{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{},"path":{"id":"path-pulse-foundations-20261018","title":{"zh-Hant":"擴展數學地圖第一批：拓撲、數論、決定問題與形式陳述","en":"First Extended Atlas Slice: Topology, Number Theory, Decision Problems, and Formal Statements"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-proposition","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-topological-space","position":2,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。","en":"A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections."}},{"id":"mko-natural-divisibility","position":3,"requirement":"optional","rationale":{"zh-Hant":"自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。","en":"A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished."}},{"id":"mko-gcd-coprime","position":4,"requirement":"optional","rationale":{"zh-Hant":"gcd(a,b) 是同時整除 a、b 且被每個公因數整除的自然數；gcd=1 時稱互質。","en":"The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1."}},{"id":"mko-prime-number","position":5,"requirement":"optional","rationale":{"zh-Hant":"自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。","en":"A prime natural number is at least 2 and has no positive divisors other than 1 and itself."}},{"id":"mko-modular-congruence","position":6,"requirement":"optional","rationale":{"zh-Hant":"固定正模數 n 時，a 與 b 模 n 同餘表示兩者除以 n 的餘數相同。","en":"For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n."}},{"id":"mko-decision-problem","position":7,"requirement":"optional","rationale":{"zh-Hant":"決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。","en":"A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer."}},{"id":"mko-lean-formal-statement","position":8,"requirement":"optional","rationale":{"zh-Hant":"形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。","en":"A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context."}}]},"title":"First Extended Atlas Slice: Topology, Number Theory, Decision Problems, and Formal Statements"}