bourbaki

Explore a mathematical conjecture - gather evidence, find counterexamples, or build toward a proof. Triggers on "explore", "investigate", "what if", "is it true that", "find pattern", "check if"

Determine convergence or divergence of series using ratio test, root test, comparison test, and integral test. Triggers on "series converges", "test for convergence", "ratio test", "root test", "comparison test", "integral test", "sum of 1/n"

Prove geometric theorems using coordinate geometry. Convert shapes to algebraic expressions, use distance formula, midpoint formula, and slope calculations. Triggers on "prove using coordinates", "analytic geometry", "distance formula", "show collinear", "prove perpendicular", "prove parallel"

Prove identities or existence using combinatorial counting - double counting, bijections, inclusion-exclusion. Triggers on "count", "number of ways", "how many", "bijection", "one-to-one correspondence", "combinatorial proof"

Prove statements directly by assuming the hypothesis and deriving the conclusion through logical steps. Triggers on "prove directly", "show that", "demonstrate", "verify", "simple proof"

Prove limit statements using epsilon-delta definitions. Triggers on "epsilon-delta", "limit", "prove continuous", "show limit equals", "formal limit proof"

Explain a proof or Lean code step-by-step for learning. Triggers on "explain", "how does this work", "teach me", "what does this mean", "walk me through", "I don't understand"

Prove results by considering minimal or maximal elements - the extreme case often has special properties. Triggers on "smallest", "largest", "minimal", "maximal", "least", "greatest", "extreme", "consider the minimum"

Convert an informal mathematical proof into verified Lean 4 code. Triggers on "formalize", "convert to Lean", "make rigorous", "verify this proof", or when user pastes informal proof text

Prove properties about group homomorphisms, kernels, images, and isomorphisms. Use the First Isomorphism Theorem and properties preserved under homomorphisms. Triggers on "homomorphism", "kernel", "image", "isomorphism theorem", "quotient group", "normal subgroup", "group map"

Prove inequalities using AM-GM, Cauchy-Schwarz, triangle inequality, and chaining techniques. Triggers on "prove inequality", "AM-GM", "Cauchy-Schwarz", "triangle inequality", "show that a <= b", "bound", "estimate"

Prove existence using the pigeonhole principle - when n+1 items go into n categories, some category has multiple items. Triggers on "must exist", "at least two", "some pair", "among any n", "prove existence of duplicate/collision"

Prove properties of polynomials including irreducibility, factorization, and minimal polynomials. Use Eisenstein's criterion, reduction mod p, and the factor theorem. Triggers on "irreducible polynomial", "Eisenstein", "minimal polynomial", "factor theorem", "reducible", "polynomial factorization"

Prove existence using probability - if a random object has the property with positive probability, such an object must exist. Triggers on "probabilistic", "random", "expected value", "exists by probability", "Erdos", "non-constructive existence"

Prove statements by assuming the negation and deriving a contradiction. Triggers on "cannot exist", "impossible", "no such", "prove X is irrational", "infinitely many", "there is no"

Prove statements about natural numbers using mathematical induction. Triggers on "prove for all n", "prove for every integer", "show that for n ≥", statements with recursive/iterative structure

Prove properties about rings, ideals, and quotient rings. Work with prime and maximal ideals, quotient constructions, and ring homomorphisms. Triggers on "ideal", "quotient ring", "prime ideal", "maximal ideal", "ring homomorphism", "R/I", "principal ideal"

Prove sequence limits using epsilon-N definition, monotone convergence, and squeeze theorem. Triggers on "sequence converges", "limit of sequence", "prove a_n tends to", "epsilon-N", "monotone convergence", "squeeze theorem", "bounded monotone"

Prove statements using strong (complete) induction where the inductive hypothesis applies to all smaller cases. Triggers on "strong induction", "complete induction", "inductive hypothesis for all k < n", "well-ordering"

Prove geometric theorems using classical compass-and-straightedge reasoning. Use auxiliary constructions, congruence (SSS, SAS, ASA, AAS), and similarity arguments. Triggers on "prove congruent", "prove similar", "construct", "auxiliary line", "classical geometry", "Euclidean proof"

Prove geometric theorems using transformations (translations, rotations, reflections, dilations). Identify invariants preserved under transformations. Triggers on "rotation", "reflection", "translation", "symmetry", "transform", "invariant", "isometry", "congruent by transformation"