lean_problems/Y2025.lean

/-
Copyright (c) 2026 Humanize/Codex. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Humanize/Codex
-/
import Mathlib

/-!
Algebraic answer claims for the IPhO 2025 problems. The declarations encode the
official closed-form answers and numerical answer intervals from the source rows.
-/

namespace Ipho20212023Archon.Problems.Y2025

noncomputable section

abbrev RangeAnswer := ℝ × ℝ

inductive Choice
  | A
  | B
  | C
  | D
  deriving DecidableEq, Repr

structure ExperimentOutcome where
  behavior : Choice
  hstar : Option RangeAnswer
  fmax : RangeAnswer

def sq (x : ℝ) : ℝ := x ^ 2

def cube (x : ℝ) : ℝ := x ^ 3

def fourth (x : ℝ) : ℝ := x ^ 4

/- Problem 1, Hydrogen and galaxies. -/

section Problem1

def bohrCircularVelocity (e eps0 me r : ℝ) : ℝ :=
  Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r))

theorem IPhO_2025_1_A_1 (e eps0 me r : ℝ) :
    bohrCircularVelocity e eps0 me r =
      Real.sqrt (sq e / (4 * Real.pi * eps0 * me * r)) := by
  sorry

def bohrOrbitRadius (hbar n alpha me c : ℝ) : ℝ :=
  sq n * hbar / (alpha * me * c)

def bohrRadius (hbar alpha me c : ℝ) : ℝ :=
  hbar / (alpha * me * c)

def bohrRadiusNumerical : ℝ := (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ)

def bohrGroundVelocity (alpha c : ℝ) : ℝ := alpha * c

theorem IPhO_2025_1_A_2 (hbar n alpha me c : ℝ) :
    (bohrOrbitRadius hbar n alpha me c, bohrRadius hbar alpha me c,
        bohrRadiusNumerical, bohrGroundVelocity alpha c) =
      (sq n * hbar / (alpha * me * c), hbar / (alpha * me * c),
        (531 : ℝ) / 100 * (10 : ℝ) ^ (-11 : ℤ), alpha * c) := by
  sorry

def bohrOrbitEnergy (e eps0 n r1 : ℝ) : ℝ :=
  -sq e / (8 * Real.pi * eps0 * sq n * r1)

def bohrGroundEnergy (alpha me c : ℝ) : ℝ :=
  -((1 : ℝ) / 2) * sq alpha * me * sq c

def bohrGroundEnergyEV : ℝ := -((136 : ℝ) / 10)

theorem IPhO_2025_1_A_3 (e eps0 n r1 alpha me c : ℝ) :
    (bohrOrbitEnergy e eps0 n r1, bohrGroundEnergy alpha me c, bohrGroundEnergyEV) =
      (-sq e / (8 * Real.pi * eps0 * sq n * r1),
        -((1 : ℝ) / 2) * sq alpha * me * sq c, -((136 : ℝ) / 10)) := by
  sorry

def orbitMagneticField (mu0 e alpha c r1 : ℝ) : ℝ :=
  mu0 * e * alpha * c / (4 * Real.pi * sq r1)

theorem IPhO_2025_1_A_4 (mu0 e alpha c r1 : ℝ) :
    orbitMagneticField mu0 e alpha c r1 =
      mu0 * e * alpha * c / (4 * Real.pi * sq r1) := by
  sorry

def fineStructureEnergyGap (alpha E1 : ℝ) : ℝ := -4 * sq alpha * E1

def hyperfineWavelength (h c me mp alpha E1 : ℝ) : ℝ :=
  -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1)

def hyperfineWavelengthCm : ℝ := 21

theorem IPhO_2025_1_A_5 (h c me mp alpha E1 : ℝ) :
    (fineStructureEnergyGap alpha E1, hyperfineWavelength h c me mp alpha E1,
        hyperfineWavelengthCm) =
      (-4 * sq alpha * E1,
        -(h * c) / (((372 : ℝ) / 100) * (me / mp) * 4 * sq alpha * E1), 21) := by
  sorry

def circularOrbitSpeedFromPotential (r dphi_dr : ℝ) : ℝ :=
  Real.sqrt (r * dphi_dr)

theorem IPhO_2025_1_B_1 (r dphi_dr : ℝ) :
    circularOrbitSpeedFromPotential r dphi_dr = Real.sqrt (r * dphi_dr) := by
  sorry

