| import OAI.Probability.RadialTransition.ThermodynamicLimit |
|
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| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators NNReal ContDiff ENNReal |
| open Filter MeasureTheory Set |
|
|
| namespace RadialTransition |
| |
| instance (d : ℝ) : IsCountablyGenerated (thermodynamicFilter d) := by |
| unfold thermodynamicFilter |
| infer_instance |
| |
| def HasBoxReplication (H : ℝ → ℕ → ℝ) : Prop := |
| ∀ (s L : ℝ) (b : ℕ), 0 < s → 0 < L → (b : ℝ) * (s + 2) ≤ L → |
| ∀ m : Fin (b ^ 3) → ℕ, |
| H L (∑ j, m j) ≤ s ^ 3 / L ^ 3 * ∑ j, H s (m j) |
| |
| theorem boxCost_adjustment {H : ℝ → ℕ → ℝ} (hrep : HasBoxReplication H) |
| {s L : ℝ} (hs : 0 < s) (hL : 0 < L) (b m q N : ℕ) |
| (hbL : (b : ℝ) * (s + 2) ≤ L) (hm : 0 < m) (hmq : m < q) |
| (hN : N ≤ b ^ 3 * q) {c K : ℝ} (hc0 : 0 < c) (hcm : c ≤ (m : ℝ)) |
| (hcq : c ≤ ((q - m : ℕ) : ℝ)) (hK : 0 ≤ K) |
| (hcost : ∀ j ≤ q, |H s j| ≤ K) : |
| H L N ≤ s ^ 3 / L ^ 3 * |
| ((b : ℝ) ^ 3 * H s m + 2 * K * (|(N : ℝ) - (b : ℝ) ^ 3 * m| / c + 1)) := by |
| obtain ⟨f, hfN, hfq, hfc⟩ := exists_integer_adjustment (b ^ 3) m q N hm hmq hN hc0 hcm hcq |
| have hr := hrep s L b hs hL hbL f |
| rw [hfN] at hr |
| have hsum := sum_changed_cost f m (H s) (hcost m hmq.le) (fun i => hcost (f i) (hfq i)) |
| simp only [Fintype.card_fin, Nat.cast_pow] at hsum hfc |
| apply hr.trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| exact hsum.trans (by gcongr) |
| |
| theorem boxCost_limsup_bound {H : ℝ → ℕ → ℝ} (hrep : HasBoxReplication H) |
| {d s c K : ℝ} (hd : 0 < d) (hs : 0 < s) (hc : 0 < c) (hK : 0 ≤ K) |
| (m q : ℕ) (hm : 0 < m) (hmq : m < q) |
| (hcm : c * s ^ 3 ≤ (m : ℝ)) (hcq : c * s ^ 3 ≤ ((q - m : ℕ) : ℝ)) |
| (hcap : d < (q : ℝ) / (s + 2) ^ 3) (hcost : ∀ j ≤ q, |H s j| ≤ K) |
| (hbelow : IsBoundedUnder (· ≥ ·) (thermodynamicFilter d) (fun z => H z.1 z.2)) : |
| limsup (fun z => H z.1 z.2) (thermodynamicFilter d) ≤ |
| (s / (s + 2)) ^ 3 * H s m + |
| 2 * K / c * |d - (s / (s + 2)) ^ 3 * ((m : ℝ) / s ^ 3)| := by |
| let _ := thermodynamicFilter_neBot hd.le |
| let F := thermodynamicFilter d |
| let b (z : ℝ × ℕ) : ℕ := ⌊z.1 / (s + 2)⌋₊ |
| let r (z : ℝ × ℕ) : ℝ := ((b z : ℝ) / z.1) ^ 3 |
| have hr : Tendsto r F (nhds ((s + 2)⁻¹ ^ 3)) := by |
| have hh := ((tendsto_nat_floor_mul_div_atTop |
| (show 0 ≤ (s + 2)⁻¹ by positivity)).pow 3).comp (thermodynamicFilter_side d) |
| simpa only [r, b, Function.comp_def, div_eq_mul_inv, mul_comm] using hh |
| have hden : Tendsto (fun z : ℝ × ℕ => (z.1 ^ 3)⁻¹) F (nhds 0) := |
| tendsto_inv_atTop_zero.comp ((tendsto_pow_atTop (by norm_num : 3 ≠ 0)).comp |
| (thermodynamicFilter_side d)) |
