{ "schema": "an04-reading-proof-index/v1", "scope": "Full receiving stationary-phase lesson and its exact earlier prerequisite proofs, relative to the declared axioms. Dependency contracts are named combinations of the linked complete proofs.", "topological_reading_order": [ "A0", "P6.0", "P6.1", "P6.3-tail", "P6.3-subtract", "P6.4-ball", "P8-root", "L2.1.10-inc", "L2.2.1", "L2.2.3", "L7.1.4", "L7.2.6", "L7.2.9-open", "L7.4.2", "D0", "P6.2", "L7.5.2", "L3.1.7", "L2.1.17", "L7.4.10", "L2.2.5", "L7.1.5", "P6.6", "L7.3.11", "L8.1.13", "P7", "P10.6-zero", "I0", "L2.3.2", "P12.1", "L2.2.11", "L7.3.9", "P9.2", "P10.1", "L7.2.19", "L7.4.11", "L7.3.12", "P9.1", "E0", "L2.3.4-upper", "L5.1.2", "P12.2-upper", "P6.5", "L3.3.7", "L7.2.22", "L7.5.5", "L7.5.11", "P6.4-complete", "L8.1.14", "L8.1.16", "L8.1.17", "R0", "P6.3-inf", "L2.3.8", "L5.1.7", "L7.6.2", "L3.3.8", "L7.4.9", "P9.4", "P9.5", "L8.1.18", "P17.1", "L10.1.14", "J0", "L2.3.5", "P1", "L5.1.8", "L5.2.1", "L7.5.6", "L10.1.2", "L10.4.1", "L10.4.2", "U001-DEF", "L2.4.5", "L5.1.10", "L5.1.11", "L5.1.13", "P12.2-criterion", "L5.2.4-positive", "L5.2.6", "L5.2.2", "F0-COMP", "L10.1.5", "L7.4.4", "P12.3", "P9.3", "L5.2.7", "P17.4-grid", "L10.1.6", "CLOSED-BALL-CONTRACTION", "P12.4", "P12.5-endpoints", "L5.5.3", "L5.5.4", "P9.2-operator", "L10.1.12", "P17.5-upper", "L7.5.12", "COMPACT-INTERVAL-INTEGRAL", "L5.3.3", "L8.2.4", "L10.1.13", "L10.1.15", "L10.2.2", "L8.2.5", "L8.2.6", "L8.2.7", "L8.3.2", "L8.3.5", "P10.2-typo", "L8.2.8", "L8.3.3", "L8.3.9", "L8.3.7", "L8.3.6", "L8.2.9", "P12.5-extension", "P10.2", "L4.2.2", "P10.5-rules", "L4.2.3", "P2", "L4.2.4", "P8-smooth", "F0-ALG", "L4.2.6", "P10.4", "P10.3", "L5.3.1", "L9.1.1", "Q5", "L8.4.1", "L8.4.6", "P5", "L8.4.2", "L8.5.1", "P10.6", "L8.5.6", "P3", "F0-DIFF", "P11.1", "L5.3.5", "P12.6", "P13.1", "P11.2-bridge", "L5.4.1", "P13.2-series", "P17.2", "L8.6.2", "P14.1", "P14.2-inverse", "L2.6.5", "P17.3", "P17.4", "P11.2-orders", "P4", "L5.4.2", "P13.2-mertens", "L10.1.19", "P18.1", "P17.5-order", "P12.7", "P14.2-powers", "P15.1", "L5.5.5", "COMPACT-RECTANGLE-FUBINI", "M1", "FTC-TAYLOR-COMPACT-PARAMETERS", "P14.3", "L5.5.2-gt1", "P19.1", "P15.2", "M2", "M3", "P19.2", "L10.3.2", "L10.3.4", "P16.1", "PARAMETER-MORSE", "P19.3", "P16.2", "L10.3.7", "P19.4", "P20.1", "P16.3", "L10.3.10", "L10.4.3", "L11.4.2", "P20.2", "L10.5.1", "EXPONENTIAL-TRIGONOMETRIC", "P20.3", "L10.5.5", "L10.5.9", "F0-CALC", "L10.5.3", "P21.2", "P20.4", "P20.5", "P18.2", "U001-A4", "U001-4.1", "P21.1", "P18.3", "Q1", "L10.7.2", "P18.4", "U001-E9.3", "P21.3", "P18.5", "F0-INT", "P21.4", "P21.5", "JORDAN-SUBSTITUTION", "Q3", "Q9", "U001-A5", "Q2", "F0-COV", "GLOBAL-INTEGRAL-Q1-Q3-Q9", "U001-2.1", "Q4", "U001-E9.5", "Q6", "U001-A1", "Q7", "U001-A2", "Q8", "E1", "U001-A3", "U001-3.1", "P21.6", "U001-3.2", "QUADRATIC-Q1-Q9-E1", "U001-5.1", "U001-6.1", "U001-E9.1", "U001-E9.2", "U001-A6", "U001-7.2", "U001-MODELS", "U001-E9.4", "U001-FULL-STATIONARY" ], "proofs": [ { "id": "A0", "kind": "axioms_and_definitions", "scope": "Ordered real field with least-upper-bound axiom, natural-number induction, finite arithmetic, sets/functions/sequences and metric/open/closed/compact definitions. No unproved construction/uniqueness theorem for R is used.", "dependencies": [], "reader": "elementary-proof-completions.html#proof-A0", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "The underlying definitions" }, { "id": "P6.0", "kind": "written_local_completion", "scope": "P6.0. The Archimedean argument", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P6.0", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.0. The Archimedean argument" }, { "id": "P6.1", "kind": "written_local_completion", "scope": "P6.1. Subsequence indices", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P6.1", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.1. Subsequence indices" }, { "id": "P6.3-tail", "kind": "written_local_completion", "scope": "First paragraph only: tail extrema bounds and increasing infima", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P6.3-tail", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.3. The omitted lower-limit" }, { "id": "P6.3-subtract", "kind": "written_local_completion", "scope": "Last paragraph only: subtraction of convergent sequences", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P6.3-subtract", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.3. The omitted lower-limit" }, { "id": "P6.4-ball", "kind": "written_local_completion", "scope": "First paragraph: closed balls", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P6.4-ball", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.4. Closed balls" }, { "id": "P8-root", "kind": "written_local_completion", "scope": "Existence, uniqueness, monotonicity and continuity only; smoothness still imports the calculus chain/product rules", "dependencies": [ "A0" ], "reader": "elementary-proof-completions.html#proof-P8-root", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P8. Positive square roots" }, { "id": "L2.1.10-inc", "kind": "existing_programme_proof", "scope": "The fully written increasing half only; decreasing half is P6.2", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_seqsandlims.html#proof-L2.1.10-inc", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_seqsandlims.html", "original_source_page_sha256": "aa52e36074581ffd0a237880570b6a31627c6ef2dc1026cc081811ba31a33e90" }, { "id": "L2.2.1", "kind": "existing_programme_proof", "scope": "Squeeze lemma", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_factslimsseqs.html#proof-L2.2.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_factslimsseqs.html", "original_source_page_sha256": "0a6fb3eccfb86c323555c9ae0cd27c06b4369b79c3a4c6f33253d7888fc02f9f" }, { "id": "L2.2.3", "kind": "existing_programme_proof", "scope": "Nonstrict inequalities pass to limits", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_factslimsseqs.html#proof-L2.2.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_factslimsseqs.html", "original_source_page_sha256": "0a6fb3eccfb86c323555c9ae0cd27c06b4369b79c3a4c6f33253d7888fc02f9f" }, { "id": "L7.1.4", "kind": "existing_programme_proof", "scope": "Real finite-dimensional Cauchy–Schwarz by sum of squares", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_metric.html#proof-L7.1.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metric.html", "original_source_page_sha256": "316fc23f24ef9bfdf955d2f156ca16be19ec878ed96439e9d4e95ab7b68a52fe" }, { "id": "L7.2.6", "kind": "existing_programme_proof", "scope": "Open unions and finite intersections", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_mettop.html#proof-L7.2.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mettop.html", "original_source_page_sha256": "ca94b6b1f75aa0b6f627703e67fbf5a4c3b74d3b1a1a3de367831a86f41f3d46" }, { "id": "L7.2.9-open", "kind": "existing_programme_proof", "scope": "Only the complete open-ball half; closed-ball half is P6.4-ball", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_mettop.html#proof-L7.2.9-open", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mettop.html", "original_source_page_sha256": "ca94b6b1f75aa0b6f627703e67fbf5a4c3b74d3b1a1a3de367831a86f41f3d46" }, { "id": "L7.4.2", "kind": "existing_programme_proof", "scope": "A convergent metric sequence is Cauchy", "dependencies": [ "A0" ], "reader": "human-prerequisites/sec_metcompact.html#proof-L7.4.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcompact.html", "original_source_page_sha256": "eff46c68c5e77865104ac9c33ebdbf25da45b5d9c97dd91f88177665177eb95b" }, { "id": "D0", "kind": "axioms_and_definitions", "scope": "Vector-space, norm, derivative, matrix and higher-regularity definitions; zero-dimensional conventions.", "dependencies": [ "A0" ], "reader": "differential-prerequisite-completions.html#proof-D0", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "We use the ordered complete real-field axioms" }, { "id": "P6.2", "kind": "written_local_completion", "scope": "P6.2. Decreasing limits", "dependencies": [ "A0", "P6.0" ], "reader": "elementary-proof-completions.html#proof-P6.2", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.2. Decreasing limits" }, { "id": "L7.5.2", "kind": "existing_programme_proof", "scope": "Metric continuity and preservation of sequence limits", "dependencies": [ "P6.0" ], "reader": "human-prerequisites/sec_metcont.html#proof-L7.5.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcont.html", "original_source_page_sha256": "30b3fcc05eec06699be2cb0764d6b2184b1efded82b88417a4e69c74100215c0" }, { "id": "L3.1.7", "kind": "existing_programme_proof", "scope": "Full function/sequential-limit equivalence.", "dependencies": [ "A0", "P6.0" ], "reader": "human-prerequisites/sec_limoffunc.html#proof-L3.1.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_limoffunc.html", "original_source_page_sha256": "92586cb89f6f8448a39e7af8bd27f6a57e7b1fc442955e583a9b3ac5f0eba503" }, { "id": "L2.1.17", "kind": "existing_programme_proof", "scope": "Subsequences preserve real limits; index induction supplied in P6.1", "dependencies": [ "A0", "P6.1" ], "reader": "human-prerequisites/sec_seqsandlims.html#proof-L2.1.17", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_seqsandlims.html", "original_source_page_sha256": "aa52e36074581ffd0a237880570b6a31627c6ef2dc1026cc081811ba31a33e90" }, { "id": "L7.4.10", "kind": "existing_programme_proof", "scope": "Lebesgue covering lemma proved directly from sequential compactness", "dependencies": [ "P6.0", "P6.1" ], "reader": "human-prerequisites/sec_metcompact.html#proof-L7.4.10", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcompact.html", "original_source_page_sha256": "eff46c68c5e77865104ac9c33ebdbf25da45b5d9c97dd91f88177665177eb95b" }, { "id": "L2.2.5", "kind": "existing_programme_proof", "scope": "Full addition, multiplication and reciprocal proofs; subtraction completed locally", "dependencies": [ "A0", "P6.3-subtract" ], "reader": "human-prerequisites/sec_factslimsseqs.html#proof-L2.2.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_factslimsseqs.html", "original_source_page_sha256": "0a6fb3eccfb86c323555c9ae0cd27c06b4369b79c3a4c6f33253d7888fc02f9f" }, { "id": "L7.1.5", "kind": "existing_programme_proof", "scope": "Euclidean distance satisfies triangle inequality; positivity and symmetry follow from its displayed sum of squares", "dependencies": [ "L7.1.4", "P8-root" ], "reader": "human-prerequisites/sec_metric.html#proof-L7.1.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metric.html", "original_source_page_sha256": "316fc23f24ef9bfdf955d2f156ca16be19ec878ed96439e9d4e95ab7b68a52fe" }, { "id": "P6.6", "kind": "written_local_completion", "scope": "Closed-set complement rules and empty compactness case, plus the full metric-majorant convergence equivalence used by Theorem 7.4.11.", "dependencies": [ "A0", "L7.2.6", "P6.0" ], "reader": "elementary-proof-completions.html#proof-P6.6", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.6. Closed-set operations" }, { "id": "L7.3.11", "kind": "existing_programme_proof", "scope": "Metric and neighbourhood definitions of convergence agree", "dependencies": [ "L7.2.9-open" ], "reader": "human-prerequisites/sec_metseqs.html#proof-L7.3.11", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metseqs.html", "original_source_page_sha256": "3e69386e7818abc32cce85ebe4ef2f2bae24a2e5863c034314c0657e1a9abbb3" }, { "id": "L8.1.13", "kind": "existing_programme_proof", "scope": "Uniqueness of basis coordinates.", "dependencies": [ "D0" ], "reader": "human-prerequisites/sec_vectorspaces.html#proof-L8.1.13", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_vectorspaces.html", "original_source_page_sha256": "76092061843e6437e0400df870dd9ba08015a7699c5556bf1ee8aa2d1cf67806" }, { "id": "P7", "kind": "local_proof_completion", "scope": "Well-defined determinant permutation parity and sign multiplication.", "dependencies": [ "D0" ], "reader": "elementary-proof-completions.html#proof-P7", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "## P7." }, { "id": "P10.6-zero", "kind": "local_proof_completion", "scope": "Trivial inverse and implicit cases in zero dimensions, without undefined norm reciprocals.", "dependencies": [ "D0" ], "reader": "differential-prerequisite-completions.html#proof-P10.6-zero", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "The inverse theorem in dimension 0" }, { "id": "I0", "kind": "axioms_and_definitions", "scope": "Darboux/Riemann definitions, vector integrals and orientation; finite-interval scope.", "dependencies": [ "A0", "D0" ], "reader": "integration-prerequisite-completions.html#proof-I0", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "The earlier topology and differential chains" }, { "id": "L2.3.2", "kind": "existing_programme_proof", "scope": "Existence and bounds of liminf/limsup, with explicit local omitted steps", "dependencies": [ "A0", "L2.1.10-inc", "P6.2", "P6.3-tail", "L2.2.3" ], "reader": "human-prerequisites/sec_bw.html#proof-L2.3.