{
  "graph": {
    "direction": "theorem-to-selected-constant-in-compiled-proof-body",
    "edges": [
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
        "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ],
      [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ]
    ],
    "kind": "compiled-proof-body-constant-occurrences",
    "node_labels": {
      "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left": "Joint probability mass",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling": "Shared-middle gluing",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure": "First pair marginal",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure": "Second pair marginal",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt": "Near-optimal plans",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling": "Coupling bounds infimum",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt": "Strict transport join",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le": "L2 Minkowski",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono": "L2 monotonicity",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle": "Joint-law L2 triangle",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23": "Endpoint coupling",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost": "Pair (1,2) cost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost": "Endpoint cost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost": "Pair (2,3) cost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost": "Cost measurability"
    },
    "nodes": [
      "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure",
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost",
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
    ],
    "scope": "induced public-theorem slice; unselected constants, private auxiliaries and Mathlib parents are omitted",
    "source_sha256_lf": {
      "AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean": "d3eff4916a04ab9473ee86b839fa13b8447fa1c4d162874a4ceddbce4003e414",
      "AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean": "df336ce4f6047441b52d075272f31b6e7d3f56e6bbeb1c8c9eb6a6e54e3cc062",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean": "943f06c49f3e8e182ab731f9c8e4e5372b65b2eae81001c781a2f6bbc69998b5",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangle.lean": "70734431850472a3a175858173dfbff49e469fb23384a5e927bbd718456193e0",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean": "182ab3f1827a46647810263ece9f118e0bc2f84f26c66873b463a90e62a33084",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean": "b899bd0ea5739e82e453a8e0300f701db85ec623cd12f09ef864e69bdb2e905b"
    },
    "statement_constant_whitelist": [
      "CompleteSpace",
      "Differentiable",
      "HasDerivAt",
      "HasDerivWithinAt",
      "HasGradientAt"
    ],
    "statement_profile_scope": "Direct constant occurrences in the unexpanded elaborated statement; not a binder-role analysis, inferred hypothesis list or full type dependency closure.",
    "statement_profiles": {
      "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost": [],
      "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost": []
    },
    "truth_boundary": "An edge records constant occurrence in this compiled proof body, including proof-term annotations. Statement profiles record only whitelisted direct type-constant occurrences, not logical requirements. Neither certifies mathematical necessity or a full dependency closure. No path in an induced slice does not imply no path in the full closure."
  },
  "graph_sha256": "f665eb72773d5a237ec140eb14578e4f24f236cce35210c0a917ff21a4fe3a8d",
  "name": "SamplingOptimisationGeometryBench compiled proof-body snapshot",
  "provenance": {
    "context_sha256": "52014c740e6c7ba03a84e97de3eb322233e19220d92074ad394f989c62e9cd6d",
    "core_binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2",
    "core_id": "sogb-018",
    "generator_sha256_lf": "2f570810651543df28853b31847a0ba5b676675231febca78269a67cb13a9229",
    "lakefile_sha256_lf": "1757b85b37f0a68972791443e3a8da33fd97ede414732fabb8e7d747b070c51c",
    "local_import_closure_sha256_lf": {
      "AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean": "d3eff4916a04ab9473ee86b839fa13b8447fa1c4d162874a4ceddbce4003e414",
      "AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean": "df336ce4f6047441b52d075272f31b6e7d3f56e6bbeb1c8c9eb6a6e54e3cc062",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean": "943f06c49f3e8e182ab731f9c8e4e5372b65b2eae81001c781a2f6bbc69998b5",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangle.lean": "70734431850472a3a175858173dfbff49e469fb23384a5e927bbd718456193e0",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean": "182ab3f1827a46647810263ece9f118e0bc2f84f26c66873b463a90e62a33084",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean": "b899bd0ea5739e82e453a8e0300f701db85ec623cd12f09ef864e69bdb2e905b"
    },
    "mathlib_manifest_sha256_lf": "b83ca83b9cf7caa85fa8b023c2ff9a3fb7a7cce7e1c332e810c348535ec49c37",
    "selection_sha256": "be6d4d3982a77f7a479a358458cda8004ef7b42669ce7506fafe00ff14f7d659",
    "source_sha256_lf": {
      "AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean": "d3eff4916a04ab9473ee86b839fa13b8447fa1c4d162874a4ceddbce4003e414",
      "AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean": "df336ce4f6047441b52d075272f31b6e7d3f56e6bbeb1c8c9eb6a6e54e3cc062",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean": "943f06c49f3e8e182ab731f9c8e4e5372b65b2eae81001c781a2f6bbc69998b5",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangle.lean": "70734431850472a3a175858173dfbff49e469fb23384a5e927bbd718456193e0",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean": "182ab3f1827a46647810263ece9f118e0bc2f84f26c66873b463a90e62a33084",
      "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean": "b899bd0ea5739e82e453a8e0300f701db85ec623cd12f09ef864e69bdb2e905b"
    },
    "toolchain": "leanprover/lean4:v4.33.0"
  },
  "rows": [
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure",
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le",
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [
        "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost"
      ],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost",
      "statement_constants": []
    },
    {
      "axioms": [
        "Classical.choice",
        "Quot.sound",
        "propext"
      ],
      "body_parents": [],
      "id": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost",
      "statement_constants": []
    }
  ],
  "schema_version": 1
}
