{"component": "clause", "props": {"groups": [{"snippet": "The r-DHI assumption holds for Qr A under the assumption that the AS scheme is IND-CPA secure for parameters (x, k, t, r + 1).", "samples": [{"hash": "4WPmY7Ihg1G", "uri": "/contracts/4WPmY7Ihg1G#proposition-5", "label": "Key Agreement Protocol", "score": 20.3002496006, "published": true}, {"hash": "G33F4HrGza", "uri": "/contracts/G33F4HrGza#proposition-5", "label": "Key Agreement Protocol", "score": 20.1759069131, "published": true}], "snippet_links": [{"key": "the-r", "type": "clause", "offset": [0, 5]}, {"key": "the-assumption", "type": "clause", "offset": [42, 56]}], "size": 2, "hash": "92c5ca5681a42154c9cd52f8aafb5e62", "id": 1}, {"snippet": "The closing of the demerger deal and legal independence creates a sense of \u201cbeing done\u201d with the demerger and moving on to normal business. However, IT dis- integration most commonly will not be completed at this time and requires complete attention and priority within the company in order to succeed.", "samples": [{"hash": "khvRoqFjGXY", "uri": "/contracts/khvRoqFjGXY#proposition-5", "label": "Distribution Agreement", "score": 21.2774898267, "published": true}], "snippet_links": [{"key": "the-closing", "type": "definition", "offset": [0, 11]}, {"key": "normal-business", "type": "definition", "offset": [123, 138]}, {"key": "in-order-to", "type": "clause", "offset": [282, 293]}], "size": 1, "hash": "906e2f7c94cd26e114012ba2960d7779", "id": 2}, {"snippet": ".1. Suppose B is a functor satisfying the hypotheses of Theorem 5.\n1. Let (CB\u00af , s) be the codensity monad of B\u00af, with distributive law \u03c4 and monad structure (CB\u00af , \u03b7, \u03bc). If Bk+1,k is an isomorphism for some k, then\n1. sk : CB\u00af Bk \u2192 Bk is the algebra induced by \u03c4 on the \ufb01nal coalgebra;\n2. if C has an initial object 0 then sk is isomorphic to \u03bc0.\n5.1 Codensity and the Companion of a Monotone Function\u200c Throughout this section, let b : L L be a monotone function on a complete lattice. By Theorem 5.1, the companion of a monotone function b (viewed as a functor on a poset category) is given by the right Kan extension of the \ufb01nal sequence \u00afb : Ordop L along itself. Using Lemma 4.1, we obtain the characteri- sation of the companion given in the Introduction (5). .\nTheorem 5.2. The companion t of b is given by t : x \u203a\u2192 bi x\u2264bi .", "samples": [{"hash": "8BgvlCUQtID", "uri": "/contracts/8BgvlCUQtID#proposition-5", "label": "End User Agreement", "score": 24.8904859685, "published": true}], "snippet_links": [{"key": "satisfying-the", "type": "clause", "offset": [27, 41]}, {"key": "theorem-5", "type": "clause", "offset": [56, 65]}, {"key": "the-right", "type": "clause", "offset": [597, 606]}, {"key": "extension-of-the", "type": "clause", "offset": [611, 627]}], "size": 1, "hash": "c935461ca1e94359e899282eb74c3b53", "id": 3}, {"snippet": "\ud83d\udf90 In the subgame perfect equilibrium for the hub-and-spoke setting with transfers, there is no payoff premium for the hub relative to the common payoff earned by the spokes. Isolated Bilateral with Transfers Hub-and-Spoke with Transfers Sing \ud83d\udf90 For 0 < a < 0.41, the CPNE is the \u2587\u2587\u2587\u2587 equilibrium for the setting in which there is an isolated bilateral agreement. \ud83d\udf90 For 0.41 < a < 0.514, the CPNE is the \u2587\u2587\u2587\u2587 equilibrium for the hub-and-spoke arrangement. \ud83d\udf90 For 0.514 < a < 1, the CPNE is the \u2587\u2587\u2587\u2587 equilibrium for the setting in which all nations stand alone in R&D production. \u25fc Suppose that global welfare is the sum of all nations\u2019 