Proof of Theorem Clause Examples
The 'Proof of Theorem' clause serves to formally demonstrate the validity of a stated theorem within a legal or academic document. It typically outlines the logical steps, supporting evidence, and reasoning that collectively establish the truth of the theorem, often referencing prior results or established principles. By providing a clear and structured argument, this clause ensures that the theorem is not merely asserted but is substantiated, thereby enhancing the credibility and rigor of the document.
Proof of Theorem. 4.1.1: We view R as an algebra over its prime subring k. Let C be an auxiliary variable that at the end of the algorithm will become equal to the desired maximal length R-module that is isomorphic to a direct summand both of M1 and of M2. At the beginning of the algorithm, we put C equal to zero. Similarly, we define an auxiliary variable f ∈ HomR(M1, M2) and initialise it at zero. We compute B, a set of k-generators of HomR(M1, M2) (or a set of EndR(M1)- module generators thereof). For each element of B, we test if it is a splitter: by Propo- sition 4.3.6, if a splitter exists, we will find one inside B. Finding a splitter b ∈ B also gives us a homomorphism c ∈ HomR(M2, M1) such that cb is not nilpotent, and a decomposition M1 = N1 ⊕ K1 and M2 = N2 ⊕ K2, where N1 = im((cb)d), K1 = ker((cb)d), N2 = im((bc)d), K2 = ker((bc)d), and d = max{lengthR(M1), lengthR(M2)} (see Proposition 4.3.5). We make the following replacements: C := C ⊕ N1 and f := f ⊕ (the restriction of b to N1), M1 := K1, M2 := K2, B := (set of k-module generators of the new HomR(M1, M2)), and we repeat the process. Note that we are all the time assuming xxx Xxxxx-Xxxxx-Schmidt Theorem (see Theorem 1.3.13), which ensures existence and uniqueness up to isomorphism of the direct summands of M1 and M2. The algorithm produces the following data: the R-module C and the R-module homomorphism f such that f : C —∼→ f (C). Put D := f (C). The algorithm also produces injections of C and D into M1 and M2 respectively, that define them as submodules. By splitting these injections (see Proposition 2.5.1), we can recover the direct complements of C and D in M1 and M2 respectively, together with the corre- sponding isomorphisms.
Proof of Theorem. In Lemma C.1, we prove succinctness, in Lemma C.2, we prove robustness, and in Lemma C.5, we prove unforgeability.
Proof of Theorem. 4.1.3 ^ Just as in the proof of Theorem 4.1.2, we break the argument into two steps, and use the same notation as before for the operator τN ٨ Γ (ω) and for the symbols γz. We also set Ω+ = {z ∈ Ω | Im z > 0}.
Step 1. Finitely many jumps. Just as before, suppose that the symbol ω has finitely- many jump-discontinuities. Write
Proof of Theorem. 4.1.2 The proof of the result will be broken down in two Steps. For brevity, we de- ^ note by Γ (N)(ω) the operator τN ٨ Γ (ω). We also recall that Ω is the set of jump- discontinuities of the symbol ω and c is the function in (4.1.10).
Step 1. Finitely many jumps. Suppose that Ω is finite. Setting γz(v) = −iγ(zv), with Σ γ being the symbol defined in (4.1.4), write ω(v) = κz(ω)γz(v) + η(v) (4.4.46) z∈Ω Σ where η is continuous on T and let Φ denote the symbol Φ(v) = κz(ω)γz(v). z∈Ω Weyl’s inequality (2.4.18) shows that for 0 < s < t one has ^ ^ n(t; Γ (N)(ω)) ≤ n(t − s; Γ (N)(Φ^)) + n(s; Γ (N)(η)), n(t; Γ (N)(ω^)) ≥ n(t + s; Γ (N)(Φ^)) − n(s; Γ (N)(η^)). Since Γ (η^) is compact, Lemma 2.4.4 shows that n(s; Γ (N)(η)) = O (1), N → ∞,s ^ ^ and so, using the definition of the fu^nctionals LDτ , LDτ we deduce that for any t > 0 LDτ (t; Γ (ω)) ≤ LDτ (t − 0; Γ (Φ)), (4.4.47) ^ ^ LDτ (t; Γ (ω)) ≥ LDτ (t + 0; Γ (Φ)). (4.4.48) ^ Σ ^ Integration by parts shows that Φ(j) = κz(ω)γz(j) z∈Ω π(j + 1)
j + 1 = −i Σ κ (ω)zj = O 1 , j → ∞ (4.4.49) z∈Ω ^
Proof of Theorem. 3.2. Recall that we have knowable sets K1, K2,..., Kn that cover Σˆ(P ) and satisfy the activity property. We wish to show that they have a nonempty intersection. The first step in adapting the proof of Theorem 3.3 is to prove an analog of Xxxxx ∩ ⊇ ⊇ ···
5.1. Recall that for a fragment τ the set of schedules that are quasi extensions of τ is denoted Qτ , and Qˆτ = ΣˆP Qτ . The sets Qˆτ play the role of .-balls. For example, for each compressed schedule φ, and for each integer w, let φ(w) denote the maximal prefix of φ whose weight (sum of block sizes) is at most w. If we consider the sequence of sets Qˆφ(w) we see that Qˆφ(1) Qˆφ(2) and that the intersection of all of them is just φ itself. This is analogous to a sequence of balls of decreasing radius around a particular point. The analog of Lemma 5.1 is the following. Lemma 6.1. Let K be a collection of knowable sets that covers Σˆ(P ). Then there is an integer w = w(K) with the property that for each schedule φ, some member of K contains Qˆφ(w) .
