openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /DegreeEnvelope.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedRates | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupDegreeEnvelope | |
| open AllFieldHistory AllFieldGroupOrbitData Filter | |
| open scoped Topology | |
| attribute [local instance] Classical.propDecidable | |
| variable {K tick : ℕ} | |
| def variableDegree (allocation : Allocation) (m : ℕ) (width : ℝ) | |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) : ℕ := | |
| ((data (K := K) (tick := tick) allocation m width sigma).orbits e).sup fun o => | |
| ((data (K := K) (tick := tick) allocation m width sigma).passing e o).sup fun v => | |
| ((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card | |
| def uniformDegree (allocation : Allocation) (m : ℕ) (width : ℝ) | |
| (sigma : Placement) : ℕ := | |
| Finset.univ.sup fun e : Targets (K := K) (tick := tick) allocation m sigma => | |
| max ((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card | |
| (variableDegree (K := K) (tick := tick) allocation m width sigma e) | |
| theorem eligibility_card_le (allocation : Allocation) (m : ℕ) (width : ℝ) | |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) : | |
| ((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card ≤ | |
| uniformDegree (K := K) (tick := tick) allocation m width sigma := | |
| (le_max_left _ _).trans (Finset.le_sup (f := fun e => | |
| max ((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card | |
| (variableDegree (K := K) (tick := tick) allocation m width sigma e)) (Finset.mem_univ e)) | |
| theorem competitors_card_le (allocation : Allocation) (m : ℕ) (width : ℝ) | |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (ho : o ∈ (data (K := K) (tick := tick) allocation m width sigma).orbits e) | |
| (v : Variable (K := K) (tick := tick) allocation m sigma) (hv : v ∈ (data (K := K) (tick := tick) allocation m width sigma).passing e o) : | |
| ((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card ≤ | |
| uniformDegree (K := K) (tick := tick) allocation m width sigma := by | |
| have hvle := Finset.le_sup (f := fun v => | |
| ((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card) hv | |
| have hole := Finset.le_sup (f := fun o => | |
| ((data (K := K) (tick := tick) allocation m width sigma).passing e o).sup fun v => | |
| ((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card) ho | |
| have hel := Finset.le_sup (f := fun e => | |
| max ((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card | |
| (variableDegree (K := K) (tick := tick) allocation m width sigma e)) (Finset.mem_univ e) | |
| exact hvle.trans (hole.trans ((le_max_right _ _).trans hel)) | |
| private theorem cast_max_le {a b : ℕ} {R : ℝ} | |
| (ha : (a : ℝ) ≤ R) (hb : (b : ℝ) ≤ R) : ((max a b : ℕ) : ℝ) ≤ R := by | |
| rcases le_total a b with hab | hba | |
| · simpa only [max_eq_right hab] using hb | |
| · simpa only [max_eq_left hba] using ha | |
| private theorem cast_sup_le {I : Type*} (s : Finset I) (f : I → ℕ) | |
| {R : ℝ} (hR : 0 ≤ R) (hs : ∀ i ∈ s, (f i : ℝ) ≤ R) : | |
| ((s.sup f : ℕ) : ℝ) ≤ R := by | |
| classical | |
| revert hs | |
| induction s using Finset.induction_on with | |
| | empty => | |
| intro _ | |
| simpa using hR | |
| | @insert a s _ ih => | |
| intro hs | |
| rw [Finset.sup_insert] | |
| exact cast_max_le (hs a (Finset.mem_insert_self a s)) | |
| (ih (fun i hi => hs i (Finset.mem_insert_of_mem hi))) | |
| theorem uniformDegree_real_le (allocation : Allocation) (m : ℕ) (width : ℝ) | |
| (sigma : Placement) {R : ℝ} (hR : 0 ≤ R) | |
| (heligibility : ∀ e : Targets (K := K) (tick := tick) allocation m sigma, | |
| (((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤ R) | |
| (hcompetitors : ∀ e : Targets (K := K) (tick := tick) allocation m sigma, | |
| ∀ o ∈ (data (K := K) (tick := tick) allocation m width sigma).orbits e, | |
| ∀ v ∈ (data (K := K) (tick := tick) allocation m width sigma).passing e o, | |
| (((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card : ℝ) ≤ R) : | |
| (uniformDegree (K := K) (tick := tick) allocation m width sigma : ℝ) ≤ R := by | |
| apply cast_sup_le _ _ hR | |
| intro e _ | |
| apply cast_max_le (heligibility e) | |
| apply cast_sup_le _ _ hR | |
| intro o ho | |
| apply cast_sup_le _ _ hR | |
| exact hcompetitors e o ho | |
| theorem eventually_log_uniformDegree_le (allocation : Allocation) (width : ℝ) | |
| (sigma : Placement) {D : ℝ} (hD : 0 ≤ D) | |
| (hcounts : ∀ᶠ m : ℕ in atTop, | |
| (∀ e : Targets (K := K) (tick := tick) allocation m sigma, | |
| (((data (K := K) (tick := tick) allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤ | |
| Real.exp ((m : ℝ) * D)) ∧ | |
| (∀ e : Targets (K := K) (tick := tick) allocation m sigma, | |
| ∀ o ∈ (data (K := K) (tick := tick) allocation m width sigma).orbits e, | |
| ∀ v ∈ (data (K := K) (tick := tick) allocation m width sigma).passing e o, | |
| (((data (K := K) (tick := tick) allocation m width sigma).competitors e o v).card : ℝ) ≤ | |
| Real.exp ((m : ℝ) * D))) : | |
| ∀ᶠ m : ℕ in atTop, | |
| Real.log (uniformDegree (K := K) (tick := tick) allocation m width sigma : ℝ) / | |
| (m : ℝ) ≤ D := by | |
| filter_upwards [hcounts, eventually_gt_atTop (0 : ℕ)] with m hm hmp | |
| by_cases hz : uniformDegree (K := K) (tick := tick) allocation m width sigma = 0 | |
| · simpa only [hz, Nat.cast_zero, Real.log_zero, zero_div] using hD | |
| have hd : (0 : ℝ) < uniformDegree (K := K) (tick := tick) allocation m width sigma := by | |
| exact_mod_cast Nat.pos_of_ne_zero hz | |
| have hb := uniformDegree_real_le (K := K) (tick := tick) allocation m width sigma | |
| (Real.exp_pos _).le hm.1 hm.2 | |
| have hl := Real.log_le_log hd hb | |
| rw [Real.log_exp] at hl | |
| have hmpos : (0 : ℝ) < m := by exact_mod_cast hmp | |
| exact (div_le_iff₀ hmpos).mpr (by simpa only [mul_comm] using hl) | |
| end MatrixMultiplication.AllFieldGroupDegreeEnvelope | |
| end | |
| end OAI | |