| let G (z : ℝ × ℕ) : ℝ := s ^ 3 * (r z * H s m + |
| 2 * K * (|(z.2 : ℝ) / z.1 ^ 3 - r z * m| / (c * s ^ 3) + (z.1 ^ 3)⁻¹)) |
| have hG : Tendsto G F (nhds (s ^ 3 * ((s + 2)⁻¹ ^ 3 * H s m + |
| 2 * K * (|d - (s + 2)⁻¹ ^ 3 * m| / (c * s ^ 3))))) := by |
| have hh := ((hr.mul_const (H s m)).add |
| (((((thermodynamicFilter_density d).sub (hr.mul_const (m : ℝ))).abs.div_const |
| (c * s ^ 3)).add hden).const_mul (2 * K))).const_mul (s ^ 3) |
| simpa only [add_zero] using hh |
| have hcaplimit : Tendsto (fun z => r z * q) F (nhds ((q : ℝ) / (s + 2) ^ 3)) := by |
| simpa only [inv_pow, div_eq_mul_inv, mul_comm] using hr.mul_const (q : ℝ) |
| obtain ⟨a, hda, haq⟩ := exists_between hcap |
| have hN : ∀ᶠ z in F, (z.2 : ℝ) / z.1 ^ 3 < r z * q := by |
| filter_upwards [(thermodynamicFilter_density d).eventually_lt_const hda, |
| hcaplimit.eventually_const_lt haq] with z hz hq |
| exact hz.trans hq |
| have hle : ∀ᶠ z in F, H z.1 z.2 ≤ G z := by |
| filter_upwards [hN, (thermodynamicFilter_side d).eventually (eventually_gt_atTop (0 : ℝ))] |
| with z hNz hL |
| have hbL : (b z : ℝ) * (s + 2) ≤ z.1 := |
| (le_div_iff₀ (by positivity : 0 < s + 2)).mp (Nat.floor_le (by positivity)) |
| have hNe : z.2 ≤ b z ^ 3 * q := by |
| have he : r z * q = ((b z : ℝ) ^ 3 * q) / z.1 ^ 3 := by dsimp [r]; ring |
| rw [he] at hNz |
| have hh := (div_lt_div_iff_of_pos_right (pow_pos hL 3)).mp hNz |
| exact_mod_cast hh.le |
| have hh := boxCost_adjustment hrep hs hL (b z) m q z.2 hbL hm hmq hNe |
| (mul_pos hc (pow_pos hs 3)) hcm hcq hK hcost |
| apply hh.trans_eq |
| have habs : |(z.2 : ℝ) / z.1 ^ 3 - r z * m| = |
| |(z.2 : ℝ) - (b z : ℝ) ^ 3 * m| / z.1 ^ 3 := by |
| dsimp only [r] |
| rw [div_pow, div_mul_eq_mul_div, ← sub_div, abs_div, abs_of_pos (pow_pos hL 3)] |
| dsimp only [G] |
| rw [habs] |
| dsimp only [r] |
| ring |
| have hh := limsup_le_limsup hle hbelow.isCobounded_le hG.isBoundedUnder_le |
| rw [hG.limsup_eq] at hh |
| apply hh.trans_eq |
| field_simp |
| |
| theorem tendsto_side_ratio : |
| Tendsto (fun s : ℝ => s / (s + 2)) atTop (nhds 1) := by |
| have hh := ((tendsto_inv_atTop_zero.comp |
| (tendsto_atTop_add_const_right atTop (2 : ℝ) tendsto_id)).const_mul 2).const_sub 1 |
| simp only [mul_zero, sub_zero] at hh |
| apply hh.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with s hs |
| simp only [Function.comp_def, id_eq] |
| field_simp |
| ring |
| |
| theorem boxCost_limit {H : ℝ → ℕ → ℝ} (hrep : HasBoxReplication H) |
| (hbnd : ∀ D : ℝ, 0 < D → ∃ K : ℝ, 0 ≤ K ∧ |
| ∀ s : ℝ, 1 ≤ s → ∀ n : ℕ, (n : ℝ) / s ^ 3 ≤ D → |H s n| ≤ K) |
| {d : ℝ} (hd : 0 < d) : |
| ∃ A : ℝ, Tendsto (fun z : ℝ × ℕ => H z.1 z.2) (thermodynamicFilter d) (nhds A) := by |
