lean/OAI/LinearAlgebra/MatrixFields/Certificates/SparseContraction.lean

import Mathlib
import OAI.LinearAlgebra.MatrixFields.Entropy.InitialEntropy

namespace OAI

namespace MatrixAllFields

open scoped BigOperators Topology Polynomial

section
noncomputable section

namespace MatrixMultiplication.AllFieldInitialEntropyCertificate
open AllFieldParameters AllFieldCertificates AllFieldCertificates.Replay
open scoped BigOperators

def exponentialCoefficients : List ℚ :=
  (List.range 17).map (fun j => 1 / (Nat.factorial j : ℚ))

lemma exponential_polynomial_cast (n : ℤ) :
    (((∑ j ∈ Finset.range 17, ((n : ℚ) / 640000) ^ j /
      Nat.factorial j : ℚ) : ℝ)) =
    polynomial (((n : ℚ) / 640000 : ℚ) : ℝ) exponentialCoefficients := by
  simp only [Rat.cast_sum, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast]
  norm_num [Finset.sum_range_succ, exponentialCoefficients, List.range_succ,
    polynomial, List.map_append]
  ring

lemma weight_enclosed_of_trace (n : ℤ) (trace : List Ball)
    (b₁ b₂ b₃ b₄ b₅ b₆ : Ball)
    (hp : checkPolynomial (.exact ((n : ℚ) / 640000))
      exponentialCoefficients trace = true)
    (h₁ : contains (.mul (traceHead trace) (traceHead trace)) b₁ = true)
    (h₂ : contains (.mul b₁ b₁) b₂ = true)
    (h₃ : contains (.mul b₂ b₂) b₃ = true)
    (h₄ : contains (.mul b₃ b₃) b₄ = true)
    (h₅ : contains (.mul b₄ b₄) b₅ = true)
    (h₆ : contains (.mul b₅ b₅) b₆ = true) :
    b₆.Encloses (weight n : ℝ) := by
  have p := checkPolynomial_sound (Ball.exact_sound ((n : ℚ) / 640000))
    exponentialCoefficients trace hp
  rw [← exponential_polynomial_cast] at p
  have p₁ := contains_sound (Ball.mul_sound p p) h₁
  have p₂ := contains_sound (Ball.mul_sound p₁ p₁) h₂
  have p₃ := contains_sound (Ball.mul_sound p₂ p₂) h₃
  have p₄ := contains_sound (Ball.mul_sound p₃ p₃) h₄
  have p₅ := contains_sound (Ball.mul_sound p₄ p₄) h₅
  have p₆ := contains_sound (Ball.mul_sound p₅ p₅) h₆
  convert p₆ using 1
  simp only [weight, Rat.cast_pow]
  ring

