Coverage Map
This page tracks major mathematical areas and important results that openmath should eventually cover. Linked entries have a written proof. Unlinked entries are either missing entirely or have no proof yet.
Foundations 60%
Sets, functions, relations, logic, induction, and the basic language used everywhere else.
- Principle of Induction proved
- Strong Induction proved
- Well-ordering Principle proved
- Schroeder-Bernstein Theorem missing
- Cantor's Theorem missing
Number Theory 64%
Divisibility, primes, congruences, arithmetic functions, and classical modular arithmetic.
- Euclidean Algorithm proved
- Bezout Identity proved
- Chinese Remainder Theorem proved
- Fundamental Theorem of Arithmetic needs proof
- Infinitely Many Primes proved
- Fermat's Little Theorem proved
- Wilson's Theorem proved
- Euler's Theorem needs proof
- Mobius Inversion missing
- Quadratic Reciprocity missing
- Dirichlet's Theorem on Arithmetic Progressions proved
Algebra 5%
Groups, rings, fields, modules, polynomials, and Galois theory.
- Lagrange's Theorem missing
- Cauchy's Theorem missing
- Cayley's Theorem missing
- Orbit-Stabilizer Theorem missing
- Class Equation missing
- First Isomorphism Theorem missing
- Second Isomorphism Theorem missing
- Third Isomorphism Theorem missing
- Correspondence Theorem missing
- Sylow Theorems missing
- Simplicity of the Alternating Group missing
- Jordan-Holder Theorem missing
- Fundamental Theorem of Finitely Generated Abelian Groups missing
- Semidirect Product Classification missing
- Solvability by Normal Series missing
- Ring Isomorphism Theorems missing
- Chinese Remainder Theorem for Rings missing
- Every Euclidean Domain is a PID missing
- Every PID is a UFD missing
- Gauss's Lemma missing
- Eisenstein's Criterion missing
- Prime Ideals and Domains needs proof
- Maximal Ideals and Fields needs proof
- Irreducible Polynomial Creates a Basis proved
- Quotient Module Homomorphism Theorem missing
- Structure Theorem for Finitely Generated Modules over a PID missing
- Rational Canonical Form missing
- Jordan Canonical Form missing
- Tower Law for Field Extensions missing
- Existence of Splitting Fields missing
- Existence of Algebraic Closures missing
- Primitive Element Theorem missing
- Classification of Finite Fields missing
- Cyclotomic Polynomial Irreducibility missing
- Fundamental Theorem of Galois Theory missing
- Solvability by Radicals missing
- Quaternion Multiplication is Associative proved
Linear Algebra 0%
Vector spaces, linear maps, matrices, bases, dimension, eigenvalues, and canonical forms.
- Basis Existence Theorem missing
- Dimension Theorem missing
- Rank-nullity Theorem missing
- Invertible Matrix Theorem missing
- Spectral Theorem missing
- Jordan Normal Form missing
Analysis 50%
Limits, continuity, compactness, differentiation, integration, sequences, series, and approximation.
- Intermediate Value Theorem proved
- Extreme Value Theorem proved
- Mean Value Theorem proved
- Fundamental Theorem of Calculus proved
- Bolzano-Weierstrass Theorem missing
- Heine-Borel Theorem missing
- Arzela-Ascoli Theorem missing
- Stone-Weierstrass Theorem missing
Vector Calculus 0%
Multivariable differentiation, vector fields, line and surface integrals, and the classical integral theorems.
- Inverse Function Theorem missing
- Implicit Function Theorem missing
- Change of Variables Theorem missing
- Fundamental Theorem for Line Integrals missing
- Green's Theorem missing
- Divergence Theorem missing
- Stokes' Theorem missing
Complex Analysis 0%
Holomorphic functions, contour integration, singularities, residues, and conformal maps.
- Cauchy-Riemann Equations missing
- Goursat's Theorem missing
- Cauchy Integral Theorem missing
- Cauchy Integral Formula missing
- Morera's Theorem missing
- Liouville's Theorem missing
- Fundamental Theorem of Algebra via Liouville's Theorem missing
- Maximum Modulus Principle missing
- Open Mapping Theorem missing
- Identity Theorem missing
- Taylor Theorem for Holomorphic Functions missing
- Laurent Series Theorem missing
- Classification of Isolated Singularities missing
- Residue Theorem missing
- Argument Principle missing
- Rouche's Theorem missing
- Schwarz Lemma missing
- Riemann Mapping Theorem missing
Topology 0%
Topological spaces, bases, products, quotients, compactness, connectedness, and separation.
- Basis Generates a Topology missing
- Continuous Functions Preserve Compactness missing
- Continuous Functions Preserve Connectedness missing
- Tychonoff's Theorem missing
- Urysohn's Lemma missing
- Seifert-van Kampen Theorem missing
Combinatorics 0%
Counting, recurrence, generating functions, extremal arguments, and discrete structures.
- Pigeonhole Principle missing
- Inclusion-Exclusion missing
- Burnside's Lemma missing
- Polya Enumeration Theorem missing
- Ramsey's Theorem missing
Graph Theory 0%
Graphs, trees, connectivity, coloring, matchings, flows, and spectral methods.
- Handshaking Lemma missing
- Euler Characterization of Trees missing
- Menger's Theorem missing
- Max-flow Min-cut Theorem missing
- Hall's Marriage Theorem missing
- Kuratowski's Theorem missing
Probability 0%
Probability spaces, random variables, expectation, convergence, and distributions.
- Linearity of Expectation missing
- Law of Total Probability missing
- Bayes' Theorem missing
- Weak Law of Large Numbers missing
- Central Limit Theorem missing
Category Theory 0%
Categories, functors, natural transformations, limits, adjunctions, and universal properties.
- Yoneda Lemma missing
- Adjoint Functor Theorem missing
- Limits as Universal Constructions missing
- Equivalence of Categories missing