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namespace OAI
section
open scoped Topology BigOperators ENNReal
open Filter MeasureTheory Set
namespace ContinuumTemperature
noncomputable def tileCenter (ℓ L : ℝ) {k : ℕ} (i : Fin 3 → Fin k) : Space :=
WithLp.toLp 2 (fun j => (ℓ+2)*((i j : ℝ)+1/2)-L/2)
noncomputable def tileBox (ℓ L : ℝ) {k : ℕ} (i : Fin 3 → Fin k) : Set Space :=
(fun x => x + tileCenter ℓ L i) '' cube ℓ
theorem isCompact_tileBox (ℓ L : ℝ) {k : ℕ} (i : Fin 3 → Fin k) :
IsCompact (tileBox ℓ L i) :=
(isCompact_cube ℓ).image (continuous_id.add continuous_const)
theorem tileBox_subset {ℓ L : ℝ} {k : ℕ} (hℓ : 0 ≤ ℓ)
(hk : (k : ℝ)*(ℓ+2) ≤ L) (i : Fin 3 → Fin k) : tileBox ℓ L i ⊆ cube L := by
rintro x ⟨y, hy, rfl⟩ j
have hyj := hy j
have hij : (i j : ℝ)+1 ≤ k := by exact_mod_cast (i j).isLt
have h0 : (0 : ℝ) ≤ (i j : ℝ) := Nat.cast_nonneg _
change y j + ((ℓ+2)*((i j : ℝ)+1/2)-L/2) ∈ Icc (-L/2) (L/2)
constructor <;> nlinarith [hyj.1, hyj.2]
theorem tileBox_separation {ℓ L : ℝ} {k : ℕ} (hℓ : 0 ≤ ℓ)
{i j : Fin 3 → Fin k} (hij : i ≠ j) {x y : Space}
(hx : x ∈ tileBox ℓ L i) (hy : y ∈ tileBox ℓ L j) : 2 ≤ ‖x-y‖ := by
obtain ⟨a, ha, rfl⟩ := hx
obtain ⟨b, hb, rfl⟩ := hy
obtain ⟨r, hr⟩ : ∃ r, i r ≠ j r := by
by_contra! h
exact hij (funext h)
have har := ha r
have hbr := hb r
have hnorm : |(a+tileCenter ℓ L i) r - (b+tileCenter ℓ L j) r| ≤
‖(a+tileCenter ℓ L i)-(b+tileCenter ℓ L j)‖ := by
simpa only [PiLp.sub_apply, Real.norm_eq_abs] using
PiLp.norm_apply_le ((a+tileCenter ℓ L i)-(b+tileCenter ℓ L j)) r
change |a r + ((ℓ+2)*((i r : ℝ)+1/2)-L/2) -
(b r + ((ℓ+2)*((j r : ℝ)+1/2)-L/2))| ≤
‖(a+tileCenter ℓ L i)-(b+tileCenter ℓ L j)‖ at hnorm
rw [abs_le] at hnorm
rcases lt_or_gt_of_ne hr with h | h
· have hh : (i r : ℝ)+1 ≤ (j r : ℝ) := by exact_mod_cast h
have hmul := mul_le_mul_of_nonneg_left hh (by linarith : 0 ≤ ℓ+2)
nlinarith [har.1, har.2, hbr.1, hbr.2]
· have hh : (j r : ℝ)+1 ≤ (i r : ℝ) := by exact_mod_cast h
have hmul := mul_le_mul_of_nonneg_left hh (by linarith : 0 ≤ ℓ+2)
nlinarith [har.1, har.2, hbr.1, hbr.2]
theorem tileBox_pairwiseDisjoint {ℓ L : ℝ} {k : ℕ} (hℓ : 0 ≤ ℓ) :
Pairwise (fun i j : Fin 3 → Fin k => Disjoint (tileBox ℓ L i) (tileBox ℓ L j)) := by
intro i j hij
apply Set.disjoint_left.mpr
intro x hx hy
have hh := tileBox_separation hℓ hij hx hy
simp only [sub_self, norm_zero] at hh
norm_num at hh
theorem partition_tiling_lower {φ : ℝ → ℝ} (hφ : Measurable (radialPotential φ))
{β B ℓ L : ℝ} (hβ : 0 ≤ β) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)
(hstable : ∀ (N : ℕ) (x : Fin N → Space),
((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x)
(hsign : ∀ r, 1 ≤ r → φ r ≤ 0)
(n : (Fin 3 → Fin ⌊L/(ℓ+2)⌋₊) → ℕ) :
(∏ i, partition φ β ℓ (n i)) ≤ partition φ β L (∑ i, n i) := by
classical
have hden : 0 < ℓ+2 := by linarith
have hk : (⌊L/(ℓ+2)⌋₊ : ℝ)*(ℓ+2) ≤ L := by
apply (le_div_iff₀ hden).mp
exact Nat.floor_le (div_nonneg hL hden.le)
have hh := setPartition_product_lower hφ hβ hstable (tileBox ℓ L) n Finset.univ
