openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompactFiberDecomposition.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.UnitSquarePoincare | |
| import OAI.Analysis.Conductivity.Walls.WallMomentIndependence | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| section | |
| open Set MeasureTheory Filter Topology | |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] | |
| def compactFiberMarginal (a b : ℝ) (f : E × ℝ → ℝ) (x : E) : ℝ := | |
| ∫ t in a..b, f (x,t) | |
| lemma compactFiberMarginal_smooth {a b : ℝ} (hab : a≤b) {f : E × ℝ → ℝ} | |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) : | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (compactFiberMarginal a b f) := by | |
| have he : compactFiberMarginal a b f=fun x => ∫ t in Icc a b, f (x,t) := by | |
| funext x | |
| rw [compactFiberMarginal,intervalIntegral.integral_of_le hab,integral_Icc_eq_integral_Ioc] | |
| rw [he] | |
| exact contDiff_compact_integral hf isCompact_Icc | |
| omit [NormedSpace ℝ E] [ProperSpace E] in | |
| lemma compactFiberMarginal_support {a b : ℝ} {f : E × ℝ → ℝ} | |
| (hs : HasCompactSupport f) : | |
| HasCompactSupport (compactFiberMarginal a b f) ∧ | |
| tsupport (compactFiberMarginal a b f) ⊆ Prod.fst '' tsupport f := by | |
| have hc := hs.image continuous_fst | |
| have hsub : Function.support (compactFiberMarginal a b f) ⊆ Prod.fst '' tsupport f := by | |
| intro x hx | |
| by_contra hn | |
| have hz : ∀ t, f (x,t)=0 := by | |
| intro t | |
| by_contra hf | |
| exact hn ⟨(x,t),subset_tsupport f hf,rfl⟩ | |
| exact hx (by simp [compactFiberMarginal,hz]) | |
| exact ⟨HasCompactSupport.of_support_subset_isCompact hc hsub,closure_minimal hsub hc.isClosed⟩ | |
| lemma exists_unit_interval_bump {a b : ℝ} (hab : a<b) : | |
| ∃ η : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) η ∧ HasCompactSupport η ∧ | |
| tsupport η ⊆ Ioo a b ∧ (∫ t in a..b, η t)=1 := by | |
| let l := (2*a+b)/3 | |
| let r := (a+2*b)/3 | |
| have hal : a<l := by dsimp [l]; linarith | |
| have hlr : l<r := by dsimp [l,r]; linarith | |
| have hrb : r<b := by dsimp [r]; linarith | |
| obtain ⟨χ,hc,hs,hn,hp,hsub⟩ := positive_interval_bump hlr | |
| have hI : 0<∫ t, χ t := by | |
| apply integral_pos_iff_support_of_nonneg (fun t => hn t) (hc.continuous.integrable_of_hasCompactSupport hs) |>.2 | |
| apply lt_of_lt_of_le ((Measure.measure_Ioo_pos volume).mpr hlr) | |
| exact measure_mono (fun t ht => (hp t ht).ne') | |
| let η := fun t => χ t/(∫ s, χ s) | |
| have hη : ContDiff ℝ (↑(⊤ : ℕ∞)) η := hc.div_const _ | |
| have hsη : HasCompactSupport η := by | |
| apply HasCompactSupport.of_support_subset_isCompact hs | |
| intro t ht | |
| apply subset_tsupport χ | |
| exact (div_ne_zero_iff.mp ht).1 | |
| have hsubη : tsupport η ⊆ Ioo a b := by | |
| change tsupport (fun t => χ t*(∫ s, χ s)⁻¹) ⊆ _ | |
| apply tsupport_mul_subset_left.trans | |
| exact hsub.trans (fun t ht => ⟨hal.trans_le ht.1,ht.2.trans_lt hrb⟩) | |
| refine ⟨η,hη,hsη,hsubη,?_⟩ | |
| rw [intervalIntegral.integral_of_le hab.le] | |
| have he : ∫ t in Ioc a b, η t=∫ t, η t := by | |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero | |
| intro t ht | |
| apply image_eq_zero_of_notMem_tsupport | |
| exact fun hh => ht ⟨(hsubη hh).1,(hsubη hh).2.le⟩ | |
| rw [he] | |
| dsimp [η] | |
| rw [integral_div,div_self hI.ne'] | |
| theorem compact_fiber_decomposition {a b : ℝ} (hab : a<b) | |
| {f : E × ℝ → ℝ} (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) | |