| let _ := thermodynamicFilter_neBot hd.le |
| obtain ⟨K, hK, hbound⟩ := hbnd (2 * d) (by positivity) |
| have hevent : ∀ᶠ z in thermodynamicFilter d, |H z.1 z.2| ≤ K := by |
| filter_upwards [(thermodynamicFilter_side d).eventually (eventually_ge_atTop (1 : ℝ)), |
| (thermodynamicFilter_density d).eventually_lt_const (show d < 2 * d by linarith)] |
| with z hz hdens |
| exact hbound z.1 hz z.2 hdens.le |
| have hbelow : IsBoundedUnder (· ≥ ·) (thermodynamicFilter d) (fun z => H z.1 z.2) := |
| ⟨-K, hevent.mono (fun z hz => (abs_le.mp hz).1)⟩ |
| have habove : IsBoundedUnder (· ≤ ·) (thermodynamicFilter d) (fun z => H z.1 z.2) := |
| ⟨K, hevent.mono (fun z hz => (abs_le.mp hz).2)⟩ |
| let A := liminf (fun z => H z.1 z.2) (thermodynamicFilter d) |
| obtain ⟨x, hxH, hx⟩ := exists_seq_tendsto_liminf habove.isCobounded_ge hbelow |
| let s (j : ℕ) := (x j).1 |
| let m (j : ℕ) := (x j).2 |
| let q (j : ℕ) : ℕ := ⌊2 * d * s j ^ 3⌋₊ |
| have hs : Tendsto s atTop atTop := (thermodynamicFilter_side d).comp hx |
| have hm : Tendsto (fun j => (m j : ℝ) / s j ^ 3) atTop (nhds d) := |
| (thermodynamicFilter_density d).comp hx |
| have hq : Tendsto (fun j => (q j : ℝ) / s j ^ 3) atTop (nhds (2 * d)) := |
| (floor_density_tendsto (show 0 ≤ 2 * d by positivity)).comp hs |
| have hratio := tendsto_side_ratio.comp hs |
| have hcap : Tendsto (fun j => (q j : ℝ) / (s j + 2) ^ 3) atTop (nhds (2 * d)) := by |
| have hh := hq.mul (hratio.pow 3) |
| simp only [one_pow, mul_one] at hh |
| apply hh.congr' |
| filter_upwards [hs.eventually (eventually_gt_atTop (0 : ℝ))] with j hj |
| simp only [Function.comp_def] |
| field_simp |
| have hdiff : Tendsto (fun j => ((q j : ℝ) - m j) / s j ^ 3) atTop (nhds d) := by |
| have hh := hq.sub hm |
| simp only [← sub_div] at hh |
| convert hh using 1 |
| congr 1 |
| ring |
| have href : ∀ᶠ j in atTop, |
| limsup (fun z => H z.1 z.2) (thermodynamicFilter d) ≤ |
| (s j / (s j + 2)) ^ 3 * H (s j) (m j) + |
| 2 * K / (d / 2) * |d - (s j / (s j + 2)) ^ 3 * ((m j : ℝ) / s j ^ 3)| := by |
| filter_upwards [hs.eventually (eventually_ge_atTop (1 : ℝ)), |
| hm.eventually_const_lt (show d / 2 < d by linarith), |
| hdiff.eventually_const_lt (show d / 2 < d by linarith), |
| hcap.eventually_const_lt (show d < 2 * d by linarith)] with j hsj hmj hdiffj hcapj |
| have hsj0 : 0 < s j := by linarith |
| have hcm : d / 2 * s j ^ 3 < (m j : ℝ) := |
| (lt_div_iff₀ (pow_pos hsj0 3)).mp hmj |
| have hcq : d / 2 * s j ^ 3 < (q j : ℝ) - m j := |
| (lt_div_iff₀ (pow_pos hsj0 3)).mp hdiffj |
| have hm0 : 0 < m j := by |
| have hh : (0 : ℝ) < m j := (by positivity : (0 : ℝ) < d / 2 * s j ^ 3).trans hcm |
