| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four0_1_7 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four0_5_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four0_7_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four0_8_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four1_0_7 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four1_7_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four2_0_6 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four2_6_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four3_0_5 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four3_5_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four4_0_4 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four4_4_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four5_0_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four5_3_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four6_0_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four6_2_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four7_0_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four7_1_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Four8_0_0 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal1_2_5 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal1_3_4 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal1_4_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal1_5_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal1_6_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal2_1_5 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal2_2_4 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal2_3_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal2_4_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal2_5_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal3_1_4 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal3_2_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal3_3_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal3_4_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal4_1_3 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal4_2_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal4_3_1 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal5_1_2 |
| import OAI.LinearAlgebra.MatrixFields.Certificates.Terminal5_2_1 |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
| namespace MatrixMultiplication.NativeNumerics |
| open AllFieldParameters AllFieldHistory AllFieldCertificates AllFieldCertificates.Replay |
| open AllFieldInitialEntropyCertificate |
| open scoped BigOperators |
| lemma B6_1_1_below : below (shape 6 1 1) = [ (shape 2 1 1), (shape 3 0 1), (shape 3 1 0), (shape 4 0 0)] := by decide +kernel |
| def B6_1_1Raw (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 2, 1, 1 => ⟨(19757482365589500963001701230393 / 20000000000000000000000000000000), (63 / 50000000000000000000000000000000)⟩ |
| | 3, 0, 1 => ⟨1, 0⟩ |
| | 3, 1, 0 => ⟨1, 0⟩ |
| | 4, 0, 0 => ⟨(19757482365589500963001701230393 / 20000000000000000000000000000000), (63 / 50000000000000000000000000000000)⟩ |
| | _, _, _ => (.exact 0) |
| lemma B6_1_1Raw_2_1_1 : (B6_1_1Raw (shape 2 1 1)).Encloses (splitWeight positiveSecond tablep (shape 6 1 1) (shape 2 1 1) : ℝ) := by |
| exact contains_sound (splitWeight_three_enclosed positiveSecond tablep (shape 6 1 1) (shape 2 1 1) |
| (-122) (0) (0) w193 w195 w195 |
| (by decide +kernel) w193_sound w195_sound w195_sound) (by decide +kernel) |
| lemma B6_1_1Raw_3_0_1 : (B6_1_1Raw (shape 3 0 1)).Encloses (splitWeight positiveSecond tablep (shape 6 1 1) (shape 3 0 1) : ℝ) := by |
| exact contains_sound (splitWeight_three_enclosed positiveSecond tablep (shape 6 1 1) (shape 3 0 1) |
| (0) (0) (0) w195 w195 w195 |
| (by decide +kernel) w195_sound w195_sound w195_sound) (by decide +kernel) |
| lemma B6_1_1Raw_3_1_0 : (B6_1_1Raw (shape 3 1 0)).Encloses (splitWeight positiveSecond tablep (shape 6 1 1) (shape 3 1 0) : ℝ) := by |
| exact contains_sound (splitWeight_three_enclosed positiveSecond tablep (shape 6 1 1) (shape 3 1 0) |
| (0) (0) (0) w195 w195 w195 |
| (by decide +kernel) w195_sound w195_sound w195_sound) (by decide +kernel) |
| lemma B6_1_1Raw_4_0_0 : (B6_1_1Raw (shape 4 0 0)).Encloses (splitWeight positiveSecond tablep (shape 6 1 1) (shape 4 0 0) : ℝ) := by |
| exact contains_sound (splitWeight_three_enclosed positiveSecond tablep (shape 6 1 1) (shape 4 0 0) |
| (-122) (0) (0) w193 w195 w195 |
| (by decide +kernel) w193_sound w195_sound w195_sound) (by decide +kernel) |
| lemma B6_1_1Raw_sound (u : Shape) (hu : u ∈ below (shape 6 1 1)) : |
| (B6_1_1Raw u).Encloses (splitWeight positiveSecond tablep (shape 6 1 1) u : ℝ) := by |
| rw [B6_1_1_below] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl | rfl | rfl | rfl |
| · exact B6_1_1Raw_2_1_1 |
| · exact B6_1_1Raw_3_0_1 |
| · exact B6_1_1Raw_3_1_0 |
| · exact B6_1_1Raw_4_0_0 |
| def B6_1_1Den : Ball := ⟨(39757482365589500963001701230393 / 10000000000000000000000000000000), (63 / 25000000000000000000000000000000)⟩ |
| lemma B6_1_1Den_sound : B6_1_1Den.Encloses (((below (shape 6 1 1)).map (splitWeight positiveSecond tablep (shape 6 1 1))).sum : ℝ) := by |
| exact contains_sound (list_cast_enclosed (below (shape 6 1 1)) B6_1_1Raw (splitWeight positiveSecond tablep (shape 6 1 1)) B6_1_1Raw_sound) (by decide +kernel) |
| lemma B6_1_1Den_positive : 0 < B6_1_1Den.lower := by decide +kernel |
| def B6_1_1Table (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 2, 1, 1 => ⟨(124237509457400902736572326499 / 500000000000000000000000000000), (3 / 6250000000000000000000000000000)⟩ |
| | 3, 0, 1 => ⟨(125762490542599097263427673501 / 500000000000000000000000000000), (17 / 100000000000000000000000000000000)⟩ |
| | 3, 1, 0 => ⟨(125762490542599097263427673501 / 500000000000000000000000000000), (17 / 100000000000000000000000000000000)⟩ |
| | 4, 0, 0 => ⟨(124237509457400902736572326499 / 500000000000000000000000000000), (3 / 6250000000000000000000000000000)⟩ |
| | _, _, _ => (.exact 0) |
| def B6_1_1Law (u : Shape) : Ball := if u ∈ below (shape 6 1 1) then B6_1_1Table u else .exact 0 |
| lemma B6_1_1Law_check_2_1_1 : contains (.mul (B6_1_1Raw (shape 2 1 1)) (.inv B6_1_1Den)) (B6_1_1Table (shape 2 1 1)) = true := by decide +kernel |
| lemma B6_1_1Law_check_3_0_1 : contains (.mul (B6_1_1Raw (shape 3 0 1)) (.inv B6_1_1Den)) (B6_1_1Table (shape 3 0 1)) = true := by decide +kernel |
| lemma B6_1_1Law_check_3_1_0 : contains (.mul (B6_1_1Raw (shape 3 1 0)) (.inv B6_1_1Den)) (B6_1_1Table (shape 3 1 0)) = true := by decide +kernel |
| lemma B6_1_1Law_check_4_0_0 : contains (.mul (B6_1_1Raw (shape 4 0 0)) (.inv B6_1_1Den)) (B6_1_1Table (shape 4 0 0)) = true := by decide +kernel |
| lemma B6_1_1Law_check (u : Shape) (hu : u ∈ below (shape 6 1 1)) : |
| contains (.mul (B6_1_1Raw u) (.inv B6_1_1Den)) (B6_1_1Table u) = true := by |
| rw [B6_1_1_below] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl | rfl | rfl | rfl |
| · exact B6_1_1Law_check_2_1_1 |
| · exact B6_1_1Law_check_3_0_1 |
| · exact B6_1_1Law_check_3_1_0 |
| · exact B6_1_1Law_check_4_0_0 |
| lemma B6_1_1Law_sound (u : Shape) : |
| (B6_1_1Law u).Encloses (stageBLaw (shape 6 1 1) u : ℝ) := by |
| by_cases hu : u ∈ below (shape 6 1 1) |
| · have h := contains_sound (Ball.mul_sound (B6_1_1Raw_sound u hu) |