def weightTrace0 : List Ball := [
  ⟨(45822469347703230916837028446061 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(95761729883054217060631522574569 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1518032241603819172753264716571 / 3125000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3261876795622354777174156430543 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4095003862446164503471876088913 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(205341225223953618638702205277 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(137176467343829037379890181127 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19626959451965561090978938867 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1228161761087584971721304317 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(54637558371312300664525013 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27340322092271316640462777 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2487117682933565567802199 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(103687562373860905549233 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(15959542323714148962659 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(285109727917665106419 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76054635388880697247 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare0_1 : Ball := ⟨(83987947884848086426829976722031 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare0_2 : Ball := ⟨(70539753899079581546921669791051 / 100000000000000000000000000000000), (3 / 50000000000000000000000000000000)⟩
def weightSquare0_3 : Ball := ⟨(49758568801427130276705189494043 / 100000000000000000000000000000000), (1 / 10000000000000000000000000000000)⟩
def weightSquare0_4 : Ball := ⟨(12379575845831786800610165591697 / 50000000000000000000000000000000), (11 / 100000000000000000000000000000000)⟩
def weightSquare0_5 : Ball := ⟨(61301559249080719838295497489 / 1000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare0_6 : Ball := ⟨(375788116636855394790068879267 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩
lemma weight_enclosed_0 : weightSquare0_6.Encloses (weight (-55839) : ℝ) := by
  exact weight_enclosed_of_trace (-55839) weightTrace0 weightSquare0_1 weightSquare0_2 weightSquare0_3 weightSquare0_4 weightSquare0_5 weightSquare0_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace1 : List Ball := [
  ⟨(47255022421610958409873827160341 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(97229191696542968020629859555673 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(49072067803428631784511439365999 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16434031762616588480302150310221 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(164802101549160184047696785151 / 4000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(825553916487359822440105826453 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(137776428077124469979954185547 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(9851051824020091619644825781 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2464686157317462422255187163 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(137012571857256431690259327 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(13708263777035553765322303 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2493473966999507375380809 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(103932227199065796269981 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3198903658586015670193 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(571385949173744347803 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76201768611384910161 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare1_1 : Ball := ⟨(89321485762678176318205827626583 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare1_2 : Ball := ⟨(1595665563770464030669190950117 / 2000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare1_3 : Ball := ⟨(31826857392536410151645281386261 / 50000000000000000000000000000000), (3 / 25000000000000000000000000000000)⟩
def weightSquare1_4 : Ball := ⟨(810359081187879632371486932441 / 2000000000000000000000000000000), (17 / 100000000000000000000000000000000)⟩
def weightSquare1_5 : Ball := ⟨(16417046011591612337241268965321 / 100000000000000000000000000000000), (7 / 50000000000000000000000000000000)⟩
def weightSquare1_6 : Ball := ⟨(2695193997467160660436803279809 / 100000000000000000000000000000000), (3 / 50000000000000000000000000000000)⟩
lemma weight_enclosed_1 : weightSquare1_6.Encloses (weight (-36137) : ℝ) := by
  exact weight_enclosed_of_trace (-36137) weightTrace1 weightSquare1_1 weightSquare1_2 weightSquare1_3 weightSquare1_4 weightSquare1_5 weightSquare1_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace2 : List Ball := [
  ⟨(19462084945085296406442948535709 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(98649101709384577877389230656831 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(24774339675450460154475682836493 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16553683043349819604053296423209 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(518006181749542381171965125597 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(103695169558496726647696312179 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(138349773272399157380263337543 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(308966483173666977262623271 / 1562500000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2472665918923664623838816993 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(137411862870134171885457943 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3436146482162016182737377 / 12500000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(2499530900421064933309269 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(208330586210581940821979 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(125217395867685532157 / 781250000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(143124148124781859341 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19085332337372655453 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare2_1 : Ball := ⟨(94693187602428936210099773718353 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare2_2 : Ball := ⟨(44833998891544005922143607529859 / 50000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare2_3 : Ball := ⟨(10050437283034845758507319035581 / 12500000000000000000000000000000), (13 / 100000000000000000000000000000000)⟩
def weightSquare2_4 : Ball := ⟨(64647225331338785478344935394587 / 100000000000000000000000000000000), (11 / 50000000000000000000000000000000)⟩
def weightSquare2_5 : Ball := ⟨(41792637430408911418759860943933 / 100000000000000000000000000000000), (29 / 100000000000000000000000000000000)⟩
def weightSquare2_6 : Ball := ⟨(2183280679237019972788253432353 / 12500000000000000000000000000000), (1 / 4000000000000000000000000000000)⟩
lemma weight_enclosed_2 : weightSquare2_6.Encloses (weight (-17449) : ℝ) := by
  exact weight_enclosed_of_trace (-17449) weightTrace2 weightSquare2_1 weightSquare2_2 weightSquare2_3 weightSquare2_4 weightSquare2_5 weightSquare2_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace3 : List Ball := [
  ⟨(99722261409261526095071612219837 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(99861066332934437727060774889811 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(49953678046044862180395545236147 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8327542284069720378328946310521 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(26027187701631258446683248593 / 625000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(832947202036967261967421765977 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27766744907223109458809337961 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(9917187015432196825563880161 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2479392505329480885756231427 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(55099313563992663293648747 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27550353220886312747669859 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2504630327938123105603659 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(208722914534389897186987 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8027927063567209908089 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(229372382000523736751 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76458344396867462343 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare3_1 : Ball := ⟨(99445294205770906125170809966217 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare3_2 : Ball := ⟨(9889366539672332554869689802191 / 10000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare3_3 : Ball := ⟨(24449892638997681165967353683633 / 25000000000000000000000000000000), (7 / 50000000000000000000000000000000)⟩
def weightSquare3_4 : Ball := ⟨(23911890002340519753388331266581 / 25000000000000000000000000000000), (7 / 25000000000000000000000000000000)⟩
def weightSquare3_5 : Ball := ⟨(45742278678722600142174067342087 / 50000000000000000000000000000000), (11 / 20000000000000000000000000000000)⟩
def weightSquare3_6 : Ball := ⟨(83694242348876807273467974770337 / 100000000000000000000000000000000), (51 / 50000000000000000000000000000000)⟩
lemma weight_enclosed_3 : weightSquare3_6.Encloses (weight (-1780) : ℝ) := by
  exact weight_enclosed_of_trace (-1780) weightTrace3 weightSquare3_1 weightSquare3_2 weightSquare3_3 weightSquare3_4 weightSquare3_5 weightSquare3_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace4 : List Ball := [
  ⟨(50911410227239288182312346605491 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(20181733256055512038055594758737 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(50302432261710796797673490870941 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8371103169920852457012116486571 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2090882745092313740543310965499 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(104481089995403957581214526993 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(34812028439039215519527827849 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19886161668670775979789845573 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(621286427138617212332161333 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(276071808085179654271380119 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27602641726950831010849701 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2508987275976284863517391 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(52264509050953876130863 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16079789645021142467363 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1148457507801906020393 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76557974095447743761 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare4_1 : Ball := ⟨(1036788676530498075661364538317 / 1000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare4_2 : Ball := ⟨(107493075978186177125241613635047 / 100000000000000000000000000000000), (1 / 12500000000000000000000000000000)⟩
def weightSquare4_3 : Ball := ⟨(14443451729065132700277449317601 / 12500000000000000000000000000000), (9 / 50000000000000000000000000000000)⟩
def weightSquare4_4 : Ball := ⟨(133512510623894125738161773133231 / 100000000000000000000000000000000), (21 / 50000000000000000000000000000000)⟩
def weightSquare4_5 : Ball := ⟨(11140994058184651120224401477951 / 6250000000000000000000000000000), (113 / 100000000000000000000000000000000)⟩
def weightSquare4_6 : Ball := ⟨(317751676427534595662029560374059 / 100000000000000000000000000000000), (101 / 25000000000000000000000000000000)⟩
lemma weight_enclosed_4 : weightSquare4_6.Encloses (weight (11561) : ℝ) := by
  exact weight_enclosed_of_trace (11561) weightTrace4 weightSquare4_1 weightSquare4_2 weightSquare4_3 weightSquare4_4 weightSquare4_5 weightSquare4_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace5 : List Ball := [
  ⟨(103600813901653687360089808813237 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(101789792272895755762256079526159 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(12648708642372832242092434813847 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16815111505055235854974961283079 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2098160259909105135099570549999 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(167654293436288924565226478201 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(17449236212407383366781122517 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4982337882474756504311598923 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(31124271600418638659055071 / 1250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(276551176671687180049383751 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27646203019880523641287881 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2512616138364183427012361 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4186741947709709968707 / 2000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8049858753420616865717 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(574892862468114940523 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76640711328831368513 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare5_1 : Ball := ⟨(13416410801356349903607034777723 / 12500000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare5_2 : Ball := ⟨(115200050426080854395202845284257 / 100000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare5_3 : Ball := ⟨(132710516181715716422850695559233 / 100000000000000000000000000000000), (9 / 50000000000000000000000000000000)⟩
def weightSquare5_4 : Ball := ⟨(176120811052174290165929815923193 / 100000000000000000000000000000000), (49 / 100000000000000000000000000000000)⟩
def weightSquare5_5 : Ball := ⟨(62037080171351355194935848106531 / 20000000000000000000000000000000), (87 / 50000000000000000000000000000000)⟩
def weightSquare5_6 : Ball := ⟨(962149829046668872508647133711623 / 100000000000000000000000000000000), (27 / 2500000000000000000000000000000)⟩
lemma weight_enclosed_5 : weightSquare5_6.Encloses (weight (22640) : ℝ) := by
  exact weight_enclosed_of_trace (22640) weightTrace5 weightSquare5_1 weightSquare5_2 weightSquare5_3 weightSquare5_4 weightSquare5_5 weightSquare5_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace6 : List Ball := [
  ⟨(52496907097213613827039425021221 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(102476628332481265185663204667291 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(50822179453251562101591990094477 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16871708672598427120010057088159 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(84152163522076882223784705499 / 2000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(840148968827547918893289383799 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(69930850267688450018787230643 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(159702309542728519984293019 / 800000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2493653488700243236021188337 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(276922068313722247860040669 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27679898912162961846196791 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(503084522465855867187777 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(26194110004123274166153 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3223024873382905372317 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(46032498729593238173 / 4000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(9588068402869008441 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare6_1 : Ball := ⟨(11023701019093908391497969098631 / 10000000000000000000000000000000), (1 / 25000000000000000000000000000000)⟩
def weightSquare6_2 : Ball := ⟨(121521984158372074423106444564011 / 100000000000000000000000000000000), (1 / 10000000000000000000000000000000)⟩
def weightSquare6_3 : Ball := ⟨(18459490792234541766581018912859 / 12500000000000000000000000000000), (1 / 4000000000000000000000000000000)⟩
def weightSquare6_4 : Ball := ⟨(218081792197498771472088623185963 / 100000000000000000000000000000000), (3 / 4000000000000000000000000000000)⟩
def weightSquare6_5 : Ball := ⟨(237798340440365180199344472936997 / 50000000000000000000000000000000), (41 / 12500000000000000000000000000000)⟩
def weightSquare6_6 : Ball := ⟨(226192202864767271537325845748747 / 10000000000000000000000000000000), (39 / 1250000000000000000000000000000)⟩
lemma weight_enclosed_6 : weightSquare6_6.Encloses (weight (31188) : ℝ) := by
  exact weight_enclosed_of_trace (31188) weightTrace6 weightSquare6_1 weightSquare6_2 weightSquare6_3 weightSquare6_4 weightSquare6_5 weightSquare6_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace7 : List Ball := [
  ⟨(13238388542478214491069816710213 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(102925303898300024346667979534741 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(25485209971305104433142398821633 / 50000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(8454286069948186469498114558423 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4214954748496148057251911685369 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(841370296775696005405701232473 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(140035844466856609575439150853 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4996131128333977017996839571 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(499213144876440899368362439 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27716305853430552915679447 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27701789454298507636100787 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2517245580029807777294919 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(41938605269701210777891 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16125129745665541975627 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1151479177264804137941 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(38372970988705270699 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare7_1 : Ball := ⟨(11216315596903581711527855028257 / 10000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare7_2 : Ball := ⟨(125805735569342550503356966593139 / 100000000000000000000000000000000), (1 / 12500000000000000000000000000000)⟩
def weightSquare7_3 : Ball := ⟨(158270831021433413898498261129081 / 100000000000000000000000000000000), (21 / 100000000000000000000000000000000)⟩
def weightSquare7_4 : Ball := ⟨(31312069940268911822779912641567 / 12500000000000000000000000000000), (67 / 100000000000000000000000000000000)⟩
def weightSquare7_5 : Ball := ⟨(125497052664869372826269098660877 / 20000000000000000000000000000000), (337 / 100000000000000000000000000000000)⟩
def weightSquare7_6 : Ball := ⟨(393737755689224923790566075091771 / 10000000000000000000000000000000), (423 / 10000000000000000000000000000000)⟩
lemma weight_enclosed_7 : weightSquare7_6.Encloses (weight (36731) : ℝ) := by
  exact weight_enclosed_of_trace (36731) weightTrace7 weightSquare7_1 weightSquare7_2 weightSquare7_3 weightSquare7_4 weightSquare7_5 weightSquare7_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace8 : List Ball := [
  ⟨(53107840245772877522502511762089 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(103076607534767528536346359585271 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(5102036178196093237793563948317 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3384196825616027372211588988827 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(131794605971792912907148072173 / 3125000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(420890623113675921432978149363 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(70047213901119971249691626091 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2498979428084996990116058961 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2496876982418733363454750399 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(34655512160242955405574419 / 12500000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(27709150025502850262332103 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(62946462422495109382441 / 2500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(41948028645634175042909 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16128493345257602942617 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(28792582462514276983 / 2500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76759847269274455947 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare8_1 : Ball := ⟨(28204426955705333720136269142093 / 25000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare8_2 : Ball := ⟨(31819587995988705552098445079421 / 25000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare8_3 : Ball := ⟨(161997788837514964265321695821339 / 100000000000000000000000000000000), (19 / 100000000000000000000000000000000)⟩