(fun i _ => (isCompact_tileBox ℓ L i).measurableSet)
(fun i _ j _ hij => tileBox_pairwiseDisjoint hℓ hij)
(fun i _ j _ hij a ha b hb => hsign _ (le_trans (by norm_num)
(tileBox_separation hℓ hij ha hb)))
(isCompact_cube L).measure_lt_top.ne (fun i _ => tileBox_subset hℓ hk i)
have hv (i : Fin 3 → Fin ⌊L/(ℓ+2)⌋₊) :
setPartition φ β (tileBox ℓ L i) (n i) = partition φ β ℓ (n i) := by
rw [tileBox, setPartition_translate, setPartition, generalPartition_eq_partition]
simp_rw [hv] at hh
simpa only [setPartition, generalPartition_eq_partition] using hh
theorem exists_concentrated_counts (K H N : ℕ) (hN : N ≤ K*H) :
∃ r : Fin K → ℕ, (∑ i, r i) = N ∧ (∀ i, r i ≤ H) ∧
(∑ i, if r i = 0 then 0 else 1) * H ≤ N+H := by
induction K generalizing N with
| zero =>
have hzero : N = 0 := by omega
subst N
exact ⟨Fin.elim0, by simp, (fun i => Fin.elim0 i), by simp⟩
| succ K ih =>
by_cases hh : N ≤ H
· refine ⟨Fin.cons N (fun _ => 0), ?_, ?_, ?_⟩
· simp [Fin.sum_univ_succ]
· intro i
refine Fin.cases hh (fun _ => Nat.zero_le _) i
· simp only [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ, ite_true, Finset.sum_const_zero,
add_zero]
split_ifs <;> omega
· have hHK : N-H ≤ K*H := by
rw [Nat.succ_mul] at hN
omega
obtain ⟨r, hr, hbound, hc⟩ := ih (N-H) hHK
have hH : 0 < H := by
by_contra hzero
have hz : H = 0 := by omega
simp only [hz, Nat.mul_zero, Nat.le_zero] at hN
omega
refine ⟨Fin.cons H r, ?_, ?_, ?_⟩
· simp only [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ, hr]
omega
· intro i
exact Fin.cases (le_refl H) hbound i
· simp only [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ, ite_eq_right hH.ne']
have hsub := Nat.sub_add_cancel (show H ≤ N by omega)
nlinarith
theorem exists_adjusted_counts (K b H N : ℕ) (hb : 0 < b) (hH : 2*b ≤ H)
(hN : N ≤ K*H) :
∃ r : Fin K → ℕ, (∑ i, r i) = N ∧ (∀ i, r i ≤ H) ∧
((∑ i, if r i = b then 0 else 1 : ℕ) : ℝ) * b ≤
|(N : ℝ) - (K : ℝ)*b| + H := by
have hbH : b ≤ H := by omega
by_cases hNb : K*b ≤ N
· have hdel : N-K*b ≤ K*(H-b) := by
rw [Nat.mul_sub_left_distrib]
omega
obtain ⟨d, hsum, hbound, hc⟩ := exists_concentrated_counts K (H-b) (N-K*b) hdel
refine ⟨fun i => b+d i, ?_, ?_, ?_⟩
· simp only [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
Fintype.card_fin, smul_eq_mul, hsum]
omega
· intro i
change b+d i ≤ H
have hh := hbound i
omega
· have heq : (∑ i, if b+d i = b then 0 else 1 : ℕ) =
∑ i, if d i = 0 then 0 else 1 := by
apply Finset.sum_congr rfl
intro i _
simp only [Nat.add_eq_left]
rw [heq]
have hineq : (∑ i, if d i = 0 then 0 else 1) * b ≤ (N-K*b)+H := by
have hh := Nat.mul_le_mul_left (∑ i, if d i = 0 then 0 else 1) (show b ≤ H-b by omega)
omega
have hsub : ((N-K*b : ℕ) : ℝ) = (N : ℝ)-(K : ℝ)*b := by
rw [Nat.cast_sub hNb, Nat.cast_mul]
have hsgn : 0 ≤ (N : ℝ)-(K : ℝ)*b := by
have hh : (K : ℝ)*b ≤ (N : ℝ) := by exact_mod_cast hNb
linarith
rw [abs_of_nonneg hsgn]
have hh : ((∑ i, if d i = 0 then 0 else 1 : ℕ) : ℝ)*b ≤ ((N-K*b : ℕ) : ℝ)+H := by
exact_mod_cast hineq
rwa [hsub] at hh
· have hNb' : N ≤ K*b := by omega