| (Ball.inv_sound B6_1_1Den_sound B6_1_1Den_positive)) (B6_1_1Law_check u hu) |
| simpa only [B6_1_1Law, hu, ite_true, stageBLaw, splitLaw, Rat.cast_div, Rat.cast_mul, Rat.cast_inv, div_eq_mul_inv] using h |
| · simp only [B6_1_1Law, hu, ite_false, stageBLaw, splitLaw, Rat.cast_zero] |
| simpa only [Rat.cast_zero] using Ball.exact_sound 0 |
|
|
| end MatrixMultiplication.NativeNumerics |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
| namespace MatrixMultiplication.NativeNumerics |
| open AllFieldParameters AllFieldHistory AllFieldCertificates AllFieldCertificates.Replay |
| open AllFieldInitialEntropyCertificate AllFieldNativeCapacity |
| open scoped BigOperators |
| def B6_1_1Binary (u : Shape) : ℚ := |
| match u 0, u 1, u 2 with |
| | 2, 1, 1 => (60257 / 1000000) |
| | _, _, _ => 0 |
| lemma B6_1_1Binary_eq (u : Shape) (hu : u ∈ shapes 4) : |
| binaryParameter (shape 6 1 1) u = B6_1_1Binary u := by |
| rw [shapes4_list] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| · decide +kernel |
| def B6_1_1LittleRat (u : Shape) (w : Fin 3) (i : Fin 6) : ℚ := |
| if u w = 2 then |
| if i = 2 then 1 - B6_1_1Binary u else if i = 3 then B6_1_1Binary u else 0 |
| else if i = singletonSlot (u w) then 1 else 0 |
|
|
| def B6_1_1Little (u : Shape) (w : Fin 3) (i : Fin 6) : Ball := |
| if u ∈ shapes 4 then .exact (B6_1_1LittleRat u w i) |
| else .exact (littleLaw (shape 6 1 1) u w i) |
|
|
| lemma B6_1_1Little_sound (u : Shape) (w : Fin 3) (i : Fin 6) : |
| (B6_1_1Little u w i).Encloses (littleLaw (shape 6 1 1) u w i : ℝ) := by |
| by_cases hu : u ∈ shapes 4 |
| · simp only [B6_1_1Little, hu, ite_true] |
| apply exact_rat_enclosed |
| simp only [B6_1_1LittleRat, littleLaw, B6_1_1Binary_eq u hu] |
| · simp only [B6_1_1Little, hu, ite_false] |
| exact Ball.exact_sound _ |
| def B6_1_1Half (w : Fin 3) (ij : PairSlot) : Ball := |
| match w.val, ij.1.val, ij.2.val with |
| | 0, 2, 5 => ⟨(2335026597000525930807493756423 / 10000000000000000000000000000000), (23 / 50000000000000000000000000000000)⟩ |
| | 0, 3, 5 => ⟨(149723592147492123923952773557 / 10000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩ |
| | 0, 4, 4 => ⟨(125762490542599097263427673501 / 250000000000000000000000000000), (17 / 50000000000000000000000000000000)⟩ |
| | 0, 5, 2 => ⟨(2335026597000525930807493756423 / 10000000000000000000000000000000), (23 / 50000000000000000000000000000000)⟩ |
| | 0, 5, 3 => ⟨(149723592147492123923952773557 / 10000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩ |
| | 1, 1, 0 => ⟨(1 / 2), (13 / 20000000000000000000000000000000)⟩ |
| | 1, 0, 1 => ⟨(1 / 2), (13 / 20000000000000000000000000000000)⟩ |
| | 2, 1, 0 => ⟨(1 / 2), (13 / 20000000000000000000000000000000)⟩ |
| | 2, 0, 1 => ⟨(1 / 2), (13 / 20000000000000000000000000000000)⟩ |
| | _, _, _ => (.exact 0) |
| lemma B6_1_1Half_check_0_0 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((0 : Fin 6),j)) (B6_1_1Half 0 (0,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_0_1 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((1 : Fin 6),j)) (B6_1_1Half 0 (1,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_0_2 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((2 : Fin 6),j)) (B6_1_1Half 0 (2,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_0_3 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((3 : Fin 6),j)) (B6_1_1Half 0 (3,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_0_4 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((4 : Fin 6),j)) (B6_1_1Half 0 (4,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_0_5 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 0) ((5 : Fin 6),j)) (B6_1_1Half 0 (5,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_0 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((0 : Fin 6),j)) (B6_1_1Half 1 (0,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_1 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((1 : Fin 6),j)) (B6_1_1Half 1 (1,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_2 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((2 : Fin 6),j)) (B6_1_1Half 1 (2,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_3 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((3 : Fin 6),j)) (B6_1_1Half 1 (3,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_4 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((4 : Fin 6),j)) (B6_1_1Half 1 (4,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_1_5 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 1) ((5 : Fin 6),j)) (B6_1_1Half 1 (5,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_0 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((0 : Fin 6),j)) (B6_1_1Half 2 (0,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_1 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((1 : Fin 6),j)) (B6_1_1Half 2 (1,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_2 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((2 : Fin 6),j)) (B6_1_1Half 2 (2,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_3 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((3 : Fin 6),j)) (B6_1_1Half 2 (3,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_4 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((4 : Fin 6),j)) (B6_1_1Half 2 (4,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_check_2_5 : ∀ j : Fin 6, |
| contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u 2) ((5 : Fin 6),j)) (B6_1_1Half 2 (5,j)) = true := by |
| decide +kernel |
| lemma B6_1_1Half_sound (w : Fin 3) (ij : PairSlot) : |
| (B6_1_1Half w ij).Encloses (halfLaw (shape 6 1 1) w ij : ℝ) := by |
| have hcheck : contains (pairLawBall (below (shape 6 1 1)) (shape 6 1 1) B6_1_1Law |
| (fun u => B6_1_1Little u w) ij) (B6_1_1Half w ij) = true := by |
| rcases ij with ⟨i,j⟩ |
| revert i w |
| apply forall_fin3 |
| · exact forall_fin6 (B6_1_1Half_check_0_0 j) (B6_1_1Half_check_0_1 j) (B6_1_1Half_check_0_2 j) (B6_1_1Half_check_0_3 j) (B6_1_1Half_check_0_4 j) (B6_1_1Half_check_0_5 j) |
| · exact forall_fin6 (B6_1_1Half_check_1_0 j) (B6_1_1Half_check_1_1 j) (B6_1_1Half_check_1_2 j) (B6_1_1Half_check_1_3 j) (B6_1_1Half_check_1_4 j) (B6_1_1Half_check_1_5 j) |
| · exact forall_fin6 (B6_1_1Half_check_2_0 j) (B6_1_1Half_check_2_1 j) (B6_1_1Half_check_2_2 j) (B6_1_1Half_check_2_3 j) (B6_1_1Half_check_2_4 j) (B6_1_1Half_check_2_5 j) |
| have h := contains_sound (pairLawBall_sound (below (shape 6 1 1)) (shape 6 1 1) |
| B6_1_1Law (fun u => (stageBLaw (shape 6 1 1) u : ℝ)) |
| (fun u => B6_1_1Little u w) (fun u i => (littleLaw (shape 6 1 1) u w i : ℝ)) |
| (fun u _ => B6_1_1Law_sound u) (fun u i => B6_1_1Little_sound u w i) ij) hcheck |
| have hp : positive (shape 6 1 1) = true := by decide +kernel |
| simpa only [halfLaw, hp, ite_true, pairLaw, positiveHalfLaw, Rat.cast_list_sum, List.map_map, Function.comp_def, Rat.cast_mul] using h |
|
|
| end MatrixMultiplication.NativeNumerics |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
| namespace MatrixMultiplication.NativeNumerics |
| open AllFieldParameters AllFieldHistory AllFieldCertificates AllFieldCertificates.Replay |
| open AllFieldInitialEntropyCertificate AllFieldNativeCapacity |
| open scoped BigOperators |
| def B6_1_1U2_1_1D : Ball := .exact (60257 / 1000000) |
| lemma B6_1_1U2_1_1D_sound : B6_1_1U2_1_1D.Encloses (binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ) := by |
