# Question batches 36,000 questions have bilingual step-by-step derivations. Browse original batches or search and filter in the question explorer. [Search all questions & derivations →](/en/explore/) [← Previous](/en/practice/?page=9#results)[1](/en/practice/?page=1#results) … [9](/en/practice/?page=9#results) 10 [11](/en/practice/?page=11#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=11#results) 360 results · Page 10 of 15 Question batch · 100 questions ## [Explicit Finite Discrete Dynamics and Invariant Subsets](/en/practice/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Discrete-time dynamical system batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations Question batch · 100 questions ## [Synthetic Proof Terms and Receipt Obligations, Not Lean Execution](/en/practice/batch-20261013-advance-20261019-r07-lean-proof-term-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Proof term batch-20261013-advance-20261019-r07-lean-proof-term-foundations Question batch · 100 questions ## [Explicit Full-Tail Resource Bounds versus Finite Benchmarks](/en/practice/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Asymptotic resource bounds batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations Question batch · 100 questions ## [Partial Correctness, Ranking Obligations, and All-Path Termination](/en/practice/batch-20261013-advance-20261019-r05-algorithm-correctness-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Algorithm correctness batch-20261013-advance-20261019-r05-algorithm-correctness-foundations Question batch · 100 questions ## [Finite Transition Models, Halt Truncation, and Stuck States](/en/practice/batch-20261013-advance-20261019-r04-computational-model-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Abstract computational model batch-20261013-advance-20261019-r04-computational-model-foundations Question batch · 100 questions ## [Complete Open-Preimage Continuity and Open-Map Contrasts](/en/practice/batch-20261013-advance-20261019-r03-topological-continuity-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Continuous maps between topological spaces batch-20261013-advance-20261019-r03-topological-continuity-foundations Question batch · 100 questions ## [Point-Qualified Neighborhoods and All Open Witnesses](/en/practice/batch-20261013-advance-20261019-r02-neighborhood-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Neighborhood batch-20261013-advance-20261019-r02-neighborhood-foundations Question batch · 100 questions ## [Open, Closed, and Clopen Sets in Complete Finite Topologies](/en/practice/batch-20261013-advance-20261019-r01-open-closed-set-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Open and closed sets batch-20261013-advance-20261019-r01-open-closed-set-foundations Question batch · 100 questions ## [Synthetic Formal-Statement Pipeline Records, Not Lean Execution](/en/practice/batch-20261013-advance-20261018-r07-lean-formal-statement-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Formal statements in a proof assistant batch-20261013-advance-20261018-r07-lean-formal-statement-foundations Question batch · 100 questions ## [Finite Decision Specifications versus Complete and Sampled Tables](/en/practice/batch-20261013-advance-20261018-r06-decision-problem-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Decision problem batch-20261013-advance-20261018-r06-decision-problem-foundations Question batch · 100 questions ## [Positive-Modulus Natural Congruence and Guarded Preservation](/en/practice/batch-20261013-advance-20261018-r05-modular-congruence-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Natural-number congruence modulo a positive modulus batch-20261013-advance-20261018-r05-modular-congruence-foundations Question batch · 100 questions ## [Prime Natural Numbers and Complete Factor Checks](/en/practice/batch-20261013-advance-20261018-r04-prime-number-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Prime natural number batch-20261013-advance-20261018-r04-prime-number-foundations Question batch · 100 questions ## [Natural GCD, Euclid Traces, and Coprimality](/en/practice/batch-20261013-advance-20261018-r03-gcd-coprime-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Greatest common divisor and coprimality batch-20261013-advance-20261018-r03-gcd-coprime-foundations Question batch · 100 questions ## [Natural Divisibility, Explicit Witnesses, and Zero Boundaries](/en/practice/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Natural-number divisibility batch-20261013-advance-20261018-r02-natural-divisibility-foundations Question batch · 100 questions ## [Complete Finite Topology Axioms and Open-Set Inventories](/en/practice/batch-20261013-advance-20261018-r01-topological-space-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Topological space batch-20261013-advance-20261018-r01-topological-space-foundations Question batch · 100 questions ## [Finite-Domain Counterexamples and Explicit Predicate Extensions](/en/practice/batch-20261012-advance-20261017-r07-counterexample-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Counterexamples and domain eligibility batch-20261012-advance-20261017-r07-counterexample-foundations Question batch · 100 questions ## [Nonempty Tree Qualification, Paths, Leaves, and New-Edge Cycles](/en/practice/batch-20261012-advance-20261017-r06-tree-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Nonempty finite trees: connectivity, cycles, and unique paths batch-20261012-advance-20261017-r06-tree-foundations Question batch · 100 questions ## [Simple Undirected Graph Inventories and Induced Subsets](/en/practice/batch-20261012-advance-20261017-r05-graph-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Finite simple undirected graphs batch-20261012-advance-20261017-r05-graph-foundations Question batch · 100 questions ## [Exact Rational Order and Closed-Interval Boundaries](/en/practice/batch-20261012-advance-20261017-r04-order-relation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Usual order on the real numbers batch-20261012-advance-20261017-r04-order-relation-foundations Question batch · 100 questions ## [Typed Relations, Projections, and Totality Guards](/en/practice/batch-20261012-advance-20261017-r03-relation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Binary relations and set projections batch-20261012-advance-20261017-r03-relation-foundations Question batch · 100 questions ## [Inverse Existence, Whole-Table Types, and Two-Sided Identities](/en/practice/batch-20261012-advance-20261017-r02-inverse-function-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Inverse functions of bijections batch-20261012-advance-20261017-r02-inverse-function-foundations Question batch · 100 questions ## [Complete Finite-Map Bijections and Cardinality Guards](/en/practice/batch-20261012-advance-20261017-r01-bijective-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Bijectivity and unique preimages batch-20261012-advance-20261017-r01-bijective-foundations Question batch · 100 questions ## [Declared Codomain Fibers, Changes, and Target Guards](/en/practice/batch-20261012-advance-20261016-r07-surjective-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Surjectivity and codomain coverage batch-20261012-advance-20261016-r07-surjective-foundations Question batch · 100 questions ## [Complete Finite-Map Collisions and Original Source Labels](/en/practice/batch-20261012-advance-20261016-r06-injective-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D10) Injectivity and distinct inputs batch-20261012-advance-20261016-r06-injective-foundations [← Previous](/en/practice/?page=9#results)[1](/en/practice/?page=1#results) … [9](/en/practice/?page=9#results) 10 [11](/en/practice/?page=11#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=11#results) 360 results · Page 10 of 15