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_bw.html", "original_source_page_sha256": "da5594230b6e256f9e98b10c39a4aff80cba27d4491a1654f335ceabee20810d" }, { "id": "P12.1", "kind": "local_proof_completion", "scope": "Supremum/infimum order, scaling, sums and cofinal-subset operations used in integral proofs.", "dependencies": [ "A0", "P6.2" ], "reader": "integration-prerequisite-completions.html#proof-P12.1", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P12.1." }, { "id": "L2.2.11", "kind": "existing_programme_proof", "scope": "For 01 part is not needed", "dependencies": [ "A0", "P6.2", "P6.1", "L2.2.5" ], "reader": "human-prerequisites/sec_factslimsseqs.html#proof-L2.2.11", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_factslimsseqs.html", "original_source_page_sha256": "0a6fb3eccfb86c323555c9ae0cd27c06b4369b79c3a4c6f33253d7888fc02f9f" }, { "id": "L7.3.9", "kind": "existing_programme_proof", "scope": "Coordinatewise and Euclidean convergence agree for every finite n", "dependencies": [ "L7.1.5", "P8-root" ], "reader": "human-prerequisites/sec_metseqs.html#proof-L7.3.9", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metseqs.html", "original_source_page_sha256": "3e69386e7818abc32cce85ebe4ef2f2bae24a2e5863c034314c0657e1a9abbb3" }, { "id": "P9.2", "kind": "local_proof_completion", "scope": "Norm metric, reverse triangle, convex balls and R-to-R^m operator norm.", "dependencies": [ "D0", "L7.1.5" ], "reader": "differential-prerequisite-completions.html#proof-P9.2", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P9.2." }, { "id": "P10.1", "kind": "local_proof_completion", "scope": "Scalar and finite-coordinate limit laws, order and squeeze; explicit Fermat sequences.", "dependencies": [ "D0", "P6.0", "L7.1.5" ], "reader": "differential-prerequisite-completions.html#proof-P10.1", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.1." }, { "id": "L7.2.19", "kind": "existing_programme_proof", "scope": "Closure is closed and contains the set", "dependencies": [ "P6.6" ], "reader": "human-prerequisites/sec_mettop.html#proof-L7.2.19", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mettop.html", "original_source_page_sha256": "ca94b6b1f75aa0b6f627703e67fbf5a4c3b74d3b1a1a3de367831a86f41f3d46" }, { "id": "L7.4.11", "kind": "existing_programme_proof", "scope": "Sequential compactness and the finite-subcover definition are equivalent", "dependencies": [ "P6.0", "P6.1", "L7.4.10", "L7.4.2", "L7.2.9-open", "P6.6" ], "reader": "human-prerequisites/sec_metcompact.html#proof-L7.4.11", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcompact.html", "original_source_page_sha256": "eff46c68c5e77865104ac9c33ebdbf25da45b5d9c97dd91f88177665177eb95b" }, { "id": "L7.3.12", "kind": "existing_programme_proof", "scope": "Closed sets contain convergent sequence limits", "dependencies": [ "L7.3.11" ], "reader": "human-prerequisites/sec_metseqs.html#proof-L7.3.12", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metseqs.html", "original_source_page_sha256": "3e69386e7818abc32cce85ebe4ef2f2bae24a2e5863c034314c0657e1a9abbb3" }, { "id": "P9.1", "kind": "local_proof_completion", "scope": "Subspaces, linearity, extension, matrix units, corrected denominator and empty lists.", "dependencies": [ "D0", "L8.1.13" ], "reader": "differential-prerequisite-completions.html#proof-P9.1", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P9.1." }, { "id": "E0", "kind": "axioms_and_definitions", "scope": "Complex coordinate operations, series and factorial definitions.", "dependencies": [ "A0", "D0", "I0" ], "reader": "exponential-prerequisite-completions.html#proof-E0", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "Identify \\(\\mathbb C\\)" }, { "id": "L2.3.4-upper", "kind": "existing_programme_proof", "scope": "Fully written limsup subsequence proof; no assumption of the omitted liminf proof", "dependencies": [ "L2.3.2", "L2.1.17", "P6.0" ], "reader": "human-prerequisites/sec_bw.html#proof-L2.3.4-upper", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_bw.html", "original_source_page_sha256": "da5594230b6e256f9e98b10c39a4aff80cba27d4491a1654f335ceabee20810d" }, { "id": "L5.1.2", "kind": "existing_programme_proof", "scope": "Darboux sum bounds and existence of their defining finite extrema.", "dependencies": [ "I0", "P12.1" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "P12.2-upper", "kind": "local_proof_completion", "scope": "Upper-refinement proof and correction of the q=0 interval-length index.", "dependencies": [ "I0", "P12.1" ], "reader": "integration-prerequisite-completions.html#proof-P12.2-upper", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "Use Lebl's definitions of a partition" }, { "id": "P6.5", "kind": "written_local_completion", "scope": "Finite geometric bound, Cauchy threshold and direct limit at the fixed point", "dependencies": [ "A0", "L2.2.11" ], "reader": "elementary-proof-completions.html#proof-P6.5", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.5. The geometric bound" }, { "id": "L3.3.7", "kind": "existing_programme_proof", "scope": "Complete bisection proof of a zero between strictly signed endpoints.", "dependencies": [ "L2.1.10-inc", "P6.2", "L2.2.11", "L2.2.5", "L2.2.3", "L7.5.2" ], "reader": "human-prerequisites/sec_minmaxint.html#proof-L3.3.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_minmaxint.html", "original_source_page_sha256": "9bb6af8b932032834ece8dc7fa4ad5462ab69179603dd49373b3441aada4460b" }, { "id": "L7.2.22", "kind": "existing_programme_proof", "scope": "Every ball about a point of the closure meets the set", "dependencies": [ "L7.2.19", "L7.2.9-open" ], "reader": "human-prerequisites/sec_mettop.html#proof-L7.2.22", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mettop.html", "original_source_page_sha256": "ca94b6b1f75aa0b6f627703e67fbf5a4c3b74d3b1a1a3de367831a86f41f3d46" }, { "id": "L7.5.5", "kind": "existing_programme_proof", "scope": "Continuous image of a compact set is compact", "dependencies": [ "L7.5.2", "L7.4.11" ], "reader": "human-prerequisites/sec_metcont.html#proof-L7.5.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcont.html", "original_source_page_sha256": "30b3fcc05eec06699be2cb0764d6b2184b1efded82b88417a4e69c74100215c0" }, { "id": "L7.5.11", "kind": "existing_programme_proof", "scope": "Continuous maps on compact spaces are uniformly continuous", "dependencies": [ "L7.4.10", "L7.4.11" ], "reader": "human-prerequisites/sec_metcont.html#proof-L7.5.11", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcont.html", "original_source_page_sha256": "30b3fcc05eec06699be2cb0764d6b2184b1efded82b88417a4e69c74100215c0" }, { "id": "P6.4-complete", "kind": "written_local_completion", "scope": "Second paragraph: a closed subset of a complete metric space is complete", "dependencies": [ "A0", "L7.3.12" ], "reader": "elementary-proof-completions.html#proof-P6.4-complete", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.4. Closed balls" }, { "id": "L8.1.14", "kind": "existing_programme_proof", "scope": "All six finite-dimensional exchange, basis and subspace assertions.", "dependencies": [ "D0", "P9.1" ], "reader": "human-prerequisites/sec_vectorspaces.html#proof-L8.1.14", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_vectorspaces.html", "original_source_page_sha256": "76092061843e6437e0400df870dd9ba08015a7699c5556bf1ee8aa2d1cf67806" }, { "id": "L8.1.16", "kind": "existing_programme_proof", "scope": "Inverse linearity, with the other four parts completed locally.", "dependencies": [ "P9.1" ], "reader": "human-prerequisites/sec_vectorspaces.html#proof-L8.1.16", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_vectorspaces.html", "original_source_page_sha256": "76092061843e6437e0400df870dd9ba08015a7699c5556bf1ee8aa2d1cf67806" }, { "id": "L8.1.17", "kind": "existing_programme_proof", "scope": "Determination and extension from basis values, including the omitted linearity check.", "dependencies": [ "L8.1.13", "P9.1" ], "reader": "human-prerequisites/sec_vectorspaces.html#proof-L8.1.17", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_vectorspaces.html", "original_source_page_sha256": "76092061843e6437e0400df870dd9ba08015a7699c5556bf1ee8aa2d1cf67806" }, { "id": "R0", "kind": "axioms_and_definitions", "scope": "Coordinate rectangles, volume and Darboux definitions; finite-vector/complex integral definitions.", "dependencies": [ "A0", "D0", "I0", "E0" ], "reader": "global-integral-prerequisite-completions.html#proof-R0", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "Rectangles, grid partitions" }, { "id": "P6.3-inf", "kind": "written_local_completion", "scope": "Second paragraph only: reflection proves the omitted liminf subsequence assertion", "dependencies": [ "A0", "L2.3.4-upper", "P6.3-tail" ], "reader": "elementary-proof-completions.html#proof-P6.3-inf", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "P6.3. The omitted lower-limit" }, { "id": "L2.3.8", "kind": "existing_programme_proof", "scope": "Bolzano–Weierstrass via the first complete proof. Alternate bisection proof also read, but not needed by this graph.", "dependencies": [ "L2.3.4-upper" ], "reader": "human-prerequisites/sec_bw.html#proof-L2.3.8", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_bw.html", "original_source_page_sha256": "da5594230b6e256f9e98b10c39a4aff80cba27d4491a1654f335ceabee20810d" }, { "id": "L5.1.7", "kind": "existing_programme_proof", "scope": "Both refinement inequalities; omitted upper half supplied by P12.2.", "dependencies": [ "L5.1.2", "P12.1", "P12.2-upper" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "L7.6.2", "kind": "existing_programme_proof", "scope": "Complete-space contraction principle with finite geometric bound and explicit limit step", "dependencies": [ "P6.5" ], "reader": "human-prerequisites/sec_metpicard.html#proof-L7.6.