| apply exact_rat_enclosed |
| decide +kernel |
| def B6_1_1U2_1_1OneD : Ball := .exact (939743 / 1000000) |
| lemma B6_1_1U2_1_1OneD_sound : B6_1_1U2_1_1OneD.Encloses (1 - (binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ)) := by |
| have he : binaryParameter (shape 6 1 1) (shape 2 1 1) = (60257 / 1000000) := by decide +kernel |
| rw [he] |
| convert Ball.exact_sound ((939743 / 1000000) : ℚ) using 1 <;> norm_num [B6_1_1U2_1_1OneD, Ball.exact] |
| def B6_1_1U2_1_1EDLogWitness : LogWitness := ⟨-5, 60257, 31250⟩ |
| def B6_1_1U2_1_1EDLog : Ball := ⟨(-70228413269685773586736117896293 / 25000000000000000000000000000000), (63527471049 / 50000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1EDLog_sound : B6_1_1U2_1_1EDLog.Encloses (Real.log ((binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ))) := by |
| exact checkLog_sound logTwoBall_sound (B6_1_1U2_1_1D_sound) (w := B6_1_1U2_1_1EDLogWitness) (by |
| simp only [checkLog, Bool.and_eq_true] |
| exact ⟨by decide +kernel, by decide +kernel⟩) |
| def B6_1_1U2_1_1ED : Ball := ⟨(4231753498391455659015958256077 / 25000000000000000000000000000000), (7655949647 / 100000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1ED_sound : B6_1_1U2_1_1ED.Encloses (Foundation.entropyTerm ((binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ))) := by |
| exact contains_sound (entropyTerm_ball (B6_1_1U2_1_1D_sound) B6_1_1U2_1_1EDLog_sound) (by decide +kernel) |
| def B6_1_1U2_1_1EOneDLogWitness : LogWitness := ⟨-1, 939743, 500000⟩ |
| def B6_1_1U2_1_1EOneDLog : Ball := ⟨(-124297690710327487232327528573 / 2000000000000000000000000000000), (21175823683 / 50000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1EOneDLog_sound : B6_1_1U2_1_1EOneDLog.Encloses (Real.log ((1 - (binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ)))) := by |
| exact checkLog_sound logTwoBall_sound (B6_1_1U2_1_1OneD_sound) (w := B6_1_1U2_1_1EOneDLogWitness) (by |
| simp only [checkLog, Bool.and_eq_true] |
| exact ⟨by decide +kernel, by decide +kernel⟩) |
| def B6_1_1U2_1_1EOneD : Ball := ⟨(5840394238059764191708458434189 / 100000000000000000000000000000000), (39799664151 / 100000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1EOneD_sound : B6_1_1U2_1_1EOneD.Encloses (Foundation.entropyTerm ((1 - (binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ)))) := by |
| exact contains_sound (entropyTerm_ball (B6_1_1U2_1_1OneD_sound) B6_1_1U2_1_1EOneDLog_sound) (by decide +kernel) |
| def B6_1_1U2_1_1BinaryEntropy : Ball := ⟨(26944105197525649278727705902131 / 100000000000000000000000000000000), (25003798507 / 50000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1BinaryEntropy_sound : B6_1_1U2_1_1BinaryEntropy.Encloses |
| (binaryEntropyRate (binaryParameter (shape 6 1 1) (shape 2 1 1))) := by |
| exact contains_sound (Ball.add_sound (Ball.add_sound B6_1_1U2_1_1EOneD_sound B6_1_1U2_1_1ED_sound) |
| (Ball.mul_sound B6_1_1U2_1_1D_sound logN2_sound)) (by decide +kernel) |
| def B6_1_1U2_1_1Leaf : Ball := ⟨(312189592457865916292919997050341 / 100000000000000000000000000000000), (30806741819 / 25000000000000000000000000000000)⟩ |
| lemma B6_1_1U2_1_1Leaf_sound : B6_1_1U2_1_1Leaf.Encloses (AllFieldTerminalRates.secondLeafRate (shape 6 1 1) (shape 2 1 1)) := by |
| have hh : shapeMax (shape 2 1 1) = 2 := by decide +kernel |
| have hp : positive (shape 2 1 1) = true := by decide +kernel |
| simp only [AllFieldTerminalRates.secondLeafRate, hh, hp, Bool.not_true, Bool.false_eq_true, show (2 : ℕ) ≠ 4 by decide, show (2 : ℕ) ≠ 3 by decide, ite_false] |
| have ht : (Ball.exact (1939743 / 1000000)).Encloses (2 - (binaryParameter (shape 6 1 1) (shape 2 1 1) : ℝ)) := by |
| have he : binaryParameter (shape 6 1 1) (shape 2 1 1) = (60257 / 1000000) := by decide +kernel |
| rw [he] |
| convert Ball.exact_sound ((1939743 / 1000000) : ℚ) using 1 |
| norm_num [Ball.exact] |
| exact contains_sound (Ball.mul_sound ht logN5_sound) (by decide +kernel) |
| def B6_1_1U2_1_1Mass : Ball := ⟨0, 0⟩ |
| lemma B6_1_1U2_1_1Mass_sound : B6_1_1U2_1_1Mass.Encloses (littleMass (shape 6 1 1) (shape 2 1 1) : ℝ) := by |
| rw [littleMass_cast] |
| exact contains_sound (Ball.mul_sound (Ball.mul_sound |
| (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) (second_6_1_1)) |
| (B6_1_1Law_sound (shape 2 1 1))) (by decide +kernel) |
| def B6_1_1U3_0_1Leaf : Ball := ⟨(115129254649702284200899544090987 / 50000000000000000000000000000000), (21175823683 / 25000000000000000000000000000000)⟩ |
| lemma B6_1_1U3_0_1Leaf_sound : B6_1_1U3_0_1Leaf.Encloses (AllFieldTerminalRates.secondLeafRate (shape 6 1 1) (shape 3 0 1)) := by |
| have hh : shapeMax (shape 3 0 1) = 3 := by decide +kernel |
| simp only [AllFieldTerminalRates.secondLeafRate, hh, show (3 : ℕ) ≠ 4 by decide, ite_false, ite_true] |
| exact logN10_sound |
| def B6_1_1U3_0_1Mass : Ball := ⟨0, 0⟩ |
| lemma B6_1_1U3_0_1Mass_sound : B6_1_1U3_0_1Mass.Encloses (littleMass (shape 6 1 1) (shape 3 0 1) : ℝ) := by |
| rw [littleMass_cast] |
| exact contains_sound (Ball.mul_sound (Ball.mul_sound |
| (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) (second_6_1_1)) |
| (B6_1_1Law_sound (shape 3 0 1))) (by decide +kernel) |
| def B6_1_1U3_1_0Leaf : Ball := ⟨(115129254649702284200899544090987 / 50000000000000000000000000000000), (21175823683 / 25000000000000000000000000000000)⟩ |
| lemma B6_1_1U3_1_0Leaf_sound : B6_1_1U3_1_0Leaf.Encloses (AllFieldTerminalRates.secondLeafRate (shape 6 1 1) (shape 3 1 0)) := by |
| have hh : shapeMax (shape 3 1 0) = 3 := by decide +kernel |
| simp only [AllFieldTerminalRates.secondLeafRate, hh, show (3 : ℕ) ≠ 4 by decide, ite_false, ite_true] |
| exact logN10_sound |
| def B6_1_1U3_1_0Mass : Ball := ⟨0, 0⟩ |
| lemma B6_1_1U3_1_0Mass_sound : B6_1_1U3_1_0Mass.Encloses (littleMass (shape 6 1 1) (shape 3 1 0) : ℝ) := by |
| rw [littleMass_cast] |
| exact contains_sound (Ball.mul_sound (Ball.mul_sound |
| (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) (second_6_1_1)) |
| (B6_1_1Law_sound (shape 3 1 0))) (by decide +kernel) |
| def B6_1_1U4_0_0Leaf : Ball := ⟨0, 0⟩ |
| lemma B6_1_1U4_0_0Leaf_sound : B6_1_1U4_0_0Leaf.Encloses (AllFieldTerminalRates.secondLeafRate (shape 6 1 1) (shape 4 0 0)) := by |
| have hh : shapeMax (shape 4 0 0) = 4 := by decide +kernel |
| simp only [AllFieldTerminalRates.secondLeafRate, hh, ite_true] |
| simpa only [B6_1_1U4_0_0Leaf, Ball.exact, Rat.cast_zero] using Ball.exact_sound 0 |
| def B6_1_1U4_0_0Mass : Ball := ⟨0, 0⟩ |
| lemma B6_1_1U4_0_0Mass_sound : B6_1_1U4_0_0Mass.Encloses (littleMass (shape 6 1 1) (shape 4 0 0) : ℝ) := by |
| rw [littleMass_cast] |
| exact contains_sound (Ball.mul_sound (Ball.mul_sound |
| (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) (second_6_1_1)) |
| (B6_1_1Law_sound (shape 4 0 0))) (by decide +kernel) |
| def B6_1_1Leaves (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 2, 1, 1 => B6_1_1U2_1_1Leaf |
| | 3, 0, 1 => B6_1_1U3_0_1Leaf |
| | 3, 1, 0 => B6_1_1U3_1_0Leaf |
| | 4, 0, 0 => B6_1_1U4_0_0Leaf |
| | _, _, _ => (.exact 0) |
| lemma B6_1_1Leaves_sound (u : Shape) (hu : u ∈ below (shape 6 1 1)) : |
| (B6_1_1Leaves u).Encloses (AllFieldTerminalRates.secondLeafRate (shape 6 1 1) u) := by |
| rw [B6_1_1_below] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl | rfl | rfl | rfl |
