tendsto (fun n => e n*K) atTop (𝓝 0) by simpa only [zero_mul] using helim.mul_const K) _ hfinite | |
| intro rs hrs ht n | |
| have hh := le_of_tendsto_of_tendsto ((tendsto_const_nhds.sub (hfinite rs hrs ht)).abs) | |
| ((tendsto_const_nhds.add helim).mul_const K) (Eventually.of_forall (hbound rs hrs ht n)) | |
| simpa only [add_zero,abs_sub_comm] using hh | |
| end MicroscopicJamming | |
| open Set Filter | |
| open scoped Topology | |
| namespace MicroscopicJamming | |
| theorem rowNonsmoothConcavity : RowNonsmoothConcavityStatement := by | |
| intro u hu Q hQ | |
| rcases hu with ⟨hc,L,hL,hlip⟩ | ⟨β,hβ,rfl⟩ | |
| · exact row_lipschitz_profile_conclusions hc hL hlip hQ | |
| · exact row_hinge_profile_conclusions hβ hQ | |
| end MicroscopicJamming | |
| open Set Filter | |
| open scoped Topology | |
| namespace MicroscopicJamming | |
| theorem rowConcavityMain : RowConcavityMainStatement := by | |
| constructor | |
| · intro u A B C κ Q hQ hu hc p q hp hq θ hθ hθ1 | |
| exact rowProfileValue_smooth_concave hQ hu hc p q hp hq hθ hθ1 | |
| · exact rowNonsmoothConcavity | |
| end MicroscopicJamming | |
| end | |
| end OAI | |