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metpicard.html", "original_source_page_sha256": "a215f3c653d26eb05da1216407f730a3c437ce2da52e55580149f281cdbdd7e9" }, { "id": "L3.3.8", "kind": "existing_programme_proof", "scope": "Both strict orientations of the intermediate value theorem.", "dependencies": [ "L3.3.7", "P10.1" ], "reader": "human-prerequisites/sec_minmaxint.html#proof-L3.3.8", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_minmaxint.html", "original_source_page_sha256": "9bb6af8b932032834ece8dc7fa4ad5462ab69179603dd49373b3441aada4460b" }, { "id": "L7.4.9", "kind": "existing_programme_proof", "scope": "Compact sets are closed and bounded, with the empty-set case explicit locally", "dependencies": [ "L7.2.9-open", "P6.4-ball", "L7.2.22", "P6.0", "P6.6" ], "reader": "human-prerequisites/sec_metcompact.html#proof-L7.4.9", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcompact.html", "original_source_page_sha256": "eff46c68c5e77865104ac9c33ebdbf25da45b5d9c97dd91f88177665177eb95b" }, { "id": "P9.4", "kind": "local_proof_completion", "scope": "Gram-Schmidt and orthonormal coordinates on the invariant complement in Q5.", "dependencies": [ "L8.1.14", "P9.1", "L7.1.5", "P8-root" ], "reader": "differential-prerequisite-completions.html#proof-P9.4", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P9.4." }, { "id": "P9.5", "kind": "local_proof_completion", "scope": "Injective dimension bound, nonzero projection kernel and invariance under bijections.", "dependencies": [ "L8.1.14", "P9.1" ], "reader": "differential-prerequisite-completions.html#proof-P9.5", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P9.5." }, { "id": "L8.1.18", "kind": "existing_programme_proof", "scope": "Injectivity iff surjectivity for finite-dimensional endomorphisms.", "dependencies": [ "L8.1.14", "L8.1.16", "L8.1.17" ], "reader": "human-prerequisites/sec_vectorspaces.html#proof-L8.1.18", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_vectorspaces.html", "original_source_page_sha256": "76092061843e6437e0400df870dd9ba08015a7699c5556bf1ee8aa2d1cf67806" }, { "id": "P17.1", "kind": "local_proof_completion", "scope": "Grid coverage and volume additivity, singleton-side and dimension-zero cases, and the omitted upper refinement.", "dependencies": [ "R0", "P12.1" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.1", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P17.1." }, { "id": "L10.1.14", "kind": "existing_programme_proof", "scope": "Full rectangle-diameter inequality.", "dependencies": [ "R0", "P8-root", "L7.1.4" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.14", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "J0", "kind": "axioms_and_definitions", "scope": "Outer measure, null sets, Jordan sets and zero extensions are explicit definitions; dimension-zero convention.", "dependencies": [ "A0", "D0", "R0" ], "reader": "change-of-variables-prerequisite-completions.html#proof-J0", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "All sets below lie" }, { "id": "L2.3.5", "kind": "existing_programme_proof", "scope": "Convergence iff the two tail limits agree", "dependencies": [ "L2.3.2", "L2.3.4-upper", "P6.3-inf", "L2.1.17", "L2.2.1" ], "reader": "human-prerequisites/sec_bw.html#proof-L2.3.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_bw.html", "original_source_page_sha256": "da5594230b6e256f9e98b10c39a4aff80cba27d4491a1654f335ceabee20810d" }, { "id": "P1", "kind": "written_local_completion", "scope": "P1. The finite-dimensional step", "dependencies": [ "L2.3.8", "L2.1.17", "L7.3.12", "L7.4.11", "P8-root", "L7.1.5" ], "reader": "prerequisite-completions.html#proof-P1", "editable_source": "prerequisite-completions.md", "editable_source_sha256": "3be6c23c56f645049ad6776b63d0f1fccab0e39938d4998f0201b9664cc155af", "locator": "P1. The finite-dimensional step" }, { "id": "L5.1.8", "kind": "existing_programme_proof", "scope": "Lower Darboux integral <= upper integral, with common-refinement proof.", "dependencies": [ "L5.1.2", "L5.1.7", "P12.1" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.8", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "L5.2.1", "kind": "existing_programme_proof", "scope": "Additivity of upper and lower integrals on adjacent intervals.", "dependencies": [ "L5.1.7", "P12.1" ], "reader": "human-prerequisites/sec_rintprop.html#proof-L5.2.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rintprop.html", "original_source_page_sha256": "160b59e87ceda09d392bf73621b75aef72344aec1f928f18640977aaa84afb58" }, { "id": "L7.5.6", "kind": "existing_programme_proof", "scope": "Continuous real functions on nonempty compact spaces attain both extrema", "dependencies": [ "L7.5.5", "L7.4.9", "P6.2", "L7.3.12" ], "reader": "human-prerequisites/sec_metcont.html#proof-L7.5.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcont.html", "original_source_page_sha256": "30b3fcc05eec06699be2cb0764d6b2184b1efded82b88417a4e69c74100215c0" }, { "id": "L10.1.2", "kind": "existing_programme_proof", "scope": "Full Darboux bounds, with the used volume exercise now proved.", "dependencies": [ "P17.1", "P12.1" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "L10.4.1", "kind": "existing_programme_proof", "scope": "Continuity iff zero oscillation; full proof.", "dependencies": [ "P12.1", "P10.1", "J0" ], "reader": "human-prerequisites/sec_riemannlebesgue.html#proof-L10.4.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_riemannlebesgue.html", "original_source_page_sha256": "702d47a48eb05b6e87894147772429916ebf92a11fbaa9ec9392eca13a6c81eb" }, { "id": "L10.4.2", "kind": "existing_programme_proof", "scope": "Closed positive oscillation level sets on a closed domain; full proof.", "dependencies": [ "P12.1", "J0", "L7.1.4" ], "reader": "human-prerequisites/sec_riemannlebesgue.html#proof-L10.4.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_riemannlebesgue.html", "original_source_page_sha256": "702d47a48eb05b6e87894147772429916ebf92a11fbaa9ec9392eca13a6c81eb" }, { "id": "U001-DEF", "kind": "axioms_and_definitions", "scope": "Uniform supports, parameter compacta, Fourier convention, symbols, ordinary densities, chart definition and dimension-zero conventions.", "dependencies": [ "A0", "D0", "I0", "R0", "J0", "E0" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-DEF", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "## 1. What a uniform estimate means" }, { "id": "L2.4.5", "kind": "existing_programme_proof", "scope": "Cauchy sequences are bounded and real Cauchy sequences converge", "dependencies": [ "L2.3.2", "L2.3.4-upper", "P6.3-inf", "L2.3.5", "P6.1" ], "reader": "human-prerequisites/sec_cauchy.html#proof-L2.4.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_cauchy.html", "original_source_page_sha256": "55d9d278e2bbc9f925a83e5a4aeb25f00e58dfc467e398d0e154b581ee6e7328" }, { "id": "L5.1.10", "kind": "existing_programme_proof", "scope": "Integral bound is the just-proved Darboux bound with equal lower/upper integrals, as Definition 5.1.9 specifies.", "dependencies": [ "L5.1.8", "I0" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.10", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "L5.1.11", "kind": "existing_programme_proof", "scope": "Exact constant integral, proved by coinciding upper/lower bounds.", "dependencies": [ "L5.1.8" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.11", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "L5.1.13", "kind": "existing_programme_proof", "scope": "Arbitrarily small Darboux gaps imply integrability.", "dependencies": [ "L5.1.8" ], "reader": "human-prerequisites/sec_rint.html#proof-L5.1.13", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rint.html", "original_source_page_sha256": "e50422b32a44b596fd10dab64510c85b0d7ed7876169fb99d008ef32cbd97c79" }, { "id": "P12.2-criterion", "kind": "local_proof_completion", "scope": "Necessary small-gap criterion and common partition for finitely many integrable functions.", "dependencies": [ "L5.1.7", "L5.1.8", "P6.2" ], "reader": "integration-prerequisite-completions.html#proof-P12.2-criterion", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "Proposition 5.1.13 proves that" }, { "id": "L5.2.4-positive", "kind": "existing_programme_proof", "scope": "The actually written nonnegative-scalar proof only; omitted negative and sum cases are supplied locally.", "dependencies": [ "P12.1", "L5.1.8" ], "reader": "human-prerequisites/sec_rintprop.html#proof-L5.2.4-positive", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rintprop.html", "original_source_page_sha256": "160b59e87ceda09d392bf73621b75aef72344aec1f928f18640977aaa84afb58" }, { "id": "L5.2.6", "kind": "existing_programme_proof", "scope": "Integral monotonicity via extrema and finite sums.", "dependencies": [ "P12.1", "L5.1.8" ], "reader": "human-prerequisites/sec_rintprop.html#proof-L5.2.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rintprop.html", "original_source_page_sha256": "160b59e87ceda09d392bf73621b75aef72344aec1f928f18640977aaa84afb58" }, { "id": "L5.2.2", "kind": "existing_programme_proof", "scope": "Integrability iff both interval restrictions are integrable; interval additivity.", "dependencies": [ "L5.2.1", "L5.1.8" ], "reader": "human-prerequisites/sec_rintprop.html#proof-L5.2.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rintprop.html", "original_source_page_sha256": "160b59e87ceda09d392bf73621b75aef72344aec1f928f18640977aaa84afb58" }, { "id": "F0-COMP", "kind": "assembled_dependency_contract", "scope": "Closed bounded Euclidean sets are compact; continuous real functions attain extrema on nonempty compact sets; continuous maps on compact sets are uniformly continuous.", "dependencies": [ "P1", "L7.5.6", "L7.5.11" ], "reader": "proof-map.html#F0-COMP", "is_dependency_contract_not_additional_proof": true }, { "id": "L10.1.5", "kind": "existing_programme_proof", "scope": "Both refinement inequalities; upper half is P17.1.", "dependencies": [ "P17.1", "L10.1.2", "P12.1" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "L7.4.4", "kind": "existing_programme_proof", "scope": "Euclidean completeness by coordinatewise real completeness", "dependencies": [ "L2.4.5", "L7.3.9" ], "reader": "human-prerequisites/sec_metcompact.html#proof-L7.4.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcompact.html", "original_source_page_sha256": "eff46c68c5e77865104ac9c33ebdbf25da45b5d9c97dd91f88177665177eb95b" }, { "id": "P12.3", "kind": "local_proof_completion", "scope": "Full scalar linearity and finite sums, restriction exercise, arbitrary orientation and base-point identity.", "dependencies": [ "L5.2.4-positive", "P12.1", "P12.2-criterion", "L5.1.13", "L5.2.2" ], "reader": "integration-prerequisite-completions.html#proof-P12.3", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P12.3." }, { "id": "P9.3", "kind": "local_proof_completion", "scope": "Norm equivalence and boundedness from any finite-dimensional normed domain into any normed target.", "dependencies": [ "L8.1.14", "P9.1", "P9.2", "F0-COMP" ], "reader": "differential-prerequisite-completions.html#proof-P9.3", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P9.3." }, { "id": "L5.2.7", "kind": "existing_programme_proof", "scope": "Continuous functions on compact intervals are Riemann integrable.", "dependencies": [ "F0-COMP", "P6.0", "L5.1.8", "L5.1.13" ], "reader": "human-prerequisites/sec_rintprop.html#proof-L5.2.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rintprop.html", "original_source_page_sha256": "160b59e87ceda09d392bf73621b75aef72344aec1f928f18640977aaa84afb58" }, { "id": "P17.4-grid", "kind": "local_proof_completion", "scope": "Construction of fine grids and handling the zero-volume/dimension cases in the continuous-integrability proof.", "dependencies": [ "P17.1", "P6.0", "F0-COMP" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.4-grid", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "Retain the complete diameter" }, { "id": "L10.1.6", "kind": "existing_programme_proof", "scope": "Complete common-refinement proof of lower <= upper integral and volume bounds.", "dependencies": [ "L10.1.5", "L10.1.2", "P12.1" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "CLOSED-BALL-CONTRACTION", "kind": "assembled_dependency_contract", "scope": "Unique fixed point of a contraction mapping a nonempty closed Euclidean ball into itself.", "dependencies": [ "L7.6.2", "L7.4.4", "P6.4-complete", "P6.4-ball" ], "reader": "proof-map.html#CLOSED-BALL-CONTRACTION", "is_dependency_contract_not_additional_proof": true }, { "id": "P12.4", "kind": "local_proof_completion", "scope": "Finite-vector norm integrability, norm bound, and convergence of integrals under a uniform error with integrable limit.", "dependencies": [ "P12.1", "P12.2-criterion", "L5.1.13", "P12.3", "L5.2.6", "L5.1.11", "L7.1.4", "P9.2" ], "reader": "integration-prerequisite-completions.html#proof-P12.4", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P12.4." }, { "id": "P12.5-endpoints", "kind": "local_proof_completion", "scope": "One-sided endpoint interpretation, strict-limit error and arbitrary-base-point details.", "dependencies": [ "I0", "P12.3", "P10.1" ], "reader": "integration-prerequisite-completions.html#proof-P12.5-endpoints", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "The programme supplies the full proofs" }, { "id": "L5.5.3", "kind": "existing_programme_proof", "scope": "Complete scalar tail identity for an infinite right endpoint.", "dependencies": [ "P12.3", "P10.1" ], "reader": "human-prerequisites/sec_impropriemann.html#proof-L5.5.