| import Mathlib |
| import OAI.Probability.CriticalPercolation.Clusters.ComponentSelf |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
|
|
| namespace CriticalPercolation.ClusterColumns |
| open CriticalPercolation.ClusterDeletion |
| open scoped Classical |
|
|
| variable {V : Type*} {n : ℕ} |
|
|
| def earlier (x : Fin n → V) (k : Fin n) : Set V := {v | ∃ j < k, x j = v} |
| def first (H : SimpleGraph V) (x : Fin n → V) (k : Fin n) : Prop := |
| x k ∉ roots H (earlier x k) |
| noncomputable def select (H : SimpleGraph V) (x : Fin n → V) (k i : Fin n) : ℝ := |
| if first H x k ∧ x i ∈ component H (x k) then 1 else 0 |
|
|
| lemma root_not_mem_earlier {x : Fin n → V} (hx : Function.Injective x) (k : Fin n) : |
| x k ∉ earlier x k := by |
| rintro ⟨j, hj, heq⟩ |
| exact (ne_of_lt hj) (hx heq) |
|
|
| lemma earlier_mono (x : Fin n → V) {k l : Fin n} (hkl : k ≤ l) : |
| earlier x k ⊆ earlier x l := by |
| rintro v ⟨j, hj, rfl⟩ |
| exact ⟨j, hj.trans_le hkl, rfl⟩ |
|
|
| lemma first_unique {H : SimpleGraph V} {x : Fin n → V} {k l i : Fin n} |
| (hk : first H x k) (hki : x i ∈ component H (x k)) |
| (hl : first H x l) (hli : x i ∈ component H (x l)) : k = l := by |
| apply le_antisymm |
| · by_contra! h |
| exact hk ⟨x l, ⟨l, h, rfl⟩, hli.trans hki.symm⟩ |
| · by_contra! h |
| exact hl ⟨x k, ⟨k, h, rfl⟩, hki.trans hli.symm⟩ |
|
|
| lemma exists_first (H : SimpleGraph V) (x : Fin n → V) (i : Fin n) : |
| ∃ k, first H x k ∧ x i ∈ component H (x k) := by |
| classical |
| let J := Finset.univ.filter (fun j => x i ∈ component H (x j)) |
| have hi : i ∈ J := by simp [J, component_self] |
| have hJ : J.Nonempty := ⟨i, hi⟩ |
| let k := J.min' hJ |
| have hkJ : k ∈ J := Finset.min'_mem J hJ |
| have hki : x i ∈ component H (x k) := (Finset.mem_filter.mp hkJ).2 |
| refine ⟨k, ?_, hki⟩ |
| rintro ⟨v, ⟨j, hj, rfl⟩, hjk⟩ |
| have hjJ : j ∈ J := Finset.mem_filter.mpr ⟨Finset.mem_univ j, hjk.trans hki⟩ |
| have hkj : k ≤ j := Finset.min'_le J j hjJ |
| exact (not_lt_of_ge hkj) hj |
|
|
| lemma select_nonneg (H : SimpleGraph V) (x : Fin n → V) (k i : Fin n) : |
| 0 ≤ select H x k i := by unfold select; positivity |
|
|
| lemma select_zero_of_lt (H : SimpleGraph V) (x : Fin n → V) {k i : Fin n} (hik : i < k) : |
| select H x k i = 0 := by |
| apply ite_eq_right |
| rintro ⟨hk, hki⟩ |
| exact hk ⟨x i, ⟨i, hik, rfl⟩, hki.symm⟩ |
|
|
| lemma select_diag (H : SimpleGraph V) (x : Fin n → V) (k : Fin n) : |
| select H x k k = if first H x k then 1 else 0 := by |
| simp only [select, component_self, and_true] |
|
|
| lemma select_mul_sum (H : SimpleGraph V) (x : Fin n → V) (i : Fin n) |
| (a : Fin n → ℝ) {k : Fin n} (hk : first H x k) |
| (hki : x i ∈ component H (x k)) : |
| ∑ l, select H x l i * a l = a k := by |
| classical |
| rw [Finset.sum_eq_single k] |
| · simp [select, hk, hki] |
| · intro l _ hlk |
| have hh : ¬ (first H x l ∧ x i ∈ component H (x l)) := by |
| rintro ⟨hl, hli⟩ |
| exact hlk (first_unique hl hli hk hki) |
| simp [select, hh] |
| · simp |
|
|
| lemma select_mass (H : SimpleGraph V) (x : Fin n → V) (i : Fin n) : |
| ∑ k, select H x k i = 1 := by |
| obtain ⟨k, hk, hki⟩ := exists_first H x i |
| simpa using select_mul_sum H x i (fun _ => 1) hk hki |
|
|
| lemma cluster_partition (H : SimpleGraph V) (x : Fin n → V) |
| (F : Set V → ℝ) (i : Fin n) : |
| F (component H (x i)) = ∑ k, select H x k i * F (component H (x k)) := by |
| obtain ⟨k, hk, hki⟩ := exists_first H x i |
| rw [select_mul_sum H x i _ hk hki, component_eq_of_mem hki] |
|
|
| lemma first_hit_not_mem_roots_iff (H : SimpleGraph V) (x : Fin n → V) |
| {j i : Fin n} (hj : first H x j) (hji : x i ∈ component H (x j)) (k : Fin n) : |
| x i ∉ roots H (earlier x k) ↔ k ≤ j := by |
| constructor |
| · intro hi |
| by_contra! hjk |
| exact hi ⟨x j, ⟨j, hjk, rfl⟩, hji⟩ |
| · rintro hkj ⟨v, ⟨l, hlk, rfl⟩, hli⟩ |
| exact hj ⟨x l, ⟨l, hlk.trans_le hkj, rfl⟩, hli.trans hji.symm⟩ |
|
|
| lemma select_truncation (H : SimpleGraph V) (x : Fin n → V) (k i : Fin n) : |
| (if x i ∉ roots H (earlier x k) then (1 : ℝ) else 0) = |
| select H x k i + ∑ l, if k < l then select H x l i else 0 := by |
| classical |
| obtain ⟨j, hj, hji⟩ := exists_first H x i |
| have he : ∀ l, select H x l i = if l = j then 1 else 0 := by |
| intro l |
| by_cases hlj : l = j |
| · subst l |
| simp [select, hj, hji] |
| · have hh : ¬ (first H x l ∧ x i ∈ component H (x l)) := by |
| rintro ⟨hl, hli⟩ |
| exact hlj (first_unique hl hli hj hji) |
| simp [select, hh, hlj] |
| simp_rw [he] |
| have hsum : (∑ l : Fin n, if k < l then if l = j then (1 : ℝ) else 0 else 0) = |
| if k < j then 1 else 0 := by |
| rw [Finset.sum_eq_single j] |
| · simp |
| · intro l _ hlj |
| simp [hlj] |
| · simp |
| rw [hsum] |
| simp only [first_hit_not_mem_roots_iff H x hj hji k] |
| rcases lt_trichotomy k j with h | h | h |
| · simp [h, ne_of_lt h, le_of_lt h] |
| · subst j |
| simp |
| · simp [ne_of_gt h, not_le_of_gt h, not_lt_of_gt h] |
|
|
| end CriticalPercolation.ClusterColumns |
|
|
| end |
| end |
|
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| end OAI |
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