def weightSquare8_4 : Ball := ⟨(52486567176488175914387270062549 / 20000000000000000000000000000000), (63 / 100000000000000000000000000000000)⟩
def weightSquare8_5 : Ball := ⟨(344354966746500746353077958219731 / 50000000000000000000000000000000), (83 / 25000000000000000000000000000000)⟩
def weightSquare8_6 : Ball := ⟨(592901715614918174080204664509561 / 12500000000000000000000000000000), (2287 / 50000000000000000000000000000000)⟩
lemma weight_enclosed_8 : weightSquare8_6.Encloses (weight (38593) : ℝ) := by
  exact weight_enclosed_of_trace (38593) weightTrace8 weightSquare8_1 weightSquare8_2 weightSquare8_3 weightSquare8_4 weightSquare8_5 weightSquare8_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace9 : List Ball := [
  ⟨(105769353286979812971192319657867 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(12857213854765865443628769642581 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(50948101186729931232361150203779 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16903024110177614037303920285779 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(842769869335374211420370584613 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(52574160131711393877303756643 / 6250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(140009649991401748572542612153 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19981255379359022718212237153 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(62392573665020423810114129 / 2500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(277126818288917152811301887 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27698497726817026655144159 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(39327679210857430103861 / 1562500000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(104835977052347421176833 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16123625401104954019783 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(287844734674196536367 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(38369860594472552971 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare9_1 : Ball := ⟨(699197255921621710400340726909 / 625000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare9_2 : Ball := ⟨(31288115372052849022082108910093 / 25000000000000000000000000000000), (1 / 12500000000000000000000000000000)⟩
def weightSquare9_3 : Ball := ⟨(783156930827911832787526073193 / 500000000000000000000000000000), (21 / 100000000000000000000000000000000)⟩
def weightSquare9_4 : Ball := ⟨(61333477830379467923854252642317 / 25000000000000000000000000000000), (67 / 100000000000000000000000000000000)⟩
def weightSquare9_5 : Ball := ⟨(601887280443143949422394537773243 / 100000000000000000000000000000000), (33 / 10000000000000000000000000000000)⟩
def weightSquare9_6 : Ball := ⟨(3622682983592438129289817987640667 / 100000000000000000000000000000000), (3973 / 100000000000000000000000000000000)⟩
lemma weight_enclosed_9 : weightSquare9_6.Encloses (weight (35898) : ℝ) := by
  exact weight_enclosed_of_trace (35898) weightTrace9 weightSquare9_1 weightSquare9_2 weightSquare9_3 weightSquare9_4 weightSquare9_5 weightSquare9_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace10 : List Ball := [
  ⟨(104643817941585644011055002692351 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(102304343486104167397858997043291 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(6345650025415076652394745911099 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16857530898254533759977249142099 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2102390764796993902426298307757 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(839678918917955610443536100733 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(6989733183295338113197086393 / 5000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3990884012883817837828762809 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(623181153966461295253418909 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(138414630349275657661540399 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(6917866971559857065203159 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2514720451697499889277579 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(209498895670714520474879 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16111270073369578912207 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(287638905637475131677 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3067543529976875711 / 4000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare10_1 : Ball := ⟨(109503286333917215300234963650589 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare10_2 : Ball := ⟨(23981939435855721532382233715047 / 20000000000000000000000000000000), (1 / 12500000000000000000000000000000)⟩
def weightSquare10_3 : Ball := ⟨(143783354776262960787841124947381 / 100000000000000000000000000000000), (1 / 5000000000000000000000000000000)⟩
def weightSquare10_4 : Ball := ⟨(206736531107167007786767430192403 / 100000000000000000000000000000000), (59 / 100000000000000000000000000000000)⟩
def weightSquare10_5 : Ball := ⟨(427399932942246318664573400014931 / 100000000000000000000000000000000), (49 / 20000000000000000000000000000000)⟩
def weightSquare10_6 : Ball := ⟨(913353513395183249684030644938607 / 50000000000000000000000000000000), (419 / 20000000000000000000000000000000)⟩
lemma weight_enclosed_10 : weightSquare10_6.Encloses (weight (29051) : ℝ) := by
  exact weight_enclosed_of_trace (29051) weightTrace10 weightSquare10_1 weightSquare10_2 weightSquare10_3 weightSquare10_4 weightSquare10_5 weightSquare10_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace11 : List Ball := [
  ⟨(51457318287274132865276567057379 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1014503403334361270220267489773 / 1000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(50482287126726561924028897888317 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(3357412966824382785429865106037 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1047680795825287522555360752867 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(418670001796633248356362121959 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(69730488017545672562413193419 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3982550427688823527492235683 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1244049333538284529198387107 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(69091744539850108590769427 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(5525893169158403393733513 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2511221947893666302852981 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(209229889795857520126909 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8046031081551623817853 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1149275510972162029753 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76608950208370987967 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare11_1 : Ball := ⟨(105914224212713475942880094061453 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare11_2 : Ball := ⟨(56089114452904707614563249114169 / 50000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare11_3 : Ball := ⟨(62919775202220875163649729179661 / 50000000000000000000000000000000), (17 / 100000000000000000000000000000000)⟩
def weightSquare11_4 : Ball := ⟨(9897245278745022430232851573857 / 6250000000000000000000000000000), (11 / 25000000000000000000000000000000)⟩
def weightSquare11_5 : Ball := ⟨(250765988115560030064146963919903 / 100000000000000000000000000000000), (7 / 5000000000000000000000000000000)⟩
def weightSquare11_6 : Ball := ⟨(314417903977865971190225770587881 / 50000000000000000000000000000000), (703 / 100000000000000000000000000000000)⟩
lemma weight_enclosed_11 : weightSquare11_6.Encloses (weight (18387) : ℝ) := by
  exact weight_enclosed_of_trace (18387) weightTrace11 weightSquare11_1 weightSquare11_2 weightSquare11_3 weightSquare11_4 weightSquare11_5 weightSquare11_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace12 : List Ball := [
  ⟨(50299565702744265076996558074139 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(50149633731710606664346630250871 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(10019941228855690837754994565271 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16691585754174909970175937448921 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2085824745654684049952040366639 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(834163685934846265222339166067 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(69503749014945907528607500637 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1241005922081361949527725719 / 6250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1240902916616263466583660047 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(55147578719850151178826729 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(13786145698506878875057519 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(250645847183754706694749 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(208863538519519027742349 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(16065898544688539931839 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1147531528860113200627 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(7650018722732584753 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare12_1 : Ball := ⟨(50600926197693743461694777077217 / 50000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare12_2 : Ball := ⟨(20483629856515592049753749622719 / 20000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare12_3 : Ball := ⟨(5244738651234212176799626520019 / 5000000000000000000000000000000), (3 / 20000000000000000000000000000000)⟩
def weightSquare12_4 : Ball := ⟨(110029134079000252453786194910311 / 100000000000000000000000000000000), (1 / 3125000000000000000000000000000)⟩
def weightSquare12_5 : Ball := ⟨(60532051730873073663919899910177 / 50000000000000000000000000000000), (71 / 100000000000000000000000000000000)⟩
def weightSquare12_6 : Ball := ⟨(29313034293992750986222098316009 / 20000000000000000000000000000000), (173 / 100000000000000000000000000000000)⟩
lemma weight_enclosed_12 : weightSquare12_6.Encloses (weight (3823) : ℝ) := by
  exact weight_enclosed_of_trace (3823) weightTrace12 weightSquare12_1 weightSquare12_2 weightSquare12_3 weightSquare12_4 weightSquare12_5 weightSquare12_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace13 : List Ball := [
  ⟨(97749308384307458995717295526927 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(386212440059808669826807686367 / 390625000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(620284354280447813487395038807 / 1250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(16572246608625983214041733343471 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(207388417688370545953590935703 / 5000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(166036382119894000421479712647 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(138438502963314708257138641127 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1978495383124961248679789739 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(309237476945452854929631139 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(274947171450935846495308899 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3437549805329453114224957 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1250233374430986093929039 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(208402594317235319799913 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8016485665633495607099 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(286334057844721094857 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(15272567399497273009 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare13_1 : Ball := ⟨(95549272896104404998317042425221 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare13_2 : Ball := ⟨(91296635509742318701678139152337 / 100000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare13_3 : Ball := ⟨(41675378276993710444793986953517 / 50000000000000000000000000000000), (7 / 50000000000000000000000000000000)⟩
def weightSquare13_4 : Ball := ⟨(34736743090610384990883814056083 / 50000000000000000000000000000000), (3 / 12500000000000000000000000000000)⟩
def weightSquare13_5 : Ball := ⟨(24132826410861366426580373032961 / 50000000000000000000000000000000), (7 / 20000000000000000000000000000000)⟩
def weightSquare13_6 : Ball := ⟨(23295732423070716039874970270233 / 100000000000000000000000000000000), (17 / 50000000000000000000000000000000)⟩
lemma weight_enclosed_13 : weightSquare13_6.Encloses (weight (-14569) : ℝ) := by
  exact weight_enclosed_of_trace (-14569) weightTrace13 weightSquare13_1 weightSquare13_2 weightSquare13_3 weightSquare13_4 weightSquare13_5 weightSquare13_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace14 : List Ball := [
  ⟨(9397419149312240765671878406229 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(96955889088939538910397682022679 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(23916046352232353441792145211 / 48828125000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3282178827443733390859812092909 / 20000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(4115406748962183626188354016423 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(412388718459236045682322206297 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(34416317034840072279521000559 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2461023229140385701538388279 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(492627540295271625210502361 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(68467532913805441181202433 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27402422922560608218237489 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2492297689867778508766373 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(207773912733602071421357 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(15988047386627809666029 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(571170154415164387469 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(38087296400995144381 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare14_1 : Ball := ⟨(88311486667860398901944158266739 / 100000000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare14_2 : Ball := ⟨(77989186774876849812044262325737 / 100000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare14_3 : Ball := ⟨(60823132538066261346053024041467 / 100000000000000000000000000000000), (3 / 25000000000000000000000000000000)⟩
def weightSquare14_4 : Ball := ⟨(1156079203669742114596957224833 / 3125000000000000000000000000000), (1 / 6250000000000000000000000000000)⟩
def weightSquare14_5 : Ball := ⟨(3421488960403622578089278590681 / 25000000000000000000000000000000), (13 / 100000000000000000000000000000000)⟩
def weightSquare14_6 : Ball := ⟨(117065867061638619900413203703 / 6250000000000000000000000000000), (1 / 25000000000000000000000000000000)⟩
lemma weight_enclosed_14 : weightSquare14_6.Encloses (weight (-39776) : ℝ) := by
  exact weight_enclosed_of_trace (-39776) weightTrace14 weightSquare14_1 weightSquare14_2 weightSquare14_3 weightSquare14_4 weightSquare14_5 weightSquare14_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace15 : List Ball := [
  ⟨(94147322024652654828833510491163 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(97044248516044896355939512038323 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(49009817859248311627512106727623 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1641837841548993622447410991039 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(823381940790235157278803689397 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(206257164476878098440948109979 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(137701235103513169711025747113 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19692689326920054765335269601 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2463638768430057590648224087 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(273920293962133881870515871 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(27406987350225809089504263 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1246339178279343493885119 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(207803214433657946794709 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(1998767697855936571617 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(114247998713503541457 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(76183390027455214293 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare15_1 : Ball := ⟨(44318591222068234326289200916567 / 50000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩
def weightSquare15_2 : Ball := ⟨(61379297747149485991007624471 / 78125000000000000000000000000), (1 / 25000000000000000000000000000000)⟩
def weightSquare15_3 : Ball := ⟨(1928918114269813742751307284463 / 3125000000000000000000000000000), (1 / 12500000000000000000000000000000)⟩
def weightSquare15_4 : Ball := ⟨(2381264058597257105709481658969 / 6250000000000000000000000000000), (11 / 100000000000000000000000000000000)⟩
def weightSquare15_5 : Ball := ⟨(14516271402923727674314000842103 / 100000000000000000000000000000000), (9 / 100000000000000000000000000000000)⟩
def weightSquare15_6 : Ball := ⟨(2107221354433412088486138736011 / 100000000000000000000000000000000), (1 / 25000000000000000000000000000000)⟩
lemma weight_enclosed_15 : weightSquare15_6.Encloses (weight (-38598) : ℝ) := by
  exact weight_enclosed_of_trace (-38598) weightTrace15 weightSquare15_1 weightSquare15_2 weightSquare15_3 weightSquare15_4 weightSquare15_5 weightSquare15_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightTrace16 : List Ball := [
  ⟨(96363397544038665735902571610653 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3926818916509623635941208316487 / 4000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(24694139865151464166961931335581 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3302690843648775656223451075217 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(165439458394777402989072424451 / 4000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(82821546947141128907996775141 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(69078641632151769912138780799 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(19749772217589986559441937981 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(154374259625197756705469647 / 6250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(274555792962808879841926837 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1373240124245016613661353 / 5000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(249749928356493009833667 / 10000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(52043563214049462922493 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(8008328294019073421673 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(228849664429048117677 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(38147293777383269777 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4779477332387385297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def weightSquare16_1 : Ball := ⟨(5803690241394023222060139461611 / 6250000000000000000000000000000), (3 / 100000000000000000000000000000000)⟩
def weightSquare16_2 : Ball := ⟨(17245604054042734355635644885209 / 20000000000000000000000000000000), (7 / 100000000000000000000000000000000)⟩
def weightSquare16_3 : Ball := ⟨(116176116870630935339684428353 / 156250000000000000000000000000), (13 / 100000000000000000000000000000000)⟩
def weightSquare16_4 : Ball := ⟨(11056652395428657424319758888073 / 20000000000000000000000000000000), (21 / 100000000000000000000000000000000)⟩
def weightSquare16_5 : Ball := ⟨(30562390548334567074915733474673 / 100000000000000000000000000000000), (3 / 12500000000000000000000000000000)⟩
def weightSquare16_6 : Ball := ⟨(2335149290072325198800662941033 / 25000000000000000000000000000000), (1 / 6250000000000000000000000000000)⟩
lemma weight_enclosed_16 : weightSquare16_6.Encloses (weight (-23708) : ℝ) := by
  exact weight_enclosed_of_trace (-23708) weightTrace16 weightSquare16_1 weightSquare16_2 weightSquare16_3 weightSquare16_4 weightSquare16_5 weightSquare16_6
    (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel) (by decide +kernel)