| (hs : HasCompactSupport f) (hv : ∀ p, f p≠0 → p.2∈Ioo a b) | |
| {η : ℝ → ℝ} (hη : ContDiff ℝ (↑(⊤ : ℕ∞)) η) (hsη : HasCompactSupport η) | |
| (hvη : tsupport η⊆Ioo a b) (hiη : (∫ t in a..b, η t)=1) : | |
| ∃ F : E × ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) F ∧ HasCompactSupport F ∧ | |
| (∀ p, wallDerivative F p+η p.2*compactFiberMarginal a b f p.1=f p) ∧ | |
| tsupport F⊆(Prod.fst '' tsupport f) ×ˢ Icc a b := by | |
| let R := compactFiberMarginal a b f | |
| have hR := compactFiberMarginal_smooth hab.le hf | |
| have hsR : HasCompactSupport R := (compactFiberMarginal_support (a := a) (b := b) hs).1 | |
| let g := fun p : E × ℝ => f p-η p.2*R p.1 | |
| have hg : ContDiff ℝ (↑(⊤ : ℕ∞)) g := hf.sub ((hη.comp contDiff_snd).mul (hR.comp contDiff_fst)) | |
| have hsg : HasCompactSupport g := by | |
| apply hs.sub | |
| apply HasCompactSupport.of_support_subset_isCompact (hsR.prod hsη) | |
| intro p hp | |
| exact ⟨subset_tsupport R (mul_ne_zero_iff.mp hp).2,subset_tsupport η (mul_ne_zero_iff.mp hp).1⟩ | |
| have hvg : ∀ p, g p≠0 → p.2∈Ioo a b := by | |
| intro p hp | |
| by_contra hn | |
| have hz : f p=0 := by by_contra hh; exact hn (hv p hh) | |
| have hzη : η p.2=0 := image_eq_zero_of_notMem_tsupport (fun hh => hn (hvη hh)) | |
| exact hp (by simp [g,hz,hzη]) | |
| have hzg : ∀ x, (∫ t in a..b, g (x,t))=0 := by | |
| intro x | |
| rw [show (fun t => g (x,t))=(fun t => f (x,t)-η t*R x) from rfl, | |
| intervalIntegral.integral_sub (f := fun t => f (x,t)) (g := fun t => η t*R x) | |
| ((hf.continuous.comp (continuous_const.prodMk continuous_id)).intervalIntegrable a b) | |
| ((hη.continuous.mul continuous_const).intervalIntegrable a b)] | |
| rw [intervalIntegral.integral_mul_const,hiη,one_mul] | |
| exact sub_self _ | |
| refine ⟨fiberPrimitive a g,fiberPrimitive_smooth a hg,(fiberPrimitive_compact hg.continuous hsg hvg hzg).1,?_,?_⟩ | |
| · intro p | |
| rw [fiberPrimitive_derivative a hg] | |
| exact sub_add_cancel _ _ | |
| · apply ((fiberPrimitive_compact hg.continuous hsg hvg hzg).2).trans | |
| intro p hp | |
| refine ⟨?_,hp.2⟩ | |
| obtain ⟨q,hq,hqp⟩ := hp.1 | |
| by_contra hn | |
| have hfg : Function.support g⊆(Prod.fst '' tsupport f) ×ˢ univ := by | |
| intro y hy | |
| refine ⟨?_,mem_univ _⟩ | |
| by_contra hyf | |
| have hz : f y=0 := by | |
| by_contra h | |
| exact hyf ⟨y,subset_tsupport f h,rfl⟩ | |
| have hzR : R y.1=0 := image_eq_zero_of_notMem_tsupport | |
| (fun h => hyf ((compactFiberMarginal_support hs).2 h)) | |
| exact hy (by simp [g,hz,hzR]) | |
| have hts := closure_minimal hfg ((hs.image continuous_fst).isClosed.prod isClosed_univ) | |
| exact hn (hqp ▸ (hts hq).1) | |
| end | |
| open Set MeasureTheory | |
| lemma smooth_unit_interval_trace {q : ℝ → ℝ} | |
| (hq : ContDiff ℝ (↑(⊤ : ℕ∞)) q) {a : ℝ} (ha : a∈Icc (0:ℝ) 1) : | |
| (q a)^2≤2*unitAverage (fun t => (q t)^2)+ | |
| 2*unitAverage (fun t => (deriv q t)^2) := by | |
| have hd := hq.continuous_deriv (by simp) | |
| have hp (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : | |
| (q a)^2≤2*(q t)^2+2*unitAverage (fun t => (deriv q t)^2) := by | |
| have hh : (q a-q t)^2≤unitAverage (fun t => (deriv q t)^2) := by | |
| rcases le_total t a with h|h | |
| · have hb := smooth_interval_poincare_point hq h | |
| have hi : (∫ s in t..a, (deriv q s)^2)≤unitAverage (fun s => (deriv q s)^2) := by | |
| rw [unitAverage_eq_interval] | |
| exact intervalIntegral.integral_mono_interval ht.1 h ha.2 | |