payoffs: Global Welfare \u2261 \u2211 Isolated Bilateral with Transfers Hub-and-Spoke with Transfers Singl \ud83d\udf90 For sufficiently small attrition rates, constrained global welfare levels improve when green R&D agreements prohibit transfers.", "samples": [{"hash": "13V9dgf590c", "uri": "/contracts/13V9dgf590c#proposition-5", "label": "International Environmental Agreements", "score": 27.7185157428, "published": true}], "snippet_links": [{"key": "the-hub", "type": "clause", "offset": [41, 48]}, {"key": "the-\u2587", "type": "clause", "offset": [274, 279]}, {"key": "bilateral-agreement", "type": "clause", "offset": [341, 360]}, {"key": "stand-alone", "type": "clause", "offset": [545, 556]}, {"key": "sum-of", "type": "clause", "offset": [613, 619]}, {"key": "sufficiently-small", "type": "definition", "offset": [736, 754]}, {"key": "rd-agreements", "type": "definition", "offset": [825, 839]}], "size": 1, "hash": "bde714f268e11908b155a5163c0dfdd4", "id": 4}, {"snippet": "3.1. Let V be a free, finite index, subgroup of E+ of rank n \u2212 1. Then \u03c3\u2208G u2(V ) = \u2211 u2(\u03c3)[V \u2236E+] \u2297 \u03c3\u22121, u2 = \u2211 u2(\u03c3) \u2297 \u03c3\u22121. \u03c3\u2208G", "samples": [{"hash": "dSglaQUei0X", "uri": "/contracts/dSglaQUei0X#proposition-5", "label": "End User License Agreement", "score": 25.3874058864, "published": true}], "snippet_links": [], "size": 1, "hash": "2a19d0304961eed175a2e0a0c3b2d093", "id": 5}, {"snippet": "9. Let A be a Noetherian ring and B = A[[x1, . . . , xn]] be the A-algebra of formal power series in n variables. Then B is faithfully flat over A.", "samples": [{"hash": "c3XMsByRNBQ", "uri": "/contracts/c3XMsByRNBQ#proposition-5", "label": "Coherent Sheaves, Chern Classes, and Superconnections on Compact Complex Analytic Manifolds", "score": 25.9952087611, "published": true}], "snippet_links": [], "size": 1, "hash": "e1bebd0f73b9710c3a9d15ee8c6bcd0e", "id": 6}, {"snippet": "Let (nn : Z\u00d7 \u2192 &\u00d7 )n\u2208Z be any sequence of weights satisfying np \u2261 nn mod p, then inside &K[[q]], If moreover nb : Z\u00d7 \u2192 &\u00d7 (En )p \u2261 En mod p. n+1 n Kb is the corresponding t-adic weight, then E b corresponds to the sequence (En )n under the isomorphism M + (0) = \u2587\u2587\u2587 \u2587 + (0)/p of Theorem 5.3.3. n \u2208Z\u22650 nb \u2190\u2212 n", "samples": [{"hash": "2CZmbrmeWtx", "uri": "/contracts/2CZmbrmeWtx#proposition-5", "label": "End User License Agreement", "score": 22.8952772074, "published": true}], "snippet_links": [], "size": 1, "hash": "b952f6cea635a2e9acdc039f20f35013", "id": 7}, {"snippet": "The function f (n, m) is multiplicative in m. Proof. Let m1 and m2 be relatively prime positive integers and let P be the set of all prime divisors of m1m2. For i = 1, 2, let Ki be a number field that attains the maximum value for fK(mi). For each prime p in P with p | mi, define Ep = Ki\u2297Qp. According to lemma 5.17, there exists a number field K such that K \u2297 Qp Ep for all p \u2208 P . For i = 1, 2 and pk | mi, we have fK(pk) = fK\u2297Q (k) = fK \u2297Q (k) = fK (pk) by proposition 5.19. By lemma 5.21, we get fK(mi) = fKi (mi) = f (n, mi) and therefore f (n, m1m2) \u2265 fK(m1m2) = f (n, m1)f (n, m2). On the other hand let K\u2032 be a number field of degree n that attains the maximum value for fK(m1m2). Then we also have the bound f (n, m1m2) = fK\u2032 (m1m2) = fK\u2032 (m1)fK\u2032 (m2) \u2264 f (n, m1)f (n, m2). Now we prove proposition 5.3, restated here for convenience. Proposition 5.3. The following equality holds for all \u2587. \u2587\u2587\u2587 sup log f (n, m) = lim sup log fEt(n, p, k) log m pk \u2192\u221e k log p log fEt(n, p, k) log