def weightBall : Fin 17 → Ball := (Matrix.vecCons (weightSquare0_6) (Matrix.vecCons (weightSquare1_6) (Matrix.vecCons (weightSquare2_6) (Matrix.vecCons (weightSquare3_6) (Matrix.vecCons (weightSquare4_6) (Matrix.vecCons (weightSquare5_6) (Matrix.vecCons (weightSquare6_6) (Matrix.vecCons (weightSquare7_6) (Matrix.vecCons (weightSquare8_6) (Matrix.vecCons (weightSquare9_6) (Matrix.vecCons (weightSquare10_6) (Matrix.vecCons (weightSquare11_6) (Matrix.vecCons (weightSquare12_6) (Matrix.vecCons (weightSquare13_6) (Matrix.vecCons (weightSquare14_6) (Matrix.vecCons (weightSquare15_6) (Matrix.vecCons (weightSquare16_6) Matrix.vecEmpty)))))))))))))))))

lemma weightBall_sound (k : Fin 17) :
    (weightBall k).Encloses (weight (initialInteger k.val) : ℝ) := by
  fin_cases k
  · exact weight_enclosed_0
  · exact weight_enclosed_1
  · exact weight_enclosed_2
  · exact weight_enclosed_3
  · exact weight_enclosed_4
  · exact weight_enclosed_5
  · exact weight_enclosed_6
  · exact weight_enclosed_7
  · exact weight_enclosed_8
  · exact weight_enclosed_9
  · exact weight_enclosed_10
  · exact weight_enclosed_11
  · exact weight_enclosed_12
  · exact weight_enclosed_13
  · exact weight_enclosed_14
  · exact weight_enclosed_15
  · exact weight_enclosed_16

def sourceWeightBall (k : ℕ) : Ball := weightBall ⟨k % 17, Nat.mod_lt _ (by decide)⟩

lemma sourceWeightBall_sound (k : ℕ) (hk : k < 17) :
    (sourceWeightBall k).Encloses (weight (initialInteger k) : ℝ) := by
  simpa only [sourceWeightBall, Nat.mod_eq_of_lt hk] using weightBall_sound ⟨k,hk⟩

def listSumBall {α : Type*} (xs : List α) (b : α → Ball) : Ball :=
  xs.foldr (fun x a => .add (b x) a) (.exact 0)

lemma listSumBall_sound {α : Type*} (xs : List α) (b : α → Ball) (v : α → ℝ)
    (h : ∀ x ∈ xs, (b x).Encloses (v x)) :
    (listSumBall xs b).Encloses ((xs.map v).sum) := by
  induction xs with
  | nil => simpa [listSumBall] using Ball.exact_sound 0
  | cons x xs ih =>
    simpa only [listSumBall, List.foldr_cons, List.map_cons, List.sum_cons] using
      Ball.add_sound (h x (by simp)) (ih (fun y hy => h y (by simp [hy])))

def finSumBall : (n : ℕ) → (Fin n → Ball) → Ball
  | 0, _ => .exact 0
  | n+1, b => .add (b 0) (finSumBall n (fun i => b i.succ))

lemma finSumBall_sound (n : ℕ) (b : Fin n → Ball) (v : Fin n → ℝ)
    (h : ∀ i, (b i).Encloses (v i)) :
    (finSumBall n b).Encloses (∑ i, v i) := by
  induction n with
  | zero => simpa [finSumBall] using Ball.exact_sound 0
  | succ n ih =>
    rw [Fin.sum_univ_succ]
    exact Ball.add_sound (h 0) (ih _ _ (fun i => h i.succ))

def initialRawBall (s : Shape) : Ball :=
  .mul (.exact (initialMultiplicity s))
    (.mul (sourceWeightBall (s 0))
      (.mul (sourceWeightBall (s 1)) (sourceWeightBall (s 2))))

lemma initial_shape_bound : ∀ s ∈ sortedInitial, ∀ i : Fin 3, s i < 17 := by
  decide +kernel

lemma initialRawBall_sound (s : Shape) (hs : s ∈ sortedInitial) :
    (initialRawBall s).Encloses (initialWeight s : ℝ) := by
  have h := Ball.mul_sound (Ball.exact_sound (initialMultiplicity s : ℚ))
    (Ball.mul_sound (sourceWeightBall_sound (s 0) (initial_shape_bound s hs 0))
      (Ball.mul_sound (sourceWeightBall_sound (s 1) (initial_shape_bound s hs 1))
        (sourceWeightBall_sound (s 2) (initial_shape_bound s hs 2))))
  simpa only [initialRawBall, initialWeight, Rat.cast_mul, Rat.cast_prod,
    Fin.prod_univ_succ, Fin.prod_univ_zero, mul_one, Fin.succ_zero_eq_one,
    Fin.succ_one_eq_two] using h

def initialDenominatorBall : Ball := listSumBall sortedInitial initialRawBall

lemma initialDenominatorBall_sound :
    initialDenominatorBall.Encloses ((sortedInitial.map initialWeight).sum : ℝ) := by
  have h := listSumBall_sound sortedInitial initialRawBall
    (fun s => (initialWeight s : ℝ)) initialRawBall_sound
  simpa only [initialDenominatorBall, Rat.cast_list_sum, List.map_map,
    Function.comp_def] using h

lemma initialDenominatorBall_positive : 0 < initialDenominatorBall.lower := by
  decide +kernel

def initialLawBall (s : Shape) : Ball :=
  .mul (initialRawBall s) (.inv initialDenominatorBall)

lemma initialLawBall_sound (s : Shape) (hs : s ∈ sortedInitial) :
    (initialLawBall s).Encloses (initialLaw s : ℝ) := by
  have h := Ball.mul_sound (initialRawBall_sound s hs)
    (Ball.inv_sound initialDenominatorBall_sound initialDenominatorBall_positive)
  simpa only [initialLawBall, initialLaw, hs, ite_true, Rat.cast_div,
    div_eq_mul_inv, Rat.cast_mul, Rat.cast_inv] using h

def initialMarginalBall (k : Fin 17) : Ball :=
  finSumBall sortedInitial.length (fun g =>
    .mul (initialLawBall (sortedInitial.get g))
      (.exact ((∑ w : Fin 3, if sortedInitial.get g w = k.val then (1 : ℚ) else 0) / 3)))

lemma initialMarginalBall_sound (k : Fin 17) :
    (initialMarginalBall k).Encloses (AllFieldInitialEntropy.nativeInitialMarginal k) := by
  apply finSumBall_sound
  intro g
  have h := Ball.mul_sound (initialLawBall_sound (sortedInitial.get g) (List.get_mem _ _))
    (Ball.exact_sound ((∑ w : Fin 3, if sortedInitial.get g w = k.val then (1 : ℚ) else 0) / 3))
  convert h using 1
  simp only [Rat.cast_div, Rat.cast_sum, apply_ite, Rat.cast_one,
    Rat.cast_zero, Rat.cast_ofNat, mul_div_assoc]

def roundedMarginal : Fin 17 → Ball := (Matrix.vecCons (⟨(37824257369353410027720060279 / 50000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(32277877294971248510635537179 / 6250000000000000000000000000000), (1 / 20000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(1311429618383774862762137339319 / 50000000000000000000000000000000), (1 / 5000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(8221633968389025907864338663411 / 100000000000000000000000000000000), (3 / 5000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(3458333166749295114800388945299 / 20000000000000000000000000000000), (1 / 781250000000000000000000000000)⟩) (Matrix.vecCons (⟨(24885489904708998392039676962479 / 100000000000000000000000000000000), (181 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(24112146157967309906901428396197 / 100000000000000000000000000000000), (91 / 50000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(3779863608787839049947557982127 / 25000000000000000000000000000000), (111 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(5764358674456721497871303174559 / 100000000000000000000000000000000), (43 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(1226948071890256520646842010821 / 100000000000000000000000000000000), (1 / 10000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(151656043521920341821121947439 / 100000000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(2239268756742155430560743707 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(484478437366253348343232821 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(3038569022328601883975409 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(7930177818229705654311 / 6250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(14180164171909061741993 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(4381963528820193732059 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩) Matrix.vecEmpty)))))))))))))))))

lemma marginal_containment_0 : contains (initialMarginalBall 0) (roundedMarginal 0) = true := by
  decide +kernel

lemma marginal_containment_1 : contains (initialMarginalBall 1) (roundedMarginal 1) = true := by
  decide +kernel

lemma marginal_containment_2 : contains (initialMarginalBall 2) (roundedMarginal 2) = true := by
  decide +kernel

lemma marginal_containment_3 : contains (initialMarginalBall 3) (roundedMarginal 3) = true := by
  decide +kernel

lemma marginal_containment_4 : contains (initialMarginalBall 4) (roundedMarginal 4) = true := by
  decide +kernel

lemma marginal_containment_5 : contains (initialMarginalBall 5) (roundedMarginal 5) = true := by
  decide +kernel

lemma marginal_containment_6 : contains (initialMarginalBall 6) (roundedMarginal 6) = true := by
  decide +kernel

lemma marginal_containment_7 : contains (initialMarginalBall 7) (roundedMarginal 7) = true := by
  decide +kernel

lemma marginal_containment_8 : contains (initialMarginalBall 8) (roundedMarginal 8) = true := by
  decide +kernel

lemma marginal_containment_9 : contains (initialMarginalBall 9) (roundedMarginal 9) = true := by
  decide +kernel

lemma marginal_containment_10 : contains (initialMarginalBall 10) (roundedMarginal 10) = true := by
  decide +kernel

lemma marginal_containment_11 : contains (initialMarginalBall 11) (roundedMarginal 11) = true := by
  decide +kernel