def ngc6946BulgeMassRange : RangeAnswer :=
  ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8)

theorem IPhO_2025_1_B_2 :
    ngc6946BulgeMassRange =
      ((675 : ℝ) / 100 * (10 : ℝ) ^ 8, (1125 : ℝ) / 100 * (10 : ℝ) ^ 8) := by
  sorry

def haloVelocityK1 (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G

def haloVelocityK2 (Cm G rm : ℝ) : ℝ := 4 * Real.pi * Cm * G * rm

def haloInteriorMass (Cm r rm : ℝ) : ℝ :=
  4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm))

def haloGravityMagnitude (Cm r rm : ℝ) : ℝ :=
  4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r

def haloVelocityProfile (Cm G r rm : ℝ) : ℝ :=
  Real.sqrt (haloVelocityK1 Cm G - haloVelocityK2 Cm G rm * Real.arctan (r / rm) / r)

def haloVelocitySmallRadius (Cm G r rm : ℝ) : ℝ :=
  Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm))

def haloVelocityLargeRadius (Cm G : ℝ) : ℝ :=
  Real.sqrt (4 * Real.pi * Cm * G)

def haloMassLargeRadius (Cm r : ℝ) : ℝ := 4 * Real.pi * Cm * r

def ngc6946VisibleMassRange : RangeAnswer :=
  (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2))

theorem IPhO_2025_1_B_3 (Cm G r rm : ℝ) :
    (haloVelocityK1 Cm G, haloVelocityK2 Cm G rm, haloInteriorMass Cm r rm,
        haloGravityMagnitude Cm r rm, haloVelocityProfile Cm G r rm,
        haloVelocitySmallRadius Cm G r rm, haloVelocityLargeRadius Cm G,
        haloMassLargeRadius Cm r, ngc6946VisibleMassRange) =
      (4 * Real.pi * Cm * G, 4 * Real.pi * Cm * G * rm,
        4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)),
        4 * Real.pi * Cm * (r - rm * Real.arctan (r / rm)) / sq r,
        Real.sqrt (4 * Real.pi * Cm * G - 4 * Real.pi * Cm * G * rm *
          Real.arctan (r / rm) / r),
        Real.sqrt (4 * Real.pi * Cm * G * sq r / (3 * sq rm)),
        Real.sqrt (4 * Real.pi * Cm * G), 4 * Real.pi * Cm * r,
        (Real.rpow 10 ((21 : ℝ) / 2), Real.rpow 10 ((23 : ℝ) / 2))) := by
  sorry

def verticalGalacticAcceleration (z z0 phi0 r r0 : ℝ) : ℝ :=
  2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0)))

def verticalSmallOscillationFrequency (phi0 z0 r r0 : ℝ) : ℝ :=
  Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|)

theorem IPhO_2025_1_C_1 (z z0 phi0 r r0 : ℝ) :
    (verticalGalacticAcceleration z z0 phi0 r r0,
        verticalSmallOscillationFrequency phi0 z0 r r0) =
      (2 * z / sq z0 * phi0 * Real.log (r / r0) * Real.exp (-(sq (z / z0))),
        Real.sqrt (2 * phi0 / sq z0 * |Real.log (r / r0)|)) := by
  sorry

inductive RadiusRegime
  | muchSmaller
  | muchLarger
  deriving DecidableEq, Repr

def flatRotationRegime : RadiusRegime := RadiusRegime.muchLarger

def flatPotentialScale (Cm G : ℝ) : ℝ := 4 * Real.pi * Cm * G

def flatCircularSpeed (phi0 : ℝ) : ℝ := Real.sqrt phi0

theorem IPhO_2025_1_C_2 (Cm G phi0 : ℝ) :
    (flatRotationRegime, flatPotentialScale Cm G, flatCircularSpeed phi0) =
      (RadiusRegime.muchLarger, 4 * Real.pi * Cm * G, Real.sqrt phi0) := by
  sorry

def radialVelocityEarthSource (vSun RSun R ell : ℝ) : ℝ :=
  vSun * (RSun / R - 1) * Real.sin ell

def sourceGalactocentricRadius (RSun vSun ell vr : ℝ) : ℝ :=
  RSun / (1 + vr / (vSun * Real.sin ell))

