「theorem」の検索結果

Fix missing theorem proof terms when analyzing Lean 4 environments via importModules. Use when: (1) ConstantInfo.value? returns none for theorems despite TheoremVal.value being Expr, (2) building code graph / dependency extractor for Lean 4 and getting 0 proof dependency edges, (3) Lean 4.30+ project where theorem proofs appear missing from loaded environment, (4) analyzing Mathlib or any Lean 4 project and proof terms are empty. Root cause: breaking change in Lean 4.30 — value? treats theorems as opaque by default.

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Generate structured proof skeletons with tactics, strategies, and intermediate lemmas for theorems in Isabelle/HOL or Coq. Use when users need to: (1) Create proof outlines for theorem statements, (2) Generate proof structure with tactic placeholders, (3) Identify key lemmas needed for a proof, (4) Plan proof strategies (induction, case analysis, forward/backward reasoning), (5) Scaffold proofs with intermediate steps and subgoals, or (6) Convert theorem statements into detailed proof templates. Supports both Isabelle/HOL and Coq equally.

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Prove geometric theorems using Euclidean axiomatic methods, coordinate geometry, or vector methods with rigorous step-by-step logical structure. Covers direct proof, proof by contradiction, coordinate proofs, vector proofs, and handling of special cases and degenerate configurations. Use when given a geometric statement to prove, verifying a conjecture, establishing a lemma, converting geometric intuition into a rigorous proof, or comparing the effectiveness of different proof methods.

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Deploy differential privacy in production systems including epsilon selection strategies, noise calibration with Laplace and Gaussian mechanisms, privacy budget tracking, composition theorems, and Python implementation patterns. Covers both central and local differential privacy models.

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Family-G proof & derivation integrity forensics: does a THIRD PARTY's written proof/derivation actually establish its theorem, or does it skip an obligation, assume its own conclusion, take an invalid step, drift a symbol's meaning, or smuggle an unstated assumption? Decides from the WRITTEN proof/derivation — verdict-bearing at L1 (the LaTeX source; PDF-extracted math is unreliable, so an L0 PDF-only run surfaces info only) — never asserts 'fabricated', only that the step shown does not hold. A fresh cross-model reviewer reads the theorem/proof + an extraction-only obligation scaffold and proposes per-obligation findings, each span-anchored to the evidence ledger (claims.json); reviewer≠adjudicator. Emits proof-derivation-forensics.findings.json; NEVER computes the verdict. dimension=proof, can be critical. Triggers: "proof forensics", "check this proof", "derivation integrity", "audit the math", "证明审计", "推导有没有漏洞".

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Calculus, series, numerical methods, and optimization for computational mathematics. Covers limits (epsilon-delta, L'Hopital, squeeze theorem), derivatives (rules, optimization, related rates), integrals (Riemann sums, Fundamental Theorem of Calculus, integration techniques), series (convergence tests, Taylor and Maclaurin expansions), differential equations (separable, linear, exact), numerical methods (Newton-Raphson, Euler method, Simpson's rule), and computational pitfalls (floating-point arithmetic, catastrophic cancellation, numerical stability). Use when computing derivatives, integrals, limits, series, solving differential equations, or analyzing numerical accuracy.

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Deep symmetries and unifying principles — gauge theory, Lie groups, Noether's theorem, and the Standard Model. Constructs field theories from symmetry requirements, derives conservation laws, and traces force unification. Use when working with gauge symmetry, Lie groups (U(1), SU(2), SU(3)), conservation laws via Noether's theorem, the Higgs mechanism, or Standard Model structure.

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Derive a theoretical result step-by-step from first principles or established theorems, with every step explicitly justified and special cases checked. Use when deriving a formula or theorem from first principles, proving a mathematical statement by logical deduction, re-deriving a textbook result for verification or adaptation, extending a known result to a more general setting, or producing a self-contained derivation for a paper or thesis.

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Analyze magnetic levitation systems by applying Earnshaw's theorem to determine whether passive static levitation is possible, then identifying the appropriate circumvention mechanism (diamagnetic, superconducting, active feedback, or spin-stabilized). Use when evaluating maglev transport, magnetic bearings, superconducting levitation, diamagnetic suspension, or Levitron-type devices. Covers force balance calculations, stability analysis in all spatial and tilting modes, and Meissner effect versus flux pinning distinctions.

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"Apply the Modigliani-Miller theorem to analyze capital structure decisions and identify when financing choices affect firm value. Use this skill when the user needs to evaluate debt-equity tradeoffs, assess the impact of leverage on firm value, understand tax shield benefits, or when they ask 'does capital structure matter', 'should we take on more debt', or 'what is the optimal leverage ratio'.".

