import Mathlib import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_1_14I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_1_14I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_1_14I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_2_13I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_2_13I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_2_13I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_3_12I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_3_12I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_3_12I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_4_11I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_4_11I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_4_11I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_5_10I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_5_10I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_5_10I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_6_9I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_6_9I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_6_9I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_7_8I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_7_8I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA1_7_8I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_2_12I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_2_12I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_2_12I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_3_11I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_3_11I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_3_11I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_4_10I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_4_10I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_4_10I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_5_9I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_5_9I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_5_9I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_6_8I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_6_8I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_6_8I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_7_7I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_7_7I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA2_7_7I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_3_10I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_3_10I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_3_10I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_4_9I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_4_9I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_4_9I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_5_8I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_5_8I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_5_8I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_6_7I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_6_7I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA3_6_7I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_4_8I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_4_8I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_4_8I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_5_7I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_5_7I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_5_7I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_6_6I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_6_6I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA4_6_6I2 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA5_5_6I0 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA5_5_6I1 import OAI.LinearAlgebra.MatrixFields.CapacityA.CapA5_5_6I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_1_6I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_1_6I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_1_6I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_2_5I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_2_5I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_2_5I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_3_4I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_3_4I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_3_4I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_4_3I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_4_3I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_4_3I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_5_2I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_5_2I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_5_2I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_6_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_6_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB1_6_1I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_1_5I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_1_5I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_1_5I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_2_4I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_2_4I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_2_4I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_3_3I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_3_3I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_3_3I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_4_2I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_4_2I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_4_2I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_5_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_5_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB2_5_1I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_1_4I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_1_4I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_1_4I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_2_3I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_2_3I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_2_3I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_3_2I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_3_2I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_3_2I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_4_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_4_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB3_4_1I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_1_3I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_1_3I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_1_3I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_2_2I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_2_2I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_2_2I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_3_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_3_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB4_3_1I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_1_2I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_1_2I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_1_2I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_2_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_2_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB5_2_1I2 