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_impropriemann.html", "original_source_page_sha256": "63b18c0708143c25e0b7df2d4c9d9ca82f678486411b3f16b0df51e1c1446610" }, { "id": "L5.5.4", "kind": "existing_programme_proof", "scope": "Complete nonnegative-supremum and divergent-endpoint subsequence proof.", "dependencies": [ "P12.3", "L5.2.6", "P12.1", "P6.2", "P10.1" ], "reader": "human-prerequisites/sec_impropriemann.html#proof-L5.5.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_impropriemann.html", "original_source_page_sha256": "63b18c0708143c25e0b7df2d4c9d9ca82f678486411b3f16b0df51e1c1446610" }, { "id": "P9.2-operator", "kind": "local_proof_completion", "scope": "Positivity, scaling bound and Lipschitz continuity after the finite norm is established.", "dependencies": [ "D0", "P9.1", "P9.3" ], "reader": "differential-prerequisite-completions.html#proof-P9.2-operator", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "For the operator norm itself" }, { "id": "L10.1.12", "kind": "existing_programme_proof", "scope": "Both directions of the small-gap criterion with supremum/infimum operations explicitly available.", "dependencies": [ "L10.1.6", "L10.1.5", "P12.1", "P6.2" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.12", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "P17.5-upper", "kind": "local_proof_completion", "scope": "Upper-sum inequality in Fubini A, boundedness of upper/lower section integrals.", "dependencies": [ "L10.1.6", "P12.1", "P17.1" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.5-upper", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "For its omitted" }, { "id": "L7.5.12", "kind": "existing_programme_proof", "scope": "Continuity of a compact integral in one parameter.", "dependencies": [ "F0-COMP", "L5.2.7", "P12.3", "P12.4" ], "reader": "human-prerequisites/sec_metcont.html#proof-L7.5.12", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_metcont.html", "original_source_page_sha256": "30b3fcc05eec06699be2cb0764d6b2184b1efded82b88417a4e69c74100215c0" }, { "id": "COMPACT-INTERVAL-INTEGRAL", "kind": "proved_contract", "scope": "Bounded-interval real/vector Riemann construction, linearity, restriction, orientation, norm bound, continuous integrability and uniform-error passage.", "dependencies": [ "L5.1.8", "L5.1.13", "L5.2.2", "P12.3", "L5.2.6", "L5.2.7", "P12.4" ], "reader": "proof-map.html#COMPACT-INTERVAL-INTEGRAL", "is_dependency_contract_not_additional_proof": true }, { "id": "L5.3.3", "kind": "existing_programme_proof", "scope": "Primitive is Lipschitz; derivative equals f at every point of continuity, one-sided at endpoints.", "dependencies": [ "L5.2.2", "L5.1.10", "L5.2.6", "P12.3", "P12.5-endpoints" ], "reader": "human-prerequisites/sec_ftc.html#proof-L5.3.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_ftc.html", "original_source_page_sha256": "1e3f3c4a7fa5d456fbc17932a1119e8f066372813accf429910ce426d1f28d79" }, { "id": "L8.2.4", "kind": "existing_programme_proof", "scope": "Source Euclidean proof plus P9.3 for its full stated normed-space generality.", "dependencies": [ "L7.1.4", "L8.1.17", "P9.3", "P9.2-operator" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "L10.1.13", "kind": "existing_programme_proof", "scope": "Restriction to a subrectangle including the locally supplied degenerate cases.", "dependencies": [ "L10.1.12", "P17.1" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.13", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "L10.1.15", "kind": "existing_programme_proof", "scope": "Complete continuous-integrability argument, with its fine-grid and zero-case omissions supplied.", "dependencies": [ "F0-COMP", "L10.1.14", "P17.4-grid", "L10.1.12" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.15", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "L10.2.2", "kind": "existing_programme_proof", "scope": "Full upper/lower compact-rectangle Fubini theorem, not just its continuous specialization.", "dependencies": [ "P17.5-upper", "L10.1.12", "P12.1", "P17.1" ], "reader": "human-prerequisites/sec_iteratedints.html#proof-L10.2.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_iteratedints.html", "original_source_page_sha256": "ec8ad5735e6fec25c2903e954c7df244c344435efa74c664336aef547e74315a" }, { "id": "L8.2.5", "kind": "existing_programme_proof", "scope": "Operator norm sum, scalar and composition inequalities and norm axioms.", "dependencies": [ "L8.2.4", "L8.1.16", "P9.2" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "L8.2.6", "kind": "existing_programme_proof", "scope": "Invertible neighbourhood and continuity of inversion in every finite-dimensional normed space.", "dependencies": [ "L8.1.18", "L8.2.5", "D0" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "L8.2.7", "kind": "existing_programme_proof", "scope": "Matrix-coordinate/operator-norm topology equivalence; full preceding Frobenius estimates.", "dependencies": [ "L7.1.4", "P9.1", "L8.2.5" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "L8.3.2", "kind": "existing_programme_proof", "scope": "Uniqueness of the total derivative.", "dependencies": [ "D0", "L8.2.5", "P10.1" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "L8.3.5", "kind": "existing_programme_proof", "scope": "Differentiability implies continuity.", "dependencies": [ "L8.2.5", "P10.1" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "P10.2-typo", "kind": "local_proof_completion", "scope": "Independent scalar remainder calculation correcting the extra closing parenthesis.", "dependencies": [ "L8.2.5", "P10.1" ], "reader": "differential-prerequisite-completions.html#proof-P10.2-typo", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "In the scalar-multiple proof of Proposition 8.3.6" }, { "id": "L8.2.8", "kind": "existing_programme_proof", "scope": "All seven determinant properties; continuity uses P10.1.", "dependencies": [ "P7", "P9.1", "P10.1", "L8.2.7" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.8", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "L8.3.3", "kind": "existing_programme_proof", "scope": "Full zero-remainder computation for a linear map.", "dependencies": [ "L8.2.4", "L8.3.2" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "L8.3.9", "kind": "existing_programme_proof", "scope": "Partial derivatives as columns of the total derivative.", "dependencies": [ "L8.3.2", "L7.3.9", "P10.1" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.9", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "L8.3.7", "kind": "existing_programme_proof", "scope": "Total chain rule, including zero intermediate increment.", "dependencies": [ "L8.3.5", "L8.2.5", "P10.1" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "L8.3.6", "kind": "existing_programme_proof", "scope": "Derivative sum and scalar rules; extra-parenthesis typo explicitly corrected in P10.2.", "dependencies": [ "L8.2.5", "P10.1", "P10.2-typo" ], "reader": "human-prerequisites/sec_svtheder.html#proof-L8.3.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svtheder.html", "original_source_page_sha256": "d60a9ddf0c5a87c998ccdd108af8b325aa209f82dcf431d249ec34bc1d873062" }, { "id": "L8.2.9", "kind": "existing_programme_proof", "scope": "Determinant multiplication, transpose invariance and invertibility criterion.", "dependencies": [ "L8.2.8", "L8.1.18" ], "reader": "human-prerequisites/sec_normsmatsdets.html#proof-L8.2.9", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_normsmatsdets.html", "original_source_page_sha256": "05441bbcb0c4e5b239398b28f967da82382d9dede5fa07a34a6454b5fc7eb348" }, { "id": "P12.5-extension", "kind": "local_proof_completion", "scope": "Continuous constant endpoint extension supplies an open-domain primitive for the chain rule.", "dependencies": [ "L5.3.3", "L5.2.7", "P12.3", "L8.3.7", "P10.1" ], "reader": "integration-prerequisite-completions.html#proof-P12.5-extension", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "In the substitution proof of Theorem 5.3.5" }, { "id": "P10.2", "kind": "local_proof_completion", "scope": "Product and quotient rules, reciprocal derivative and scalar-multiple typo correction.", "dependencies": [ "L8.2.5", "P10.1", "L8.3.6", "L8.3.7" ], "reader": "differential-prerequisite-completions.html#proof-P10.2", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.2." }, { "id": "L4.2.2", "kind": "existing_programme_proof", "scope": "Scalar Fermat theorem at an interior extremum.", "dependencies": [ "L3.1.7", "L2.2.3", "L8.3.6", "P10.1" ], "reader": "human-prerequisites/sec_mvt.html#proof-L4.2.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvt.html", "original_source_page_sha256": "5121cdc5675cc3140a884db86695128a5a495a65e149d2ba13f151f7ece1f707" }, { "id": "P10.5-rules", "kind": "local_proof_completion", "scope": "Induction proving finite C^k sum/product/composition closure, smooth reciprocal and polynomials; excludes its later P2/IFT applications.", "dependencies": [ "L8.2.7", "L8.3.3", "L8.3.5", "L8.3.6", "L8.3.7", "P10.1", "P10.2" ], "reader": "differential-prerequisite-completions.html#proof-P10.5-rules", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.5." }, { "id": "L4.2.3", "kind": "existing_programme_proof", "scope": "Rolle theorem on a nondegenerate closed interval.", "dependencies": [ "F0-COMP", "L4.2.2", "L8.3.3" ], "reader": "human-prerequisites/sec_mvt.html#proof-L4.2.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvt.html", "original_source_page_sha256": "5121cdc5675cc3140a884db86695128a5a495a65e149d2ba13f151f7ece1f707" }, { "id": "P2", "kind": "local_proof_completion", "scope": "Cofactor identity, smooth inverse and its differential.", "dependencies": [ "L8.2.9", "P7", "P10.2", "P10.5-rules" ], "reader": "prerequisite-completions.html#proof-P2", "editable_source": "prerequisite-completions.md", "editable_source_sha256": "3be6c23c56f645049ad6776b63d0f1fccab0e39938d4998f0201b9664cc155af", "locator": "## P2." }, { "id": "L4.2.4", "kind": "existing_programme_proof", "scope": "Scalar mean-value theorem by subtraction of the affine secant.", "dependencies": [ "L4.2.3", "L8.3.3", "L8.3.6" ], "reader": "human-prerequisites/sec_mvt.html#proof-L4.2.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvt.html", "original_source_page_sha256": "5121cdc5675cc3140a884db86695128a5a495a65e149d2ba13f151f7ece1f707" }, { "id": "P8-smooth", "kind": "local_proof_completion", "scope": "Positive square-root derivative and smoothness; existence/continuity imported separately.", "dependencies": [ "P8-root", "P10.1", "P10.2", "P10.5-rules", "P2" ], "reader": "elementary-proof-completions.html#proof-P8-smooth", "editable_source": "elementary-proof-completions.md", "editable_source_sha256": "d978a2c8dd8001f19681b9d424126b51139bf436409a6d7aecd8afee7c8ee155", "locator": "## P8." }, { "id": "F0-ALG", "kind": "proved_contract", "scope": "Determinant multiplication, cofactor inversion and smooth matrix dependence.", "dependencies": [ "L8.2.9", "P2" ], "reader": "proof-map.html#F0-ALG", "is_dependency_contract_not_additional_proof": true }, { "id": "L4.2.6", "kind": "existing_programme_proof", "scope": "Zero derivative implies constant on an interval.", "dependencies": [ "L4.2.4", "L8.3.5" ], "reader": "human-prerequisites/sec_mvt.html#proof-L4.2.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvt.html", "original_source_page_sha256": "5121cdc5675cc3140a884db86695128a5a495a65e149d2ba13f151f7ece1f707" }, { "id": "P10.4", "kind": "local_proof_completion", "scope": "Zero endpoint-difference and equal-point cases for vector mean value.", "dependencies": [ "L4.2.4", "P9.2", "L7.1.4" ], "reader": "differential-prerequisite-completions.html#proof-P10.4", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.4." }, { "id": "P10.3", "kind": "local_proof_completion", "scope": "Vector-valued n=1 base case and domain/zero increments in continuous-partials induction.", "dependencies": [ "P9.2", "P10.1", "L4.2.4" ], "reader": "differential-prerequisite-completions.html#proof-P10.3", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.3." }, { "id": "L5.3.1", "kind": "existing_programme_proof", "scope": "Integral of an integrable derivative equals the endpoint difference; full stated generality.", "dependencies": [ "L4.2.4", "L5.1.8", "I0" ], "reader": "human-prerequisites/sec_ftc.html#proof-L5.3.