{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-equation","semantic_family_id":"sf-equation","type":"definition","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"方程式","en":"Equation"},
  "summary":{"zh-Hant":"方程式要求兩個表達式在指定變數範圍內取相等值；解是使等式成立的允許值。"},
  "statement":{"zh-Hant":"給定兩個具有共同語境的表達式 L(x)、R(x)，方程式 L(x)=R(x) 的解集由所有使兩側皆有定義且值相等的允許 x 組成。"},
  "formula":{"tex":"x+1=3","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>=</mo><mn>3</mn></mrow></math>","semantic_ast":{"type":"equation","lhs":{"type":"addition","operands":[{"type":"symbol","name":"x","glyph":"x"},{"type":"number","value":1}]},"rhs":{"type":"number","value":3}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"6b2d3ac44f1b1ed3fc37104aa3cde350ea2c375a34a98f390bb790ad933c312f"}},
  "assumptions":[{"id":"a1","zh-Hant":"變數的允許範圍、表達式型別與運算規則已明示。"}],
  "symbols":[{"token":"x","role":"unknown_variable","role_zh":"待求或驗證的變數","scope":"representative_equation"},{"token":"=","role":"equality_relation","role_zh":"兩側值相等的條件","scope":"equation"}],
  "dependencies":[{"id":"mko-variable-expression","title_zh":"變數與代數式","reason_zh":"方程式以兩個有共同語境的表達式構成。"}],
  "explanation":{"paragraphs_zh":["方程式不是單一表達式；等號左右各有一個表達式，並且必須在共同的變數範圍中解讀。","驗證候選解時，需把值代入原方程式並分別計算兩側。兩側都定義且相等，該值才屬於解集。","對兩側加減相同量，或乘以相同量，可把原等式的解帶到新等式；若要保持完全相同的解集，乘數必須非零。除以同一量時也必須先證明該量非零；一次方程規則不能自動外推到所有方程。"],"common_misconception_zh":"常見誤解是只化簡一側、乘零後仍宣稱解集不變、除以可能為零的式子、把表達式當方程式，或求得候選值後不代回原式檢查。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"整數係數的一元線性方程可做精確有限重算，但不構成所有方程求解算法的完備性或正確性證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"建立帶明示論域的等式、解集與等價變形聲明，分離非零除數等前提。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-equation.md 整理；待人工與形式化審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-24"}
}