obtain ⟨d, hsum, hbound, hc⟩ := exists_concentrated_counts K b (K*b-N) (Nat.sub_le _ _)
refine ⟨fun i => b-d i, ?_, ?_, ?_⟩
· have hh : (∑ i : Fin K, (b - d i)) + (∑ i : Fin K, d i) = K*b := by
rw [← Finset.sum_add_distrib]
simp only [Nat.sub_add_cancel (hbound _), Finset.sum_const, Finset.card_univ,
Fintype.card_fin, smul_eq_mul]
change (∑ i : Fin K, (b - d i)) = N
rw [hsum] at hh
omega
· intro i
exact (Nat.sub_le b _).trans hbH
· have heq : (∑ i, if b-d i = b then 0 else 1 : ℕ) =
∑ i, if d i = 0 then 0 else 1 := by
apply Finset.sum_congr rfl
intro i _
have hh : b-d i = b ↔ d i = 0 := by omega
simp only [hh]
rw [heq]
have hineq : (∑ i, if d i = 0 then 0 else 1) * b ≤ (K*b-N)+H := by omega
have hsub : ((K*b-N : ℕ) : ℝ) = (K : ℝ)*b-N := by
rw [Nat.cast_sub hNb', Nat.cast_mul]
have hsgn : (N : ℝ)-(K : ℝ)*b ≤ 0 := by
have hh : (N : ℝ) ≤ (K : ℝ)*b := by exact_mod_cast hNb'
linarith
rw [abs_of_nonpos hsgn]
have hh : ((∑ i, if d i = 0 then 0 else 1 : ℕ) : ℝ)*b ≤ ((K*b-N : ℕ) : ℝ)+H := by
exact_mod_cast hineq
rw [hsub] at hh
linarith
noncomputable def tileCount (ℓ L : ℝ) : ℕ := ⌊L/(ℓ+2)⌋₊ ^ 3
theorem sum_baseline_le {K b H : ℕ} {r : Fin K → ℕ} {a : ℕ → ℝ} {M : ℝ}
(hr : ∀ i, r i ≤ H) (ha : ∀ n, n ≤ H → a n ≤ M) (hb : -M ≤ a b) :
∑ i, a (r i) ≤ K*a b + 2*M*(∑ i, if r i = b then 0 else 1 : ℕ) := by
classical
calc
∑ i, a (r i) ≤ ∑ i, (a b + 2*M*(if r i = b then 0 else 1 : ℕ)) := by
apply Finset.sum_le_sum
intro i _
by_cases hh : r i = b
· simp [hh]
· simp only [ite_eq_right hh, Nat.cast_one, mul_one]
linarith [ha (r i) (hr i)]
_ = _ := by
rw [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
nsmul_eq_mul, ← Finset.mul_sum, Nat.cast_sum]
theorem adjusted_box_comparison (a : ℝ → ℕ → ℝ) {ℓ L M : ℝ} {b H N : ℕ}
(hℓ : 0 < ℓ) (hL : 0 < L) (hM : 0 ≤ M) (hb : 0 < b) (hH : 2*b ≤ H)
(hN : N ≤ tileCount ℓ L * H)
(hbox : ∀ r : Fin (tileCount ℓ L) → ℕ, (∑ i, r i) = N →
a L N ≤ ℓ^3/L^3 * ∑ i, a ℓ (r i))
(hbound : ∀ n, n ≤ H → a ℓ n ≤ M) (hbase : -M ≤ a ℓ b) :
a L N ≤ ((tileCount ℓ L : ℝ)/L^3)*ℓ^3*a ℓ b +
(2*M*ℓ^3/(b : ℝ)) * (|(N : ℝ)/L^3 - ((tileCount ℓ L : ℝ)/L^3)*b| + H/L^3) := by
classical
obtain ⟨r, hr, hcounts, hchange⟩ := exists_adjusted_counts (tileCount ℓ L) b H N hb hH hN
have hs := sum_baseline_le hcounts hbound hbase
have hm : 0 ≤ ℓ^3/L^3 := div_nonneg (pow_nonneg hℓ.le _) (pow_nonneg hL.le _)
have hbR : (0 : ℝ) < b := by exact_mod_cast hb
have hc : (∑ i, if r i = b then 0 else 1 : ℕ) ≤
(|(N : ℝ) - (tileCount ℓ L : ℝ)*b|+H)/(b : ℝ) :=
(le_div_iff₀ hbR).2 hchange
calc
a L N ≤ ℓ^3/L^3 * ∑ i, a ℓ (r i) := hbox r hr
_ ≤ ℓ^3/L^3 * ((tileCount ℓ L : ℝ)*a ℓ b +
2*M*((|(N : ℝ)-(tileCount ℓ L : ℝ)*b|+H)/(b : ℝ))) := by
apply mul_le_mul_of_nonneg_left _ hm
exact hs.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left hc (show 0 ≤ 2*M by positivity)))
_ = _ := by
have heq : (N : ℝ)/L^3 - ((tileCount ℓ L : ℝ)/L^3)*b =
((N : ℝ)-(tileCount ℓ L : ℝ)*b)/L^3 := by ring
rw [heq, abs_div, abs_of_pos (pow_pos hL 3)]
ring
end ContinuumTemperature
end
end OAI
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