{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"knowledge","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":{"page":"4"},"kind":"knowledge","total":302,"pages":13,"page":4,"entries":[{"id":"mko-function-limit","title":{"zh-Hant":"去心實數論域的函數極限","en":"Function limits on punctured real domains"},"summary":{"zh-Hant":"在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。","en":"An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["去心實數論域的函數極限","AI候選","待形式化"]},{"id":"mko-continuity","title":{"zh-Hant":"實函數的點值與連續","en":"Point values and continuity of real functions"},"summary":{"zh-Hant":"在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。","en":"Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實函數的點值與連續","AI候選","待形式化"]},{"id":"mko-derivative","title":{"zh-Hant":"實函數在一點的導數與差商","en":"Real derivatives at a point and finite difference quotients"},"summary":{"zh-Hant":"導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。","en":"A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["實函數在一點的導數與差商","AI候選","待形式化"]},{"id":"mko-derivative-rules","title":{"zh-Hant":"多項式的和、乘積與合成微分","en":"Differentiating polynomial sums, products, and compositions"},"summary":{"zh-Hant":"四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。","en":"Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多項式的和、乘積與合成微分","AI候選","待形式化"]},{"id":"mko-antiderivative","title":{"zh-Hant":"多項式原函數與常數選擇","en":"Polynomial antiderivatives and the choice of constant"},"summary":{"zh-Hant":"原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。","en":"An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["多項式原函數與常數選擇","AI候選","待形式化"]},{"id":"mko-definite-integral","title":{"zh-Hant":"有限區間上的多項式定積分","en":"Signed polynomial definite integrals on finite intervals"},"summary":{"zh-Hant":"連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。","en":"Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["有限區間上的多項式定積分","AI候選","待形式化"]},{"id":"mko-fundamental-theorem-calculus","title":{"zh-Hant":"微積分基本定理：連續前提與有向端點","en":"Fundamental theorem of calculus: continuity and oriented endpoints"},"summary":{"zh-Hant":"在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。","en":"With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference."},"type":"theorem","domain":"analysis","formalization":"not_formalized","tags":["微積分基本定理：連續前提與有向端點","AI候選","待形式化"]},{"id":"mko-tree","title":{"zh-Hant":"非空有限樹：連通、無環與唯一道路","en":"Nonempty finite trees: connectivity, cycles, and unique paths"},"summary":{"zh-Hant":"非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。","en":"A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["非空有限樹：連通、無環與唯一道路","AI候選","待形式化"]},{"id":"mko-topological-space","title":{"zh-Hant":"拓撲空間","en":"Topological space"},"summary":{"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."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲空間","AI候選","待形式化"]},{"id":"mko-natural-divisibility","title":{"zh-Hant":"自然數整除","en":"Natural-number divisibility"},"summary":{"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."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數整除","AI候選","待形式化"]},{"id":"mko-gcd-coprime","title":{"zh-Hant":"最大公因數與互質","en":"Greatest common divisor and coprimality"},"summary":{"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."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["最大公因數與互質","AI候選","待形式化"]},{"id":"mko-prime-number","title":{"zh-Hant":"自然數質數","en":"Prime natural number"},"summary":{"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."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數質數","AI候選","待形式化"]},{"id":"mko-modular-congruence","title":{"zh-Hant":"自然數模同餘","en":"Natural-number congruence modulo a positive modulus"},"summary":{"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."},"type":"definition","domain":"number_theory","formalization":"not_formalized","tags":["自然數模同餘","AI候選","待形式化"]},{"id":"mko-decision-problem","title":{"zh-Hant":"決定問題","en":"Decision problem"},"summary":{"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."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["決定問題","AI候選","待形式化"]},{"id":"mko-lean-formal-statement","title":{"zh-Hant":"證明助理中的形式陳述","en":"Formal statements in a proof assistant"},"summary":{"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."},"type":"definition","domain":"formalized_mathematics","formalization":"not_formalized","tags":["證明助理中的形式陳述","AI候選","待形式化"]},{"id":"mko-open-closed-set","title":{"zh-Hant":"開集與閉集","en":"Open and closed sets"},"summary":{"zh-Hant":"在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。","en":"In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["開集與閉集","AI候選","待形式化"]},{"id":"mko-neighborhood","title":{"zh-Hant":"鄰域","en":"Neighborhood"},"summary":{"zh-Hant":"集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。","en":"A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["鄰域","AI候選","待形式化"]},{"id":"mko-topological-continuity","title":{"zh-Hant":"拓撲連續映射","en":"Continuous maps between topological spaces"},"summary":{"zh-Hant":"拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。","en":"A map is topologically continuous when every open set in its target has an open preimage in its source."},"type":"definition","domain":"topology","formalization":"not_formalized","tags":["拓撲連續映射","AI候選","待形式化"]},{"id":"mko-computational-model","title":{"zh-Hant":"抽象計算模型","en":"Abstract computational model"},"summary":{"zh-Hant":"有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。","en":"A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["抽象計算模型","AI候選","待形式化"]},{"id":"mko-algorithm-correctness","title":{"zh-Hant":"算法正確性","en":"Algorithm correctness"},"summary":{"zh-Hant":"算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。","en":"Algorithm correctness separates partial correctness from termination; total correctness requires both."},"type":"concept","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["算法正確性","AI候選","待形式化"]},{"id":"mko-asymptotic-resource-bound","title":{"zh-Hant":"漸近資源界","en":"Asymptotic resource bounds"},"summary":{"zh-Hant":"漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。","en":"Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail."},"type":"definition","domain":"computer_science_mathematics","formalization":"not_formalized","tags":["漸近資源界","AI候選","待形式化"]},{"id":"mko-lean-proof-term","title":{"zh-Hant":"證明項","en":"Proof term"},"summary":{"zh-Hant":"證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。","en":"A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel."},"type":"concept","domain":"formalized_mathematics","formalization":"not_formalized","tags":["證明項","AI候選","待形式化"]},{"id":"mko-discrete-dynamical-system","title":{"zh-Hant":"離散時間動力系統","en":"Discrete-time dynamical system"},"summary":{"zh-Hant":"離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。","en":"A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability."},"type":"definition","domain":"dynamical_systems","formalization":"not_formalized","tags":["離散時間動力系統","AI候選","待形式化"]},{"id":"mko-ordinary-differential-equation","title":{"zh-Hant":"常微分方程","en":"Ordinary differential equation"},"summary":{"zh-Hant":"常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。","en":"An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems."},"type":"definition","domain":"differential_equations","formalization":"not_formalized","tags":["常微分方程","AI候選","待形式化"]}],"title":"知識物件"}