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB6_1_1I0 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB6_1_1I1 import OAI.LinearAlgebra.MatrixFields.CapacityB.CapB6_1_1I2 namespace OAI 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 capacityABalls (s : Shape) (i : Fin 3) : Ball := match s 0, s 1, s 2 with | 1, 1, 14 => ((Matrix.vecCons (CA1_1_14I0H) (Matrix.vecCons (CA1_1_14I1H) (Matrix.vecCons (CA1_1_14I2H) Matrix.vecEmpty))) i) | 1, 2, 13 => ((Matrix.vecCons (CA1_2_13I0H) (Matrix.vecCons (CA1_2_13I1H) (Matrix.vecCons (CA1_2_13I2H) Matrix.vecEmpty))) i) | 1, 3, 12 => ((Matrix.vecCons (CA1_3_12I0H) (Matrix.vecCons (CA1_3_12I1H) (Matrix.vecCons (CA1_3_12I2H) Matrix.vecEmpty))) i) | 1, 4, 11 => ((Matrix.vecCons (CA1_4_11I0H) (Matrix.vecCons (CA1_4_11I1H) (Matrix.vecCons (CA1_4_11I2H) Matrix.vecEmpty))) i) | 1, 5, 10 => ((Matrix.vecCons (CA1_5_10I0H) (Matrix.vecCons (CA1_5_10I1H) (Matrix.vecCons (CA1_5_10I2H) Matrix.vecEmpty))) i) | 1, 6, 9 => ((Matrix.vecCons (CA1_6_9I0H) (Matrix.vecCons (CA1_6_9I1H) (Matrix.vecCons (CA1_6_9I2H) Matrix.vecEmpty))) i) | 1, 7, 8 => ((Matrix.vecCons (CA1_7_8I0H) (Matrix.vecCons (CA1_7_8I1H) (Matrix.vecCons (CA1_7_8I2H) Matrix.vecEmpty))) i) | 2, 2, 12 => ((Matrix.vecCons (CA2_2_12I0H) (Matrix.vecCons (CA2_2_12I1H) (Matrix.vecCons (CA2_2_12I2H) Matrix.vecEmpty))) i) | 2, 3, 11 => ((Matrix.vecCons (CA2_3_11I0H) (Matrix.vecCons (CA2_3_11I1H) (Matrix.vecCons (CA2_3_11I2H) Matrix.vecEmpty))) i) | 2, 4, 10 => ((Matrix.vecCons (CA2_4_10I0H) (Matrix.vecCons (CA2_4_10I1H) (Matrix.vecCons (CA2_4_10I2H) Matrix.vecEmpty))) i) | 2, 5, 9 => ((Matrix.vecCons (CA2_5_9I0H) (Matrix.vecCons (CA2_5_9I1H) (Matrix.vecCons (CA2_5_9I2H) Matrix.vecEmpty))) i) | 2, 6, 8 => ((Matrix.vecCons (CA2_6_8I0H) (Matrix.vecCons (CA2_6_8I1H) (Matrix.vecCons (CA2_6_8I2H) Matrix.vecEmpty))) i) | 2, 7, 7 => ((Matrix.vecCons (CA2_7_7I0H) (Matrix.vecCons (CA2_7_7I1H) (Matrix.vecCons (CA2_7_7I2H) Matrix.vecEmpty))) i) | 3, 3, 10 => ((Matrix.vecCons (CA3_3_10I0H) (Matrix.vecCons (CA3_3_10I1H) (Matrix.vecCons (CA3_3_10I2H) Matrix.vecEmpty))) i) | 3, 4, 9 => ((Matrix.vecCons (CA3_4_9I0H) (Matrix.vecCons (CA3_4_9I1H) (Matrix.vecCons (CA3_4_9I2H) Matrix.vecEmpty))) i) | 3, 5, 8 => ((Matrix.vecCons (CA3_5_8I0H) (Matrix.vecCons (CA3_5_8I1H) (Matrix.vecCons (CA3_5_8I2H) Matrix.vecEmpty))) i) | 3, 6, 7 => ((Matrix.vecCons (CA3_6_7I0H) (Matrix.vecCons (CA3_6_7I1H) (Matrix.vecCons (CA3_6_7I2H) Matrix.vecEmpty))) i) | 4, 4, 8 => ((Matrix.vecCons (CA4_4_8I0H) (Matrix.vecCons (CA4_4_8I1H) (Matrix.vecCons (CA4_4_8I2H) Matrix.vecEmpty))) i) | 4, 5, 7 => ((Matrix.vecCons (CA4_5_7I0H) (Matrix.vecCons (CA4_5_7I1H) (Matrix.vecCons (CA4_5_7I2H) Matrix.vecEmpty))) i) | 4, 6, 6 => ((Matrix.vecCons (CA4_6_6I0H) (Matrix.vecCons (CA4_6_6I1H) (Matrix.vecCons (CA4_6_6I2H) Matrix.vecEmpty))) i) | 5, 5, 6 => ((Matrix.vecCons (CA5_5_6I0H) (Matrix.vecCons (CA5_5_6I1H) (Matrix.vecCons (CA5_5_6I2H) Matrix.vecEmpty))) i) | _, _, _ => (.exact 0) lemma capacityABalls_sound (s : Shape) (hs : s ∈ positiveInitial) (i : Fin 3) : (capacityABalls s i).Encloses (stageACapacity s i) := by rw [positiveInitial_list] at hs simp only [List.mem_cons, List.not_mem_nil, or_false] at hs rcases hs with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl · revert i exact forall_fin3 CA1_1_14I0_sound CA1_1_14I1_sound CA1_1_14I2_sound · revert i exact forall_fin3 