lemma marginal_containment_12 : contains (initialMarginalBall 12) (roundedMarginal 12) = true := by
  decide +kernel

lemma marginal_containment_13 : contains (initialMarginalBall 13) (roundedMarginal 13) = true := by
  decide +kernel

lemma marginal_containment_14 : contains (initialMarginalBall 14) (roundedMarginal 14) = true := by
  decide +kernel

lemma marginal_containment_15 : contains (initialMarginalBall 15) (roundedMarginal 15) = true := by
  decide +kernel

lemma marginal_containment_16 : contains (initialMarginalBall 16) (roundedMarginal 16) = true := by
  decide +kernel

lemma roundedMarginal_sound (k : Fin 17) :
    (roundedMarginal k).Encloses (AllFieldInitialEntropy.nativeInitialMarginal k) := by
  apply contains_sound (initialMarginalBall_sound k)
  fin_cases k
  · exact marginal_containment_0
  · exact marginal_containment_1
  · exact marginal_containment_2
  · exact marginal_containment_3
  · exact marginal_containment_4
  · exact marginal_containment_5
  · exact marginal_containment_6
  · exact marginal_containment_7
  · exact marginal_containment_8
  · exact marginal_containment_9
  · exact marginal_containment_10
  · exact marginal_containment_11
  · exact marginal_containment_12
  · exact marginal_containment_13
  · exact marginal_containment_14
  · exact marginal_containment_15
  · exact marginal_containment_16

def logTwoTrace : List Ball := [
  ⟨(20794415416798359282516957915099 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(35748693755926167713263106179463 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(5434560950833877354841988903793 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3128838846003916954862320107311 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3056579874686922885058244064393 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2007375097745844772419357263631 / 20000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4256848790190598288526447522607 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(7392508992661538424245286176159 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1633145233488461454551893896359 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(1463012983749094267437633302527 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(662505795291976835048297229793 / 12500000000000000000000000000000), (1 / 50000000000000000000000000000000)⟩,
  ⟨(2421637201939737490310128843981 / 50000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4459034852306579173408406148187 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(4131313670759212560675655333689 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(240530606468723732046722791867 / 6250000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(450240571609440969944458529469 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(84626577284090520093573740141 / 2500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3192840549545314506413819178039 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(3021279231622116272010086888069 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2867188760274722123766457668297 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(340971970693677773852649492219 / 12500000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2598762377749677766171251244647 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(2458628841607565011820330969267 / 100000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩,
  ⟨(531914893617021276595744680851 / 25000000000000000000000000000000), (1 / 100000000000000000000000000000000)⟩]
def logTwoBall : Ball := ⟨(6931471805599453094172319305033 / 10000000000000000000000000000000), (21175823683 / 100000000000000000000000000000000)⟩
lemma logTwoBall_sound : logTwoBall.Encloses (Real.log 2) := by
  have h := normalizedLogBall_sound (m := 2) (trace := logTwoTrace)
    (by norm_num) (by norm_num) (by decide +kernel)
  exact contains_sound h (by decide +kernel)
def marginalLogWitness : Fin 17 → LogWitness := (Matrix.vecCons (⟨-11, 37824257369353410027720060279, 24414062500000000000000000000⟩) (Matrix.vecCons (⟨-8, 32277877294971248510635537179, 24414062500000000000000000000⟩) (Matrix.vecCons (⟨-6, 1311429618383774862762137339319, 781250000000000000000000000000⟩) (Matrix.vecCons (⟨-4, 8221633968389025907864338663411, 6250000000000000000000000000000⟩) (Matrix.vecCons (⟨-3, 3458333166749295114800388945299, 2500000000000000000000000000000⟩) (Matrix.vecCons (⟨-3, 24885489904708998392039676962479, 12500000000000000000000000000000⟩) (Matrix.vecCons (⟨-3, 24112146157967309906901428396197, 12500000000000000000000000000000⟩) (Matrix.vecCons (⟨-3, 3779863608787839049947557982127, 3125000000000000000000000000000⟩) (Matrix.vecCons (⟨-5, 5764358674456721497871303174559, 3125000000000000000000000000000⟩) (Matrix.vecCons (⟨-7, 1226948071890256520646842010821, 781250000000000000000000000000⟩) (Matrix.vecCons (⟨-10, 151656043521920341821121947439, 97656250000000000000000000000⟩) (Matrix.vecCons (⟨-14, 2239268756742155430560743707, 1220703125000000000000000000⟩) (Matrix.vecCons (⟨-18, 484478437366253348343232821, 381469726562500000000000000⟩) (Matrix.vecCons (⟨-23, 3038569022328601883975409, 2980232238769531250000000⟩) (Matrix.vecCons (⟨-30, 31720711272918822617244, 23283064365386962890625⟩) (Matrix.vecCons (⟨-33, 28360328343818123483986, 23283064365386962890625⟩) (Matrix.vecCons (⟨-35, 35055708230561549856472, 23283064365386962890625⟩) Matrix.vecEmpty)))))))))))))))))
def marginalLogBall : Fin 17 → Ball := (Matrix.vecCons (⟨(-359341382912849805381766834083483 / 50000000000000000000000000000000), (254109886839 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-526595466063060246707895328811773 / 100000000000000000000000000000000), (38116482823 / 20000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-182045257575976223204856091760809 / 50000000000000000000000000000000), (9264422909 / 6250000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-249840121706468106425374731856733 / 100000000000000000000000000000000), (13234889893 / 12500000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-175494554427421837074873290309581 / 100000000000000000000000000000000), (5293955967 / 6250000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-139088528712395736536901400948077 / 100000000000000000000000000000000), (84703295459 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-27782314111222943517224434997 / 19531250000000000000000000000), (84703295487 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-7556751592875787652219540186071 / 4000000000000000000000000000000), (42351647733 / 50000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-142673814156440611744929044763629 / 50000000000000000000000000000000), (31763735711 / 25000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-440064034235271909031072697214261 / 100000000000000000000000000000000), (169406590279 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-649131037985310617044853658780989 / 100000000000000000000000000000000), (29116757729 / 12500000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-113716727348098189960178186941083 / 12500000000000000000000000000000), (317637364177 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-305940195463298290399109426509441 / 25000000000000000000000000000000), (6286575881 / 1562500000000000000000000000000)⟩) (Matrix.vecCons (⟨(-1592299969457533467596519841341019 / 100000000000000000000000000000000), (508227995949 / 100000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-81940687367260669107912929811931 / 4000000000000000000000000000000), (32861933139 / 5000000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-2267659192418886369231475949409093 / 100000000000000000000000000000000), (90878763719 / 12500000000000000000000000000000)⟩) (Matrix.vecCons (⟨(-1192546955237440450086597482595127 / 50000000000000000000000000000000), (392575236187 / 50000000000000000000000000000000)⟩) Matrix.vecEmpty)))))))))))))))))
lemma marginalLogBall_sound (k : Fin 17) :
    (marginalLogBall k).Encloses (Real.log (AllFieldInitialEntropy.nativeInitialMarginal k)) := by
  apply checkLog_sound logTwoBall_sound (roundedMarginal_sound k)
    (w := marginalLogWitness k)
  fin_cases k <;> decide +kernel
def totalMarginalBall : Ball := ⟨(99999999999999999999999999999989 / 100000000000000000000000000000000), (3 / 400000000000000000000000000000)⟩
lemma totalMarginalBall_sound :
    totalMarginalBall.Encloses (∑ k, AllFieldInitialEntropy.nativeInitialMarginal k) := by
  apply contains_sound (finSumBall_sound 17 roundedMarginal
    AllFieldInitialEntropy.nativeInitialMarginal roundedMarginal_sound)
  decide +kernel
def totalLogWitness : LogWitness := ⟨-1, 99999999999999999999999999999989, 50000000000000000000000000000000⟩
def totalLogBall : Ball := ⟨(-11 / 100000000000000000000000000000000), (10587912029 / 25000000000000000000000000000000)⟩
lemma totalLogBall_sound :
    totalLogBall.Encloses (Real.log (∑ k, AllFieldInitialEntropy.nativeInitialMarginal k)) := by
  exact checkLog_sound logTwoBall_sound totalMarginalBall_sound
    (w := totalLogWitness) (by decide +kernel)

def entropySumBall : Ball :=
  finSumBall 17 (fun k => .neg (.mul (roundedMarginal k) (marginalLogBall k)))

lemma entropySumBall_sound :
    entropySumBall.Encloses (MatrixMultiplication.Foundation.finiteEntropy
      AllFieldInitialEntropy.nativeInitialMarginal) := by
  apply finSumBall_sound
  intro k
  have h := Ball.neg_sound (Ball.mul_sound (roundedMarginal_sound k)
    (marginalLogBall_sound k))
  convert h using 1
  unfold MatrixMultiplication.Foundation.entropyTerm
  ring

def initialEntropyRawBall : Ball := .add
  (.mul totalMarginalBall totalLogBall) entropySumBall

lemma initialEntropyRawBall_sound :
    initialEntropyRawBall.Encloses AllFieldInitialEntropy.nativeH0 := by
  exact Ball.add_sound (Ball.mul_sound totalMarginalBall_sound totalLogBall_sound)
    entropySumBall_sound
def initialEntropyBall : Ball := ⟨(92057368710548199998533979957297 / 50000000000000000000000000000000), (2107346571 / 1562500000000000000000000000000)⟩
theorem nativeH0_enclosed : initialEntropyBall.Encloses AllFieldInitialEntropy.nativeH0 := by
  exact contains_sound initialEntropyRawBall_sound (by decide +kernel)
theorem nativeH0_strict_bounds :
    (184114737421096399996 : ℝ) / 100000000000000000000 < AllFieldInitialEntropy.nativeH0 ∧
      AllFieldInitialEntropy.nativeH0 < (184114737421096399998 : ℝ) / 100000000000000000000 := by
  obtain ⟨hl,hu⟩ := Ball.bounds nativeH0_enclosed
  norm_num [initialEntropyBall, Ball.lower, Ball.upper] at hl hu ⊢
  constructor <;> linarith

