| import OAI.Probability.FreeIsing.Trees.RestrictedCutCardinality | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| private theorem isChain_append_iff_last {α : Type*} {R : α → α → Prop} | |
| (p q : List α) (hp : p ≠ []) : | |
| List.IsChain R (p ++ q) ↔ | |
| List.IsChain R p ∧ List.IsChain R (p.getLast hp :: q) := by | |
| induction p with | |
| | nil => exact False.elim (hp rfl) | |
| | cons a p ih => | |
| cases p with | |
| | nil => simp | |
| | cons b p => | |
| simpa only [List.cons_append, List.isChain_cons_cons, List.getLast_cons_cons, | |
| and_assoc] using and_congr_right (fun _ : R a b => ih (by simp)) | |
| def cutLastColor {d k : ℕ} (v : TreeShell d (k + 1)) : Fin d := | |
| v.1.1.getLast (by intro h; have hv := v.2; simp [h] at hv) | |
| def appendCutContinuation {d k L : ℕ} (v : TreeShell d (k + 1)) | |
| (w : ReducedTail (cutLastColor v) L) : TreeShell d (k + 1 + L) := | |
| ⟨⟨v.1.1 ++ w.1, | |
| (isChain_append_iff_last v.1.1 w.1 (by intro h; have hv := v.2; simp [h] at hv)).mpr | |
| ⟨v.1.2, w.2.2⟩⟩, by simp [v.2, w.2.1]⟩ | |
| def cutPrefix {d k L : ℕ} (v : TreeShell d (k + 1 + L)) : TreeShell d (k + 1) := | |
| ⟨⟨v.1.1.take (k + 1), v.1.2.take _⟩, by simp [List.length_take, v.2]⟩ | |
| @[simp] theorem cutPrefix_appendCutContinuation {d k L : ℕ} | |
| (v : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor v) L) : | |
| cutPrefix (appendCutContinuation v w) = v := by | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| simp [cutPrefix, appendCutContinuation, v.2] | |
| def cutSuffix {d k L : ℕ} (v : TreeShell d (k + 1 + L)) : | |
| ReducedTail (cutLastColor (cutPrefix v)) L := by | |
| refine ⟨v.1.1.drop (k + 1), ?_, ?_⟩ | |
| · simp [List.length_drop, v.2] | |
| · have h : List.IsChain (fun a b : Fin d => a ≠ b) | |
| (v.1.1.take (k + 1) ++ v.1.1.drop (k + 1)) := by | |
| rw [List.take_append_drop] | |
| exact v.1.2 | |
| exact ((isChain_append_iff_last _ _ | |
| (by intro heq; have := (cutPrefix v).2; simp [cutPrefix, heq] at this)).mp h).2 | |
| @[simp] theorem appendCutContinuation_prefix_suffix {d k L : ℕ} | |
| (v : TreeShell d (k + 1 + L)) : | |
| appendCutContinuation (cutPrefix v) (cutSuffix v) = v := by | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| exact List.take_append_drop _ _ | |
| def cutContinuationEquiv (d k L : ℕ) : | |
| (Σ v : TreeShell d (k + 1), ReducedTail (cutLastColor v) L) ≃ | |
| TreeShell d (k + 1 + L) := | |
| Equiv.ofBijective (fun p => appendCutContinuation p.1 p.2) (by | |
| constructor | |
| · rintro ⟨v, w⟩ ⟨v', w'⟩ heq | |
| have hv : v = v' := by | |
| have := congrArg (@cutPrefix d k L) heq | |
| simpa only [cutPrefix_appendCutContinuation] using this | |
| subst v' | |
| have hw : w = w' := by | |
| apply Subtype.ext | |
| have hlist := congrArg (fun x : TreeShell d (k + 1 + L) => x.1.1) heq | |
| exact List.append_cancel_left hlist | |
| subst w' | |
| rfl | |
| · intro v | |
| exact ⟨⟨cutPrefix v, cutSuffix v⟩, appendCutContinuation_prefix_suffix v⟩) | |