| exact_mod_cast hh |
| have hmq : m j < q j := by |
| have hh : (m j : ℝ) < q j := by linarith [show 0 < d / 2 * s j ^ 3 by positivity] |
| exact_mod_cast hh |
| apply boxCost_limsup_bound hrep hd hsj0 (show 0 < d / 2 by positivity) hK (m j) (q j) |
| hm0 hmq hcm.le |
| · rw [Nat.cast_sub hmq.le] |
| exact hcq.le |
| · exact hcapj |
| · intro n hn |
| apply hbound (s j) hsj n |
| apply (div_le_iff₀ (pow_pos hsj0 3)).mpr |
| exact (show (n : ℝ) ≤ q j by exact_mod_cast hn).trans (Nat.floor_le (by positivity)) |
| · exact hbelow |
| have hlim : Tendsto (fun j => (s j / (s j + 2)) ^ 3 * H (s j) (m j) + |
| 2 * K / (d / 2) * |d - (s j / (s j + 2)) ^ 3 * ((m j : ℝ) / s j ^ 3)|) |
| atTop (nhds A) := by |
| have hh := ((hratio.pow 3).mul hxH).add |
| ((((hratio.pow 3).mul hm).const_sub d).abs.const_mul (2 * K / (d / 2))) |
| simpa only [one_pow, one_mul, sub_self, abs_zero, mul_zero, add_zero, Function.comp_def, s, m, A] using hh |
| refine ⟨A, tendsto_of_le_liminf_of_limsup_le le_rfl ?_ habove hbelow⟩ |
| exact ge_of_tendsto hlim href |
| |
| theorem boxCost_limit_with_bounds {H : ℝ → ℕ → ℝ} (hrep : HasBoxReplication H) |
| {l u : ℝ → ℝ} (hl : Continuous l) (hu : Continuous u) (hzero : l 0 = u 0) |
| (hb : ∀ s : ℝ, 1 ≤ s → ∀ n : ℕ, |
| l ((n : ℝ) / s ^ 3) ≤ H s n ∧ H s n ≤ u ((n : ℝ) / s ^ 3)) |
| {d : ℝ} (hd : 0 ≤ d) : |
| ∃ A : ℝ, Tendsto (fun z : ℝ × ℕ => H z.1 z.2) (thermodynamicFilter d) (nhds A) ∧ |
| l d ≤ A ∧ A ≤ u d := by |
| let _ := thermodynamicFilter_neBot hd |
| have hlo : Tendsto (fun z : ℝ × ℕ => l ((z.2 : ℝ) / z.1 ^ 3)) |
| (thermodynamicFilter d) (nhds (l d)) := hl.continuousAt.tendsto.comp (thermodynamicFilter_density d) |
| have hup : Tendsto (fun z : ℝ × ℕ => u ((z.2 : ℝ) / z.1 ^ 3)) |
| (thermodynamicFilter d) (nhds (u d)) := hu.continuousAt.tendsto.comp (thermodynamicFilter_density d) |
| have hevent : ∀ᶠ z in thermodynamicFilter d, |
| l ((z.2 : ℝ) / z.1 ^ 3) ≤ H z.1 z.2 ∧ H z.1 z.2 ≤ u ((z.2 : ℝ) / z.1 ^ 3) := by |
| filter_upwards [(thermodynamicFilter_side d).eventually (eventually_ge_atTop (1 : ℝ))] with z hz |
| exact hb z.1 hz z.2 |
| obtain rfl | hdpos := hd.eq_or_lt |
| · refine ⟨l 0, ?_, le_rfl, hzero.le⟩ |
| rw [← hzero] at hup |
| exact tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo hup |
| (hevent.mono (fun _ hz => hz.1)) (hevent.mono (fun _ hz => hz.2)) |
| · have hlocal : ∀ D : ℝ, 0 < D → ∃ K : ℝ, 0 ≤ K ∧ |
| ∀ s : ℝ, 1 ≤ s → ∀ n : ℕ, (n : ℝ) / s ^ 3 ≤ D → |H s n| ≤ K := by |
| intro D _ |
| obtain ⟨Cl, hCl⟩ := isCompact_Icc.exists_bound_of_continuousOn (s := Icc (0 : ℝ) D) hl.continuousOn |
| obtain ⟨Cu, hCu⟩ := isCompact_Icc.exists_bound_of_continuousOn (s := Icc (0 : ℝ) D) hu.continuousOn |