end MatrixMultiplication.AllFieldInitialEntropyCertificate

end
end

end MatrixAllFields

namespace MatrixAllFields

open scoped BigOperators Topology Polynomial

section
noncomputable section

namespace MatrixMultiplication.AllFieldVolumeCertificate
open AllFieldParameters AllFieldCertificates AllFieldCertificates.Replay
open AllFieldInitialEntropyCertificate AllFieldTerminalRates
open scoped BigOperators

def zeroLeafShape (k : Fin 9) : Shape := shape 0 k.val (16-k.val)
def zeroLeafCoefficient (k : Fin 9) : ℕ :=
  Contractions.coefficients[shapeMax (zeroLeafShape k)]?.getD 0

lemma zeroLeafShape_mem (k : Fin 9) : zeroLeafShape k ∈ sortedInitial := by
  revert k
  decide +kernel

lemma initialZeroShapes : sortedInitial.filter (fun g => !positive g) =
    List.ofFn zeroLeafShape := by
  decide +kernel

lemma nativeS0_eq_sum : S0 = ∑ k : Fin 9,
    (initialLaw (zeroLeafShape k) : ℝ) * initialRate (zeroLeafShape k) := by
  rw [S0, initialZeroShapes, List.map_ofFn, List.sum_ofFn]
  rfl

def leafLogWitness0 : LogWitness := ⟨0, 1, 1⟩
def leafRateBall0 : Ball := ⟨0, (10587911841 / 50000000000000000000000000000000)⟩
lemma leafRateBall0_sound : leafRateBall0.Encloses
    (initialRate (zeroLeafShape 0)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 0 : ℚ)).Encloses
      (zeroLeafCoefficient 0 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 0 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness0) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness1 : LogWitness := ⟨5, 5, 4⟩
def leafRateBall1 : Ball := ⟨(368887945411393630285245474282633 / 100000000000000000000000000000000), (127054942097 / 100000000000000000000000000000000)⟩
lemma leafRateBall1_sound : leafRateBall1.Encloses
    (initialRate (zeroLeafShape 1)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 1 : ℚ)).Encloses
      (zeroLeafCoefficient 1 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 1 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness1) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness2 : LogWitness := ⟨9, 177, 128⟩
def leafRateBall2 : Ball := ⟨(164061102342342994021143953374117 / 25000000000000000000000000000000), (21175823683 / 10000000000000000000000000000000)⟩
lemma leafRateBall2_sound : leafRateBall2.Encloses
    (initialRate (zeroLeafShape 2)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 2 : ℚ)).Encloses
      (zeroLeafCoefficient 2 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 2 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness2) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness3 : LogWitness := ⟨12, 455, 256⟩
def leafRateBall3 : Ball := ⟨(889288614119073165342852571257363 / 100000000000000000000000000000000), (275285707879 / 100000000000000000000000000000000)⟩
lemma leafRateBall3_sound : leafRateBall3.Encloses
    (initialRate (zeroLeafShape 3)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 3 : ℚ)).Encloses
      (zeroLeafCoefficient 3 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 3 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness3) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness4 : LogWitness := ⟨15, 23989, 16384⟩
def leafRateBall4 : Ball := ⟨(269462446287246686158044077687313 / 25000000000000000000000000000000), (21175823683 / 6250000000000000000000000000000)⟩
lemma leafRateBall4_sound : leafRateBall4.Encloses
    (initialRate (zeroLeafShape 4)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 4 : ℚ)).Encloses
      (zeroLeafCoefficient 4 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 4 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness4) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness5 : LogWitness := ⟨17, 26355, 16384⟩
def leafRateBall5 : Ball := ⟨(9577230336694643113727840367767 / 781250000000000000000000000000), (190582413147 / 50000000000000000000000000000000)⟩
lemma leafRateBall5_sound : leafRateBall5.Encloses
    (initialRate (zeroLeafShape 5)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 5 : ℚ)).Encloses
      (zeroLeafCoefficient 5 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 5 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness5) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness6 : LogWitness := ⟨19, 38941, 32768⟩
def leafRateBall6 : Ball := ⟨(1334239168137038775559946496265297 / 100000000000000000000000000000000), (423516473659 / 100000000000000000000000000000000)⟩
lemma leafRateBall6_sound : leafRateBall6.Encloses
    (initialRate (zeroLeafShape 6)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 6 : ℚ)).Encloses
      (zeroLeafCoefficient 6 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 6 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness6) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness7 : LogWitness := ⟨20, 152675, 131072⟩
def leafRateBall7 : Ball := ⟨(43798463436784910719256790204483 / 3125000000000000000000000000000), (222346148671 / 50000000000000000000000000000000)⟩
lemma leafRateBall7_sound : leafRateBall7.Encloses
    (initialRate (zeroLeafShape 7)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 7 : ℚ)).Encloses
      (zeroLeafCoefficient 7 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 7 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness7) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafLogWitness8 : LogWitness := ⟨20, 1542195, 1048576⟩
def leafRateBall8 : Ball := ⟨(1424871728425409722017918230467949 / 100000000000000000000000000000000), (222346148671 / 50000000000000000000000000000000)⟩
lemma leafRateBall8_sound : leafRateBall8.Encloses
    (initialRate (zeroLeafShape 8)) := by
  have hc : (Ball.exact (zeroLeafCoefficient 8 : ℚ)).Encloses
      (zeroLeafCoefficient 8 : ℝ) := by
    simpa using Ball.exact_sound (zeroLeafCoefficient 8 : ℚ)
  exact checkLog_sound logTwoBall_sound hc (w := leafLogWitness8) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)

def leafRateBall : Fin 9 → Ball := (Matrix.vecCons (leafRateBall0) (Matrix.vecCons (leafRateBall1) (Matrix.vecCons (leafRateBall2) (Matrix.vecCons (leafRateBall3) (Matrix.vecCons (leafRateBall4) (Matrix.vecCons (leafRateBall5) (Matrix.vecCons (leafRateBall6) (Matrix.vecCons (leafRateBall7) (Matrix.vecCons (leafRateBall8) Matrix.vecEmpty)))))))))
lemma leafRateBall_sound (k : Fin 9) :
    (leafRateBall k).Encloses (initialRate (zeroLeafShape k)) := by
  fin_cases k
  · exact leafRateBall0_sound
  · exact leafRateBall1_sound
  · exact leafRateBall2_sound
  · exact leafRateBall3_sound
  · exact leafRateBall4_sound
  · exact leafRateBall5_sound
  · exact leafRateBall6_sound
  · exact leafRateBall7_sound
  · exact leafRateBall8_sound
def initialVolumeRawBall : Ball := finSumBall 9 (fun k =>
  .mul (initialLawBall (zeroLeafShape k)) (leafRateBall k))

lemma initialVolumeRawBall_sound : initialVolumeRawBall.Encloses S0 := by
  rw [nativeS0_eq_sum]
  apply finSumBall_sound
  intro k
  exact Ball.mul_sound (initialLawBall_sound _ (zeroLeafShape_mem k))
    (leafRateBall_sound k)
def initialVolumeBall : Ball := ⟨(3170407140796466915468826036527 / 100000000000000000000000000000000), (249845433 / 25000000000000000000000000000000)⟩
theorem nativeS0_enclosed : initialVolumeBall.Encloses S0 := by
  exact contains_sound initialVolumeRawBall_sound (by decide +kernel)
theorem nativeS0_strict_bounds :
    (3170407140796466915 : ℝ) / 100000000000000000000 < S0 ∧ S0 < (3170407140796466916 : ℝ) / 100000000000000000000 := by
  obtain ⟨hl,hu⟩ := Ball.bounds nativeS0_enclosed
  norm_num [initialVolumeBall, Ball.lower, Ball.upper] at hl hu ⊢
  constructor <;> linarith
end MatrixMultiplication.AllFieldVolumeCertificate

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
open scoped BigOperators

lemma splitWeight_three_enclosed (parents : List Shape) (data : List ℤ)
    (s u : Shape) (n₀ n₁ n₂ : ℤ) (b₀ b₁ b₂ : Ball)
    (hsig : splitSignature parents data s u = (Matrix.vecCons (n₀) (Matrix.vecCons (n₁) (Matrix.vecCons (n₂) Matrix.vecEmpty))))
    (h₀ : b₀.Encloses (weight n₀ : ℝ))
    (h₁ : b₁.Encloses (weight n₁ : ℝ))
    (h₂ : b₂.Encloses (weight n₂ : ℝ)) :
    (Ball.mul b₀ (Ball.mul b₁ b₂)).Encloses (splitWeight parents data s u : ℝ) := by
  convert Ball.mul_sound h₀ (Ball.mul_sound h₁ h₂) using 1
  simp [splitWeight, hsig, Fin.prod_univ_succ]

lemma list_cast_enclosed {A : Type*} (xs : List A) (b : A → Ball) (v : A → ℚ)
    (h : ∀ a ∈ xs, (b a).Encloses (v a : ℝ)) :
    (listSumBall xs b).Encloses ((xs.map v).sum : ℝ) := by
  simpa only [Rat.cast_list_sum, List.map_map, Function.comp_def] using
    listSumBall_sound xs b (fun a => (v a : ℝ)) h

lemma product_fin6_enclosed (bs : Fin 6 → Fin 6 → Ball) (v : PairSlot → ℝ)
    (h : ∀ i j, (bs i j).Encloses (v (i,j))) :
    (finSumBall 6 (fun i => finSumBall 6 (bs i))).Encloses (∑ s : PairSlot, v s) := by
  rw [Fintype.sum_prod_type]
  exact finSumBall_sound 6 _ _ (fun i => finSumBall_sound 6 _ _ (h i))

lemma entropyTerm_ball {p : ℝ} {b l : Ball} (hb : b.Encloses p)
    (hl : l.Encloses (Real.log p)) :
    (Ball.mul (Ball.neg b) l).Encloses (Foundation.entropyTerm p) :=
  Ball.mul_sound (Ball.neg_sound hb) hl

lemma entropyTerm_zero_ball {p : ℝ} (hp : (Ball.exact 0).Encloses p) :
    (Ball.exact 0).Encloses (Foundation.entropyTerm p) := by
  have hb := Ball.bounds hp
  have h : p = 0 := by norm_num [Ball.lower, Ball.upper, Ball.exact] at hb; linarith
  simp only [h, Foundation.entropyTerm, neg_zero, zero_mul]
  simpa only [Rat.cast_zero] using Ball.exact_sound 0