| · exact B6_1_1U2_1_1Leaf_sound |
| · exact B6_1_1U3_0_1Leaf_sound |
| · exact B6_1_1U3_1_0Leaf_sound |
| · exact B6_1_1U4_0_0Leaf_sound |
| def B6_1_1Masses (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 2, 1, 1 => B6_1_1U2_1_1Mass |
| | 3, 0, 1 => B6_1_1U3_0_1Mass |
| | 3, 1, 0 => B6_1_1U3_1_0Mass |
| | 4, 0, 0 => B6_1_1U4_0_0Mass |
| | _, _, _ => (.exact 0) |
| lemma B6_1_1Masses_sound (u : Shape) (hu : u ∈ below (shape 6 1 1)) : |
| (B6_1_1Masses u).Encloses ((fun u => (littleMass (shape 6 1 1) u : ℝ)) u) := by |
| rw [B6_1_1_below] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl | rfl | rfl | rfl |
| · exact B6_1_1U2_1_1Mass_sound |
| · exact B6_1_1U3_0_1Mass_sound |
| · exact B6_1_1U3_1_0Mass_sound |
| · exact B6_1_1U4_0_0Mass_sound |
| def B6_1_1Binaries (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 2, 1, 1 => B6_1_1U2_1_1BinaryEntropy |
| | _, _, _ => (.exact 0) |
| lemma B6_1_1Binaries_sound (u : Shape) (hu : u ∈ (below (shape 6 1 1)).filter positive) : |
| (B6_1_1Binaries u).Encloses ((fun u => binaryEntropyRate (binaryParameter (shape 6 1 1) u)) u) := by |
| have hl : (below (shape 6 1 1)).filter positive = [(shape 2 1 1)] := by decide +kernel |
| rw [hl] at hu |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at hu |
| rcases hu with rfl |
| · exact B6_1_1U2_1_1BinaryEntropy_sound |
| def B6_1_1S2 : Ball := ⟨0, 0⟩ |
| lemma B6_1_1S2_sound : B6_1_1S2.Encloses |
| (((below (shape 6 1 1)).map fun u => (littleMass (shape 6 1 1) u : ℝ) * AllFieldTerminalRates.secondLeafRate (shape 6 1 1) u).sum) := by |
| exact contains_sound (listSumBall_sound (below (shape 6 1 1)) |
| (fun u => .mul (B6_1_1Masses u) (B6_1_1Leaves u)) _ |
| (fun u hu => Ball.mul_sound (B6_1_1Masses_sound u hu) (B6_1_1Leaves_sound u hu))) (by decide +kernel) |
|
|
| end MatrixMultiplication.NativeNumerics |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
| namespace MatrixMultiplication.NativeNumerics |
| open AllFieldParameters AllFieldHistory AllFieldCertificates AllFieldCertificates.Replay |
| open AllFieldInitialEntropyCertificate AllFieldNativeCapacity |
| open scoped BigOperators |
| open AllFieldTerminalRates AllFieldVolumeCertificate |
| def fourRateBalls (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 0, 0, 8 => Z0_0_8Four |
| | 0, 1, 7 => Z0_1_7Four |
| | 0, 2, 6 => Z0_2_6Four |
| | 0, 3, 5 => Z0_3_5Four |
| | 0, 4, 4 => Z0_4_4Four |
| | 0, 5, 3 => Z0_5_3Four |
| | 0, 6, 2 => Z0_6_2Four |
| | 0, 7, 1 => Z0_7_1Four |
| | 0, 8, 0 => Z0_8_0Four |
| | 1, 0, 7 => Z1_0_7Four |
| | 1, 7, 0 => Z1_7_0Four |
| | 2, 0, 6 => Z2_0_6Four |
| | 2, 6, 0 => Z2_6_0Four |
| | 3, 0, 5 => Z3_0_5Four |
| | 3, 5, 0 => Z3_5_0Four |
| | 4, 0, 4 => Z4_0_4Four |
| | 4, 4, 0 => Z4_4_0Four |
| | 5, 0, 3 => Z5_0_3Four |
| | 5, 3, 0 => Z5_3_0Four |
| | 6, 0, 2 => Z6_0_2Four |
| | 6, 2, 0 => Z6_2_0Four |
| | 7, 0, 1 => Z7_0_1Four |
| | 7, 1, 0 => Z7_1_0Four |
| | 8, 0, 0 => Z8_0_0Four |
| | _, _, _ => (.exact 0) |
| lemma fourRateBalls_sound (t : Shape) (ht : t ∈ zeroSecond) : |
| (fourRateBalls t).Encloses (AllFieldZeroLeafRates.fourRate t) := by |
| rw [zeroSecond_list] at ht |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at ht |
| rcases ht with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact Z0_0_8Four_sound |
| · exact Z0_1_7Four_sound |
| · exact Z0_2_6Four_sound |
| · exact Z0_3_5Four_sound |
| · exact Z0_4_4Four_sound |
| · exact Z0_5_3Four_sound |
| · exact Z0_6_2Four_sound |
| · exact Z0_7_1Four_sound |
| · exact Z0_8_0Four_sound |
| · exact Z1_0_7Four_sound |
| · exact Z1_7_0Four_sound |
| · exact Z2_0_6Four_sound |
| · exact Z2_6_0Four_sound |
| · exact Z3_0_5Four_sound |
| · exact Z3_5_0Four_sound |
| · exact Z4_0_4Four_sound |
| · exact Z4_4_0Four_sound |
| · exact Z5_0_3Four_sound |
| · exact Z5_3_0Four_sound |
| · exact Z6_0_2Four_sound |
| · exact Z6_2_0Four_sound |
| · exact Z7_0_1Four_sound |
| · exact Z7_1_0Four_sound |
| · exact Z8_0_0Four_sound |
| def nativeS1Ball : Ball := ⟨(88236223598149624413852945550479 / 100000000000000000000000000000000), (32131158713 / 100000000000000000000000000000000)⟩ |
| lemma nativeS1Ball_sound : nativeS1Ball.Encloses S1 := by |
| exact contains_sound (listSumBall_sound zeroSecond |
| (fun t => .mul (secondProbs t) (fourRateBalls t)) _ |
| (fun t ht => Ball.mul_sound (secondProbs_sound t (List.mem_filter.mp ht).1) |
| (fourRateBalls_sound t ht))) (by decide +kernel) |
| def s2ParentBalls (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 1, 1, 6 => B1_1_6S2 |
| | 1, 2, 5 => B1_2_5S2 |
| | 1, 3, 4 => B1_3_4S2 |
| | 1, 4, 3 => B1_4_3S2 |
| | 1, 5, 2 => B1_5_2S2 |
| | 1, 6, 1 => B1_6_1S2 |
| | 2, 1, 5 => B2_1_5S2 |
| | 2, 2, 4 => B2_2_4S2 |
| | 2, 3, 3 => B2_3_3S2 |
| | 2, 4, 2 => B2_4_2S2 |
| | 2, 5, 1 => B2_5_1S2 |
| | 3, 1, 4 => B3_1_4S2 |
| | 3, 2, 3 => B3_2_3S2 |
| | 3, 3, 2 => B3_3_2S2 |
| | 3, 4, 1 => B3_4_1S2 |
| | 4, 1, 3 => B4_1_3S2 |
| | 4, 2, 2 => B4_2_2S2 |
| | 4, 3, 1 => B4_3_1S2 |
| | 5, 1, 2 => B5_1_2S2 |
| | 5, 2, 1 => B5_2_1S2 |
| | 6, 1, 1 => B6_1_1S2 |
| | _, _, _ => (.exact 0) |
| lemma s2ParentBalls_sound (t : Shape) (ht : t ∈ positiveSecond) : |
| (s2ParentBalls t).Encloses (((below t).map fun u => |
| (littleMass t u : ℝ) * secondLeafRate t u).sum) := by |
| rw [positiveSecond_list] at ht |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at ht |
| rcases ht with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact B1_1_6S2_sound |
| · exact B1_2_5S2_sound |
| · exact B1_3_4S2_sound |
| · exact B1_4_3S2_sound |
| · exact B1_5_2S2_sound |
| · exact B1_6_1S2_sound |
| · exact B2_1_5S2_sound |
| · exact B2_2_4S2_sound |
| · exact B2_3_3S2_sound |
| · exact B2_4_2S2_sound |
| · exact B2_5_1S2_sound |
| · exact B3_1_4S2_sound |
| · exact B3_2_3S2_sound |
| · exact B3_3_2S2_sound |
| · exact B3_4_1S2_sound |
| · exact B4_1_3S2_sound |
| · exact B4_2_2S2_sound |
| · exact B4_3_1S2_sound |
| · exact B5_1_2S2_sound |
| · exact B5_2_1S2_sound |
| · exact B6_1_1S2_sound |
| def nativeS2Ball : Ball := ⟨(233614080284104308981345775925091 / 20000000000000000000000000000000), (7783352389 / 1562500000000000000000000000000)⟩ |
| lemma nativeS2Ball_sound : nativeS2Ball.Encloses S2 := by |
| exact contains_sound (listSumBall_sound positiveSecond s2ParentBalls _ |
| s2ParentBalls_sound) (by decide +kernel) |
| def nativeSBall : Ball := ⟨(1259477032159467636236050651212461 / 100000000000000000000000000000000), (531265093341 / 100000000000000000000000000000000)⟩ |
| lemma nativeSBall_sound : nativeSBall.Encloses S := by |
| exact contains_sound (Ball.add_sound (Ball.add_sound nativeS0_enclosed |
| nativeS1Ball_sound) nativeS2Ball_sound) (by decide +kernel) |
|
|
| theorem native_volume_positive : 0 < S := by |
| have hl := (Ball.bounds nativeSBall_sound).1 |
| have hp : (0 : ℝ) < nativeSBall.lower := by norm_num [nativeSBall, Ball.lower] |
| exact hp.trans_le hl |
|
|
| theorem nativeS1_strict_bounds : |
| (882362235981496244 : ℝ)/10^18 < S1 ∧ S1 < (882362235981496245 : ℝ)/10^18 := by |
| have hb := Ball.bounds nativeS1Ball_sound |
| norm_num [nativeS1Ball, Ball.lower, Ball.upper] at hb ⊢ |
| constructor <;> linarith |
| theorem nativeS2_strict_bounds : |
| (11680704014205215449 : ℝ)/10^18 < S2 ∧ S2 < (11680704014205215450 : ℝ)/10^18 := by |
| have hb := Ball.bounds nativeS2Ball_sound |
