openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /AmbientDegree.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Population | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.TypeCounts | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointAmbientDegree | |
| open MatrixMultiplication.Foundation JointPopulation | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {H : Type*} [Fintype H] [DecidableEq H] | |
| (counts : H → Shape → ℕ) | |
| def localWord (t : JointCoarseHashing.Triple (Position counts)) (h : H) : | |
| Positions counts h → Shape := | |
| fun i => (t.1 ⟨h, i⟩, t.2.1 ⟨h, i⟩, t.2.2 ⟨h, i⟩) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem localWord_injective : Function.Injective (localWord counts) := by | |
| intro t u he | |
| apply Prod.ext | |
| · funext p | |
| exact congrArg (fun w => (w p.1 p.2).1) he | |
| · apply Prod.ext | |
| · funext p | |
| exact congrArg (fun w => (w p.1 p.2).2.1) he | |
| · funext p | |
| exact congrArg (fun w => (w p.1 p.2).2.2) he | |
| def localAllowed (sum : H → ℕ) (h : H) (w : Positions counts h → Shape) : Prop := | |
| (∀ i, (w i).1.val + (w i).2.1.val + (w i).2.2.val = sum h) ∧ | |
| ∀ s a, wordPopulation (shapeSide s ∘ w) a = | |
| wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a | |
| theorem localAllowed_of_mem_ambient (sum : H → ℕ) | |
| (t : JointCoarseHashing.Triple (Position counts)) | |
| (ht : t ∈ ambientSet counts sum) (h : H) : | |
| localAllowed counts sum h (localWord counts t h) := by | |
| obtain ⟨hs, hm⟩ := (Finset.mem_filter.mp ht).2 | |
| exact ⟨fun i => hs ⟨h, i⟩, fun s a => hm h s a⟩ | |
| def sameSide (s : Fin 3) (t e : JointCoarseHashing.Triple (Position counts)) : Prop := | |
| ∀ h, sideWord counts s t h = sideWord counts s e h | |
| def sharedAmbient (sum : H → ℕ) (s : Fin 3) | |
| (e : JointCoarseHashing.Triple (Position counts)) : | |
| Finset (JointCoarseHashing.Triple (Position counts)) := | |
| (ambientSet counts sum).filter (sameSide counts s · e) | |
| def classBound (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts)) | |
| (h : H) (entropyCap : ℝ) : ℝ := | |
| ((Fintype.card (Positions counts h) : ℝ) + 1) ^ Fintype.card Shape * | |
| Real.exp ((Fintype.card (Positions counts h) : ℝ) * | |
| (entropyCap - finiteEntropy (fun a => | |
| (wordPopulation (sideWord counts s e h) a : ℝ) / | |
| Fintype.card (Positions counts h))) + | |
| ((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) * | |
| (1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ))) | |
| def classExponent (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts)) | |
| (h : H) (entropyCap : ℝ) : ℝ := | |
| (Fintype.card (Positions counts h) : ℝ) * | |
| (entropyCap - finiteEntropy (fun a => | |
| (wordPopulation (sideWord counts s e h) a : ℝ) / | |
| Fintype.card (Positions counts h))) + | |
| ((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) * | |
| (1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ)) + | |
| (Fintype.card Shape : ℝ) * Real.log (Fintype.card (Positions counts h) + 1 : ℕ) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem classBound_eq_exp (s : Fin 3) | |
| (e : JointCoarseHashing.Triple (Position counts)) (h : H) (entropyCap : ℝ) : | |
| classBound counts s e h entropyCap = Real.exp (classExponent counts s e h entropyCap) := by | |
| unfold classBound classExponent | |
