| import OAI.Algebra.DepthFive.LowerSchedule |
| import OAI.Algebra.DepthFive.ScheduleWeightBridge |
| import OAI.Algebra.DepthFive.WordSum |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace Problem335.LowerParameters |
|
|
| /-- The full graph-word estimate for the actual balanced layer partition. |
| All path-graph, endpoint, flip, and cardinality premises are discharged. -/ |
| lemma balanced_word_sum_bound_of_product {n : ℕ} (hn : 4 ≤ n + 1) |
| (A B E : ℝ) (hA : 1 ≤ A) (hAN : A ≤ Real.sqrt ((n + 1 : ℕ) : ℝ)) |
| (hB : 0 ≤ B) (hE : E ≤ Real.sqrt ((n + 1 : ℕ) : ℝ)) |
| (hprod : ∀ w : Fin (n + 1) → Bool, |
| A ^ ((Finset.univ.filter (fun i => balancedLayer hn i = true)).filter |
| (fun i => w i = true)).card * |
| B ^ ((Finset.univ \ Finset.univ.filter (fun i => balancedLayer hn i = true)).filter |
| (fun i => w i = true)).card ≤ |
| Real.exp E * ((n + 1 : ℕ) : ℝ) ^ WordRuns.runCount w) : |
| (∑ w : Fin (n + 1) → Bool, |
| IsolatedWeight.weight (BalancedSchedule.internalEdges (n + 1)) |
| (Finset.univ.filter (fun i => balancedLayer hn i = true)) |
| BalancedSchedule.adjacent A B (((n + 1 : ℕ) : ℝ)⁻¹) w) ≤ |
| Real.exp (8 * Real.sqrt ((n + 1 : ℕ) : ℝ)) := by |
| classical |
| let V := Finset.univ.filter (fun i => balancedLayer hn i = true) |
| let first : Fin (n + 1) := ⟨0, by omega⟩ |
| let S := V.erase first |
| have hfirst : first ∈ V := by |
| apply Finset.mem_filter.mpr |
| exact ⟨Finset.mem_univ _, balancedLayer_first hn⟩ |
| apply WordSum.sum_actual_weight_le_exp_of_endpoints n A B E |
| (BalancedSchedule.internalEdges (n + 1)) V S BalancedSchedule.adjacent |
| · exact Finset.erase_subset _ _ |
| · intro i hi j hij hj |
| exact balancedLayer_independent hn (Finset.mem_filter.mp hi).2 |
| (Finset.mem_filter.mp hj).2 hij |
| · intro e he |
| exact BalancedSchedule.internalEdges_adjacent he |
| · intro i hi |
| obtain ⟨hifirst, hiV⟩ := Finset.mem_erase.mp hi |
| have hival : i.val ≠ 0 := by |
| intro h |
| exact hifirst (Fin.ext h) |
| have hlast : i.val + 1 < n + 1 := by |
| by_contra hf |
| have heq : i = ⟨(n + 1) - 1, by omega⟩ := by |
| apply Fin.ext |
| change i.val = (n + 1) - 1 |
| have := i.isLt |
| omega |
| have hitrue := (Finset.mem_filter.mp hiV).2 |
| rw [heq, balancedLayer_final hn] at hitrue |
| cases hitrue |
| rw [BalancedSchedule.endpointDegree_internalEdges] |
| exact BalancedSchedule.card_internalEnds_eq_two i (by omega) hlast |
| · intro w |
| rw [BalancedSchedule.internalEdges_eq_pathEdges, |
| SwitchRestoration.switchCount_pathEdges] |
| simp [WordSwitches.switches, Finset.card_filter] |
| · change (V \ V.erase first).card ≤ 1 |
| rw [Finset.sdiff_erase hfirst] |
| simp |
| · have hcard : V.card = k (n + 1) := balancedLayer_card_true hn |
| rw [hcard] |
| have hle : 2 * k (n + 1) ≤ n + 1 := by |
| have h := two_mul_k_add_s (n + 1) |
| omega |
| exact_mod_cast hle |
| · exact hA |
| · exact hAN |
| · exact hB |
| · exact hE |
| · exact hprod |
|
|
| /-- The canonical balanced schedule satisfies the complete corrected-word bound. |
| The only premises are the scalar parameter inequalities. -/ |
| lemma balanced_word_sum_bound {n : ℕ} (hn : 4 ≤ n) {α β ε : ℝ} |
| (hα : 0 < α) (hα1 : α ≤ 1) (hαN : α⁻¹ ≤ Real.sqrt (n : ℝ)) |
| (hβ : 0 ≤ β) (hβupper : β ≤ α ^ rho n * Real.exp ε) |
| (hscale : α ^ (-(1 + rho n)) ≤ (n : ℝ)) |
| (hε : 0 ≤ ε) (hεN : ε * (n : ℝ) ≤ Real.sqrt (n : ℝ)) : |
| (∑ w : Fin n → Bool, |
| IsolatedWeight.weight (BalancedSchedule.internalEdges n) |
| (Finset.univ.filter (fun i => balancedLayer hn i = true)) |
| BalancedSchedule.adjacent α⁻¹ β ((n : ℝ)⁻¹) w) ≤ |
| Real.exp (8 * Real.sqrt (n : ℝ)) := by |
| classical |
| cases n with |
| | zero => omega |
| | succ n => |
| apply balanced_word_sum_bound_of_product hn α⁻¹ β (ε * ((n + 1 : ℕ) : ℝ)) |
| ((one_le_inv₀ hα).2 hα1) hαN hβ hεN |
| intro w |
| apply RunLayerWeights.count_weight_le_exp_mul_runCount w |
| (Finset.univ.filter (fun i => balancedLayer hn i = true)) |
| hα hα1 hβ hε hβupper hscale |
| intro b hb |
| simpa using (balancedLayer_infix_discrepancy hn b hb).le |
|
|
| end Problem335.LowerParameters |
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| end |
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| end OAI |
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