| norm_num [nativeS2Ball, Ball.lower, Ball.upper] at hb ⊢ |
| constructor <;> linarith |
|
|
| end MatrixMultiplication.NativeNumerics |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
| namespace MatrixMultiplication.NativeNumerics |
| open AllFieldParameters AllFieldHistory AllFieldCertificates AllFieldCertificates.Replay |
| open AllFieldInitialEntropyCertificate AllFieldNativeCapacity |
| open scoped BigOperators |
| lemma interior_fold (f : Shape × Shape → ℝ) : |
| (interiorPairs.map f).sum = (positiveSecond.map (fun t => |
| (((below t).filter positive).map (fun u => f (t,u))).sum)).sum := by |
| have h (xs : List Shape) : |
| ((xs.flatMap (fun t => ((below t).filter positive).map (fun u => (t,u)))).map f).sum = |
| (xs.map (fun t => (((below t).filter positive).map (fun u => f (t,u))).sum)).sum := by |
| induction xs with |
| | nil => simp |
| | cons t xs ih => simp only [List.flatMap_cons, List.map_append, List.sum_append, List.map_map, Function.comp_def, List.map_cons, List.sum_cons, ih] |
| exact h positiveSecond |
| def B1_1_6Ttotal : Ball := ⟨(3710278844513467307488141523 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩ |
| lemma B1_1_6Ttotal_sound : B1_1_6Ttotal.Encloses |
| ((((below (shape 1 1 6)).filter positive).map (fun u => (littleMass (shape 1 1 6) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 1 6)).filter positive) |
| B1_1_6Masses _ (fun u hu => B1_1_6Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_1_6Btotal : Ball := ⟨(10292390592644662403454216507 / 100000000000000000000000000000000), (1178709 / 12500000000000000000000000000000)⟩ |
| lemma B1_1_6Btotal_sound : B1_1_6Btotal.Encloses |
| ((((below (shape 1 1 6)).filter positive).map (fun u => (littleMass (shape 1 1 6) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 1 6) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 1 6)).filter positive) |
| (fun u => .mul (B1_1_6Masses u) (.add (.mul (.exact 2) logN2) (B1_1_6Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_1_6Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_1_6Binaries_sound u hu)))) (by decide +kernel) |
| def B1_2_5Ttotal : Ball := ⟨(4755163798532963526151893931759 / 100000000000000000000000000000000), (57 / 100000000000000000000000000000000)⟩ |
| lemma B1_2_5Ttotal_sound : B1_2_5Ttotal.Encloses |
| ((((below (shape 1 2 5)).filter positive).map (fun u => (littleMass (shape 1 2 5) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 2 5)).filter positive) |
| B1_2_5Masses _ (fun u hu => B1_2_5Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_2_5Btotal : Ball := ⟨(263770662946704051420489805631 / 4000000000000000000000000000000), (4032951479 / 100000000000000000000000000000000)⟩ |
| lemma B1_2_5Btotal_sound : B1_2_5Btotal.Encloses |
| ((((below (shape 1 2 5)).filter positive).map (fun u => (littleMass (shape 1 2 5) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 2 5) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 2 5)).filter positive) |
| (fun u => .mul (B1_2_5Masses u) (.add (.mul (.exact 2) logN2) (B1_2_5Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_2_5Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_2_5Binaries_sound u hu)))) (by decide +kernel) |
| def B1_3_4Ttotal : Ball := ⟨(7480325224810645241814099259701 / 25000000000000000000000000000000), (151 / 50000000000000000000000000000000)⟩ |
| lemma B1_3_4Ttotal_sound : B1_3_4Ttotal.Encloses |
| ((((below (shape 1 3 4)).filter positive).map (fun u => (littleMass (shape 1 3 4) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 3 4)).filter positive) |
| B1_3_4Masses _ (fun u hu => B1_3_4Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_3_4Btotal : Ball := ⟨(10623575146720639793993756588557 / 25000000000000000000000000000000), (38286699949 / 100000000000000000000000000000000)⟩ |
| lemma B1_3_4Btotal_sound : B1_3_4Btotal.Encloses |
| ((((below (shape 1 3 4)).filter positive).map (fun u => (littleMass (shape 1 3 4) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 3 4) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 3 4)).filter positive) |
| (fun u => .mul (B1_3_4Masses u) (.add (.mul (.exact 2) logN2) (B1_3_4Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_3_4Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_3_4Binaries_sound u hu)))) (by decide +kernel) |
| def B1_4_3Ttotal : Ball := ⟨(6944605000406853487412318907203 / 50000000000000000000000000000000), (159 / 100000000000000000000000000000000)⟩ |
| lemma B1_4_3Ttotal_sound : B1_4_3Ttotal.Encloses |
| ((((below (shape 1 4 3)).filter positive).map (fun u => (littleMass (shape 1 4 3) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 4 3)).filter positive) |
| B1_4_3Masses _ (fun u hu => B1_4_3Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_4_3Btotal : Ball := ⟨(19909829730949708038768839279537 / 100000000000000000000000000000000), (17824415679 / 100000000000000000000000000000000)⟩ |
| lemma B1_4_3Btotal_sound : B1_4_3Btotal.Encloses |
| ((((below (shape 1 4 3)).filter positive).map (fun u => (littleMass (shape 1 4 3) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 4 3) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 4 3)).filter positive) |
| (fun u => .mul (B1_4_3Masses u) (.add (.mul (.exact 2) logN2) (B1_4_3Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_4_3Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_4_3Binaries_sound u hu)))) (by decide +kernel) |
| def B1_5_2Ttotal : Ball := ⟨(403356547772450802156337181231 / 100000000000000000000000000000000), (9 / 100000000000000000000000000000000)⟩ |
| lemma B1_5_2Ttotal_sound : B1_5_2Ttotal.Encloses |
| ((((below (shape 1 5 2)).filter positive).map (fun u => (littleMass (shape 1 5 2) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 5 2)).filter positive) |
| B1_5_2Masses _ (fun u hu => B1_5_2Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_5_2Btotal : Ball := ⟨(571743387204363003226748332669 / 100000000000000000000000000000000), (516090839 / 100000000000000000000000000000000)⟩ |
| lemma B1_5_2Btotal_sound : B1_5_2Btotal.Encloses |
| ((((below (shape 1 5 2)).filter positive).map (fun u => (littleMass (shape 1 5 2) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 5 2) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 5 2)).filter positive) |
| (fun u => .mul (B1_5_2Masses u) (.add (.mul (.exact 2) logN2) (B1_5_2Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_5_2Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_5_2Binaries_sound u hu)))) (by decide +kernel) |
| def B1_6_1Ttotal : Ball := ⟨(77881891746725088607012689 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩ |
| lemma B1_6_1Ttotal_sound : B1_6_1Ttotal.Encloses |
| ((((below (shape 1 6 1)).filter positive).map (fun u => (littleMass (shape 1 6 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 6 1)).filter positive) |
| B1_6_1Masses _ (fun u hu => B1_6_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B1_6_1Btotal : Ball := ⟨(619013508687338257205025789 / 50000000000000000000000000000000), (1031081 / 100000000000000000000000000000000)⟩ |
| lemma B1_6_1Btotal_sound : B1_6_1Btotal.Encloses |
| ((((below (shape 1 6 1)).filter positive).map (fun u => (littleMass (shape 1 6 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 1 6 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 1 6 1)).filter positive) |