CA1_2_13I0_sound CA1_2_13I1_sound CA1_2_13I2_sound · revert i exact forall_fin3 CA1_3_12I0_sound CA1_3_12I1_sound CA1_3_12I2_sound · revert i exact forall_fin3 CA1_4_11I0_sound CA1_4_11I1_sound CA1_4_11I2_sound · revert i exact forall_fin3 CA1_5_10I0_sound CA1_5_10I1_sound CA1_5_10I2_sound · revert i exact forall_fin3 CA1_6_9I0_sound CA1_6_9I1_sound CA1_6_9I2_sound · revert i exact forall_fin3 CA1_7_8I0_sound CA1_7_8I1_sound CA1_7_8I2_sound · revert i exact forall_fin3 CA2_2_12I0_sound CA2_2_12I1_sound CA2_2_12I2_sound · revert i exact forall_fin3 CA2_3_11I0_sound CA2_3_11I1_sound CA2_3_11I2_sound · revert i exact forall_fin3 CA2_4_10I0_sound CA2_4_10I1_sound CA2_4_10I2_sound · revert i exact forall_fin3 CA2_5_9I0_sound CA2_5_9I1_sound CA2_5_9I2_sound · revert i exact forall_fin3 CA2_6_8I0_sound CA2_6_8I1_sound CA2_6_8I2_sound · revert i exact forall_fin3 CA2_7_7I0_sound CA2_7_7I1_sound CA2_7_7I2_sound · revert i exact forall_fin3 CA3_3_10I0_sound CA3_3_10I1_sound CA3_3_10I2_sound · revert i exact forall_fin3 CA3_4_9I0_sound CA3_4_9I1_sound CA3_4_9I2_sound · revert i exact forall_fin3 CA3_5_8I0_sound CA3_5_8I1_sound CA3_5_8I2_sound · revert i exact forall_fin3 CA3_6_7I0_sound CA3_6_7I1_sound CA3_6_7I2_sound · revert i exact forall_fin3 CA4_4_8I0_sound CA4_4_8I1_sound CA4_4_8I2_sound · revert i exact forall_fin3 CA4_5_7I0_sound CA4_5_7I1_sound CA4_5_7I2_sound · revert i exact forall_fin3 CA4_6_6I0_sound CA4_6_6I1_sound CA4_6_6I2_sound · revert i exact forall_fin3 CA5_5_6I0_sound CA5_5_6I1_sound CA5_5_6I2_sound def nativeLABall (i : Fin 3) : Ball := (Matrix.vecCons (⟨(117985629820354604769702009255473 / 100000000000000000000000000000000), (25286853793 / 25000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(14553362614718917509713695395419 / 12500000000000000000000000000000), (495362055017 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(13508474414138305071843256242429 / 12500000000000000000000000000000), (467443928227 / 100000000000000000000000000000000)⟩) Matrix.vecEmpty))) i lemma nativeLABall_sound_0 : (nativeLABall 0).Encloses (nativeLA 0) := by exact contains_sound (listSumBall_sound positiveInitial (fun s => .mul (initialProbs s) (capacityABalls s 0)) _ (fun s hs => Ball.mul_sound (initialProbs_sound s (List.mem_filter.mp hs).1) (capacityABalls_sound s hs 0))) (by decide +kernel) lemma nativeLABall_sound_1 : (nativeLABall 1).Encloses (nativeLA 1) := by exact contains_sound (listSumBall_sound positiveInitial (fun s => .mul (initialProbs s) (capacityABalls s 1)) _ (fun s hs => Ball.mul_sound (initialProbs_sound s (List.mem_filter.mp hs).1) (capacityABalls_sound s hs 1))) (by decide +kernel) lemma nativeLABall_sound_2 : (nativeLABall 2).Encloses (nativeLA 2) := by exact contains_sound (listSumBall_sound positiveInitial (fun s => .mul (initialProbs s) (capacityABalls s 2)) _ (fun s hs => Ball.mul_sound (initialProbs_sound s (List.mem_filter.mp hs).1) (capacityABalls_sound s hs 2))) (by decide +kernel) lemma nativeLABall_sound (i : Fin 3) : (nativeLABall i).Encloses (nativeLA i) := by revert i exact forall_fin3 nativeLABall_sound_0 nativeLABall_sound_1 nativeLABall_sound_2 theorem native_stageA_positive (i : Fin 3) : 0 < nativeLA i := by have hp : 0 < (nativeLABall i).lower := by fin_cases i <;> decide +kernel have hp' : (0 : ℝ) < (nativeLABall i).lower := by exact_mod_cast hp exact hp'.trans_le (Ball.bounds (nativeLABall_sound i)).1 def capacityBBalls (s : Shape) (i : Fin 3) : Ball := match s 0, s 1, s 2 with | 1, 1, 6 => ((Matrix.vecCons (CB1_1_6I0H) (Matrix.vecCons (CB1_1_6I1H) (Matrix.vecCons (CB1_1_6I2H) Matrix.vecEmpty))) i) | 1, 2, 5 => ((Matrix.vecCons (CB1_2_5I0H) (Matrix.vecCons (CB1_2_5I1H) (Matrix.vecCons (CB1_2_5I2H) Matrix.vecEmpty))) i) | 1, 3, 4 => ((Matrix.vecCons (CB1_3_4I0H) (Matrix.vecCons (CB1_3_4I1H) (Matrix.vecCons (CB1_3_4I2H) Matrix.vecEmpty))) i) | 1, 4, 3 => ((Matrix.vecCons (CB1_4_3I0H) (Matrix.vecCons (CB1_4_3I1H) (Matrix.vecCons (CB1_4_3I2H) Matrix.vecEmpty))) i) | 1, 5, 2 => ((Matrix.vecCons (CB1_5_2I0H) (Matrix.vecCons (CB1_5_2I1H) (Matrix.vecCons (CB1_5_2I2H) Matrix.vecEmpty))) i) | 1, 6, 1 => ((Matrix.vecCons (CB1_6_1I0H) (Matrix.vecCons (CB1_6_1I1H) (Matrix.vecCons (CB1_6_1I2H) Matrix.vecEmpty))) i) | 2, 1, 5 => ((Matrix.vecCons (CB2_1_5I0H) (Matrix.vecCons (CB2_1_5I1H) (Matrix.vecCons (CB2_1_5I2H) Matrix.vecEmpty))) i) | 2, 2, 4 => ((Matrix.vecCons (CB2_2_4I0H) (Matrix.vecCons (CB2_2_4I1H) (Matrix.vecCons (CB2_2_4I2H) Matrix.vecEmpty))) i) | 2, 3, 3 => ((Matrix.vecCons (CB2_3_3I0H) (Matrix.vecCons (CB2_3_3I1H) (Matrix.vecCons (CB2_3_3I2H) Matrix.vecEmpty))) i) | 2, 4, 2 => ((Matrix.vecCons (CB2_4_2I0H) (Matrix.vecCons (CB2_4_2I1H) (Matrix.vecCons (CB2_4_2I2H) Matrix.vecEmpty))) i) | 2, 5, 1 => ((Matrix.vecCons (CB2_5_1I0H) (Matrix.vecCons (CB2_5_1I1H) (Matrix.vecCons (CB2_5_1I2H) Matrix.vecEmpty))) i) | 3, 1, 4 => ((Matrix.vecCons (CB3_1_4I0H) (Matrix.vecCons (CB3_1_4I1H) (Matrix.vecCons (CB3_1_4I2H) Matrix.vecEmpty))) i) | 3, 2, 3 => ((Matrix.vecCons (CB3_2_3I0H) (Matrix.vecCons (CB3_2_3I1H) (Matrix.vecCons (CB3_2_3I2H) Matrix.vecEmpty))) i) | 3, 3, 2 => ((Matrix.vecCons (CB3_3_2I0H) (Matrix.vecCons (CB3_3_2I1H) (Matrix.vecCons (CB3_3_2I2H) Matrix.vecEmpty))) i) | 3, 4, 1 => ((Matrix.vecCons (CB3_4_1I0H) (Matrix.vecCons (CB3_4_1I1H) (Matrix.vecCons (CB3_4_1I2H) Matrix.vecEmpty))) i) | 4, 1, 3 => ((Matrix.vecCons (CB4_1_3I0H) (Matrix.vecCons (CB4_1_3I1H) (Matrix.vecCons (CB4_1_3I2H) Matrix.vecEmpty))) i) | 4, 2, 2 => ((Matrix.vecCons (CB4_2_2I0H) (Matrix.vecCons (CB4_2_2I1H) (Matrix.vecCons (CB4_2_2I2H) Matrix.vecEmpty))) i) | 4, 3, 1 => ((Matrix.vecCons (CB4_3_1I0H) (Matrix.vecCons (CB4_3_1I1H) (Matrix.vecCons (CB4_3_1I2H) Matrix.vecEmpty))) i) | 5, 1, 2 => ((Matrix.vecCons (CB5_1_2I0H) (Matrix.vecCons (CB5_1_2I1H) (Matrix.vecCons (CB5_1_2I2H) Matrix.vecEmpty))) i) | 5, 2, 1 => ((Matrix.vecCons (CB5_2_1I0H) (Matrix.vecCons (CB5_2_1I1H) (Matrix.vecCons (CB5_2_1I2H) Matrix.vecEmpty))) i) | 6, 1, 1 => ((Matrix.vecCons (CB6_1_1I0H) (Matrix.vecCons (CB6_1_1I1H) (Matrix.vecCons (CB6_1_1I2H) Matrix.vecEmpty))) i) | _, _, _ => (.exact 0) lemma capacityBBalls_sound (s : Shape) (hs : s ∈ positiveSecond) (i : Fin 3) : (capacityBBalls s i).Encloses (stageBCapacity s i) := by rw [positiveSecond_list] at hs simp only [List.mem_cons, List.not_mem_nil, or_false] at hs rcases hs with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl · revert i exact forall_fin3 CB1_1_6I0_sound CB1_1_6I1_sound CB1_1_6I2_sound · revert i exact forall_fin3 CB1_2_5I0_sound CB1_2_5I1_sound CB1_2_5I2_sound · revert i exact forall_fin3 CB1_3_4I0_sound CB1_3_4I1_sound CB1_3_4I2_sound · revert i exact forall_fin3 CB1_4_3I0_sound CB1_4_3I1_sound CB1_4_3I2_sound · revert i exact forall_fin3 CB1_5_2I0_sound CB1_5_2I1_sound CB1_5_2I2_sound · revert i exact forall_fin3 CB1_6_1I0_sound CB1_6_1I1_sound CB1_6_1I2_sound · revert i exact forall_fin3 CB2_1_5I0_sound CB2_1_5I1_sound CB2_1_5I2_sound · revert i exact forall_fin3 CB2_2_4I0_sound CB2_2_4I1_sound CB2_2_4I2_sound · revert i exact forall_fin3 CB2_3_3I0_sound