| (Filter.Eventually.of_forall (fun _ => sq_nonneg _)) ((hd.pow 2).intervalIntegrable 0 1) | |
| have hpos : 0≤∫ s in t..a, (deriv q s)^2 := | |
| intervalIntegral.integral_nonneg h (fun _ _ => sq_nonneg _) | |
| have hlen : a-t≤1 := by linarith [ht.1,ha.2] | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hlen) hpos] | |
| · have hb := smooth_interval_poincare_point hq h | |
| have hi : (∫ s in a..t, (deriv q s)^2)≤unitAverage (fun s => (deriv q s)^2) := by | |
| rw [unitAverage_eq_interval] | |
| exact intervalIntegral.integral_mono_interval ha.1 h ht.2 | |
| (Filter.Eventually.of_forall (fun _ => sq_nonneg _)) ((hd.pow 2).intervalIntegrable 0 1) | |
| have hpos : 0≤∫ s in a..t, (deriv q s)^2 := | |
| intervalIntegral.integral_nonneg h (fun _ _ => sq_nonneg _) | |
| have hlen : t-a≤1 := by linarith [ha.1,ht.2] | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hlen) hpos] | |
| nlinarith [sq_nonneg (q a-2*q t)] | |
| have hh := unitAverage_mono (continuous_const : Continuous (fun _ : ℝ => (q a)^2)) | |
| ((continuous_const.mul (hq.continuous.pow 2)).add continuous_const) hp | |
| change unitAverage (fun _ : ℝ => (q a)^2)≤ | |
| unitAverage (fun t => 2*(q t)^2+2*unitAverage (fun t => (deriv q t)^2)) at hh | |
| rw [unitAverage_const,unitAverage_add (f := fun t => 2*(q t)^2) | |
| (g := fun _ => 2*unitAverage (fun t => (deriv q t)^2)) | |
| (continuous_const.mul (hq.continuous.pow 2)) continuous_const, | |
| unitAverage_mul,unitAverage_const] at hh | |
| exact hh | |
| theorem smooth_unit_face_trace {f : Box3 → ℝ} | |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) {a : ℝ} (ha : a∈Icc (0:ℝ) 1) : | |
| unitSquareAverage (fun y => (f (y,a))^2)≤ | |
| 2*unitCubeAverage (fun z => (f z)^2)+ | |
| 2*unitCubeAverage (fun z => (cubePartial f ((0,0),1) z)^2) := by | |
| have hd := continuous_cubePartial hf ((0,0),1) | |
| have hp (y : ℝ×ℝ) : (f (y,a))^2≤2*unitAverage (fun t => (f (y,t))^2)+ | |
| 2*unitAverage (fun t => (cubePartial f ((0,0),1) (y,t))^2) := by | |
| have hh := smooth_unit_interval_trace | |
| (hf.comp (contDiff_const.prodMk contDiff_id) : | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun t : ℝ => f (y,t))) ha | |
| have he (t : ℝ) : deriv (fun s : ℝ => f (y,s)) t=cubePartial f ((0,0),1) (y,t) := by | |
| exact ((((hf.differentiable (by simp)) (y,t)).hasFDerivAt).comp_hasDerivAt t | |
| ((hasDerivAt_const t y).prodMk (hasDerivAt_id t))).deriv | |
| change (f (y,a))^2≤2*unitAverage (fun t => (f (y,t))^2)+ | |
| 2*unitAverage (fun t => (deriv (fun s => f (y,s)) t)^2) at hh | |
| simpa only [he] using hh | |
| have hh := unitSquareAverage_mono | |
| ((hf.continuous.comp (continuous_id.prodMk continuous_const)).pow 2) | |
| ((continuous_const.mul (continuous_unitAverage (hf.continuous.pow 2))).add | |
| (continuous_const.mul (continuous_unitAverage (hd.pow 2)))) | |
| (fun y _ x _ => hp (y,x)) | |
| change unitSquareAverage (fun y => (f (y,a))^2)≤ | |
| unitSquareAverage (fun y => 2*unitAverage (fun t => (f (y,t))^2)+ | |
| 2*unitAverage (fun t => (cubePartial f ((0,0),1) (y,t))^2)) at hh | |
| rw [unitSquareAverage_add | |
| (f := fun y => 2*unitAverage (fun t => (f (y,t))^2)) | |
| (g := fun y => 2*unitAverage (fun t => (cubePartial f ((0,0),1) (y,t))^2)) | |
| (continuous_const.mul (continuous_unitAverage (hf.continuous.pow 2))) | |
| (continuous_const.mul (continuous_unitAverage (hd.pow 2))), | |
| unitSquareAverage_mul,unitSquareAverage_mul] at hh | |
| exact hh | |
| end ScalarConductivity | |
| end | |
| end OAI | |