| (fun u => .mul (B1_6_1Masses u) (.add (.mul (.exact 2) logN2) (B1_6_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B1_6_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B1_6_1Binaries_sound u hu)))) (by decide +kernel) |
| def B2_1_5Ttotal : Ball := ⟨(535927124963214472545786930867 / 25000000000000000000000000000000), (29 / 100000000000000000000000000000000)⟩ |
| lemma B2_1_5Ttotal_sound : B2_1_5Ttotal.Encloses |
| ((((below (shape 2 1 5)).filter positive).map (fun u => (littleMass (shape 2 1 5) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 1 5)).filter positive) |
| B2_1_5Masses _ (fun u hu => B2_1_5Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B2_1_5Btotal : Ball := ⟨(1486547718243755829261660863071 / 50000000000000000000000000000000), (681006439 / 25000000000000000000000000000000)⟩ |
| lemma B2_1_5Btotal_sound : B2_1_5Btotal.Encloses |
| ((((below (shape 2 1 5)).filter positive).map (fun u => (littleMass (shape 2 1 5) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 2 1 5) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 1 5)).filter positive) |
| (fun u => .mul (B2_1_5Masses u) (.add (.mul (.exact 2) logN2) (B2_1_5Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B2_1_5Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B2_1_5Binaries_sound u hu)))) (by decide +kernel) |
| def B2_2_4Ttotal : Ball := ⟨(20957935808074261818840608407343 / 50000000000000000000000000000000), (113 / 25000000000000000000000000000000)⟩ |
| lemma B2_2_4Ttotal_sound : B2_2_4Ttotal.Encloses |
| ((((below (shape 2 2 4)).filter positive).map (fun u => (littleMass (shape 2 2 4) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 2 4)).filter positive) |
| B2_2_4Masses _ (fun u hu => B2_2_4Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B2_2_4Btotal : Ball := ⟨(61259431592020676095074322312553 / 100000000000000000000000000000000), (54088775631 / 100000000000000000000000000000000)⟩ |
| lemma B2_2_4Btotal_sound : B2_2_4Btotal.Encloses |
| ((((below (shape 2 2 4)).filter positive).map (fun u => (littleMass (shape 2 2 4) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 2 2 4) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 2 4)).filter positive) |
| (fun u => .mul (B2_2_4Masses u) (.add (.mul (.exact 2) logN2) (B2_2_4Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B2_2_4Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B2_2_4Binaries_sound u hu)))) (by decide +kernel) |
| def B2_3_3Ttotal : Ball := ⟨(31969784226000540926113589340621 / 50000000000000000000000000000000), (153 / 25000000000000000000000000000000)⟩ |
| lemma B2_3_3Ttotal_sound : B2_3_3Ttotal.Encloses |
| ((((below (shape 2 3 3)).filter positive).map (fun u => (littleMass (shape 2 3 3) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 3 3)).filter positive) |
| B2_3_3Masses _ (fun u hu => B2_3_3Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B2_3_3Btotal : Ball := ⟨(97155091397592012920594654935737 / 100000000000000000000000000000000), (8346926389 / 10000000000000000000000000000000)⟩ |
| lemma B2_3_3Btotal_sound : B2_3_3Btotal.Encloses |
| ((((below (shape 2 3 3)).filter positive).map (fun u => (littleMass (shape 2 3 3) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 2 3 3) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 3 3)).filter positive) |
| (fun u => .mul (B2_3_3Masses u) (.add (.mul (.exact 2) logN2) (B2_3_3Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B2_3_3Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B2_3_3Binaries_sound u hu)))) (by decide +kernel) |
| def B2_4_2Ttotal : Ball := ⟨(4469292299776841640211258871473 / 50000000000000000000000000000000), (121 / 100000000000000000000000000000000)⟩ |
| lemma B2_4_2Ttotal_sound : B2_4_2Ttotal.Encloses |
| ((((below (shape 2 4 2)).filter positive).map (fun u => (littleMass (shape 2 4 2) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 4 2)).filter positive) |
| B2_4_2Masses _ (fun u hu => B2_4_2Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B2_4_2Btotal : Ball := ⟨(13184565128477558357944470754293 / 100000000000000000000000000000000), (11573072161 / 100000000000000000000000000000000)⟩ |
| lemma B2_4_2Btotal_sound : B2_4_2Btotal.Encloses |
| ((((below (shape 2 4 2)).filter positive).map (fun u => (littleMass (shape 2 4 2) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 2 4 2) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 4 2)).filter positive) |
| (fun u => .mul (B2_4_2Masses u) (.add (.mul (.exact 2) logN2) (B2_4_2Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B2_4_2Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B2_4_2Binaries_sound u hu)))) (by decide +kernel) |
| def B2_5_1Ttotal : Ball := ⟨(31645095109351479784189369397 / 50000000000000000000000000000000), (1 / 25000000000000000000000000000000)⟩ |
| lemma B2_5_1Ttotal_sound : B2_5_1Ttotal.Encloses |
| ((((below (shape 2 5 1)).filter positive).map (fun u => (littleMass (shape 2 5 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 5 1)).filter positive) |
| B2_5_1Masses _ (fun u hu => B2_5_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B2_5_1Btotal : Ball := ⟨(17929724936204266029568343331 / 20000000000000000000000000000000), (20239527 / 25000000000000000000000000000000)⟩ |
| lemma B2_5_1Btotal_sound : B2_5_1Btotal.Encloses |
| ((((below (shape 2 5 1)).filter positive).map (fun u => (littleMass (shape 2 5 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 2 5 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 2 5 1)).filter positive) |
| (fun u => .mul (B2_5_1Masses u) (.add (.mul (.exact 2) logN2) (B2_5_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B2_5_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B2_5_1Binaries_sound u hu)))) (by decide +kernel) |
| def B3_1_4Ttotal : Ball := ⟨(907559782261289135363984197323 / 12500000000000000000000000000000), (11 / 12500000000000000000000000000000)⟩ |
| lemma B3_1_4Ttotal_sound : B3_1_4Ttotal.Encloses |
| ((((below (shape 3 1 4)).filter positive).map (fun u => (littleMass (shape 3 1 4) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 1 4)).filter positive) |
| B3_1_4Masses _ (fun u hu => B3_1_4Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B3_1_4Btotal : Ball := ⟨(103669017590719797550293135229 / 1000000000000000000000000000000), (4652487271 / 50000000000000000000000000000000)⟩ |
| lemma B3_1_4Btotal_sound : B3_1_4Btotal.Encloses |
| ((((below (shape 3 1 4)).filter positive).map (fun u => (littleMass (shape 3 1 4) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 3 1 4) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 1 4)).filter positive) |
| (fun u => .mul (B3_1_4Masses u) (.add (.mul (.exact 2) logN2) (B3_1_4Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B3_1_4Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B3_1_4Binaries_sound u hu)))) (by decide +kernel) |
| def B3_2_3Ttotal : Ball := ⟨(1150430302257331587531431126831 / 3125000000000000000000000000000), (79 / 20000000000000000000000000000000)⟩ |
| lemma B3_2_3Ttotal_sound : B3_2_3Ttotal.Encloses |
| ((((below (shape 3 2 3)).filter positive).map (fun u => (littleMass (shape 3 2 3) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 2 3)).filter positive) |