| theorem card_cutContinuations {d k : ℕ} (v : TreeShell d (k + 1)) (L : ℕ) : | |
| Fintype.card (ReducedTail (cutLastColor v) L) = (d - 1) ^ L := | |
| card_reducedTail _ _ | |
| theorem sum_cutContinuations {M : Type*} [AddCommMonoid M] | |
| (d k L : ℕ) (f : TreeShell d (k + 1 + L) → M) : | |
| (∑ v : TreeShell d (k + 1 + L), f v) = | |
| ∑ p : TreeShell d (k + 1), ∑ w : ReducedTail (cutLastColor p) L, | |
| f (appendCutContinuation p w) := by | |
| rw [← (cutContinuationEquiv d k L).sum_comp f, Fintype.sum_sigma] | |
| rfl | |
| def cutDescendants {d k : ℕ} (I : Finset (TreeShell d (k + 1))) (L : ℕ) : | |
| Finset (TreeShell d (k + 1 + L)) := by | |
| classical | |
| exact Finset.univ.filter (fun v => cutPrefix v ∈ I) | |
| @[simp] theorem mem_cutDescendants {d k L : ℕ} | |
| (I : Finset (TreeShell d (k + 1))) (v : TreeShell d (k + 1 + L)) : | |
| v ∈ cutDescendants I L ↔ cutPrefix v ∈ I := by | |
| classical | |
| simp [cutDescendants] | |
| theorem appendCutContinuation_descendant {d k L : ℕ} | |
| (v : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor v) L) : | |
| v.1.1 <+: (appendCutContinuation v w).1.1 := | |
| List.prefix_append _ _ | |
| theorem sum_cutDescendants {M : Type*} [AddCommMonoid M] {d k : ℕ} | |
| (I : Finset (TreeShell d (k + 1))) (L : ℕ) | |
| (f : TreeShell d (k + 1 + L) → M) : | |
| (∑ v ∈ cutDescendants I L, f v) = | |
| ∑ p ∈ I, ∑ w : ReducedTail (cutLastColor p) L, f (appendCutContinuation p w) := by | |
| classical | |
| have h := sum_cutContinuations d k L (fun v => if cutPrefix v ∈ I then f v else 0) | |
| simpa [cutDescendants, Finset.sum_filter] using h | |
| @[simp] theorem card_cutDescendants {d k : ℕ} | |
| (I : Finset (TreeShell d (k + 1))) (L : ℕ) : | |
| (cutDescendants I L).card = I.card * (d - 1) ^ L := by | |
| classical | |
| have h := sum_cutDescendants I L (fun _ => (1 : ℕ)) | |
| simpa using h | |
| theorem average_cutDescendants {d k : ℕ} | |
| (I : Finset (TreeShell d (k + 1))) (L : ℕ) | |
| (f : TreeShell d (k + 1 + L) → ℝ) : | |
| (∑ v ∈ cutDescendants I L, f v) / ((cutDescendants I L).card : ℝ) = | |
| (∑ p ∈ I, (∑ w : ReducedTail (cutLastColor p) L, f (appendCutContinuation p w)) / | |
| (((d - 1) ^ L : ℕ) : ℝ)) / (I.card : ℝ) := by | |
| rw [sum_cutDescendants, card_cutDescendants, Nat.cast_mul, | |
| ← Finset.sum_div, div_div, mul_comm] | |
| def rootPathVertex {d : ℕ} (v : TreeVertex d) (i : ℕ) : TreeVertex d := | |
| ⟨v.1.take i, v.2.take i⟩ | |
| def rootPathProduct {d : ℕ} (a : TreeVertex d → ℝ) (v : TreeVertex d) : ℝ := | |
| ∏ i ∈ Finset.range v.1.length, a (rootPathVertex v (i + 1)) | |
| def cutContinuationProduct {d k L : ℕ} (a : TreeVertex d → ℝ) | |
| (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) : ℝ := | |
| ∏ j ∈ Finset.range L, | |
| a (rootPathVertex (appendCutContinuation p w).1 (k + 1 + (j + 1))) | |
| theorem rootPathProduct_le_cutContinuationProduct {d k L : ℕ} | |
| (a : TreeVertex d → ℝ) (ha : ∀ v, 0 ≤ a v ∧ a v ≤ 1) | |
| (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) : | |