theorem IPhO_2025_1_C_3 (vSun RSun R ell vr : ℝ) :
    (radialVelocityEarthSource vSun RSun R ell, sourceGalactocentricRadius RSun vSun ell vr) =
      (vSun * (RSun / R - 1) * Real.sin ell,
        RSun / (1 + vr / (vSun * Real.sin ell))) := by
  sorry

def sourceVelocityRange1 : RangeAnswer := ((632 : ℝ) / 100, (634 : ℝ) / 100)

def sourceVelocityRange2 : RangeAnswer := ((316 : ℝ) / 10, (318 : ℝ) / 10)

def sourceVelocityRange3 : RangeAnswer := ((548 : ℝ) / 10, 55)

def sourceRadiusRange1 : RangeAnswer := ((94 : ℝ) / 100, (96 : ℝ) / 100)

def sourceRadiusRange2 : RangeAnswer := ((77 : ℝ) / 100, (79 : ℝ) / 100)

def sourceRadiusRange3 : RangeAnswer := ((66 : ℝ) / 100, (68 : ℝ) / 100)

theorem IPhO_2025_1_C_4 :
    (sourceVelocityRange1, sourceVelocityRange2, sourceVelocityRange3,
        sourceRadiusRange1, sourceRadiusRange2, sourceRadiusRange3) =
      (((632 : ℝ) / 100, (634 : ℝ) / 100), ((316 : ℝ) / 10, (318 : ℝ) / 10),
        ((548 : ℝ) / 10, 55), ((94 : ℝ) / 100, (96 : ℝ) / 100),
        ((77 : ℝ) / 100, (79 : ℝ) / 100), ((66 : ℝ) / 100, (68 : ℝ) / 100)) := by
  sorry

def newtonTullyFischerEta (R G : ℝ) : ℝ := R / G

def newtonTullyFischerGamma : ℝ := 2

def observedTullyFischerGammaRange : RangeAnswer := ((35 : ℝ) / 10, 4)

theorem IPhO_2025_1_D_1 (R G : ℝ) :
    (newtonTullyFischerEta R G, newtonTullyFischerGamma,
        observedTullyFischerGammaRange) =
      (R / G, 2, ((35 : ℝ) / 10, 4)) := by
  sorry

def outerGalaxyAcceleration (vc R : ℝ) : ℝ := sq vc / R

def outerGalaxyAccelerationRange : RangeAnswer :=
  (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))

theorem IPhO_2025_1_D_2 (vc R : ℝ) :
    (outerGalaxyAcceleration vc R, outerGalaxyAccelerationRange) =
      (sq vc / R, (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by
  sorry

def mondTullyFischerGamma : ℝ := 4

def mondAccelerationScale (vcInf G Mtot : ℝ) : ℝ := fourth vcInf / (G * Mtot)

def mondAccelerationScaleRange : RangeAnswer :=
  (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))

theorem IPhO_2025_1_D_3 (vcInf G Mtot : ℝ) :
    (mondTullyFischerGamma, mondAccelerationScale vcInf G Mtot, mondAccelerationScaleRange) =
      (4, fourth vcInf / (G * Mtot),
        (Real.rpow 10 (-((21 : ℝ) / 2)), Real.rpow 10 (-((19 : ℝ) / 2)))) := by
  sorry

def mondVelocityExterior (G M a0 r : ℝ) : ℝ :=
  Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M))))

def mondVelocityInterior (G M a0 r Rb : ℝ) : ℝ :=
  Real.sqrt
    (G * M / (2 * r) * cube (r / Rb) *
      (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r))))

theorem IPhO_2025_1_D_4 (G M a0 r Rb : ℝ) :
    (mondVelocityExterior G M a0 r, mondVelocityInterior G M a0 r Rb) =
      (Real.sqrt (G * M / (2 * r) * (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M)))),
        Real.sqrt
          (G * M / (2 * r) * cube (r / Rb) *
            (1 + Real.sqrt (1 + 4 * a0 * sq r / (G * M) * cube (Rb / r))))) := by
  sorry

