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    "plainLanguage": "Suppose each index i is assigned a finite set t i. One can choose an element f i from every t i with no two indices choosing the same element exactly when, for every finite collection of indices, the union of their assigned sets has at least as many elements as that collection.",
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      "title": "Hall's Marriage Theorem at a glance",
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      "transcript": "COMBINATORICS · DISTINCT REPRESENTATIVES\nHALL'S MARRIAGE THEOREM\ni ↦ t(i)\n∀ finite s, |s| ≤ |⋃ i∈s t(i)|\n⇔\n∃ injective f with f(i) ∈ t(i)\nEVERY MEMBER SET t(i) IS FINITE\nTEST EVERY FINITE SUBFAMILY\nENOUGH ELEMENTS IN EACH UNION\nONE DISTINCT ALLOWED CHOICE PER INDEX\nARBITRARY INDEX TYPE ι\nEXACT SCOPE\nThe representative function is injective, not necessarily onto α."
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      "caption": "A system of distinct representatives appears as one selected edge at every left and right endpoint.",
      "description": "Two columns contain exactly six vertices each. Six thick horizontal gold edges pair the rows one to one, while faint diagonal candidate edges and nested cobalt contours suggest the surrounding family of choices and neighborhood conditions.",
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      "title": "Hall matching schematic",
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      "introduction": "Mathlib's arbitrary-index Hall theorem builds finite local matching spaces, organizes their restrictions into an inverse system, uses compactness to select a coherent section, extracts a global injective representative function, and proves the converse by cardinality comparison.",
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          "alt": "Ivory cards pair green index medallions with distinct blue candidate beads, while a finite fan shows several valid local matchings.",
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          "caption": "Every finite subfamily has a finite, nonempty space of injective representative choices.",
          "description": "For a finite index set ι', hallMatchingsOn stores functions into α that are injective and choose inside each assigned finite set. The Hall inequality feeds the finite theorem to prove nonemptiness, while encoding a matching into the finite union proves that this local matching space is finite.",
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          "title": "Build finite local matchings",
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          "title": "Restrict larger matchings",
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          "description": "Because each object of hallMatchingsFunctor is nonempty and finite, nonempty_sections_of_finite_inverse_system supplies a section u together with compatibility hu. This coherence, rather than an arbitrary independent choice on each finite set, is the bridge to one global function.",
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          "alt": "Four green indices map to four distinct selected blue beads, which sit inside a larger outlined union containing additional unused candidates.",
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          "description": "Injectivity rewrites the size of s as the size of s.image f. Membership of every representative places that image inside s.biUnion t, and card_le_card gives the desired inequality. This closes the reverse implication and therefore the exact equivalence.",
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          "title": "Recover the Hall inequalities",
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      "title": "From finite matchings to one global choice"
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  "nonClaims": [
    "Proof Atlas did not originate Hall's theorem or Mathlib's declaration.",
    "The selected declaration includes an arbitrary index type and a finite-subfamily condition.",
    "The explanatory matching diagram is not a proof of the equivalence."
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    "locallyReproduced": "Exact upstream declaration replayed",
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  "summary": "A family of finite sets has distinct representatives exactly when every finite subfamily has a union at least as large as the subfamily.",
  "targetId": "target.library.mathlib.hall-marriage-theorem.v001",
  "title": "Hall's Marriage Theorem",
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