| refine ⟨max 0 (max Cl Cu), le_max_left _ _, ?_⟩ |
| intro s hs n hn |
| have hds : (n : ℝ) / s ^ 3 ∈ Icc (0 : ℝ) D := ⟨by positivity, hn⟩ |
| have hll : |l ((n : ℝ) / s ^ 3)| ≤ max 0 (max Cl Cu) := by |
| have hh := hCl _ hds |
| rw [Real.norm_eq_abs] at hh |
| exact hh.trans ((le_max_left _ _).trans (le_max_right _ _)) |
| have huu : |u ((n : ℝ) / s ^ 3)| ≤ max 0 (max Cl Cu) := by |
| have hh := hCu _ hds |
| rw [Real.norm_eq_abs] at hh |
| exact hh.trans ((le_max_right _ _).trans (le_max_right _ _)) |
| exact abs_le.mpr ⟨(abs_le.mp hll).1.trans (hb s hs n).1, |
| (hb s hs n).2.trans (abs_le.mp huu).2⟩ |
| obtain ⟨A, hA⟩ := boxCost_limit hrep hlocal hdpos |
| exact ⟨A, hA, le_of_tendsto_of_tendsto hlo hA (hevent.mono (fun _ hz => hz.1)), |
| le_of_tendsto_of_tendsto hA hup (hevent.mono (fun _ hz => hz.2))⟩ |
| |
| theorem finiteCanonical_limit {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {t β c B K : ℝ} (ht : 0 < t) (hβ : 0 ≤ β) |
| (hc : ∀ r : ℝ≥0, 2 ≤ (r : ℝ) → φ r ≤ 0) |
| (hlow : ∀ L : ℝ, 1 ≤ L → ∀ (N : ℕ) (x : Fin N → Position), |
| x ∈ configurationCube L N → c * (N : ℝ)^2 / L^3 - B * N ≤ energy φ x) |
| (hmean : ∀ L : ℝ, 1 ≤ L → ∀ N : ℕ, |
| (∫ x : Fin N → Position, energy φ x |
| ∂MeasureTheory.Measure.pi (fun _ : Fin N => cubeProbability L)) ≤ K * (N : ℝ)^2 / L^3) |
| {d : ℝ} (hd : 0 ≤ d) : |
| ∃ A : ℝ, Tendsto (fun z : ℝ × ℕ => finiteCanonical φ t z.1 z.2 β) |
| (thermodynamicFilter d) (nhds A) ∧ |
| β / t * (c * d ^ 2 - B * d) + (d * Real.log d - d) / t ≤ A ∧ |
| A ≤ β / t * K * d ^ 2 + d * Real.log d / t := by |
| apply boxCost_limit_with_bounds (H := fun L N => finiteCanonical φ t L N β) (d := d) |
| (fun s L b hs hL hbL m => finiteCanonical_replication hφ hb hβ ht hs hL hc b hbL m) |
| (l := fun d => β / t * (c * d ^ 2 - B * d) + (d * Real.log d - d) / t) |
| (u := fun d => β / t * K * d ^ 2 + d * Real.log d / t) |
| · fun_prop (disch := exact Real.continuous_mul_log) |
| · fun_prop (disch := exact Real.continuous_mul_log) |
| · simp |
| · intro L hL N |
| exact finiteCanonical_bounds hφ hb (by linarith) ht hβ N (hlow L hL N) (hmean L hL N) |
| · exact hd |
| |
| def canonicalThermo (φ : Potential) (t β d : ℝ) : ℝ := |
| liminf (fun z : ℝ × ℕ => finiteCanonical φ t z.1 z.2 β) (thermodynamicFilter d) |
| |
| theorem canonicalThermo_spec {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {t β c B K : ℝ} (ht : 0 < t) (hβ : 0 ≤ β) |
| (hc : ∀ r : ℝ≥0, 2 ≤ (r : ℝ) → φ r ≤ 0) |
| (hlow : ∀ L : ℝ, 1 ≤ L → ∀ (N : ℕ) (x : Fin N → Position), |
| x ∈ configurationCube L N → c * (N : ℝ)^2 / L^3 - B * N ≤ energy φ x) |
| (hmean : ∀ L : ℝ, 1 ≤ L → ∀ N : ℕ, |