f (n, pk) lim sup pk \u2192\u221e = lim sup k log p pk \u2192\u221e log pk = x. The set of prime powers is a subset of the integers, so we can bound \u2264 log f (n, m) log m Hence, if x is infinite, we are done. Assume x is finite. Then for all \u01eb > 0 we have lim f (n,pk) pk \u2192\u221e pk(x+\u01eb)", "samples": [{"hash": "re1Dn1ViMJ", "uri": "/contracts/re1Dn1ViMJ#proposition-5", "label": "Doctoral Thesis", "score": 19.0, "published": true}], "snippet_links": [{"key": "the-function", "type": "clause", "offset": [0, 12]}, {"key": "a-number", "type": "definition", "offset": [181, 189]}, {"key": "maximum-value", "type": "definition", "offset": [213, 226]}, {"key": "each-prime", "type": "definition", "offset": [243, 253]}, {"key": "according-to", "type": "definition", "offset": [293, 305]}, {"key": "all-p", "type": "clause", "offset": [372, 377]}, {"key": "the-other-hand", "type": "clause", "offset": [593, 607]}, {"key": "field-of", "type": "definition", "offset": [627, 635]}, {"key": "for-convenience", "type": "clause", "offset": [828, 843]}], "size": 1, "hash": "80b122e30beb7f80e0bb51d77f3c38f8", "id": 8}, {"snippet": "For some constant probability p, any domain V , and any integer itr, V,p,itr can be UC-realized with statistical security in the a-smt-hybrid model, in constant rounds and in the presence of an adaptive and malicious t-adversary, provided t < n/3.", "samples": [{"hash": "3Iab5IdLAzZ", "uri": "/contracts/3Iab5IdLAzZ#proposition-5", "label": "Byzantine Agreement", "score": 32.9769988642, "published": true}], "snippet_links": [{"key": "hybrid-model", "type": "clause", "offset": [135, 147]}, {"key": "and-in-the-presence-of", "type": "clause", "offset": [168, 190]}], "size": 1, "hash": "a64f051ffbc3579db0772e15e236f451", "id": 9}, {"snippet": "Let M be a submodule of W. Then M has a generating set X with the following property: No subset Y of X, whose image \u03c4 (Y ) in T is a chain with respect to the partial order \u2122, can have more than min ad \u2212 am + am, \u03c7 RF(Kj) elements, where \u03c7 = \u03c7(K, V \u2217) denotes the number of orbits of K on the nonzero elements of V . Before proving Proposition 5.4.9, we need a preliminary lemma.", "samples": [{"hash": "4vrUs0IIhpV", "uri": "/contracts/4vrUs0IIhpV#proposition-5", "label": "Library Declaration and Deposit Agreement", "score": 22.7850787132, "published": true}], "snippet_links": [{"key": "with-respect-to", "type": "clause", "offset": [139, 154]}, {"key": "partial-order", "type": "definition", "offset": [159, 172]}, {"key": "number-of", "type": "clause", "offset": [264, 273]}], "size": 1, "hash": "6ea85ad45d864174d276170e7910b719", "id": 10}], "next_curs": "ClYSUGoVc35sYXdpbnNpZGVyY29udHJhY3RzcjILEhZDbGF1c2VTbmlwcGV0R3JvdXBfdjU2IhZwcm9wb3NpdGlvbi01IzAwMDAwMDBhDKIBAmVuGAAgAA==", "clause": {"children": [], "title": "Proposition 5", "parents": [["formal-appendix-no-coalitions-possible-round", "Formal Appendix No Coalitions Possible Round"], ["formal-appendix-a-main-model", "Formal Appendix A: Main Model"]], "size": 15, "id": "proposition-5", "related": [["preconstruction-conference", "Preconstruction Conference", "Preconstruction Conference"], ["suggestions-and-feedback", "Suggestions and Feedback", "Suggestions and Feedback"], ["commencement-and-completion", "Commencement and Completion", "Commencement and Completion"], ["removal-after-your-tax-filing-deadline", "Removal After Your Tax Filing Deadline", "Removal After Your Tax Filing Deadline"], ["constructability-review", "Constructability Review", "Constructability Review"]], "related_snippets": [], "updated": "2025-07-07T12:37:48+00:00"}, "json": true, "cursor": ""}}