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_ftc.html", "original_source_page_sha256": "1e3f3c4a7fa5d456fbc17932a1119e8f066372813accf429910ce426d1f28d79" }, { "id": "L9.1.1", "kind": "existing_programme_proof", "scope": "Compact one-parameter integral derivative with uniform-continuity/MVT proof and explicit integral-error bound.", "dependencies": [ "F0-COMP", "L4.2.4", "L5.2.7", "P12.3", "P12.4", "L7.5.12", "P12.5-endpoints" ], "reader": "human-prerequisites/sec_diffunderint.html#proof-L9.1.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_diffunderint.html", "original_source_page_sha256": "3ee8441db8c926c1b30b3c6821b432e6a283eb829f741ef012604b893d408ad5" }, { "id": "Q5", "kind": "local_proof_completion", "scope": "Full finite-dimensional real spectral theorem and determinant formula.", "dependencies": [ "F0-COMP", "P10.1", "P10.2", "L8.3.7", "L4.2.2", "P8-smooth", "P9.4", "L8.2.9" ], "reader": "quadratic-stationary-phase.html#proof-Q5", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q5." }, { "id": "L8.4.1", "kind": "existing_programme_proof", "scope": "Full vector mean-value inequality with zero case supplied.", "dependencies": [ "L4.2.4", "L8.3.3", "L8.3.6", "L8.3.7", "P9.2", "L7.1.4", "P10.4" ], "reader": "human-prerequisites/sec_svthedercont.html#proof-L8.4.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svthedercont.html", "original_source_page_sha256": "567962c82e740e92808b4be596a0c6da2ed7496ec0c89af8bf5cd8f945e25ace" }, { "id": "L8.4.6", "kind": "existing_programme_proof", "scope": "C1 iff all partial derivatives exist and are continuous.", "dependencies": [ "L8.3.9", "L8.2.7", "L4.2.4", "P10.3" ], "reader": "human-prerequisites/sec_svthedercont.html#proof-L8.4.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svthedercont.html", "original_source_page_sha256": "567962c82e740e92808b4be596a0c6da2ed7496ec0c89af8bf5cd8f945e25ace" }, { "id": "P5", "kind": "local_proof_completion", "scope": "Signature invariance under congruence and the Morse Jacobian determinant factor.", "dependencies": [ "Q5", "P9.5", "L8.2.9", "P8-root" ], "reader": "prerequisite-completions.html#proof-P5", "editable_source": "prerequisite-completions.md", "editable_source_sha256": "3be6c23c56f645049ad6776b63d0f1fccab0e39938d4998f0201b9664cc155af", "locator": "## P5." }, { "id": "L8.4.2", "kind": "existing_programme_proof", "scope": "Derivative bound implies Lipschitz bound on a convex open domain.", "dependencies": [ "L8.4.1", "L8.3.7", "L8.3.3", "L8.2.5", "P10.4" ], "reader": "human-prerequisites/sec_svthedercont.html#proof-L8.4.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svthedercont.html", "original_source_page_sha256": "567962c82e740e92808b4be596a0c6da2ed7496ec0c89af8bf5cd8f945e25ace" }, { "id": "L8.5.1", "kind": "existing_programme_proof", "scope": "C1 inverse theorem, open image, derivative formula and continuity.", "dependencies": [ "L8.2.6", "L8.3.3", "L8.3.5", "L8.3.6", "L8.3.7", "L8.4.2", "P9.2", "P10.1", "CLOSED-BALL-CONTRACTION", "P10.6-zero" ], "reader": "human-prerequisites/sec_svinvfuncthm.html#proof-L8.5.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svinvfuncthm.html", "original_source_page_sha256": "f0f62555d86e2345e7c70dec2d4f867d49499d8d68eeb0195e9ba0abf7c4c8a8" }, { "id": "P10.6", "kind": "local_proof_completion", "scope": "Explicit slice/product-neighbourhood restriction and uniqueness in the implicit proof.", "dependencies": [ "L8.5.1", "L8.3.3", "L8.3.7", "P10.1", "P9.2" ], "reader": "differential-prerequisite-completions.html#proof-P10.6", "editable_source": "differential-prerequisite-completions.md", "editable_source_sha256": "609bd40f74a5680083b4c7232741454f1b4e6221061d28d59f476f74bebce1ff", "locator": "### P10.6." }, { "id": "L8.5.6", "kind": "existing_programme_proof", "scope": "C1 implicit theorem on a genuine product neighbourhood, with P10.6 domain correction.", "dependencies": [ "L8.5.1", "L8.1.18", "L8.2.9", "L8.3.7", "P10.6" ], "reader": "human-prerequisites/sec_svinvfuncthm.html#proof-L8.5.6", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_svinvfuncthm.html", "original_source_page_sha256": "f0f62555d86e2345e7c70dec2d4f867d49499d8d68eeb0195e9ba0abf7c4c8a8" }, { "id": "P3", "kind": "local_proof_completion", "scope": "Inverse and implicit C^r upgrade for all finite r>=1 and smooth maps, jointly in parameters.", "dependencies": [ "L8.5.1", "L8.5.6", "P2", "P10.5-rules" ], "reader": "prerequisite-completions.html#proof-P3", "editable_source": "prerequisite-completions.md", "editable_source_sha256": "3be6c23c56f645049ad6776b63d0f1fccab0e39938d4998f0201b9664cc155af", "locator": "## P3." }, { "id": "F0-DIFF", "kind": "proved_contract", "scope": "Finite-dimensional differential rules, mean value, smooth roots and C^r inverse/implicit maps; excludes integration/Taylor/exponential/COV.", "dependencies": [ "L8.3.2", "L8.3.5", "L8.3.6", "L8.3.7", "L8.3.9", "P10.2", "L4.2.4", "L4.2.6", "L8.4.1", "L8.4.2", "L8.4.6", "P10.5-rules", "P8-smooth", "P3" ], "reader": "proof-map.html#F0-DIFF", "is_dependency_contract_not_additional_proof": true }, { "id": "P11.1", "kind": "local_proof_completion", "scope": "Recursive total-derivative C^r agrees with continuous ordered partials, including vector-valued maps.", "dependencies": [ "F0-DIFF", "L8.4.6", "L8.3.9" ], "reader": "integration-prerequisite-completions.html#proof-P11.1", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P11.1." }, { "id": "L5.3.5", "kind": "existing_programme_proof", "scope": "Oriented one-variable substitution, including noninjective g and endpoint range values.", "dependencies": [ "L5.3.1", "L5.3.3", "L5.2.7", "P12.5-extension", "P12.3", "F0-DIFF" ], "reader": "human-prerequisites/sec_ftc.html#proof-L5.3.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_ftc.html", "original_source_page_sha256": "1e3f3c4a7fa5d456fbc17932a1119e8f066372813accf429910ce426d1f28d79" }, { "id": "P12.6", "kind": "local_proof_completion", "scope": "Integration by parts, finite integral Taylor remainder for every N>=1, norm estimate and all-sign scaled Morse identity.", "dependencies": [ "F0-DIFF", "L5.3.1", "L5.2.7", "P12.3", "P12.4" ], "reader": "integration-prerequisite-completions.html#proof-P12.6", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P12.6." }, { "id": "P13.1", "kind": "local_proof_completion", "scope": "Complex field, multiplicative modulus, all sequence limit laws and real-parameter smooth algebraic rules.", "dependencies": [ "E0", "P8-root", "L7.1.4", "P10.1", "F0-DIFF", "P12.3", "P12.4" ], "reader": "exponential-prerequisite-completions.html#proof-P13.1", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P13.1." }, { "id": "P11.2-bridge", "kind": "local_proof_completion", "scope": "Closed rectangle/equal-index cases and the conditional limit argument once the source difference-quotient identities are supplied.", "dependencies": [ "P11.1", "P10.1", "P9.2" ], "reader": "integration-prerequisite-completions.html#proof-P11.2-bridge", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "The programme contains the full double-mean-value proof" }, { "id": "L5.4.1", "kind": "existing_programme_proof", "scope": "Parts (i)–(iv) only: logarithm integral, derivative, strict increase, full range, endpoint limits, product law and uniqueness. Rational powers in (v) excluded.", "dependencies": [ "F0-DIFF", "L5.2.7", "L5.3.3", "L5.3.5", "P12.3", "L4.2.4", "L5.2.6", "L5.1.11", "P6.0", "L3.3.8", "L5.3.1" ], "reader": "human-prerequisites/sec_logandexp.html#proof-L5.4.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_logandexp.html", "original_source_page_sha256": "9c0bb440f05bd6eb694bcfdc2b25f6e9c7748e4e66b22ac83950e37e488581cf" }, { "id": "P13.2-series", "kind": "local_proof_completion", "scope": "Complex Cauchy criterion, absolute convergence, null series terms and bounded convergent sequences.", "dependencies": [ "P13.1", "L7.4.4", "P6.1", "L2.4.5" ], "reader": "exponential-prerequisite-completions.html#proof-P13.2-series", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "The metric completeness" }, { "id": "P17.2", "kind": "local_proof_completion", "scope": "Full omitted linearity and monotonicity; product/modulus closure; all finite-vector/complex norm and uniform-error bounds.", "dependencies": [ "L10.1.6", "L10.1.12", "P12.1", "L7.1.4", "P13.1" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.2", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P17.2." }, { "id": "L8.6.2", "kind": "existing_programme_proof", "scope": "Full two-order mixed-partial identity, with explicit definition/domain/limit bridges.", "dependencies": [ "L4.2.4", "L8.3.6", "P11.1", "P11.2-bridge" ], "reader": "human-prerequisites/sec_mvhighordders.html#proof-L8.6.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvhighordders.html", "original_source_page_sha256": "bce630ed8adbacc726a23ecb4dbe8699abfc1984eab01d4ec26b01170071fe5a" }, { "id": "P14.1", "kind": "local_proof_completion", "scope": "Exact IVT/logarithm binding with endpoint values and orientation explained; no unused rational-root import.", "dependencies": [ "L3.3.8", "L5.4.1" ], "reader": "exponential-prerequisite-completions.html#proof-P14.1", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P14.1." }, { "id": "P14.2-inverse", "kind": "local_proof_completion", "scope": "Use the already proved P3 inverse theorem at each point and identify each smooth local inverse with the global inverse of L.", "dependencies": [ "L5.4.1", "P3", "F0-DIFF" ], "reader": "exponential-prerequisite-completions.html#proof-P14.2-inverse", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "Retain Proposition 5.4.2" }, { "id": "L2.6.5", "kind": "existing_programme_proof", "scope": "Full written Mertens proof; real comparisons only on moduli, with the complex extension stated explicitly in P13.2.", "dependencies": [ "P13.1", "P13.2-series", "P6.1", "P6.2", "L2.4.5" ], "reader": "human-prerequisites/sec_moreonseries.html#proof-L2.6.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_moreonseries.html", "original_source_page_sha256": "25a4e33ae0e02ead2786e85912828edaf9a2b208fed4965e840717139b642f4a" }, { "id": "P17.3", "kind": "local_proof_completion", "scope": "Finite-grid integral additivity, null coordinate faces, full extension-by-zero exercise and independence of the containing rectangle.", "dependencies": [ "L10.1.13", "L10.1.12", "P17.1", "P17.2" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.3", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P17.3." }, { "id": "P17.4", "kind": "local_proof_completion", "scope": "Continuous/smooth compact-support extension, support of derivatives, and finite-parameter continuity of compact rectangle integrals.", "dependencies": [ "L10.1.15", "P17.2", "F0-COMP", "L7.5.5", "L7.4.9" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.4", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "For Exercise 10.1.1" }, { "id": "P11.2-orders", "kind": "local_proof_completion", "scope": "Every permutation of a finite derivative list up to the stated regularity, by adjacent swaps.", "dependencies": [ "L8.6.2", "P11.1" ], "reader": "integration-prerequisite-completions.html#proof-P11.2-orders", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "For a \\(C^k\\) function" }, { "id": "P4", "kind": "local_proof_completion", "scope": "Schur-complement Hessian and determinant identity with smooth mixed-partial symmetry.", "dependencies": [ "P3", "F0-DIFF", "L8.6.2", "L8.2.9" ], "reader": "prerequisite-completions.html#proof-P4", "editable_source": "prerequisite-completions.md", "editable_source_sha256": "3be6c23c56f645049ad6776b63d0f1fccab0e39938d4998f0201b9664cc155af", "locator": "## P4." }, { "id": "L5.4.2", "kind": "existing_programme_proof", "scope": "Parts (i)–(iv) and full uniqueness proof, with smooth inverse supplied by P3. Rational-power part (v) excluded.", "dependencies": [ "L5.4.1", "P14.2-inverse", "L4.2.4", "L4.2.6", "F0-DIFF" ], "reader": "human-prerequisites/sec_logandexp.html#proof-L5.4.