end Problem1

/- Problem 2, The water column. -/

section Problem2

def waterPressureAtTubeTop (P0 rho g h : ℝ) : ℝ := P0 - rho * g * h

def tubeHoldingForceScalar (m rho S h g : ℝ) : ℝ := (m + rho * S * h) * g

theorem IPhO_2025_2_A_1 (P0 rho m S h g : ℝ) :
    (waterPressureAtTubeTop P0 rho g h, tubeHoldingForceScalar m rho S h g) =
      (P0 - rho * g * h, (m + rho * S * h) * g) := by
  sorry

def tubeExperimentOutcomes : List ExperimentOutcome :=
  [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) },
    { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) },
    { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10),
      fmax := (5, (52 : ℝ) / 10) } ]

theorem IPhO_2025_2_A_2 :
    tubeExperimentOutcomes =
      [ { behavior := Choice.A, hstar := none, fmax := ((146 : ℝ) / 10, 15) },
        { behavior := Choice.A, hstar := none, fmax := (14, (145 : ℝ) / 10) },
        { behavior := Choice.B, hstar := some (2, (22 : ℝ) / 10),
          fmax := (5, (52 : ℝ) / 10) } ] := by
  sorry

def vaporPressureRelativeError (Psat P0 m g S : ℝ) : ℝ :=
  Psat / (P0 + m * g / S)

def vaporPressureRelativeErrorRange : RangeAnswer :=
  ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ))

theorem IPhO_2025_2_A_3 (Psat P0 m g S : ℝ) :
    (vaporPressureRelativeError Psat P0 m g S, vaporPressureRelativeErrorRange) =
      (Psat / (P0 + m * g / S), ((10 : ℝ) ^ (-6 : ℤ), 2 * (10 : ℝ) ^ (-6 : ℤ))) := by
  sorry

def addedMercuryMass (rho St ht Sb zl : ℝ) : ℝ :=
  rho * (St * ht + Sb * (zl - ht))

theorem IPhO_2025_2_B_1 (rho St ht Sb zl : ℝ) :
    addedMercuryMass rho St ht Sb zl = rho * (St * ht + Sb * (zl - ht)) := by
  sorry

def addedMassSegmentSlopes (rho Sb St : ℝ) : ℝ × ℝ × ℝ × ℝ :=
  (rho * Sb, rho * St, -rho * (Sb - St), 0)

def addedMassAngularPoints (zStar Hb : ℝ) : ℝ × ℝ × ℝ :=
  (0, zStar - Hb, zStar)

theorem IPhO_2025_2_B_2 (rho Sb St zStar Hb : ℝ) :
    (addedMassSegmentSlopes rho Sb St, addedMassAngularPoints zStar Hb) =
      ((rho * Sb, rho * St, -rho * (Sb - St), 0), (0, zStar - Hb, zStar)) := by
  sorry

def addedMassAmplitude (Sb A g : ℝ) : ℝ := Sb * A / g

def addedMassAmplitudeRange : RangeAnswer := (1, (11 : ℝ) / 10)

theorem IPhO_2025_2_B_3 (Sb A g : ℝ) :
    (addedMassAmplitude Sb A g, addedMassAmplitudeRange) =
      (Sb * A / g, (1, (11 : ℝ) / 10)) := by
  sorry

def restThresholdXi : ℝ := 2

theorem IPhO_2025_2_C_1 :
    restThresholdXi = 2 := by
  sorry

def cableTensionScalar (Sc Sb St rho g X : ℝ) : ℝ :=
  4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X

def cableTensionApproxScalar (Sc Sb rho g X : ℝ) : ℝ :=
  4 * Sb * Sc / (Sb + Sc) * rho * g * X

theorem IPhO_2025_2_C_2 (Sc Sb St rho g X : ℝ) :
    (cableTensionScalar Sc Sb St rho g X, cableTensionApproxScalar Sc Sb rho g X) =
      (4 * Sc * (Sb - St) / (Sb + Sc - St) * rho * g * X,
        4 * Sb * Sc / (Sb + Sc) * rho * g * X) := by
  sorry

def firstRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda > 2

def secondRegimeCondition (xi lambda : ℝ) : Prop := xi + 2 * lambda < 2

theorem IPhO_2025_2_C_3 (xi lambda : ℝ) :
    (firstRegimeCondition xi lambda ↔ xi + 2 * lambda > 2) ∧
      (secondRegimeCondition xi lambda ↔ xi + 2 * lambda < 2) := by
  sorry

def springForceLimit (A Sc : ℝ) : ℝ := A * Sc / 2

def displacementLimit (A rho g : ℝ) : ℝ := A / (4 * rho * g)