| (∫ x : Fin N → Position, energy φ x |
| ∂MeasureTheory.Measure.pi (fun _ : Fin N => cubeProbability L)) ≤ K * (N : ℝ)^2 / L^3) |
| {d : ℝ} (hd : 0 ≤ d) : |
| Tendsto (fun z : ℝ × ℕ => finiteCanonical φ t z.1 z.2 β) |
| (thermodynamicFilter d) (nhds (canonicalThermo φ t β d)) ∧ |
| β / t * (c * d ^ 2 - B * d) + (d * Real.log d - d) / t ≤ canonicalThermo φ t β d ∧ |
| canonicalThermo φ t β d ≤ β / t * K * d ^ 2 + d * Real.log d / t := by |
| let _ := thermodynamicFilter_neBot hd |
| obtain ⟨A, hA, hlo, hhi⟩ := finiteCanonical_limit hφ hb ht hβ hc hlow hmean hd |
| have he : canonicalThermo φ t β d = A := hA.liminf_eq |
| rw [he] |
| exact ⟨hA, hlo, hhi⟩ |
| |
| theorem familyCanonicalThermo_spec {t β d : ℝ} {a R : ℕ → ℝ} (ht : 1 ≤ t) |
| (ha : ∀ n, 0 ≤ a n) (hR : ∀ n, 1 ≤ R n) (hs : Summable a) |
| (hcoerc : 2197 * t * bumpIntegral⁻¹ * (∑' n, a n) / 2 ≤ t ^ 2 / 16) |
| (hβ : 0 ≤ β) (hd : 0 ≤ d) : |
| Tendsto (fun z : ℝ × ℕ => finiteCanonical (familyPotential t a R) t z.1 z.2 β) |
| (thermodynamicFilter d) (nhds (canonicalThermo (familyPotential t a R) t β d)) ∧ |
| β / t * (t ^ 2 / 3456 * d ^ 2 - t ^ 2 / 2 * d) + |
| (d * Real.log d - d) / t ≤ canonicalThermo (familyPotential t a R) t β d ∧ |
| canonicalThermo (familyPotential t a R) t β d ≤ |
| β / t * coreIntegral t * d ^ 2 + d * Real.log d / t := by |
| apply canonicalThermo_spec (continuous_familyPotential ha hR hs t) |
| (bounded_familyPotential ha hR hs t) (by linarith) hβ |
| (fun _ hr => familyPotential_nonpos_outer ha hR ht hr) |
| · intro L hL N x hx |
| exact family_density_stability (by linarith) hL ha hR hs hcoerc x hx |
| · intro L hL N |
| exact probability_energy_family_upper ht ha hR hs (by linarith) N |
| · exact hd |
| |
| theorem canonicalThermo_concave {φ : Potential} (hφ : Continuous φ) (hb : IsBoundedPotential φ) |
| {t d : ℝ} (ht : 0 < t) (hd : 0 ≤ d) |
| (hlim : ∀ β : ℝ, 0 < β → Tendsto (fun z : ℝ × ℕ => finiteCanonical φ t z.1 z.2 β) |
| (thermodynamicFilter d) (nhds (canonicalThermo φ t β d))) : |
| ConcaveOn ℝ (Ioi 0) (fun β => canonicalThermo φ t β d) := by |
| let _ := thermodynamicFilter_neBot hd |
| refine ⟨convex_Ioi _, ?_⟩ |
| intro β hβ γ hγ a b ha hb' hab |
| have hmix : 0 < a * β + b * γ := by |
| exact (convex_Ioi (0 : ℝ)) hβ hγ ha hb' hab |
| have hl := (hlim β hβ).const_mul a |>.add ((hlim γ hγ).const_mul b) |
| have hr := hlim (a * β + b * γ) hmix |
| apply le_of_tendsto_of_tendsto hl hr |
| filter_upwards [(thermodynamicFilter_side d).eventually (eventually_gt_atTop (0 : ℝ))] with z hz |
| exact (finiteCanonical_concave hφ hb ht hz z.2).2 (mem_univ β) (mem_univ γ) ha hb' hab |
| |
| end RadialTransition |
| |
| end |
| |
| end OAI |
| |