| rootPathProduct a (appendCutContinuation p w).1 ≤ cutContinuationProduct a p w := by | |
| unfold rootPathProduct | |
| rw [(appendCutContinuation p w).2, Finset.prod_range_add] | |
| have hpre : (∏ i ∈ Finset.range (k + 1), | |
| a (rootPathVertex (appendCutContinuation p w).1 (i + 1))) ≤ 1 := | |
| Finset.prod_le_one₀ (fun i _ => (ha _).1) (fun i _ => (ha _).2) | |
| have htail : 0 ≤ (∏ j ∈ Finset.range L, | |
| a (rootPathVertex (appendCutContinuation p w).1 (k + 1 + j + 1))) := | |
| Finset.prod_nonneg (fun j _ => (ha _).1) | |
| have h := mul_le_mul_of_nonneg_right hpre htail | |
| simpa only [one_mul, cutContinuationProduct, Nat.add_assoc] using h | |
| theorem average_rootPathProduct_le_continuation_average {d k : ℕ} | |
| (a : TreeVertex d → ℝ) (ha : ∀ v, 0 ≤ a v ∧ a v ≤ 1) | |
| (I : Finset (TreeShell d (k + 1))) (L : ℕ) : | |
| (∑ v ∈ cutDescendants I L, rootPathProduct a v.1) / | |
| ((cutDescendants I L).card : ℝ) ≤ | |
| (∑ p ∈ I, (∑ w : ReducedTail (cutLastColor p) L, | |
| cutContinuationProduct a p w) / (((d - 1) ^ L : ℕ) : ℝ)) / (I.card : ℝ) := by | |
| rw [average_cutDescendants] | |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) | |
| apply Finset.sum_le_sum | |
| intro p hp | |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) | |
| exact Finset.sum_le_sum (fun w _ => rootPathProduct_le_cutContinuationProduct a ha p w) | |
| theorem cutContinuation_vertex_descendant {d k L : ℕ} | |
| (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) (j : ℕ) : | |
| p.1.1 <+: | |
| (rootPathVertex (appendCutContinuation p w).1 (k + 1 + (j + 1))).1 := by | |
| change p.1.1 <+: (p.1.1 ++ w.1).take (k + 1 + (j + 1)) | |
| have htake : p.1.1.take (k + 1 + (j + 1)) = p.1.1 := | |
| List.take_of_length_le (by rw [p.2]; omega) | |
| rw [List.take_append, htake] | |
| exact List.prefix_append _ _ | |
| theorem cutContinuationProduct_congr_on_descendants {d k L : ℕ} | |
| {a b : TreeVertex d → ℝ} (p : TreeShell d (k + 1)) | |
| (w : ReducedTail (cutLastColor p) L) | |
| (hab : ∀ v : TreeVertex d, p.1.1 <+: v.1 → a v = b v) : | |
| cutContinuationProduct a p w = cutContinuationProduct b p w := by | |
| apply Finset.prod_congr rfl | |
| intro j hj | |
| exact hab _ (cutContinuation_vertex_descendant p w j) | |
| private theorem shell_head_eq_first {d k : ℕ} (p : TreeShell d (k + 1)) : | |
| p.1.1.head? = some (treeShellSuccEquiv d k p).1 := by | |
| rcases p with ⟨⟨p, hp⟩, hlen⟩ | |
| cases p with | |
| | nil => simp at hlen | |
| | cons a p => rfl | |
| theorem mem_descendants_avoidingTreeCut {d k L : ℕ} | |
| (forbidden : Finset (Fin d)) (v : TreeShell d (k + 1 + L)) : | |
| v ∈ cutDescendants (avoidingTreeCut forbidden k) L ↔ | |
| ∀ c ∈ v.1.1.head?, c ∉ forbidden := by | |
| classical | |
| rw [mem_cutDescendants, mem_avoidingTreeCut] | |
| have hhead : v.1.1.head? = (cutPrefix v).1.1.head? := by | |
| change v.1.1.head? = (v.1.1.take (k + 1)).head? | |
| cases v.1.1 <;> rfl | |
| rw [hhead, shell_head_eq_first] | |
| simp | |
| end Problem367 | |
| end | |
| end OAI | |