def maximumRecoverableWork (A Sc rho g : ℝ) : ℝ := sq A * Sc / (2 * rho * g)

def maximumRecoverableWorkRange : RangeAnswer := (19, 21)

theorem IPhO_2025_2_C_4 (A Sc rho g : ℝ) :
    (springForceLimit A Sc, displacementLimit A rho g, maximumRecoverableWork A Sc rho g,
        maximumRecoverableWorkRange) =
      (A * Sc / 2, A / (4 * rho * g), sq A * Sc / (2 * rho * g), (19, 21)) := by
  sorry

def practicalMaximumWork (Sc A rho g : ℝ) : ℝ := Sc * sq A / (rho * g)

def recoverableToPracticalWorkRatio : ℝ := (1 : ℝ) / 2

theorem IPhO_2025_2_C_5 (Sc A rho g : ℝ) :
    (practicalMaximumWork Sc A rho g, recoverableToPracticalWorkRatio) =
      (Sc * sq A / (rho * g), (1 : ℝ) / 2) := by
  sorry

end Problem2

/- Problem 3, Champagne. -/

section Problem3

def bubblePressure (P0 sigma a : ℝ) : ℝ := P0 + 2 * sigma / a

theorem IPhO_2025_3_A_1 (P0 sigma a : ℝ) :
    bubblePressure P0 sigma a = P0 + 2 * sigma / a := by
  sorry

def criticalBubbleRadius (sigma P0 cl c0 : ℝ) : ℝ :=
  2 * sigma / (P0 * (cl / c0 - 1))

def criticalBubbleRadiusMicrometer : ℝ := (3 : ℝ) / 10

theorem IPhO_2025_3_A_2 (sigma P0 cl c0 : ℝ) :
    (criticalBubbleRadius sigma P0 cl c0, criticalBubbleRadiusMicrometer) =
      (2 * sigma / (P0 * (cl / c0 - 1)), (3 : ℝ) / 10) := by
  sorry

def bubbleMoles (a P0 R T0 : ℝ) : ℝ :=
  (4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0)

def diffusionModelRadiusSquared (a0 D R T0 P0 cl c0 t : ℝ) : ℝ :=
  sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t

def transferModelRadius (a0 K R T0 P0 cl c0 t : ℝ) : ℝ :=
  a0 + (K * R * T0 / P0) * (cl - c0) * t

def selectedBubbleGrowthModel : Choice := Choice.B

def bubbleGrowthKRange : RangeAnswer :=
  ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ), (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ))

theorem IPhO_2025_3_A_3 (a P0 R T0 a0 D K cl c0 t : ℝ) :
    (bubbleMoles a P0 R T0, diffusionModelRadiusSquared a0 D R T0 P0 cl c0 t,
        transferModelRadius a0 K R T0 P0 cl c0 t, selectedBubbleGrowthModel,
        bubbleGrowthKRange) =
      ((4 : ℝ) / 3 * Real.pi * cube a * P0 / (R * T0),
        sq a0 + (2 * D * R * T0 / P0) * (cl - c0) * t,
        a0 + (K * R * T0 / P0) * (cl - c0) * t, Choice.B,
        ((9 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ),
          (11 : ℝ) / 10 * (10 : ℝ) ^ (-4 : ℤ))) := by
  sorry

def bubbleBuoyancyForce (a rhoL g0 : ℝ) : ℝ := (4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0

def bubbleStokesDrag (eta a v : ℝ) : ℝ := 6 * Real.pi * eta * a * v

def bubbleTerminalVelocity (eta a rhoL g0 : ℝ) : ℝ :=
  2 / (9 * eta) * sq a * rhoL * g0

def champagneViscosityRange : RangeAnswer :=
  ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ))

theorem IPhO_2025_3_A_4 (a rhoL g0 eta v : ℝ) :
    (bubbleBuoyancyForce a rhoL g0, bubbleStokesDrag eta a v,
        bubbleTerminalVelocity eta a rhoL g0, champagneViscosityRange) =
      ((4 : ℝ) / 3 * Real.pi * cube a * rhoL * g0, 6 * Real.pi * eta * a * v,
        2 / (9 * eta) * sq a * rhoL * g0,
        ((10 : ℝ) ^ (-3 : ℤ), 4 * (10 : ℝ) ^ (-3 : ℤ))) := by
  sorry

