{
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  "category": "Number theory and algebra",
  "claimBoundary": "This target indexes Mathlib's ring equivalence between integers modulo mn and pairs modulo m and n when m and n are coprime. It is the two-modulus ZMod form, not a finite-family formulation, an integer representative algorithm, or a new proof.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "def ZMod.chineseRemainder {m n : ℕ} (h : m.Coprime n) : ZMod (m * n) ≃+* ZMod m × ZMod n",
  "family": "Modular arithmetic",
  "id": "library.mathlib.chinese-remainder-theorem.v001",
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    "completeFormalType": "def ZMod.chineseRemainder {m n : ℕ} (h : m.Coprime n) : ZMod (m * n) ≃+* ZMod m × ZMod n",
    "concepts": [
      "Chinese remainder theorem",
      "coprime moduli",
      "residue rings",
      "ring equivalence"
    ],
    "plainLanguage": "When two moduli are coprime, one residue modulo their product carries exactly the same information as one ordered pair of component residues, and the correspondence respects both addition and multiplication.",
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      "alt": "Editorial theorem poster connects a gold product-modulus wheel to emerald and cobalt component wheels, states the ring equivalence in words, and gives a four-step reading guide.",
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      "caption": "The two-modulus ZMod correspondence is bijective and preserves addition and multiplication.",
      "description": "The retained poster presents coprimality, the three rings, the forward and reverse reading, ring-operation preservation, and the exact two-modulus boundary of ZMod.chineseRemainder.",
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      "sha256": "sha256:8145cc384142c0f9aa922421e3ce2480dd55c050ceb9e9927074276ecdd9fdf5",
      "title": "Chinese Remainder Theorem — exact two-modulus ring form",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "MODULAR ARITHMETIC · TWO COPRIME MODULI\nCHINESE REMAINDER THEOREM\nm.Coprime n\nZMod (m · n)\nZMod m\nZMod n\nZMod (m · n) IS RING-EQUIVALENT TO ZMod m × ZMod n\nONE RESIDUE MODULO m · n\nONE ORDERED PAIR MODULO m AND n\nADDITION AND MULTIPLICATION PRESERVED\nHOW TO READ IT\n1 · Choose coprime moduli m and n\n2 · Reduce one residue modulo each\n3 · Obtain one ordered residue pair\n4 · Reconstruct the product residue\nEXACT SCOPE\nTwo-modulus ZMod ring equivalence. Not a finite-family or representative algorithm statement."
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      "caption": "Coprimality makes one residue modulo m*n equivalent to one ordered pair modulo m and n.",
      "description": "The construction reduces a product residue to both component moduli, reconstructs it from the pair, and preserves the two ring operations.",
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      "title": "One product residue, two component residues",
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    "walkthrough": {
      "introduction": "Mathlib constructs a ring equivalence between ZMod (m * n) and ZMod m × ZMod n for coprime natural moduli. The source builds the forward reduction map and a case-split inverse, proves both inverse laws, and packages the correspondence with its additive and multiplicative structure.",
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          "alt": "One antique-gold product-residue wheel sends a selected ivory bead along two fine threads to matching positions on an emerald wheel and a cobalt wheel.",
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          "caption": "Reduction sends one product residue to its ordered pair of component residues.",
          "description": "The forward map to_fun is ZMod.castHom into the product ring. Coprimality identifies the least common multiple with the product, so one residue modulo m*n determines both component residues while preserving addition and multiplication.",
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          "title": "Reduce one product residue to a pair",
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          "equation": "to_fun : ZMod (m * n) → ZMod m × ZMod n",
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        {
          "alt": "The same three residue wheels remain aligned while the component positions reconnect to one product position, with a quiet lower alcove separating the degenerate branch.",
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          "caption": "The inverse reconstructs a product residue, with the zero-product edge cases kept separate from the nonzero CRT branch.",
          "description": "The inverse map branches on m*n=0. Coprimality reduces that edge case to a trivial factor, while the nonzero branch applies Nat.chineseRemainder to the canonical values of the two component residues.",
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          "sha256": "sha256:2becabebe1671d57dee3a3a542c2b92fc1fddefbc4ec38b6a68313b67278a52a",
          "title": "Construct the inverse in both product cases",
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          "equation": "inv_fun : ZMod m × ZMod n → ZMod (m * n)",
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          "alt": "The product bead and its two component beads close into a continuous correspondence while translucent cross-bands isolate their matching positions and two finite populations balance below.",
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          "description": "Nat.modEq_and_modEq_iff_modEq_mul turns agreement modulo both coprime factors into agreement modulo their product, proving the left inverse. rightInverse_of_card_le then uses the matching finite cardinalities to obtain the right inverse.",
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          "title": "Prove the two maps are inverse",
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          "caption": "The reversible correspondence preserves addition and multiplication, completing the two-modulus ring equivalence.",
          "description": "The final structure packages to_fun, inv_fun, map_add, map_mul, and the two inverse laws. The endpoint is exactly a ring equivalence between the product modulus and the ordered pair of component moduli.",
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          "sha256": "sha256:bd189bf0aad143a47d1d55c98e5fb05d3b0dec07a71aa822e209062223399558",
          "title": "Assemble the ring equivalence",
          "width": 1536,
          "equation": "ZMod (m * n) ≃+* ZMod m × ZMod n",
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      "title": "From one product residue to an ordered residue pair"
    },
    "whyLandmark": "The Chinese Remainder Theorem turns a modular problem into independent coprime components. It is foundational in number theory, algebra, algorithms, and cryptography."
  },
  "nonClaims": [
    "The selected declaration covers exactly two coprime natural moduli.",
    "It states a ZMod ring equivalence, not an exposed integer-representative algorithm.",
    "ProofAtlas is indexing an existing Mathlib construction, not claiming a new theorem or proof."
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  "statementId": "statement.library.mathlib.chinese-remainder-theorem.v001",
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    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "2 of 3 presentation reviews recorded",
    "upstreamIndexed": "Pinned source bytes verified locally"
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  "summary": "For coprime natural moduli m and n, residues modulo m*n form a ring equivalent to ordered pairs of residues modulo m and modulo n.",
  "targetId": "target.library.mathlib.chinese-remainder-theorem.v001",
  "title": "Chinese Remainder Theorem",
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