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_logandexp.html", "original_source_page_sha256": "9c0bb440f05bd6eb694bcfdc2b25f6e9c7748e4e66b22ac83950e37e488581cf" }, { "id": "P13.2-mertens", "kind": "local_proof_completion", "scope": "Full complex Mertens theorem with one absolutely convergent factor; human proof retained with hypotheses and error choice explicit.", "dependencies": [ "L2.6.5", "P13.2-series" ], "reader": "exponential-prerequisite-completions.html#proof-P13.2-mertens", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "Retain the complete proof of Mertens" }, { "id": "L10.1.19", "kind": "existing_programme_proof", "scope": "Independence of support-containing rectangle, with the exact zero-extension exercise and empty-support case supplied in P17.3.", "dependencies": [ "L10.1.15", "P17.3" ], "reader": "human-prerequisites/sec_rirect.html#proof-L10.1.19", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_rirect.html", "original_source_page_sha256": "137eba850c027b3297d03220ed1389d2b6fa6d986b3283b3648c1c885d3812a2" }, { "id": "P18.1", "kind": "local_proof_completion", "scope": "Absolute improper integral in every finite dimension, vector Cauchy limit, all real radii, vanishing tails, rectangle-exhaustion independence and half-lines.", "dependencies": [ "P17.2", "P17.3", "L5.5.3", "L5.5.4", "L7.4.4", "P6.0", "P12.1" ], "reader": "global-integral-prerequisite-completions.html#proof-P18.1", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P18.1." }, { "id": "P17.5-order", "kind": "local_proof_completion", "scope": "Full version B by exact grid relabelling; continuous vector/complex iteration in every order and product factorization.", "dependencies": [ "L10.2.2", "P17.4", "L10.1.15", "P17.2" ], "reader": "global-integral-prerequisite-completions.html#proof-P17.5-order", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "For the reverse order" }, { "id": "P12.7", "kind": "local_proof_completion", "scope": "Full joint finite-parameter C^r/C-infinity compact integral rule, with vector values and zero cases.", "dependencies": [ "L9.1.1", "P11.1", "P11.2-orders", "F0-COMP", "L5.2.7", "P12.3", "P12.4" ], "reader": "integration-prerequisite-completions.html#proof-P12.7", "editable_source": "integration-prerequisite-completions.md", "editable_source_sha256": "1d3673d38089cd69241bd4db54251b12348f7f1d28cb64b10bbad3b66c1962a3", "locator": "### P12.7." }, { "id": "P14.2-powers", "kind": "local_proof_completion", "scope": "All real derivatives, positive real powers and falling-product derivative formula.", "dependencies": [ "L5.4.2", "L5.4.1", "F0-DIFF", "P8-root" ], "reader": "exponential-prerequisite-completions.html#proof-P14.2-powers", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "For \\(u>0\\)" }, { "id": "P15.1", "kind": "local_proof_completion", "scope": "Absolute and compact-uniform exponential series; all real partials via compact FTC; equality with real exponential and path chain rule.", "dependencies": [ "P13.1", "P13.2-series", "P6.5", "L2.2.11", "P12.4", "L5.3.1", "L5.3.3", "P11.1", "L5.4.2", "F0-DIFF" ], "reader": "exponential-prerequisite-completions.html#proof-P15.1", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P15.1." }, { "id": "L5.5.5", "kind": "existing_programme_proof", "scope": "Full scalar comparison proof, with Cauchy-to-arbitrary-endpoint passage explicitly supplied in P18.1.", "dependencies": [ "L5.5.3", "L5.5.4", "P18.1", "P12.3", "L5.2.6" ], "reader": "human-prerequisites/sec_impropriemann.html#proof-L5.5.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_impropriemann.html", "original_source_page_sha256": "63b18c0708143c25e0b7df2d4c9d9ca82f678486411b3f16b0df51e1c1446610" }, { "id": "COMPACT-RECTANGLE-FUBINI", "kind": "proved_contract", "scope": "Full compact rectangle construction, properties and upper/lower-integral Fubini in both orders.", "dependencies": [ "L10.1.12", "P17.2", "P17.3", "L10.1.15", "P17.5-order" ], "reader": "proof-map.html#COMPACT-RECTANGLE-FUBINI", "is_dependency_contract_not_additional_proof": true }, { "id": "M1", "kind": "local_proof_completion", "scope": "Full smooth parameter Morse normal form, local signed chart, signature and critical-point Jacobian, including dimension zero.", "dependencies": [ "P3", "P4", "P5", "F0-ALG", "F0-DIFF", "P12.6", "P12.7", "P11.2-orders" ], "reader": "parameter-morse-reduction.html#proof-M1", "editable_source": "parameter-morse-reduction.md", "editable_source_sha256": "f4c72c29dd54f9da35deef5026133bd05b22a7f9fe23e21ac42011e06ac59f70", "locator": "## M1." }, { "id": "FTC-TAYLOR-COMPACT-PARAMETERS", "kind": "proved_contract", "scope": "Both fundamental-theorem forms, one-variable substitution, integral Taylor remainder and full smooth parameter integration on a compact interval.", "dependencies": [ "L5.3.1", "L5.3.3", "L5.3.5", "P12.6", "P12.7" ], "reader": "proof-map.html#FTC-TAYLOR-COMPACT-PARAMETERS", "is_dependency_contract_not_additional_proof": true }, { "id": "P14.3", "kind": "local_proof_completion", "scope": "Polynomial/exponential and Gaussian boundary decay; complete smooth flat-cutoff exercise. No improper-integral existence claim.", "dependencies": [ "L5.4.2", "P12.6", "F0-DIFF", "P6.0", "P14.2-powers" ], "reader": "exponential-prerequisite-completions.html#proof-P14.3", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P14.3." }, { "id": "L5.5.2-gt1", "kind": "existing_programme_proof", "scope": "The actually written p>1 infinite-right-endpoint case only; other p-test cases excluded.", "dependencies": [ "P14.2-powers", "L5.4.1", "L5.4.2", "L5.3.1", "P10.1" ], "reader": "human-prerequisites/sec_impropriemann.html#proof-L5.5.2-gt1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_impropriemann.html", "original_source_page_sha256": "63b18c0708143c25e0b7df2d4c9d9ca82f678486411b3f16b0df51e1c1446610" }, { "id": "P19.1", "kind": "local_proof_completion", "scope": "Nonnegative countable sums and double enumeration; monotonicity, finite-cover equivalence, countable subadditivity and null modifications.", "dependencies": [ "J0", "P6.0", "P6.5", "P8-root", "P14.2-powers" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P19.1", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P19.1." }, { "id": "P15.2", "kind": "local_proof_completion", "scope": "Binomial induction and retained Mertens proof establish addition; conjugation, nonvanishing, reciprocal and exact modulus.", "dependencies": [ "P15.1", "P13.2-mertens", "P13.1", "L5.4.2" ], "reader": "exponential-prerequisite-completions.html#proof-P15.2", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P15.2." }, { "id": "M2", "kind": "local_proof_completion", "scope": "Finite precompact parameter cover and derivative bounds for charts and inverses on their compact images.", "dependencies": [ "M1", "F0-COMP", "L7.5.5" ], "reader": "parameter-morse-reduction.html#proof-M2", "editable_source": "parameter-morse-reduction.md", "editable_source_sha256": "f4c72c29dd54f9da35deef5026133bd05b22a7f9fe23e21ac42011e06ac59f70", "locator": "## M2." }, { "id": "M3", "kind": "local_proof_completion", "scope": "Exact curved family, global chart and inverse, critical point, Hessian, Jacobian and degeneration example.", "dependencies": [ "M1", "F0-DIFF", "F0-ALG", "F0-COMP", "P8-smooth" ], "reader": "parameter-morse-reduction.html#proof-M3", "editable_source": "parameter-morse-reduction.md", "editable_source_sha256": "f4c72c29dd54f9da35deef5026133bd05b22a7f9fe23e21ac42011e06ac59f70", "locator": "## M3." }, { "id": "P19.2", "kind": "local_proof_completion", "scope": "Outer measure of closed/open rectangles and finite disjoint open rectangle unions; exact finite-cover/grid argument.", "dependencies": [ "P19.1", "P1", "P17.1", "P10.1" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P19.2", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P19.2." }, { "id": "L10.3.2", "kind": "existing_programme_proof", "scope": "Full small-ball characterization of null sets, with finite subdivision and countable regrouping supplied.", "dependencies": [ "P19.1", "P17.1", "L10.1.14", "P6.0" ], "reader": "human-prerequisites/sec_outermeasure.html#proof-L10.3.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_outermeasure.html", "original_source_page_sha256": "0b9988d7cea5257a15b91186b93ff3e5aef5e726c0337b743a4081419be07582" }, { "id": "L10.3.4", "kind": "existing_programme_proof", "scope": "Countable union theorem with the nonnegative double-series exercise completed in P19.1.", "dependencies": [ "P19.1" ], "reader": "human-prerequisites/sec_outermeasure.html#proof-L10.3.4", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_outermeasure.html", "original_source_page_sha256": "0b9988d7cea5257a15b91186b93ff3e5aef5e726c0337b743a4081419be07582" }, { "id": "P16.1", "kind": "local_proof_completion", "scope": "All complex sine/cosine series and squared/addition identities, real derivatives and norm bounds; all used exercises completed.", "dependencies": [ "P15.1", "P15.2", "P13.2-series", "L4.2.4" ], "reader": "exponential-prerequisite-completions.html#proof-P16.1", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P16.1." }, { "id": "PARAMETER-MORSE", "kind": "proved_contract", "scope": "Complete M1/M2/M3 proof chain, relative to explicit axioms; no Fourier or improper-integral closure.", "dependencies": [ "M1", "M2", "M3" ], "reader": "proof-map.html#PARAMETER-MORSE", "is_dependency_contract_not_additional_proof": true }, { "id": "P19.3", "kind": "local_proof_completion", "scope": "Compact small-ball subcover exercise; coordinate hyperplane and rectangle-face nullity with all union inputs proved.", "dependencies": [ "L10.3.2", "P1", "P19.1", "P10.1" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P19.3", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P19.3." }, { "id": "P16.2", "kind": "local_proof_completion", "scope": "First-zero supremum argument, pi, all return times, and the least positive period of each individual trigonometric function.", "dependencies": [ "P16.1", "P15.2", "P6.0", "P6.2", "P10.1", "L4.2.4" ], "reader": "exponential-prerequisite-completions.html#proof-P16.2", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P16.2." }, { "id": "L10.3.7", "kind": "existing_programme_proof", "scope": "Finite open rectangle and small-ball covers of compact null sets; omitted ball case supplied.", "dependencies": [ "P19.3", "L10.3.2", "P1" ], "reader": "human-prerequisites/sec_outermeasure.html#proof-L10.3.7", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_outermeasure.html", "original_source_page_sha256": "0b9988d7cea5257a15b91186b93ff3e5aef5e726c0337b743a4081419be07582" }, { "id": "P19.4", "kind": "local_proof_completion", "scope": "Complete C1 image theorem for arbitrary null subsets of open sets, including nonclosed relatively compact sets and compact exhaustion.", "dependencies": [ "P19.1", "L10.3.2", "P19.3", "L8.4.2", "P10.4", "F0-COMP", "P9.2-operator" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P19.4", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P19.4." }, { "id": "P20.1", "kind": "local_proof_completion", "scope": "Finite-grid assignments, face covers, Archimedean countable levels and zero-side cases for the full Riemann integrability criterion.", "dependencies": [ "L10.4.1", "L10.4.2", "P17.1", "P19.3", "P19.1", "P6.0", "L10.1.12" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P20.1", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P20.1." }, { "id": "P16.3", "kind": "local_proof_completion", "scope": "All quadrants and axes; bijective angle, smooth arctangent, unique smooth right-half-plane root and both signed boundary phases.", "dependencies": [ "P16.1", "P16.2", "L3.3.8", "P3", "F0-DIFF", "P8-smooth", "P13.1", "P15.2" ], "reader": "exponential-prerequisite-completions.html#proof-P16.3", "editable_source": "exponential-prerequisite-completions.md", "editable_source_sha256": "996a77ee37b3226895e13617205594d44fc9ef2fdcb3887a292a80179329d1f9", "locator": "### P16.3." }, { "id": "L10.3.10", "kind": "existing_programme_proof", "scope": "Full null-image theorem; the omitted noncompact case and positive ball-bound convention are P19.4.", "dependencies": [ "P19.4", "L10.3.7" ], "reader": "human-prerequisites/sec_outermeasure.html#proof-L10.3.10", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_outermeasure.html", "original_source_page_sha256": "0b9988d7cea5257a15b91186b93ff3e5aef5e726c0337b743a4081419be07582" }, { "id": "L10.4.3", "kind": "existing_programme_proof", "scope": "Full iff theorem: bounded Riemann integrability equals null discontinuity set; both directions and omitted grid/zero cases supplied.", "dependencies": [ "P20.1", "L10.4.1", "L10.4.2", "L10.3.7", "L10.1.12", "P19.1", "P1" ], "reader": "human-prerequisites/sec_riemannlebesgue.html#proof-L10.4.