lemma log_exact_one_ball {p : ℝ} (hp : (Ball.exact 1).Encloses p) :
    (Ball.exact 0).Encloses (Real.log p) := by
  have hb := Ball.bounds hp
  have h : p = 1 := by norm_num [Ball.lower, Ball.upper, Ball.exact] at hb; linarith
  rw [h, Real.log_one]
  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

lemma forall_fin6 {P : Fin 6 → Prop} (h0 : P 0) (h1 : P 1) (h2 : P 2)
    (h3 : P 3) (h4 : P 4) (h5 : P 5) : ∀ i, P i := by
  intro i
  fin_cases i <;> assumption

lemma forall_fin3 {P : Fin 3 → Prop} (h0 : P 0) (h1 : P 1) (h2 : P 2) : ∀ i, P i := by
  intro i
  fin_cases i <;> assumption

lemma exact_rat_enclosed {q r : ℚ} (h : q = r) : (Ball.exact q).Encloses (r : ℝ) := by
  rw [h]
  exact Ball.exact_sound r

def pairSumBall (bs : PairSlot → Ball) : Ball :=
  finSumBall 6 (fun i => finSumBall 6 (fun j => bs (i,j)))

lemma pairSumBall_sound (bs : PairSlot → Ball) (v : PairSlot → ℝ)
    (h : ∀ a, (bs a).Encloses (v a)) :
    (pairSumBall bs).Encloses (∑ a, v a) := by
  exact product_fin6_enclosed _ _ (fun i j => h (i,j))

def doublePairSumBall (bs : PairSlot × PairSlot → Ball) : Ball :=
  pairSumBall fun i => pairSumBall fun j => bs (i,j)

lemma doublePairSumBall_sound (bs : PairSlot × PairSlot → Ball)
    (v : PairSlot × PairSlot → ℝ) (h : ∀ a, (bs a).Encloses (v a)) :
    (doublePairSumBall bs).Encloses (∑ a, v a) := by
  rw [Fintype.sum_prod_type]
  exact pairSumBall_sound _ _ (fun i => pairSumBall_sound _ _ (fun j => h (i,j)))

lemma entropyFin_enclosed (n : ℕ) (p : Fin n → ℝ) (bs : Fin n → Ball)
    (h : ∀ i, (bs i).Encloses (Foundation.entropyTerm (p i))) :
    (finSumBall n bs).Encloses (Foundation.finiteEntropy p) :=
  finSumBall_sound n _ _ h

lemma entropyPair_enclosed (p : PairSlot → ℝ) (bs : PairSlot → Ball)
    (h : ∀ i, (bs i).Encloses (Foundation.entropyTerm (p i))) :
    (pairSumBall bs).Encloses (Foundation.finiteEntropy p) :=
  pairSumBall_sound _ _ h

lemma entropyDoublePair_enclosed (p : PairSlot × PairSlot → ℝ)
    (bs : PairSlot × PairSlot → Ball)
    (h : ∀ i, (bs i).Encloses (Foundation.entropyTerm (p i))) :
    (doublePairSumBall bs).Encloses (Foundation.finiteEntropy p) :=
  doublePairSumBall_sound _ _ h

lemma homogeneous_enclosed {A : Type*} [Fintype A] (p : A → ℝ)
    {b e : Ball} (hb : b.Encloses (Foundation.entropyTerm (∑ a, p a)))
    (he : e.Encloses (Foundation.finiteEntropy p)) :
    (Ball.add (Ball.neg b) e).Encloses (homogeneousEntropy p) := by
  convert Ball.add_sound (Ball.neg_sound hb) he using 1
  simp only [homogeneousEntropy, Foundation.entropyTerm, neg_mul, neg_neg]

instance designatedDecidable (priority : Placement) (position : Fin 3) (u : Shape) :
    Decidable (designated priority position u) := by
  unfold designated
  infer_instance

def marginalBall (support : List Shape) (p : Shape → Ball) (side : Fin 3) (k : Fin 17) : Ball :=
  listSumBall support fun u => if u side = k.val then p u else .exact 0

lemma marginalBall_sound (support : List Shape) (bs : Shape → Ball) (p : Shape → ℝ)
    (h : ∀ u ∈ support, (bs u).Encloses (p u)) (side : Fin 3) (k : Fin 17) :
    (marginalBall support bs side k).Encloses (marginal support p side k) := by
  apply listSumBall_sound
  intro u hu
  by_cases hk : u side = k.val
  · simpa only [hk, ite_true] using h u hu
  · simpa only [hk, ite_false, Rat.cast_zero] using Ball.exact_sound 0

def pairLawBall {A : Type*} (support : List Shape) (s : Shape) (p : Shape → Ball)
    (q : Shape → A → Ball) (ij : A × A) : Ball :=
  listSumBall support fun u => .mul (.mul (p u) (q u ij.1)) (q (complement s u) ij.2)

lemma pairLawBall_sound {A : Type*} [Fintype A] (support : List Shape) (s : Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (bq : Shape → A → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (hq : ∀ u a, (bq u a).Encloses (q u a)) (ij : A × A) :
    (pairLawBall support s bs bq ij).Encloses (pairLaw support s p q ij) := by
  apply listSumBall_sound
  intro u hu
  exact Ball.mul_sound (Ball.mul_sound (hp u hu) (hq u ij.1)) (hq _ ij.2)

lemma pairLawBall_sound_on {A : Type*} [Fintype A] (support : List Shape) (s : Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (bq : Shape → A → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (hq : ∀ u ∈ support, ∀ a, (bq u a).Encloses (q u a))
    (hcq : ∀ u ∈ support, ∀ a, (bq (complement s u) a).Encloses (q (complement s u) a))
    (ij : A × A) :
    (pairLawBall support s bs bq ij).Encloses (pairLaw support s p q ij) := by
  apply listSumBall_sound
  intro u hu
  exact Ball.mul_sound (Ball.mul_sound (hp u hu) (hq u hu ij.1)) (hcq u hu ij.2)

def residualBall {A : Type*} (support : List Shape) (p : Shape → Ball)
    (priority : Placement) (position : Fin 3) (q : Shape → A → Ball)
    (k : Fin 17) (a : A) : Ball :=
  listSumBall support fun u =>
    if ¬ designated priority position u ∧ u (priority position) = k.val
    then .mul (p u) (q u a) else .exact 0

lemma residualBall_sound {A : Type*} [Fintype A] (support : List Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (priority : Placement) (position : Fin 3)
    (bq : Shape → A → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (hq : ∀ u a, (bq u a).Encloses (q u a)) (k : Fin 17) (a : A) :
    (residualBall support bs priority position bq k a).Encloses
      (residualLaw support p priority position q k a) := by
  apply listSumBall_sound
  intro u hu
  by_cases hk : ¬ designated priority position u ∧ u (priority position) = k.val
  · simpa only [ite_eq_left hk] using Ball.mul_sound (hp u hu) (hq u a)
  · simpa only [hk, ite_false, Rat.cast_zero] using Ball.exact_sound 0

lemma residualBall_sound_on {A : Type*} [Fintype A] (support : List Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (priority : Placement) (position : Fin 3)
    (bq : Shape → A → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (hq : ∀ u ∈ support, ∀ a, (bq u a).Encloses (q u a)) (k : Fin 17) (a : A) :
    (residualBall support bs priority position bq k a).Encloses
      (residualLaw support p priority position q k a) := by
  apply listSumBall_sound
  intro u hu
  by_cases hk : ¬ designated priority position u ∧ u (priority position) = k.val
  · simpa only [ite_eq_left hk] using Ball.mul_sound (hp u hu) (hq u hu a)
  · simpa only [hk, ite_false, Rat.cast_zero] using Ball.exact_sound 0

def singletonBall (support : List Shape) (p : Shape → Ball)
    (priority : Placement) (position : Fin 3) (e : Shape → Ball) : Ball :=
  listSumBall support fun u =>
    if designated priority position u then .mul (p u) (e u) else .exact 0

