/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing
import AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals
import Mathlib.Tactic

noncomputable section
open MeasureTheory
open scoped ENNReal
open AutoSamplingTheory.TechnicalLemmas.Measure
open AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore
open AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals
variable {E : Type*} [NormedAddCommGroup E]
  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
  [StandardBorelSpace E] [Nonempty E]

example
    (μ₁ μ₂ μ₃ : Measure E)
    [IsProbabilityMeasure μ₁] [IsProbabilityMeasure μ₂] [IsProbabilityMeasure μ₃]
    {r₁₂ r₂₃ : ℝ≥0∞}
    (hr₁₂ : WassersteinSpace.wassersteinDistance μ₁ μ₂ < r₁₂)
    (hr₂₃ : WassersteinSpace.wassersteinDistance μ₂ μ₃ < r₂₃) :
    WassersteinSpace.wassersteinDistance μ₁ μ₃ < r₁₂ + r₂₃ := by
  -- Replace this comment with your proof.