CB2_3_3I1_sound CB2_3_3I2_sound · revert i exact forall_fin3 CB2_4_2I0_sound CB2_4_2I1_sound CB2_4_2I2_sound · revert i exact forall_fin3 CB2_5_1I0_sound CB2_5_1I1_sound CB2_5_1I2_sound · revert i exact forall_fin3 CB3_1_4I0_sound CB3_1_4I1_sound CB3_1_4I2_sound · revert i exact forall_fin3 CB3_2_3I0_sound CB3_2_3I1_sound CB3_2_3I2_sound · revert i exact forall_fin3 CB3_3_2I0_sound CB3_3_2I1_sound CB3_3_2I2_sound · revert i exact forall_fin3 CB3_4_1I0_sound CB3_4_1I1_sound CB3_4_1I2_sound · revert i exact forall_fin3 CB4_1_3I0_sound CB4_1_3I1_sound CB4_1_3I2_sound · revert i exact forall_fin3 CB4_2_2I0_sound CB4_2_2I1_sound CB4_2_2I2_sound · revert i exact forall_fin3 CB4_3_1I0_sound CB4_3_1I1_sound CB4_3_1I2_sound · revert i exact forall_fin3 CB5_1_2I0_sound CB5_1_2I1_sound CB5_1_2I2_sound · revert i exact forall_fin3 CB5_2_1I0_sound CB5_2_1I1_sound CB5_2_1I2_sound · revert i exact forall_fin3 CB6_1_1I0_sound CB6_1_1I1_sound CB6_1_1I2_sound def nativeLBBall (i : Fin 3) : Ball := (Matrix.vecCons (⟨(36915911542259422191174414642177 / 25000000000000000000000000000000), (143554434849 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(32908978604932824344186869459233 / 20000000000000000000000000000000), (610647767219 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(64721008843697680778152495140253 / 50000000000000000000000000000000), (102799797137 / 20000000000000000000000000000000)⟩) Matrix.vecEmpty))) i lemma nativeLBBall_sound_0 : (nativeLBBall 0).Encloses (nativeLB 0) := by exact contains_sound (listSumBall_sound positiveSecond (fun s => .mul (secondProbs s) (capacityBBalls s 0)) _ (fun s hs => Ball.mul_sound (secondProbs_sound s (List.mem_filter.mp hs).1) (capacityBBalls_sound s hs 0))) (by decide +kernel) lemma nativeLBBall_sound_1 : (nativeLBBall 1).Encloses (nativeLB 1) := by exact contains_sound (listSumBall_sound positiveSecond (fun s => .mul (secondProbs s) (capacityBBalls s 1)) _ (fun s hs => Ball.mul_sound (secondProbs_sound s (List.mem_filter.mp hs).1) (capacityBBalls_sound s hs 1))) (by decide +kernel) lemma nativeLBBall_sound_2 : (nativeLBBall 2).Encloses (nativeLB 2) := by exact contains_sound (listSumBall_sound positiveSecond (fun s => .mul (secondProbs s) (capacityBBalls s 2)) _ (fun s hs => Ball.mul_sound (secondProbs_sound s (List.mem_filter.mp hs).1) (capacityBBalls_sound s hs 2))) (by decide +kernel) lemma nativeLBBall_sound (i : Fin 3) : (nativeLBBall i).Encloses (nativeLB i) := by revert i exact forall_fin3 nativeLBBall_sound_0 nativeLBBall_sound_1 nativeLBBall_sound_2 theorem native_stageB_positive (i : Fin 3) : 0 < nativeLB i := by have hp : 0 < (nativeLBBall i).lower := by fin_cases i <;> decide +kernel have hp' : (0 : ℝ) < (nativeLBBall i).lower := by exact_mod_cast hp exact hp'.trans_le (Ball.bounds (nativeLBBall_sound i)).1 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 AllFieldInitialEntropy def nativeCstarBall : Ball := ⟨(23573978636551704730344303484969 / 6250000000000000000000000000000), (4390100049 / 500000000000000000000000000000)⟩ lemma nativeCstarBall_sound : nativeCstarBall.Encloses nativeCstar := by have hb := contains_sound (Ball.add_sound nativeBBall_sound (Ball.mul_sound (finSumBall_sound 3 (fun i => .add (nativeLABall i) (nativeLBBall i)) (fun i => nativeLA i + nativeLB i) (fun i => Ball.add_sound (nativeLABall_sound i) (nativeLBBall_sound i))) (show (Ball.exact (1/3)).Encloses (1/(3 : ℝ)) by convert Ball.exact_sound (1/3) using 1 norm_num))) (show contains (.add nativeBBall (.mul (finSumBall 3 (fun i => .add (nativeLABall i) (nativeLBBall i))) (.exact (1/3)))) nativeCstarBall = true by decide +kernel) simpa only [nativeCstar, Staggering.totalCapacity, one_div, div_eq_mul_inv, one_mul] using hb def nativeCBall (i : Fin 3) : Ball := (Matrix.vecCons (⟨(111534382195434982151109187935323 / 100000000000000000000000000000000), (1122721859821 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(96211864242411813886864945299987 / 100000000000000000000000000000000), (496007458009 / 25000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(69836922592162736777228907769783 / 50000000000000000000000000000000), (29054108183 / 1562500000000000000000000000000)⟩) Matrix.vecEmpty))) i lemma nativeCBall_sound (i : Fin 3) : (nativeCBall i).Encloses (nativeCstar - (nativeLA i + nativeLB i)) := by have hcheck : contains (.add nativeCstarBall (.neg (.add (nativeLABall i) (nativeLBBall i)))) (nativeCBall i) = true := by fin_cases i <;> decide +kernel simpa only [sub_eq_add_neg] using contains_sound (Ball.add_sound nativeCstarBall_sound (Ball.neg_sound (Ball.add_sound (nativeLABall_sound i) (nativeLBBall_sound i)))) hcheck theorem native_stageC_feasible (i : Fin 3) : nativeCstar - (nativeLA i + nativeLB i) < nativeHigh := by have hp : (nativeCBall i).upper < nativeHighBall.lower := by fin_cases i <;> decide +kernel have hp' : ((nativeCBall i).upper : ℝ) < nativeHighBall.lower := by exact_mod_cast hp exact (Ball.bounds (nativeCBall_sound i)).2.trans_lt (hp'.trans_le (Ball.bounds nativeHighBall_sound).1) theorem native_stageC_balanced_positive (i : Fin 3) : 0 < nativeLC nativeLambda i := by rw [nativeLC_balanced native_denominator_gap] have hp : 0 < (nativeCBall i).lower := by fin_cases i <;> decide +kernel have hp' : (0 : ℝ) < (nativeCBall i).lower := by exact_mod_cast hp exact hp'.trans_le (Ball.bounds (nativeCBall_sound i)).1 def nativeGapBall : Ball := ⟨(132303397031313521 / 100000000000000000000000000000000), (970498095253 / 50000000000000000000000000000000)⟩ lemma nativeGapBall_sound : nativeGapBall.Encloses (nativeH0 + nativeCstar + ((2371054886006746 : ℝ)/10^15)*S/3 - 8*Real.log 7) := by have ht : (Ball.exact (2371054886006746 / 10^15)).Encloses ((2371054886006746 : ℝ)/10^15) := by convert Ball.exact_sound (2371054886006746 / 10^15) using 1 norm_num have hi : (Ball.exact (1/3)).Encloses (1/(3 : ℝ)) := by convert Ball.exact_sound (1/3) using 1 norm_num have he : (Ball.exact 8).Encloses (8 : ℝ) := by simpa only [Rat.cast_ofNat] using Ball.exact_sound 8 have hb := contains_sound (Ball.add_sound (Ball.add_sound (Ball.add_sound nativeH0_enclosed nativeCstarBall_sound) (Ball.mul_sound (Ball.mul_sound ht nativeSBall_sound) hi)) (Ball.neg_sound (Ball.mul_sound he logN7_sound))) (show contains (.add (.add (.add initialEntropyBall nativeCstarBall) (.mul (.mul (.exact (2371054886006746 / 10^15)) nativeSBall) (.exact (1/3)))) (.neg (.mul (.exact 8) logN7))) nativeGapBall = true by decide +kernel) simpa only [one_div, div_eq_mul_inv, one_mul, sub_eq_add_neg] using hb theorem native_sharp_gap : 0 < nativeH0 + nativeCstar + ((2371054886006746 : ℝ)/10^15)*S/3 - 8*Real.log 7 := by have hp : (0 : ℝ) < nativeGapBall.lower := by norm_num [nativeGapBall, Ball.lower] exact hp.trans_le (Ball.bounds nativeGapBall_sound).1 theorem native_sharp_rate_bound : 3 * (8 * Real.log 7 - (nativeH0 + nativeCstar)) / S < (2371054886006746 : ℝ)/10^15 := by apply (div_lt_iff₀ native_volume_positive).mpr have hg := native_sharp_gap linarith end MatrixMultiplication.NativeNumerics end end end MatrixAllFields end OAI