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_riemannlebesgue.html", "original_source_page_sha256": "702d47a48eb05b6e87894147772429916ebf92a11fbaa9ec9392eca13a6c81eb" }, { "id": "L11.4.2", "kind": "existing_programme_proof", "scope": "Proposition and retained argument completed by P16.1–P16.3, including its actual exercises, first-zero supremum, return-time existence issue and omitted axes. The following arc-length assertion is excluded.", "dependencies": [ "P16.1", "P16.2", "P16.3" ], "reader": "human-prerequisites/sec_complexexp.html#proof-L11.4.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_complexexp.html", "original_source_page_sha256": "de1190803425074d3671d59c250feb14d69da1db01d0a51ad24be168e5a10e10" }, { "id": "P20.2", "kind": "local_proof_completion", "scope": "All used algebra/max/min/norm, a.e. equality/order, closed-null modification and diffeomorphism composition exercises.", "dependencies": [ "L10.4.3", "L10.3.10", "P17.2", "P19.2", "P19.1", "P9.2" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P20.2", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P20.2." }, { "id": "L10.5.1", "kind": "existing_programme_proof", "scope": "Jordan measurability iff null boundary, using an interior containing rectangle.", "dependencies": [ "L10.4.3", "P17.3", "J0" ], "reader": "human-prerequisites/sec_jordansets.html#proof-L10.5.1", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_jordansets.html", "original_source_page_sha256": "458617d9f14cb70f87bff3bafb46c662f2d31e4e1cd2c194f427b2edf3ec4221" }, { "id": "EXPONENTIAL-TRIGONOMETRIC", "kind": "proved_contract", "scope": "Real and complex exponential, trigonometric functions, pi and the exact Gaussian root phase.", "dependencies": [ "L5.4.2", "P14.2-powers", "P15.2", "L11.4.2", "P16.3" ], "reader": "proof-map.html#EXPONENTIAL-TRIGONOMETRIC", "is_dependency_contract_not_additional_proof": true }, { "id": "P20.3", "kind": "local_proof_completion", "scope": "All Jordan closure/interior/union/intersection/difference exercises, null-boundary volume equality, and zero-cell case in volume/outer measure.", "dependencies": [ "L10.5.1", "P20.2", "P19.1", "P19.2", "P17.1" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P20.3", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P20.3." }, { "id": "L10.5.5", "kind": "existing_programme_proof", "scope": "Every bounded continuous function on a bounded Jordan set is Riemann integrable; full proof.", "dependencies": [ "L10.5.1", "L10.4.3", "P17.3" ], "reader": "human-prerequisites/sec_jordansets.html#proof-L10.5.5", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_jordansets.html", "original_source_page_sha256": "458617d9f14cb70f87bff3bafb46c662f2d31e4e1cd2c194f427b2edf3ec4221" }, { "id": "L10.5.9", "kind": "existing_programme_proof", "scope": "Full compact Jordan image theorem, with the local inverse theorem already closed; source f|V is the map g|V.", "dependencies": [ "L7.5.5", "L7.4.9", "L8.5.1", "P3", "L10.5.1", "P19.4" ], "reader": "human-prerequisites/sec_jordansets.html#proof-L10.5.9", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_jordansets.html", "original_source_page_sha256": "458617d9f14cb70f87bff3bafb46c662f2d31e4e1cd2c194f427b2edf3ec4221" }, { "id": "F0-CALC", "kind": "proved_contract", "scope": "The exact elementary differential/FTC/Taylor/exponential/root inputs of Q1–Q9 and the smooth flat cutoff, relative to declared axioms.", "dependencies": [ "F0-DIFF", "FTC-TAYLOR-COMPACT-PARAMETERS", "EXPONENTIAL-TRIGONOMETRIC", "P14.3" ], "reader": "proof-map.html#F0-CALC", "is_dependency_contract_not_additional_proof": true }, { "id": "L10.5.3", "kind": "existing_programme_proof", "scope": "Full equality of Jordan volume and outer measure, with the finite-disjoint-rectangle and empty-family steps supplied.", "dependencies": [ "P20.3", "P19.2", "P17.1", "L10.1.12" ], "reader": "human-prerequisites/sec_jordansets.html#proof-L10.5.3", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_jordansets.html", "original_source_page_sha256": "458617d9f14cb70f87bff3bafb46c662f2d31e4e1cd2c194f427b2edf3ec4221" }, { "id": "P21.2", "kind": "local_proof_completion", "scope": "Restrict to nonzero-Jacobian neighbourhood before covers; glue inverse charts; prove compact Jordan covers, norm-open sets and balanced refinements.", "dependencies": [ "P3", "L8.5.1", "P19.4", "P20.3", "P9.2", "P9.3", "F0-COMP", "P6.0", "L10.1.14", "F0-ALG" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.2", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.2." }, { "id": "P20.4", "kind": "local_proof_completion", "scope": "Complete Jordan-integral properties, finite additivity including null overlaps, and compact-Jordan pullback version of Exercise 10.5.7.", "dependencies": [ "P20.2", "P20.3", "L10.5.5", "L10.4.3", "P19.4", "P17.2", "P17.3" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P20.4", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P20.4." }, { "id": "P20.5", "kind": "local_proof_completion", "scope": "Boundary-image inclusion and translation invariance of Jordan volume directly from translated Darboux grids.", "dependencies": [ "L10.5.9", "P17.1", "P20.3", "P12.1" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P20.5", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P20.5." }, { "id": "P18.2", "kind": "local_proof_completion", "scope": "Global linearity, norm/comparison bounds, exact compact-uniform limit rule and the full stated differentiation rule of Q1.", "dependencies": [ "P18.1", "P17.2", "L5.5.5", "F0-CALC", "F0-COMP" ], "reader": "global-integral-prerequisite-completions.html#proof-P18.2", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P18.2." }, { "id": "U001-A4", "kind": "local_proof_completion", "scope": "Flat bump, subordinate finite partition, zero extension and product-parameter localization, without a partition theorem import.", "dependencies": [ "P14.3", "P1", "P17.4", "F0-CALC", "F0-COMP" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A4", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.4. Finite smooth cutoffs" }, { "id": "U001-4.1", "kind": "local_proof_completion", "scope": "Smooth critical branch, signed full parameter Morse coordinates, determinant relation and compact inverse-image bounds.", "dependencies": [ "M1", "M2", "P3", "P4", "P5", "P12.7", "F0-ALG", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-4.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Lemma 4.1 (Morse coordinates with parameters)." }, { "id": "P21.1", "kind": "local_proof_completion", "scope": "Full determinant-volume exercise: shear and dilation sections, permutations, elementary matrix factorization, singular maps and all rectangle boundary conventions.", "dependencies": [ "P20.5", "P20.3", "P20.2", "P17.5-order", "P19.4", "P19.3", "P9.4", "P9.1", "L8.1.18", "L8.2.8", "L8.2.9", "P7" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.1", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.1." }, { "id": "P18.3", "kind": "local_proof_completion", "scope": "Product weights, sharp s>n rectangular-shell decay, Schwartz and Gaussian majorants; no annulus volume formula assumed.", "dependencies": [ "L5.5.2-gt1", "P18.1", "P18.2", "P17.5-order", "P14.3", "P14.2-powers", "P6.5" ], "reader": "global-integral-prerequisite-completions.html#proof-P18.3", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P18.3." }, { "id": "Q1", "kind": "local_proof_completion", "scope": "The entire compact-uniform dominated limit and parameter-differentiation argument, with all used global integral inputs closed.", "dependencies": [ "P18.2", "F0-CALC", "F0-COMP" ], "reader": "quadratic-stationary-phase.html#proof-Q1", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q1." }, { "id": "L10.7.2", "kind": "existing_programme_proof", "scope": "Full compact Jordan change-of-variables theorem for Riemann-integrable amplitudes. The neighbourhood restriction and every used exercise are supplied by P19–P21.2.", "dependencies": [ "P21.1", "P21.2", "P20.4", "P20.5", "L5.3.1", "P12.5-endpoints", "P12.4", "F0-ALG", "L10.1.12" ], "reader": "human-prerequisites/sec_mvchangeofvars.html#proof-L10.7.2", "editable_source": "human-prerequisites/source-selections.json", "human_author": "Jiří Lebl", "licence": "CC-BY-SA-4.0", "free_source_url": "https://www.jirka.org/ra/html/sec_mvchangeofvars.html", "original_source_page_sha256": "ffdb891e94888fd91da33334e29fbada7376a5f42da566c155c77cb795879f60" }, { "id": "P18.4", "kind": "local_proof_completion", "scope": "Global continuous Fubini with an explicit integrable product majorant, all section integrals, both orders and actual Gaussian/Schwartz applications.", "dependencies": [ "P18.1", "P18.2", "P18.3", "P17.5-order", "P17.4", "P15.1" ], "reader": "global-integral-prerequisite-completions.html#proof-P18.4", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P18.4." }, { "id": "U001-E9.3", "kind": "local_proof_completion", "scope": "Uniform compact convergence at t=lambda^-2 and contradiction to an unqualified uniform bound.", "dependencies": [ "Q1", "P17.2", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-E9.3", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Exercise 9.3 (failure of uniform nondegeneracy; intermediate)." }, { "id": "P21.3", "kind": "local_proof_completion", "scope": "Complete retained substitution argument with exact normalized error-box estimate, null-overlap summation, reverse inequality and complex/vector extension.", "dependencies": [ "L10.7.2", "P21.1", "P21.2", "P20.4", "P20.5", "L5.3.1", "P12.5-endpoints", "P12.4", "F0-ALG" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.3", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.3." }, { "id": "P18.5", "kind": "local_proof_completion", "scope": "Exact Fourier differentiation/one-coordinate integration-by-parts passages and compact-support nonstationary integrations.", "dependencies": [ "P18.2", "P18.3", "P18.4", "P17.4", "P17.5-order", "F0-CALC" ], "reader": "global-integral-prerequisite-completions.html#proof-P18.5", "editable_source": "global-integral-prerequisite-completions.md", "editable_source_sha256": "b147b72a5d5983caae9e74f2c4d670396211bbe30300a227c9ecd04df45b8d09", "locator": "### P18.5." }, { "id": "F0-INT", "kind": "proved_contract", "scope": "The compact and absolutely convergent improper integral/norm/tail/iteration inputs used by Q1–Q9. Global interchanges have the exact product-majorant hypotheses in P18.4, not an unsupported all-sections assertion.", "dependencies": [ "P18.1", "P18.2", "P18.3", "P18.4" ], "reader": "proof-map.html#F0-INT", "is_dependency_contract_not_additional_proof": true }, { "id": "P21.4", "kind": "local_proof_completion", "scope": "Absolute-improper Jordan exhaustion, all invertible affine changes, and compact chart substitution with zero extension outside the chosen chart.", "dependencies": [ "P21.3", "P18.1", "P18.2", "P20.4", "P21.2", "P17.4", "L7.5.5", "P9.2" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.4", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.4." }, { "id": "P21.5", "kind": "local_proof_completion", "scope": "Polar injectivity/Jacobian, sector substitution, missing-ray/origin/infinity exhaustion and positive Gaussian normalization.", "dependencies": [ "P21.3", "P20.3", "P20.4", "P20.5", "P17.5-order", "P18.1", "P18.3", "P18.4", "P16.2", "P16.3", "P14.3", "P8-root", "L5.3.1", "P12.5-endpoints" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.5", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.5." }, { "id": "JORDAN-SUBSTITUTION", "kind": "proved_contract", "scope": "Full compact Jordan C1 substitution with arbitrary Riemann-integrable amplitudes and all used null-set/Jordan prerequisites.", "dependencies": [ "P21.3", "L10.5.3" ], "reader": "proof-map.html#JORDAN-SUBSTITUTION", "is_dependency_contract_not_additional_proof": true }, { "id": "Q3", "kind": "local_proof_completion", "scope": "Full Schwartz Fourier estimates, signs, seminorm continuity and sharp stated weighted L1 threshold.", "dependencies": [ "Q1", "P18.3", "P18.4", "P18.5", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-Q3", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q3." }, { "id": "Q9", "kind": "local_proof_completion", "scope": "Full nonstationary integration-by-parts estimate, support, finite-order family bounds and parameter costs.", "dependencies": [ "P18.5", "P17.2", "P17.4", "P17.5-order", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-Q9", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q9." }, { "id": "U001-A5", "kind": "local_proof_completion", "scope": "Critical Hessian and quotient form, ordinary density algebra, exact-sequence lift/frame invariance, chart-independent integration and Euclidean normal formula.", "dependencies": [ "U001-DEF", "U001-A4", "P2", "P5", "P8-smooth", "P9.4", "L8.2.8", "L8.2.9", "P21.4", "P20.4", "F0-CALC", "F0-COMP" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A5", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.5. Hessians and quotient densities in full" }, { "id": "Q2", "kind": "local_proof_completion", "scope": "Full right-half-plane complex Gaussian by path and real-frequency differential identities, without an analytic-continuation import.", "dependencies": [ "Q1", "P21.5", "P18.3", "P18.1", "P16.3", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-Q2", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q2." }, { "id": "F0-COV", "kind": "proved_contract", "scope": "All actual affine, orthogonal, dilation, compact-chart and polar changes used by the reconstruction.", "dependencies": [ "P21.4", "P21.5" ], "reader": "proof-map.html#F0-COV", "is_dependency_contract_not_additional_proof": true }, { "id": "GLOBAL-INTEGRAL-Q1-Q3-Q9", "kind": "proved_contract", "scope": "Exact selected chain for Q1, Q3 and Q9; Q5 was closed earlier.", "dependencies": [ "Q1", "Q3", "Q9" ], "reader": "proof-map.html#GLOBAL-INTEGRAL-Q1-Q3-Q9", "is_dependency_contract_not_additional_proof": true }, { "id": "U001-2.1", "kind": "local_proof_completion", "scope": "Full transpose with divergence, all M,r estimates and M+2r amplitude derivative count.", "dependencies": [ "U001-DEF", "Q9", "P17.5-order", "P12.7", "F0-CALC", "F0-COMP" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-2.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Proposition 2.1 (nonstationary estimate)." }, { "id": "Q4", "kind": "local_proof_completion", "scope": "Full Schwartz Fourier inversion, including Gaussian approximate identity, first moment, affine substitutions and derivatives.", "dependencies": [ "Q1", "Q2", "Q3", "P21.4", "P18.3", "P18.4", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-Q4", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q4." }, { "id": "U001-E9.5", "kind": "local_proof_completion", "scope": "Noncritical extra variable and all-order compact integration-by-parts estimate.", "dependencies": [ "U001-2.1", "P17.5-order", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-E9.5", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Exercise 9.5 (an absent critical direction; advanced)." }, { "id": "Q6", "kind": "local_proof_completion", "scope": "Exact quadratic Fourier identity with both regularization limits, all root phases, orthogonal changes and absolute majorants.", "dependencies": [ "Q1", "Q2", "Q3", "Q4", "Q5", "P21.4", "P16.3", "P18.4", "F0-ALG" ], "reader": "quadratic-stationary-phase.html#proof-Q6", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q6." }, { "id": "U001-A1", "kind": "local_proof_completion", "scope": "Monomial and radial Schwartz seminorm equivalence, Gaussian normalization, Fourier Schwartz bounds, full inversion and all actual Fubini/tail justifications.", "dependencies": [ "U001-DEF", "Q1", "Q2", "Q3", "Q4", "P18.3", "P18.4", "P21.4", "P21.5", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.1. Gaussian integral and Fourier inversion" }, { "id": "Q7", "kind": "local_proof_completion", "scope": "Every stated coefficient and finite-order remainder with the original sharp M>N+n/2 threshold.", "dependencies": [ "Q3", "Q4", "Q6", "P12.6", "F0-CALC", "P9.2" ], "reader": "quadratic-stationary-phase.html#proof-Q7", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q7." }, { "id": "U001-A2", "kind": "local_proof_completion", "scope": "Plancherel on Schwartz space, damped Gaussian pairing, continuous transposed Fourier transform and exact tempered identity.", "dependencies": [ "U001-A1", "Q2", "Q5", "Q6", "Q1", "P18.3", "P18.4", "P21.4", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A2", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.2. Plancherel and the pairing calculation" }, { "id": "Q8", "kind": "local_proof_completion", "scope": "Full normalized smooth parameter/Hessian family and fixed-order differentiated remainder scope.", "dependencies": [ "Q1", "Q3", "Q4", "Q6", "Q7", "P2", "F0-COMP", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-Q8", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## Q8." }, { "id": "E1", "kind": "local_proof_completion", "scope": "Exact Gaussian model, branch identity, complete moment/derivative inductions and all stated normalized error bounds.", "dependencies": [ "Q2", "Q7", "P16.3", "P18.1", "F0-CALC" ], "reader": "quadratic-stationary-phase.html#proof-E1", "editable_source": "quadratic-stationary-phase.md", "editable_source_sha256": "f1a1be3656bb1bd712c1fb3082a653ff74f3f8104f395aab239292e842950114", "locator": "## E1." }, { "id": "U001-A3", "kind": "local_proof_completion", "scope": "Integral Cauchy–Schwarz including zero-norm case, sharp 2N+ell derivative bound, compact outer-measure support factor and all parameter derivatives.", "dependencies": [ "U001-A1", "U001-A2", "P18.1", "P18.2", "P18.3", "P19.1", "P19.2", "P1", "P20.3", "P20.4", "P8-root", "F0-ALG" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A3", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.3. The weighted estimate with its precise derivative count" }, { "id": "U001-3.1", "kind": "local_proof_completion", "scope": "Full regularized Fresnel identity, nonvanishing-candidate ODE argument, both signs, distributional limit and every determinant factor.", "dependencies": [ "U001-A1", "U001-A2", "Q1", "Q2", "Q5", "Q6", "P16.3", "P21.4", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-3.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Lemma 3.1 (Fresnel transform)." }, { "id": "P21.6", "kind": "local_proof_completion", "scope": "Exact closure of Q1–Q9 and E1; preserves the unresolved full original U001 lesson and AN04 course scope.", "dependencies": [ "Q1", "Q2", "Q3", "Q4", "Q5", "Q6", "Q7", "Q8", "Q9", "E1" ], "reader": "change-of-variables-prerequisite-completions.html#proof-P21.6", "editable_source": "change-of-variables-prerequisite-completions.md", "editable_source_sha256": "25b2b49007764eb3842d5bc6a648cd30e3fe71839287aaf454d16dc88ad57e54", "locator": "### P21.6." }, { "id": "U001-3.2", "kind": "local_proof_completion", "scope": "All N>=0 coefficients and sharp 2N+ell amplitude-derivative remainder, ell>d/2, exact inverse-Hessian power and fixed compact support.", "dependencies": [ "U001-3.1", "U001-A1", "U001-A2", "U001-A3", "P12.6", "P9.2-operator", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-3.2", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Theorem 3.2 (quadratic expansion)." }, { "id": "QUADRATIC-Q1-Q9-E1", "kind": "proved_contract", "scope": "Complete selected quadratic module; the full original stationary-phase lesson and course remain open.", "dependencies": [ "P21.6" ], "reader": "proof-map.html#QUADRATIC-Q1-Q9-E1", "is_dependency_contract_not_additional_proof": true }, { "id": "U001-5.1", "kind": "local_proof_completion", "scope": "Full isolated stationary expansion, C_j derivative order, normalized all-parameter remainders, finite local coordinate gluing and coefficient uniqueness.", "dependencies": [ "U001-2.1", "U001-3.2", "U001-4.1", "U001-A4", "P21.4", "P17.4", "P12.7", "M2", "F0-COMP", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-5.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Theorem 5.1 (uniform real stationary phase)." }, { "id": "U001-6.1", "kind": "local_proof_completion", "scope": "All real symbol orders, all scale/parameter derivatives, explicit differentiated exponential remainder, far-part control and coefficient convolution.", "dependencies": [ "U001-5.1", "U001-A3", "P12.6", "P14.2-powers", "Q1", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-6.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Corollary 6.1 (symbol remainder)." }, { "id": "U001-E9.1", "kind": "local_proof_completion", "scope": "Quartic perturbation, inverse Taylor coefficients and first correction including negative kappa support restriction.", "dependencies": [ "U001-3.2", "U001-5.1", "P3", "P12.6", "P8-smooth", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-E9.1", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Exercise 9.1 (first correction; introductory)." }, { "id": "U001-E9.2", "kind": "local_proof_completion", "scope": "Exact parameter derivative of the critical-value factor and normalized remainder comparison.", "dependencies": [ "U001-5.1", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-E9.2", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Exercise 9.2 (parameter derivatives; intermediate)." }, { "id": "U001-A6", "kind": "local_proof_completion", "scope": "Finite critical components on compact support, constant phase/signature, cutoffs after Morse refinement, tangential integration, zero-normal case and uniform parameter/symbol bounds.", "dependencies": [ "U001-A4", "U001-A5", "U001-2.1", "U001-3.2", "U001-4.1", "U001-5.1", "U001-6.1", "P17.5-order", "P12.7", "P21.4", "P17.4", "L5.3.1", "P12.5-endpoints", "F0-COMP", "M2" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-A6", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "### A.6. Finite clean localization and its uniform bounds" }, { "id": "U001-7.2", "kind": "local_proof_completion", "scope": "Full clean expansion on each component, intrinsic arbitrary-density leading term, normal-dimension exponent, finite sum and differentiated normalized uniform estimates.", "dependencies": [ "U001-A5", "U001-A6", "U001-3.2", "U001-6.1" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-7.2", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Theorem 7.2 (clean stationary phase)." }, { "id": "U001-MODELS", "kind": "local_proof_completion", "scope": "Indefinite critical point, stationary line, two approaching critical points, exact signs/constants and loss of uniformity.", "dependencies": [ "U001-3.2", "U001-5.1", "U001-7.2", "F0-CALC", "F0-ALG" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-MODELS", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "## 8. Three models worth remembering" }, { "id": "U001-E9.4", "kind": "local_proof_completion", "scope": "Positive normal family leading density, exact coordinate Jacobian and lambda^-3/2 remainder.", "dependencies": [ "U001-7.2", "U001-3.2", "P21.4", "P8-smooth", "F0-CALC" ], "reader": "stationary-phase-and-critical-manifolds.html#proof-U001-E9.4", "editable_source": "stationary-phase-and-critical-manifolds.md", "editable_source_sha256": "ea60b5824238fd1aa0e8a97244b4711899b1bd046aa12cc601d1082c583da88a", "locator": "**Exercise 9.4 (clean density; advanced)." }, { "id": "U001-FULL-STATIONARY", "kind": "proved_contract", "scope": "Entire receiving lesson Sections 1–9 and Appendices A.1–A.6, including all stated sharp, parameter, symbol and clean-density results and solved examples/exercises, relative to declared axioms and the exact selected earlier programme proofs.", "dependencies": [ "U001-A1", "U001-A2", "U001-A3", "U001-A4", "U001-2.1", "U001-3.1", "U001-3.2", "U001-4.1", "U001-5.1", "U001-6.1", "U001-A5", "U001-A6", "U001-7.2", "U001-MODELS", "U001-E9.1", "U001-E9.2", "U001-E9.3", "U001-E9.4", "U001-E9.5" ], "reader": "proof-map.html#U001-FULL-STATIONARY", "is_dependency_contract_not_additional_proof": true } ], "public_release_authorized": false }