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Verify and write proofs using the Acorn theorem prover for mathematical and cryptographic formalization. Use when working with Acorn proof files (.ac), verifying theorems, formalizing mathematical or cryptographic protocols, or writing proofs in the Acorn language. Triggers on: (1) Creating or editing .ac files, (2) Running acorn verify commands, (3) Formalizing math or crypto proofs, (4) Questions about Acorn syntax or standard library.

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Expert-level distributed systems theory covering consistency models, consensus algorithms, fault tolerance, replication, distributed transactions, and the CAP theorem.

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Expert-level complex analysis knowledge. Use when working with complex functions, holomorphic functions, Cauchy's theorem, contour integration, Laurent series, residues, conformal mappings, or analytic continuation. Also use when the user mentions 'holomorphic', 'analytic', 'Cauchy theorem', 'residue', 'contour integral', 'Laurent series', 'conformal map', 'Riemann mapping', 'Mobius transformation', 'harmonic function', 'entire function', or 'meromorphic'.

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Expert-level mathematical logic knowledge. Use when working with propositional logic, first-order logic, proof theory, model theory, computability theory, Godel incompleteness theorems, set theory, type theory, or formal verification. Also use when the user mentions 'formal proof', 'Godel', 'incompleteness', 'Turing machine', 'decidability', 'Zermelo-Fraenkel', 'axiom of choice', 'model theory', 'completeness theorem', 'computability', 'formal language', or 'type theory'.

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Expert-level number theory knowledge. Use when working with divisibility, prime numbers, modular arithmetic, congruences, Diophantine equations, cryptography, quadratic residues, or analytic number theory. Also use when the user mentions 'prime', 'divisibility', 'modular arithmetic', 'congruence', 'GCD', 'Euler totient', 'Fermat little theorem', 'Chinese remainder theorem', 'quadratic residue', 'Diophantine equation', 'RSA', or 'prime factorization'.

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Expert en systèmes distribués (CAP theorem, consensus, partitions, vector clocks)

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Structure any communication artifact — documents, reports, plans, issue comments, PR descriptions, emails, presentations, slides, charts, and dashboards — using Jean-luc Doumont's "Trees, Maps, and Theorems" method. Use whenever the goal is to get a message across to a busy audience, when asked to improve clarity or structure of writing or slides, or when a deliverable will be judged by whether readers act on it, not just read it.

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严格自然语言数学证明。用于证明或审查 theorem/lemma/proposition、补齐证明草稿、构建证明义务与依赖图、寻找反例、检查量词/常数/边界情况，或判断命题是否必须削弱。

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Develop and verify a mathematical proof in Lean, continue an incomplete Lean project, or audit whether it proves the original statement. Connect actual inputs to intermediate lemmas, assemble the target theorem, check its transitive axioms, and provide a reproducible handoff. Use when Lean is requested or a specific proof obligation benefits from formal verification; use proof-writer for ordinary mathematical drafting.

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Writes rigorous mathematical proofs for ML/AI theory. Use when asked to prove a theorem, lemma, proposition, or corollary, fill in missing proof steps, formalize a proof sketch, 补全证明, 写证明, 证明某个命题, or determine whether a claimed proof can actually be completed under the stated assumptions.

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Structures and derives research formulas when the user wants to 推导公式, build a theory line, organize assumptions, turn scattered equations into a coherent derivation, or rewrite theory notes into a paper-ready formula document. Use when the derivation target is not yet fully fixed, the main object still needs to be chosen, or the user needs a coherent derivation package rather than a finished theorem proof.

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Writes and reviews structured Lean 4 proofs and designs Lean libraries following Mathlib conventions. Use when proving theorems in Lean, formalizing mathematics or specifications in Lean 4, defining new types or definitions in a Lean library, reviewing Lean proofs for readability and maintainability, refactoring long tactic proofs into lemmas, filling in sorry placeholders in a Lean development, setting up CI or linters for a Lean project, diagnosing slow proofs or maxHeartbeats timeouts, or writing custom tactics, macros, or linters.

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Formal theorem proving with research, testing, and verification phases

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Use when working with Lean 4 (.lean files), writing mathematical proofs, seeing "failed to synthesize instance" errors, managing sorry/axiom elimination, or searching mathlib for lemmas - provides build-first workflow, haveI/letI patterns, compiler-guided repair, and LSP integration

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