Qedra

Partial fraction decomposition of a rational function with respect to 'var': 1/(x^2-1) gives 1/(2(x-1)) - 1/(2(x+1)). Not a rational function in 'var' (e.g. contains sin(x)): status "error" or unchanged input. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Query the Lisp source of the CAS library (lib/) instead of reading whole files; for developers and agents extending the library. kind: def (location, signature, doc comment and source of NAME), sig (one-line signature + module + first doc line; builtins too), refs (where NAME is used), callers / callees (direct, depth for transitive), module (exports and module dependencies), outline (definitions of a file with line numbers), search (top-10 definitions by name/doc words, Russian or English), deps (external names a file uses, by module), unused (definitions never referenced). Names may be qualified: module::name. Result: plain text (default, compact) or JSON text with json=true. An unknown name returns an error with similar names.

List the active declarations of the session, one "x ∈ D" per line (also given in 'assumptions' of every response). Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Declare a property of a symbol for the whole session: its domain or interval. It then affects simplify (sqrt(x^2) becomes x for x in positive), solve, integrate and the signs of expressions. Declarations of one symbol accumulate (they intersect): call again to add a constraint. Returns "x ∈ D"; 'assumptions' lists all active declarations. Undeclared symbols are complex numbers. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Differentiate an expression; the result is simplified. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Check whether two expressions are equal. result is "true", "false" or "unknown" (status "answer"). true: proved symbolically (the difference simplifies to 0); holds where both expressions are defined. false: a counterexample is given in 'note' (a point where the values differ; sample points respect declarations, undeclared symbols are sampled as real numbers). unknown: not proved, and the numeric checks at random points agreed (this is NOT a proof); verify by other means or say that you could not decide. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Evaluate Lisp (Scheme-like) code in the persistent session; one or more forms, the value of each is returned (one line per form). On error, 'stack' holds the Lisp backtrace. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Expand products and integer powers of sums into a sum of monomials (collects like terms). Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Factor a polynomial over the integers into irreducible factors (also a rational function: numerator and denominator). Irreducible or non-polynomial input is returned unchanged with status "answer". Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Reduced Groebner basis of the ideal generated by polynomials, as a vector of monic polynomials. 'order' is the monomial order: lex (default; useful for elimination and solving), grlex, grevlex (usually fastest). Large systems (e.g. cyclic-6) can be slow: use timeout_s; status "timeout" means no answer, not an empty basis. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Return the on-demand LLM guide (markdown). Without 'topic': the short core (input syntax, how to read the response: status / conditions / certificates, which tool when) and the list of topic cards (the same text is the MCP resource lisp://guide). With 'topic' (e.g. "integration", "ode", "linear-algebra", "sums"): a card with 5-15 functions with exact signatures, runnable examples and pitfalls. A name from other systems (Mathematica, Maple) ("DSolve", "FactorInteger", "Eigenvalues", ...) or a tool name is accepted and returns the right card; topic "synonyms" is the whole table. An unknown topic returns an error with suggestions. Read the core once before the first calculation.

Integrate an expression: indefinite, or definite when both 'from' and 'to' are given. If no antiderivative is found, status is "unknown" and result is the unevaluated integral; conditions list parameter assumptions used. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Limit of an expression as var approaches point (a number, or ∞ / -∞). Without 'side' it is the two-sided limit. If the limit is not found (it may exist, or may not, e.g. sin(1/x) at 0), status is "unknown" and result is the unevaluated limit(...); do not guess the value. Result may be ∞, -∞ or undefined. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Exact linear algebra on a matrix (step LA.4): Jordan normal form, matrix exponential exp(A·t), symbolic matrix power A^n, a function of a matrix f(A), diagonalization, eigenvalues/eigenvectors. The matrix is given as "[[2,1],[0,2]]" in the math language. fn selects the operation: jordan_form (returns [P, J] with A = P·J·P⁻¹), jordan_blocks, mat_exp (var defaults to the built-in t; verified by d/dt E = A·E and E(0) = I), mat_power (var defaults to n; nilpotent Jordan blocks at λ=0 are written as zeros with the condition n ≥ s in 'conditions'), mat_fn (func, e.g. "exp" or a lambda), diagonalize ([P, D]), is_diagonalizable, eigenvalues, eigenvectors. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Linear and integer linear programming (step LP.3) with exact rational arithmetic: the two-phase simplex method (Bland's rule) for LP, branch and bound for ILP. Maximize or minimize a linear objective cᵀx over linear constraints. The answer always carries a certificate verified by lp-check (independent: only multiplications and comparisons): optimal — x, value and the dual prices y with cᵀx = value = bᵀy; infeasible — a Farkas lemma certificate; unbounded — a feasible point and a ray; integer — re-solving the branch-and-bound tree. 'integer' is a vector of variable classes "int"/"bin"/"free"; when absent the problem is a plain LP. answers: (optimal x value y), (infeasible y), (unbounded x d), (unknown x value nil) when the node limit is exceeded. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Run a program in the CAS math language in the persistent session: statements separated by ';' or newlines, assignments 'a := 2', functions 'f(x) := x^2', domain declarations 'x :∈ ℝ', and all CAS functions (simplify, expand, factor, diff, integrate, solve, limit, series, ...). result/pretty contain one line per statement; print() output goes to 'output'. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Reset the session: forget all variables, functions and domain declarations. Returns status "answer".

Taylor/Laurent series of an expression around a point, up to the given order (the O(...) remainder is included in the result). Series that the CAS cannot compute (essential singularities, unsupported functions): status "unknown". Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Simplify an expression to canonical form. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Solve an equation or a system. Returns the SET of solutions (a system solution is a vector of equations x = ...); status "no-solution" means proven unsolvable; 'conditions' lists the generic-parameter conditions (e.g. a ≠ 0) under which the answer holds. Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Symbolic summation (step 21l.7): Faulhaber for polynomials, Gosper/Abramov for hypergeometric and rational terms, Zeilberger + rsolve for binomial sums (sum(C(n,k),k,0,n) = 2^n). The answer is a closed form, or a recurrence with initial values when no hypergeometric closed form exists (Apéry numbers); the certificate verifies the answer by direct summation. Lower bound 'from' defaults to 0; upper bound 'to' is required (usually n). Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Combine a sum of fractions into one fraction numerator/denominator in lowest terms: 1/x + 1/y gives (x + y)/(x*y). Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.

Remove all declarations of a symbol (it becomes an arbitrary complex number again). Returns a JSON object: status (answer | unknown | no-solution | nonelementary | error | timeout), result (S-expression), pretty (math-language form), note, conditions (when the answer is valid, e.g. a ≠ 0), assumptions, certificate (only with certificate=true), steps, warnings, output, error.