def freeSurfaceBubbleRadius (qa eta H rhoL g0 : ℝ) : ℝ :=
  Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3)

def freeSurfaceBubbleRadiusMmRange : RangeAnswer := ((36 : ℝ) / 100, (49 : ℝ) / 100)

theorem IPhO_2025_3_A_5 (qa eta H rhoL g0 : ℝ) :
    (freeSurfaceBubbleRadius qa eta H rhoL g0, freeSurfaceBubbleRadiusMmRange) =
      (Real.rpow (27 * qa * eta * H / (2 * rhoL * g0)) ((1 : ℝ) / 3),
        ((36 : ℝ) / 100, (49 : ℝ) / 100)) := by
  sorry

def concentrationDecayCoefficient (Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ :=
  18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl)

def concentrationDecayResidual (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : ℝ :=
  dc_dt + concentrationDecayCoefficient Nb fb eta K H rhoL g0 Vl * (cL - c0)

def concentrationDecayEquation (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) : Prop :=
  concentrationDecayResidual dc_dt cL c0 Nb fb eta K H rhoL g0 Vl = 0

def concentrationDecayTime (rhoL g0 Vl Nb fb eta K H : ℝ) : ℝ :=
  rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H)

theorem IPhO_2025_3_A_6 (dc_dt cL c0 Nb fb eta K H rhoL g0 Vl : ℝ) :
    (concentrationDecayEquation dc_dt cL c0 Nb fb eta K H rhoL g0 Vl ↔
        dc_dt + 18 * Real.pi * Nb * fb * eta * K * H / (rhoL * g0 * Vl) *
          (cL - c0) = 0) ∧
      concentrationDecayTime rhoL g0 Vl Nb fb eta K H =
        rhoL * g0 * Vl / (18 * Real.pi * Nb * fb * eta * K * H) := by
  sorry

def filmRetractionSpeed (sigma rhoL h : ℝ) : ℝ :=
  Real.sqrt (2 * sigma / (rhoL * h))

theorem IPhO_2025_3_B_1 (sigma rhoL h : ℝ) :
    filmRetractionSpeed sigma rhoL h = Real.sqrt (2 * sigma / (rhoL * h)) := by
  sorry

def pistonOscillationFrequency (S P0 gamma mp V0 : ℝ) : ℝ :=
  1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0))

theorem IPhO_2025_3_B_2 (S P0 gamma mp V0 : ℝ) :
    pistonOscillationFrequency S P0 gamma mp V0 =
      1 / (2 * Real.pi) * Real.sqrt (sq S * P0 * gamma / (mp * V0)) := by
  sorry

def champagneFilmRadiusMmRange : RangeAnswer := ((5 : ℝ) / 10, (6 : ℝ) / 10)

def champagneFilmThicknessMicrometer : ℝ := (37 : ℝ) / 10

theorem IPhO_2025_3_B_3 :
    (champagneFilmRadiusMmRange, champagneFilmThicknessMicrometer) =
      (((5 : ℝ) / 10, (6 : ℝ) / 10), (37 : ℝ) / 10) := by
  sorry

def initialBottlePressure6CBar : ℝ := (481 : ℝ) / 100

def initialBottlePressure20CBar : ℝ := (776 : ℝ) / 100

theorem IPhO_2025_3_C_1 :
    (initialBottlePressure6CBar, initialBottlePressure20CBar) =
      ((481 : ℝ) / 100, (776 : ℝ) / 100) := by
  sorry

def finalGasTemperature6C : ℝ := (1953 : ℝ) / 10

def finalGasTemperature20C : ℝ := (1843 : ℝ) / 10

def trueFogStatements : List Choice := [Choice.A, Choice.D]

theorem IPhO_2025_3_C_2 :
    (finalGasTemperature6C, finalGasTemperature20C, trueFogStatements) =
      ((1953 : ℝ) / 10, (1843 : ℝ) / 10, [Choice.A, Choice.D]) := by
  sorry

def corkRiseHeight6C : ℝ := (77 : ℝ) / 10

theorem IPhO_2025_3_C_3 :
    corkRiseHeight6C = (77 : ℝ) / 10 := by
  sorry

end Problem3

end

end Ipho20212023Archon.Problems.Y2025