| conv_rhs => rw [Real.exp_add, Real.exp_nat_mul, Real.exp_log (by positivity)] | |
| simp only [Nat.cast_add, Nat.cast_one] | |
| exact mul_comm _ _ | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem classExponent_target_eq (s : Fin 3) (e : Target counts) (h : H) | |
| (entropyCap : ℝ) : | |
| classExponent counts s (triple counts e) h entropyCap = | |
| classExponent counts s (triple counts (canonicalTarget counts)) h entropyCap := by | |
| simp_rw [classExponent, marginal_population_eq counts e (canonicalTarget counts)] | |
| theorem sharedAmbient_card_le (sum : H → ℕ) (s : Fin 3) | |
| (e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ) | |
| (hentropy : ∀ h w, localAllowed counts sum h w → | |
| finiteEntropy (fun a => (wordPopulation w a : ℝ) / | |
| Fintype.card (Positions counts h)) ≤ entropyCap h) : | |
| ((sharedAmbient counts sum s e).card : ℝ) ≤ | |
| ∏ h, classBound counts s e h (entropyCap h) := by | |
| classical | |
| let Allowed := fun h (w : Positions counts h → Shape) => | |
| localAllowed counts sum h w ∧ ∀ i, shapeSide s (w i) = sideWord counts s e h i | |
| let Local := fun h => {w : Positions counts h → Shape // Allowed h w} | |
| let encode : {t // t ∈ sharedAmbient counts sum s e} → ∀ h, Local h := fun t h => | |
| ⟨localWord counts t.val h, localAllowed_of_mem_ambient counts sum t.val | |
| (Finset.mem_filter.mp t.property).1 h, | |
| fun i => congrFun ((Finset.mem_filter.mp t.property).2 h) i⟩ | |
| have hinj : Function.Injective encode := by | |
| intro t u he | |
| apply Subtype.ext | |
| apply localWord_injective counts | |
| funext h | |
| exact congrArg Subtype.val (congrFun he h) | |
| have hcard : (sharedAmbient counts sum s e).card ≤ ∏ h, Fintype.card (Local h) := by | |
| simpa only [Fintype.card_coe, Fintype.card_pi] using | |
| Fintype.card_le_of_injective encode hinj | |
| have hlocal (h : H) : (Fintype.card (Local h) : ℝ) ≤ | |
| classBound counts s e h (entropyCap h) := by | |
| exact JointTypeCounts.card_allowed_over_coarse_le (shapeSide s) | |
| (sideWord counts s e h) (Allowed h) (entropyCap h) | |
| (fun w hw => hw.2) (fun w hw => hentropy h w hw.1) | |
| calc | |
| ((sharedAmbient counts sum s e).card : ℝ) ≤ | |
| ∏ h, (Fintype.card (Local h) : ℝ) := by exact_mod_cast hcard | |
| _ ≤ _ := Finset.prod_le_prod₀ (fun _ _ => Nat.cast_nonneg _) (fun h _ => hlocal h) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem sharesSide_iff (t e : JointCoarseHashing.Triple (Position counts)) : | |
| JointCoarseHashing.SharesSide t e ↔ ∃ s : Fin 3, sameSide counts s t e := by | |
| constructor | |
| · rintro (hx | hy | hz) | |
| · refine ⟨0, fun h => ?_⟩ | |
| funext i | |
| change t.1 ⟨h, i⟩ = e.1 ⟨h, i⟩ | |
| exact congrFun hx.symm ⟨h, i⟩ | |
| · refine ⟨1, fun h => ?_⟩ | |
| funext i | |
| change t.2.1 ⟨h, i⟩ = e.2.1 ⟨h, i⟩ | |
| exact congrFun hy.symm ⟨h, i⟩ | |
| · refine ⟨2, fun h => ?_⟩ | |
| funext i | |
| change t.2.2 ⟨h, i⟩ = e.2.2 ⟨h, i⟩ | |
| exact congrFun hz.symm ⟨h, i⟩ | |
| · rintro ⟨s, hs⟩ | |
| fin_cases s | |
| · apply Or.inl | |
| funext p | |
| simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm | |
| · apply Or.inr ∘ Or.inl | |
| funext p | |
| simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm | |
| · apply Or.inr ∘ Or.inr | |
| funext p | |
| simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm | |
| def neighbors (sum : H → ℕ) (e : JointCoarseHashing.Triple (Position counts)) : | |
| Finset (JointCoarseHashing.Triple (Position counts)) := | |
| (ambientSet counts sum).filter (fun t => t ≠ e ∧ JointCoarseHashing.SharesSide t e) | |
| theorem neighbors_card_le_sum (sum : H → ℕ) | |
| (e : JointCoarseHashing.Triple (Position counts)) : | |
| (neighbors counts sum e).card ≤ ∑ s : Fin 3, (sharedAmbient counts sum s e).card := by | |
| classical | |
| have hsub : neighbors counts sum e ⊆ | |
| (Finset.univ : Finset (Fin 3)).biUnion (fun s => sharedAmbient counts sum s e) := by | |
| intro t ht | |
| obtain ⟨ha, _, hs⟩ := Finset.mem_filter.mp ht | |
| obtain ⟨s, hs⟩ := (sharesSide_iff counts t e).mp hs | |
| exact Finset.mem_biUnion.mpr | |
| ⟨s, Finset.mem_univ _, Finset.mem_filter.mpr ⟨ha, hs⟩⟩ | |
| exact (Finset.card_le_card hsub).trans Finset.card_biUnion_le | |
| theorem neighbors_card_le (sum : H → ℕ) | |
| (e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ) | |
| (hentropy : ∀ h w, localAllowed counts sum h w → | |
| finiteEntropy (fun a => (wordPopulation w a : ℝ) / | |
| Fintype.card (Positions counts h)) ≤ entropyCap h) : | |
| ((neighbors counts sum e).card : ℝ) ≤ | |
| ∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) := by | |
| calc | |
| ((neighbors counts sum e).card : ℝ) ≤ | |
| ∑ s : Fin 3, ((sharedAmbient counts sum s e).card : ℝ) := by | |
| exact_mod_cast neighbors_card_le_sum counts sum e | |
| _ ≤ _ := Finset.sum_le_sum fun s _ => sharedAmbient_card_le counts sum s e entropyCap hentropy | |
| theorem neighbors_card_le_three_exp (sum : H → ℕ) | |
| (e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ) | |
| (hentropy : ∀ h w, localAllowed counts sum h w → | |
| finiteEntropy (fun a => (wordPopulation w a : ℝ) / | |
| Fintype.card (Positions counts h)) ≤ entropyCap h) | |
| (exponent : ℝ) | |
| (hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent) : | |
| ((neighbors counts sum e).card : ℝ) ≤ 3 * Real.exp exponent := by | |
| have hprod (s : Fin 3) : (∏ h, classBound counts s e h (entropyCap h)) ≤ | |
| Real.exp exponent := by | |
| simp_rw [classBound_eq_exp] | |
| rw [← Real.exp_sum] | |
| exact Real.exp_le_exp.mpr (hupper s) | |
| calc | |
| ((neighbors counts sum e).card : ℝ) ≤ | |
| ∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) := | |
| neighbors_card_le counts sum e entropyCap hentropy | |
| _ ≤ ∑ _s : Fin 3, Real.exp exponent := Finset.sum_le_sum fun s _ => hprod s | |
| _ = _ := by simp | |
| theorem log_neighbors_card_le (sum : H → ℕ) | |
| (e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ) | |
| (hentropy : ∀ h w, localAllowed counts sum h w → | |
| finiteEntropy (fun a => (wordPopulation w a : ℝ) / | |
| Fintype.card (Positions counts h)) ≤ entropyCap h) | |
| (exponent : ℝ) | |
| (hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent) | |
| (hne : 0 < (neighbors counts sum e).card) : | |
| Real.log ((neighbors counts sum e).card : ℝ) ≤ Real.log 3 + exponent := by | |
| have hb := Real.log_le_log (by exact_mod_cast hne : | |
| (0 : ℝ) < (neighbors counts sum e).card) | |
| (neighbors_card_le_three_exp counts sum e entropyCap hentropy exponent hupper) | |
| simpa only [Real.log_mul (by norm_num : (3 : ℝ) ≠ 0) (Real.exp_ne_zero _), | |
| Real.log_exp] using hb | |
| end MatrixMultiplication.JointAmbientDegree | |
| end | |
| end OAI | |