lemma singletonBall_sound {A : Type*} [Fintype A] (support : List Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (priority : Placement) (position : Fin 3)
    (be : Shape → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (he : ∀ u ∈ support, designated priority position u →
      (be u).Encloses (homogeneousEntropy (q u))) :
    (singletonBall support bs priority position be).Encloses
      (singletonEntropy support p priority position q) := by
  apply listSumBall_sound
  intro u hu
  by_cases hk : designated priority position u
  · simpa only [hk, ite_true] using Ball.mul_sound (hp u hu) (he u hu hk)
  · simpa only [hk, ite_false, 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 logN1 : Ball := .exact 0
lemma logN1_sound : logN1.Encloses (Real.log ((1 : ℝ))) := by
  exact log_exact_one_ball ((by convert Ball.exact_sound (1 : ℚ) using 1; norm_num))
def logN2Witness : LogWitness := ⟨1, 1, 1⟩
def logN2 : Ball := ⟨(6931471805599453094172319305033 / 10000000000000000000000000000000), (8470329473 / 20000000000000000000000000000000)⟩
lemma logN2_sound : logN2.Encloses (Real.log ((2 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (2 : ℚ) using 1; norm_num)) (w := logN2Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN4Witness : LogWitness := ⟨2, 1, 1⟩
def logN4 : Ball := ⟨(6931471805599453094172319305033 / 5000000000000000000000000000000), (7940933881 / 12500000000000000000000000000000)⟩
lemma logN4_sound : logN4.Encloses (Real.log ((4 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (4 : ℚ) using 1; norm_num)) (w := logN4Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN5Witness : LogWitness := ⟨2, 5, 4⟩
def logN5 : Ball := ⟨(40235947810852509365018973782911 / 25000000000000000000000000000000), (63527471049 / 100000000000000000000000000000000)⟩
lemma logN5_sound : logN5.Encloses (Real.log ((5 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (5 : ℚ) using 1; norm_num)) (w := logN5Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN7Witness : LogWitness := ⟨2, 7, 4⟩
def logN7 : Ball := ⟨(194591014905531330510535236152357 / 100000000000000000000000000000000), (7940933881 / 12500000000000000000000000000000)⟩
lemma logN7_sound : logN7.Encloses (Real.log ((7 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (7 : ℚ) using 1; norm_num)) (w := logN7Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN10Witness : LogWitness := ⟨3, 5, 4⟩
def logN10 : Ball := ⟨(115129254649702284200899544090987 / 50000000000000000000000000000000), (21175823683 / 25000000000000000000000000000000)⟩
lemma logN10_sound : logN10.Encloses (Real.log ((10 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (10 : ℚ) using 1; norm_num)) (w := logN10Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN20Witness : LogWitness := ⟨4, 5, 4⟩
def logN20 : Ball := ⟨(18723326709712443708970142577019 / 6250000000000000000000000000000), (21175823683 / 20000000000000000000000000000000)⟩
lemma logN20_sound : logN20.Encloses (Real.log ((20 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (20 : ℚ) using 1; norm_num)) (w := logN20Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN25Witness : LogWitness := ⟨4, 25, 16⟩
def logN25 : Ball := ⟨(321887582486820074920151790263287 / 100000000000000000000000000000000), (52939559207 / 50000000000000000000000000000000)⟩
lemma logN25_sound : logN25.Encloses (Real.log ((25 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (25 : ℚ) using 1; norm_num)) (w := logN25Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN50Witness : LogWitness := ⟨5, 25, 16⟩
def logN50 : Ball := ⟨(391202300542814605861874983313617 / 100000000000000000000000000000000), (127054942097 / 100000000000000000000000000000000)⟩
lemma logN50_sound : logN50.Encloses (Real.log ((50 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (50 : ℚ) using 1; norm_num)) (w := logN50Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN100Witness : LogWitness := ⟨6, 25, 16⟩
def logN100 : Ball := ⟨(460517018598809136803598176363947 / 100000000000000000000000000000000), (7411538289 / 5000000000000000000000000000000)⟩
lemma logN100_sound : logN100.Encloses (Real.log ((100 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (100 : ℚ) using 1; norm_num)) (w := logN100Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN250Witness : LogWitness := ⟨7, 125, 64⟩
def logN250 : Ball := ⟨(552146091786224643321950874544283 / 100000000000000000000000000000000), (21175823683 / 12500000000000000000000000000000)⟩
lemma logN250_sound : logN250.Encloses (Real.log ((250 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (250 : ℚ) using 1; norm_num)) (w := logN250Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def logN625Witness : LogWitness := ⟨9, 625, 512⟩
def logN625 : Ball := ⟨(321887582486820074920151780715543 / 50000000000000000000000000000000), (211758236829 / 100000000000000000000000000000000)⟩
lemma logN625_sound : logN625.Encloses (Real.log ((625 : ℝ))) := by
  exact checkLog_sound logTwoBall_sound ((by convert Ball.exact_sound (625 : ℚ) using 1; norm_num)) (w := logN625Witness) (by
    simp only [checkLog, Bool.and_eq_true]
    exact ⟨by decide +kernel, by decide +kernel⟩)
def multLogs (a : PairSlot) : Ball :=
  match a.1.val, a.2.val with
  | 0, 0 => logN1
  | 0, 1 => logN10
  | 0, 2 => logN25
  | 0, 3 => logN2
  | 0, 4 => logN10
  | 0, 5 => logN1
  | 1, 0 => logN10
  | 1, 1 => logN100
  | 1, 2 => logN250
  | 1, 3 => logN20
  | 1, 4 => logN100
  | 1, 5 => logN10
  | 2, 0 => logN25
  | 2, 1 => logN250
  | 2, 2 => logN625
  | 2, 3 => logN50
  | 2, 4 => logN250
  | 2, 5 => logN25
  | 3, 0 => logN2
  | 3, 1 => logN20
  | 3, 2 => logN50
  | 3, 3 => logN4
  | 3, 4 => logN20
  | 3, 5 => logN2
  | 4, 0 => logN10
  | 4, 1 => logN100
  | 4, 2 => logN250
  | 4, 3 => logN20
  | 4, 4 => logN100
  | 4, 5 => logN10
  | 5, 0 => logN1
  | 5, 1 => logN10
  | 5, 2 => logN25
  | 5, 3 => logN2
  | 5, 4 => logN10
  | 5, 5 => logN1
  | _, _ => (.exact 0)
lemma multLogs_sound (a : PairSlot) :
    (multLogs a).Encloses (Real.log (AllFieldParameters.multiplicity a.1 * AllFieldParameters.multiplicity a.2 : ℕ)) := by
  rcases a with ⟨i,j⟩
  revert j i
  apply forall_fin6
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 0 = 1 := by decide +kernel
      change (multLogs (0,0)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN1_sound
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 1 = 10 := by decide +kernel
      change (multLogs (0,1)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 2 = 25 := by decide +kernel
      change (multLogs (0,2)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN25_sound
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 3 = 2 := by decide +kernel
      change (multLogs (0,3)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN2_sound
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 4 = 10 := by decide +kernel
      change (multLogs (0,4)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 5 = 1 := by decide +kernel
      change (multLogs (0,5)).Encloses (Real.log (AllFieldParameters.multiplicity 0 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN1_sound
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 0 = 10 := by decide +kernel
      change (multLogs (1,0)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 1 = 100 := by decide +kernel
      change (multLogs (1,1)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN100_sound
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 2 = 250 := by decide +kernel
      change (multLogs (1,2)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN250_sound
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 3 = 20 := by decide +kernel
      change (multLogs (1,3)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN20_sound
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 4 = 100 := by decide +kernel
      change (multLogs (1,4)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN100_sound
    · have he : AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 5 = 10 := by decide +kernel
      change (multLogs (1,5)).Encloses (Real.log (AllFieldParameters.multiplicity 1 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 0 = 25 := by decide +kernel
      change (multLogs (2,0)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN25_sound
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 1 = 250 := by decide +kernel
      change (multLogs (2,1)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN250_sound
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 2 = 625 := by decide +kernel
      change (multLogs (2,2)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN625_sound
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 3 = 50 := by decide +kernel
      change (multLogs (2,3)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN50_sound
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 4 = 250 := by decide +kernel
      change (multLogs (2,4)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN250_sound
    · have he : AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 5 = 25 := by decide +kernel
      change (multLogs (2,5)).Encloses (Real.log (AllFieldParameters.multiplicity 2 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN25_sound
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 0 = 2 := by decide +kernel
      change (multLogs (3,0)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN2_sound
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 1 = 20 := by decide +kernel
      change (multLogs (3,1)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN20_sound
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 2 = 50 := by decide +kernel
      change (multLogs (3,2)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN50_sound
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 3 = 4 := by decide +kernel
      change (multLogs (3,3)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN4_sound
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 4 = 20 := by decide +kernel
      change (multLogs (3,4)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN20_sound
    · have he : AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 5 = 2 := by decide +kernel
      change (multLogs (3,5)).Encloses (Real.log (AllFieldParameters.multiplicity 3 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN2_sound
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 0 = 10 := by decide +kernel
      change (multLogs (4,0)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 1 = 100 := by decide +kernel
      change (multLogs (4,1)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN100_sound
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 2 = 250 := by decide +kernel
      change (multLogs (4,2)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN250_sound
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 3 = 20 := by decide +kernel
      change (multLogs (4,3)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN20_sound
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 4 = 100 := by decide +kernel
      change (multLogs (4,4)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN100_sound
    · have he : AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 5 = 10 := by decide +kernel
      change (multLogs (4,5)).Encloses (Real.log (AllFieldParameters.multiplicity 4 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
  · apply forall_fin6
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 0 = 1 := by decide +kernel
      change (multLogs (5,0)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 0 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN1_sound
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 1 = 10 := by decide +kernel
      change (multLogs (5,1)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 1 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 2 = 25 := by decide +kernel
      change (multLogs (5,2)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 2 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN25_sound
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 3 = 2 := by decide +kernel
      change (multLogs (5,3)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 3 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN2_sound
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 4 = 10 := by decide +kernel
      change (multLogs (5,4)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 4 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN10_sound
    · have he : AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 5 = 1 := by decide +kernel
      change (multLogs (5,5)).Encloses (Real.log (AllFieldParameters.multiplicity 5 * AllFieldParameters.multiplicity 5 : ℕ))
      rw [he]
      simpa only [multLogs, Fin.coe_ofNat_eq_mod, Nat.reduceMod, Nat.cast_ofNat, Nat.cast_one] using logN1_sound

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 sparseTripleBall (a b c : Ball) : Ball :=
  if b = .exact 0 ∨ c = .exact 0 then .exact 0 else .mul (.mul a b) c

lemma sparseTripleBall_eq (a b c : Ball) :
    sparseTripleBall a b c = .mul (.mul a b) c := by
  unfold sparseTripleBall
  split_ifs with h
  · rcases h with h | h
    · subst b
      cases a
      cases c
      simp [Ball.mul, Ball.exact]
    · subst c
      cases a
      cases b
      simp [Ball.mul, Ball.exact]
  · rfl

def sparsePairLawBall {A : Type*} (support : List Shape) (s : Shape) (p : Shape → Ball)
    (q : Shape → A → Ball) (ij : A × A) : Ball :=
  listSumBall support fun u => sparseTripleBall (p u) (q u ij.1) (q (complement s u) ij.2)

lemma sparsePairLawBall_eq {A : Type*} (support : List Shape) (s : Shape)
    (p : Shape → Ball) (q : Shape → A → Ball) (ij : A × A) :
    sparsePairLawBall support s p q ij = pairLawBall support s p q ij := by
  simp only [sparsePairLawBall, sparseTripleBall_eq, pairLawBall]

lemma sparsePairLawBall_sound_on {A : Type*} [Fintype A] (support : List Shape) (s : Shape)
    (bs : Shape → Ball) (p : Shape → ℝ) (bq : Shape → A → Ball) (q : Shape → A → ℝ)
    (hp : ∀ u ∈ support, (bs u).Encloses (p u))
    (hq : ∀ u ∈ support, ∀ a, (bq u a).Encloses (q u a))
    (hcq : ∀ u ∈ support, ∀ a, (bq (complement s u) a).Encloses (q (complement s u) a))
    (ij : A × A) :
    (sparsePairLawBall support s bs bq ij).Encloses (pairLaw support s p q ij) := by
  rw [sparsePairLawBall_eq]
  exact pairLawBall_sound_on support s bs p bq q hp hq hcq ij

end MatrixMultiplication.NativeNumerics

end
end

end MatrixAllFields

end OAI