| B3_2_3Masses _ (fun u hu => B3_2_3Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B3_2_3Btotal : Ball := ⟨(55952151483018865110434847256319 / 100000000000000000000000000000000), (12015665981 / 25000000000000000000000000000000)⟩ |
| lemma B3_2_3Btotal_sound : B3_2_3Btotal.Encloses |
| ((((below (shape 3 2 3)).filter positive).map (fun u => (littleMass (shape 3 2 3) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 3 2 3) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 2 3)).filter positive) |
| (fun u => .mul (B3_2_3Masses u) (.add (.mul (.exact 2) logN2) (B3_2_3Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B3_2_3Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B3_2_3Binaries_sound u hu)))) (by decide +kernel) |
| def B3_3_2Ttotal : Ball := ⟨(8758499276755162714699464520641 / 50000000000000000000000000000000), (209 / 100000000000000000000000000000000)⟩ |
| lemma B3_3_2Ttotal_sound : B3_3_2Ttotal.Encloses |
| ((((below (shape 3 3 2)).filter positive).map (fun u => (littleMass (shape 3 3 2) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 3 2)).filter positive) |
| B3_3_2Masses _ (fun u hu => B3_3_2Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B3_3_2Btotal : Ball := ⟨(13315199039969247677694047381629 / 50000000000000000000000000000000), (11435850877 / 50000000000000000000000000000000)⟩ |
| lemma B3_3_2Btotal_sound : B3_3_2Btotal.Encloses |
| ((((below (shape 3 3 2)).filter positive).map (fun u => (littleMass (shape 3 3 2) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 3 3 2) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 3 2)).filter positive) |
| (fun u => .mul (B3_3_2Masses u) (.add (.mul (.exact 2) logN2) (B3_3_2Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B3_3_2Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B3_3_2Binaries_sound u hu)))) (by decide +kernel) |
| def B3_4_1Ttotal : Ball := ⟨(564009300736610125456854593833 / 100000000000000000000000000000000), (11 / 100000000000000000000000000000000)⟩ |
| lemma B3_4_1Ttotal_sound : B3_4_1Ttotal.Encloses |
| ((((below (shape 3 4 1)).filter positive).map (fun u => (littleMass (shape 3 4 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 4 1)).filter positive) |
| B3_4_1Masses _ (fun u hu => B3_4_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B3_4_1Btotal : Ball := ⟨(202057355425613495053354661181 / 25000000000000000000000000000000), (144745553 / 20000000000000000000000000000000)⟩ |
| lemma B3_4_1Btotal_sound : B3_4_1Btotal.Encloses |
| ((((below (shape 3 4 1)).filter positive).map (fun u => (littleMass (shape 3 4 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 3 4 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 3 4 1)).filter positive) |
| (fun u => .mul (B3_4_1Masses u) (.add (.mul (.exact 2) logN2) (B3_4_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B3_4_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B3_4_1Binaries_sound u hu)))) (by decide +kernel) |
| def B4_1_3Ttotal : Ball := ⟨(2233079177302622918447096359641 / 100000000000000000000000000000000), (3 / 10000000000000000000000000000000)⟩ |
| lemma B4_1_3Ttotal_sound : B4_1_3Ttotal.Encloses |
| ((((below (shape 4 1 3)).filter positive).map (fun u => (littleMass (shape 4 1 3) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 1 3)).filter positive) |
| B4_1_3Masses _ (fun u hu => B4_1_3Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B4_1_3Btotal : Ball := ⟨(64014500780045741535225896199 / 2000000000000000000000000000000), (2865674341 / 100000000000000000000000000000000)⟩ |
| lemma B4_1_3Btotal_sound : B4_1_3Btotal.Encloses |
| ((((below (shape 4 1 3)).filter positive).map (fun u => (littleMass (shape 4 1 3) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 4 1 3) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 1 3)).filter positive) |
| (fun u => .mul (B4_1_3Masses u) (.add (.mul (.exact 2) logN2) (B4_1_3Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B4_1_3Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B4_1_3Binaries_sound u hu)))) (by decide +kernel) |
| def B4_2_2Ttotal : Ball := ⟨(3767097545009369306050857272327 / 100000000000000000000000000000000), (27 / 50000000000000000000000000000000)⟩ |
| lemma B4_2_2Ttotal_sound : B4_2_2Ttotal.Encloses |
| ((((below (shape 4 2 2)).filter positive).map (fun u => (littleMass (shape 4 2 2) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 2 2)).filter positive) |
| B4_2_2Masses _ (fun u hu => B4_2_2Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B4_2_2Btotal : Ball := ⟨(5557262538143694155705675547867 / 100000000000000000000000000000000), (4877621647 / 100000000000000000000000000000000)⟩ |
| lemma B4_2_2Btotal_sound : B4_2_2Btotal.Encloses |
| ((((below (shape 4 2 2)).filter positive).map (fun u => (littleMass (shape 4 2 2) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 4 2 2) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 2 2)).filter positive) |
| (fun u => .mul (B4_2_2Masses u) (.add (.mul (.exact 2) logN2) (B4_2_2Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B4_2_2Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B4_2_2Binaries_sound u hu)))) (by decide +kernel) |
| def B4_3_1Ttotal : Ball := ⟨(99503874082605676510083297769 / 25000000000000000000000000000000), (9 / 100000000000000000000000000000000)⟩ |
| lemma B4_3_1Ttotal_sound : B4_3_1Ttotal.Encloses |
| ((((below (shape 4 3 1)).filter positive).map (fun u => (littleMass (shape 4 3 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 3 1)).filter positive) |
| B4_3_1Masses _ (fun u hu => B4_3_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B4_3_1Btotal : Ball := ⟨(570401113257114350350352476201 / 100000000000000000000000000000000), (510740473 / 100000000000000000000000000000000)⟩ |
| lemma B4_3_1Btotal_sound : B4_3_1Btotal.Encloses |
| ((((below (shape 4 3 1)).filter positive).map (fun u => (littleMass (shape 4 3 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 4 3 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 4 3 1)).filter positive) |
| (fun u => .mul (B4_3_1Masses u) (.add (.mul (.exact 2) logN2) (B4_3_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B4_3_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B4_3_1Binaries_sound u hu)))) (by decide +kernel) |
| def B5_1_2Ttotal : Ball := ⟨(47463286326287333481085364697 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩ |
| lemma B5_1_2Ttotal_sound : B5_1_2Ttotal.Encloses |
| ((((below (shape 5 1 2)).filter positive).map (fun u => (littleMass (shape 5 1 2) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 5 1 2)).filter positive) |
| B5_1_2Masses _ (fun u hu => B5_1_2Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B5_1_2Btotal : Ball := ⟨(67347402449400564801926883563 / 100000000000000000000000000000000), (60713289 / 100000000000000000000000000000000)⟩ |
| lemma B5_1_2Btotal_sound : B5_1_2Btotal.Encloses |
| ((((below (shape 5 1 2)).filter positive).map (fun u => (littleMass (shape 5 1 2) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 5 1 2) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 5 1 2)).filter positive) |
| (fun u => .mul (B5_1_2Masses u) (.add (.mul (.exact 2) logN2) (B5_1_2Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B5_1_2Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B5_1_2Binaries_sound u hu)))) (by decide +kernel) |
| def B5_2_1Ttotal : Ball := ⟨(408525735543637671783520101 / 2500000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩ |
| lemma B5_2_1Ttotal_sound : B5_2_1Ttotal.Encloses |
| ((((below (shape 5 2 1)).filter positive).map (fun u => (littleMass (shape 5 2 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 5 2 1)).filter positive) |
| B5_2_1Masses _ (fun u hu => B5_2_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B5_2_1Btotal : Ball := ⟨(23171910014948039061690049419 / 100000000000000000000000000000000), (20911247 / 100000000000000000000000000000000)⟩ |
| lemma B5_2_1Btotal_sound : B5_2_1Btotal.Encloses |
| ((((below (shape 5 2 1)).filter positive).map (fun u => (littleMass (shape 5 2 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 5 2 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 5 2 1)).filter positive) |
| (fun u => .mul (B5_2_1Masses u) (.add (.mul (.exact 2) logN2) (B5_2_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B5_2_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B5_2_1Binaries_sound u hu)))) (by decide +kernel) |
| def B6_1_1Ttotal : Ball := ⟨0, 0⟩ |
| lemma B6_1_1Ttotal_sound : B6_1_1Ttotal.Encloses |
| ((((below (shape 6 1 1)).filter positive).map (fun u => (littleMass (shape 6 1 1) u : ℝ))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 6 1 1)).filter positive) |
| B6_1_1Masses _ (fun u hu => B6_1_1Masses_sound u (List.mem_filter.mp hu).1)) (by decide +kernel) |
| def B6_1_1Btotal : Ball := ⟨0, 0⟩ |
| lemma B6_1_1Btotal_sound : B6_1_1Btotal.Encloses |
| ((((below (shape 6 1 1)).filter positive).map (fun u => (littleMass (shape 6 1 1) u : ℝ) * (2 * Real.log 2 + binaryEntropyRate (binaryParameter (shape 6 1 1) u)))).sum) := by |
| exact contains_sound (listSumBall_sound ((below (shape 6 1 1)).filter positive) |
| (fun u => .mul (B6_1_1Masses u) (.add (.mul (.exact 2) logN2) (B6_1_1Binaries u))) _ |
| (fun u hu => Ball.mul_sound (B6_1_1Masses_sound u (List.mem_filter.mp hu).1) |
| (Ball.add_sound (Ball.mul_sound (by simpa only [Rat.cast_ofNat] using Ball.exact_sound 2) |
| logN2_sound) (B6_1_1Binaries_sound u hu)))) (by decide +kernel) |
| def TParentBalls (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 1, 1, 6 => B1_1_6Ttotal |
| | 1, 2, 5 => B1_2_5Ttotal |
| | 1, 3, 4 => B1_3_4Ttotal |
| | 1, 4, 3 => B1_4_3Ttotal |
| | 1, 5, 2 => B1_5_2Ttotal |
| | 1, 6, 1 => B1_6_1Ttotal |
| | 2, 1, 5 => B2_1_5Ttotal |
| | 2, 2, 4 => B2_2_4Ttotal |
| | 2, 3, 3 => B2_3_3Ttotal |
| | 2, 4, 2 => B2_4_2Ttotal |
| | 2, 5, 1 => B2_5_1Ttotal |
| | 3, 1, 4 => B3_1_4Ttotal |
| | 3, 2, 3 => B3_2_3Ttotal |
| | 3, 3, 2 => B3_3_2Ttotal |
| | 3, 4, 1 => B3_4_1Ttotal |
| | 4, 1, 3 => B4_1_3Ttotal |
| | 4, 2, 2 => B4_2_2Ttotal |
| | 4, 3, 1 => B4_3_1Ttotal |
| | 5, 1, 2 => B5_1_2Ttotal |
| | 5, 2, 1 => B5_2_1Ttotal |
| | 6, 1, 1 => B6_1_1Ttotal |
| | _, _, _ => (.exact 0) |
| lemma TParentBalls_sound (t : Shape) (ht : t ∈ positiveSecond) : |
| (TParentBalls t).Encloses ((((below t).filter positive).map |
| (fun u => interiorMass (t,u))).sum) := by |
| rw [positiveSecond_list] at ht |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at ht |
| rcases ht with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact B1_1_6Ttotal_sound |
| · exact B1_2_5Ttotal_sound |
| · exact B1_3_4Ttotal_sound |
| · exact B1_4_3Ttotal_sound |
| · exact B1_5_2Ttotal_sound |
| · exact B1_6_1Ttotal_sound |
| · exact B2_1_5Ttotal_sound |
| · exact B2_2_4Ttotal_sound |
| · exact B2_3_3Ttotal_sound |
| · exact B2_4_2Ttotal_sound |
| · exact B2_5_1Ttotal_sound |
| · exact B3_1_4Ttotal_sound |
| · exact B3_2_3Ttotal_sound |
| · exact B3_3_2Ttotal_sound |
| · exact B3_4_1Ttotal_sound |
| · exact B4_1_3Ttotal_sound |
| · exact B4_2_2Ttotal_sound |
| · exact B4_3_1Ttotal_sound |
| · exact B5_1_2Ttotal_sound |
| · exact B5_2_1Ttotal_sound |
| · exact B6_1_1Ttotal_sound |
| def BParentBalls (s : Shape) : Ball := |
| match s 0, s 1, s 2 with |
| | 1, 1, 6 => B1_1_6Btotal |
| | 1, 2, 5 => B1_2_5Btotal |
| | 1, 3, 4 => B1_3_4Btotal |
| | 1, 4, 3 => B1_4_3Btotal |
| | 1, 5, 2 => B1_5_2Btotal |
| | 1, 6, 1 => B1_6_1Btotal |
| | 2, 1, 5 => B2_1_5Btotal |
| | 2, 2, 4 => B2_2_4Btotal |
| | 2, 3, 3 => B2_3_3Btotal |
| | 2, 4, 2 => B2_4_2Btotal |
| | 2, 5, 1 => B2_5_1Btotal |
| | 3, 1, 4 => B3_1_4Btotal |
| | 3, 2, 3 => B3_2_3Btotal |
| | 3, 3, 2 => B3_3_2Btotal |
| | 3, 4, 1 => B3_4_1Btotal |
| | 4, 1, 3 => B4_1_3Btotal |
| | 4, 2, 2 => B4_2_2Btotal |
| | 4, 3, 1 => B4_3_1Btotal |
| | 5, 1, 2 => B5_1_2Btotal |
| | 5, 2, 1 => B5_2_1Btotal |
| | 6, 1, 1 => B6_1_1Btotal |
| | _, _, _ => (.exact 0) |
| lemma BParentBalls_sound (t : Shape) (ht : t ∈ positiveSecond) : |
| (BParentBalls t).Encloses ((((below t).filter positive).map |
| (fun u => (fun tu => interiorMass tu * (2 * Real.log 2 + interiorEntropy tu)) (t,u))).sum) := by |
| rw [positiveSecond_list] at ht |
| simp only [List.mem_cons, List.not_mem_nil, or_false] at ht |
| rcases ht with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact B1_1_6Btotal_sound |
| · exact B1_2_5Btotal_sound |
| · exact B1_3_4Btotal_sound |
| · exact B1_4_3Btotal_sound |
| · exact B1_5_2Btotal_sound |
| · exact B1_6_1Btotal_sound |
| · exact B2_1_5Btotal_sound |
| · exact B2_2_4Btotal_sound |
| · exact B2_3_3Btotal_sound |
| · exact B2_4_2Btotal_sound |
| · exact B2_5_1Btotal_sound |
| · exact B3_1_4Btotal_sound |
| · exact B3_2_3Btotal_sound |
| · exact B3_3_2Btotal_sound |
| · exact B3_4_1Btotal_sound |
| · exact B4_1_3Btotal_sound |
| · exact B4_2_2Btotal_sound |
| · exact B4_3_1Btotal_sound |
| · exact B5_1_2Btotal_sound |
| · exact B5_2_1Btotal_sound |
| · exact B6_1_1Btotal_sound |
| def nativeTBall : Ball := ⟨(7331109571865792290176153698167 / 3125000000000000000000000000000), (637 / 25000000000000000000000000000000)⟩ |
| lemma nativeTBall_sound : nativeTBall.Encloses nativeT := by |
| rw [nativeT, interior_fold] |
| exact contains_sound (listSumBall_sound positiveSecond TParentBalls _ TParentBalls_sound) |
| (by decide +kernel) |
| def nativeBBall : Ball := ⟨(926453577659126052246485196733 / 800000000000000000000000000000), (100635147743 / 100000000000000000000000000000000)⟩ |
| lemma nativeBBall_sound : nativeBBall.Encloses nativeB := by |
| rw [nativeB, interior_fold] |
| have hb := contains_sound (Ball.mul_sound |
| (listSumBall_sound positiveSecond BParentBalls _ BParentBalls_sound) |
| (show (Ball.exact (1/3)).Encloses (1/(3 : ℝ)) by |
| convert Ball.exact_sound (1/3) using 1 |
| norm_num)) |
| (show contains (.mul (listSumBall positiveSecond BParentBalls) (.exact (1/3))) nativeBBall = true by decide +kernel) |
| simpa only [one_div, div_eq_mul_inv, one_mul] using hb |
| def nativeHighBall : Ball := ⟨(81304606881836826911861830543749 / 50000000000000000000000000000000), (99355063329 / 100000000000000000000000000000000)⟩ |
| lemma nativeHighBall_sound : nativeHighBall.Encloses nativeHigh := by |
| exact contains_sound (Ball.mul_sound nativeTBall_sound logN2_sound) (by decide +kernel) |
|
|
| theorem native_denominator_gap : nativeB < nativeHigh := by |
| have hb := (Ball.bounds nativeBBall_sound).2 |
| have ht := (Ball.bounds nativeHighBall_sound).1 |
| have hi : (nativeBBall.upper : ℝ) < nativeHighBall.lower := by |
| norm_num [nativeBBall, nativeHighBall, Ball.lower, Ball.upper] |
| exact hb.trans_lt (hi.trans_le ht) |
|
|
| end MatrixMultiplication.NativeNumerics |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