Proof of Theorem. 10 First, we introduce some alternate notations for the minimum bisection problem in order to ease the transition to the Sherringkton- Xxxxxxxxxxx formalism. Denote by G an undirected weighted complete graph with n vertices. The problem consists in finding a bisection of the graph (a partition in two subsets of equal size) of minimum cost. More formally, define by gij the weight assigned to the edge between vertices i and j (gij = gji). Σ Denote by ci ∈ {−1, 1} an indicator of the suΣbset containing vertex i. We need to find c 1, 1 n such that the sum of the weights of cut edges R(c, X) = ci= cj i<j i ci = 0 (balance condition) and
Proof of Theorem. We will prove that all the honest users will agree on the same Θ, that no malicious user can build a valid certificate fo.r a diffeΣrent Θˆ , and that the honest users will be able 5+3χ h to produce a certificate for Θ within time t ℓ, 2 + λ. To conclude the proof it is sufficient to note that, since t(s+1) = t(s) + 2λ for all s ≥ 3, we have that: . Σ 5+3χ h 5+3χ h t ℓ, 2 = t(1) + Σℓ, 2 . Σ t(s) − t(s−1) (13) 5+3χ h = t(1) + . Σ t(2) − t(1) + . Σ t(3) − t(2) + Σℓ, 2 . t(s) − t(s−1) 5+3χ h = Ω + Λ + λ + Λ + λ + Σℓ, 2 2λ = Ω + 2Λ + 2λ + . Σ 2 + 3χℓ, h 2λ = Ω + 2Λ + 6 + 6χℓ, h
Proof of Theorem. 4. Let H ∈ (DR(G)∪ DR(G′))/R and s, s¯ ∈ H. We have fin ˜ ˜ ∀H˜ ∈ (DR(G) ∪ DR(G′))/R, ∀A ∈ NL , s →A P H˜ ⇔ s¯ →P H˜. Since this equality ˜ ˜ is valid for all s, s¯ ∈ H, we can rewrite it as H →P H and denote PMA(H, H) = The same holds for ′DR(G ). ˜ ˜ PMA(s, H) = PMA(s¯, H). The transitions from the states of DR(G) always lead to those from the same set, hence, ∀s ∈ DR(G), PMA(s, H) = PMA(s, H ∩ DR(G)). An A1 Let Σ = A1 · · · An be a derived step trace of G and G′. Then ∃H0, . . . , Hn ∈ → → H (DR(G) ∪ DR(G′))/R , H0 P1 A2 1 P2 · · · → Pn Hn . Let us prove that the sum of probabilities of all the paths starting in every s0 ∈ H0 and going through the states from H1, . . . , Hn is equal to the product of P1, . . . , Pn: ∑ ∏PT (Υi, si−1) =∏PMAi (Hi−1, Hi). Υ1 Υn
i=1 i=1
(1≤ i≤n)} ∏n+1 PT (Υ , s ) = ∑ ∏ → ∑{Υ Υn+1 |s s , L(Υ )=A n 0 →··· → n i i i i , s ∈H , s ∈H } ∑
i=1 PT (Υi, si−1)PT (Υn+1, sn) = [∏n PT (Υ , s ) 0 →··· → n {Υ ,...,Υ |s Υns , L(Υ )=A , s ∈H
Proof of Theorem. 5.1. We are able to closely follow the security proof for ITK in [AJM22]. The proof is divided into two parts. First a sequence of hybrids shows that a version of the MLS protocol (MLS∗) that has a different parent hash securely realizes CGKA with a slightly different version of the safe predicate denoted safe∗. The second step consists in showing that using the actual parent hash guarantees and the actual safe predicate satisfy the following property: if safe(c) = true, then no node in the ratchet tree of the epoch that corresponds to c contains an exposed key pk. We argue that both steps can be adapted to the case of MLS-Cutoff. We consider the following hybrids: • Hybrid H0: Real world. • Hybrid H1: Instead of the protocol as in the real world, now we consider a dummy functionality FDummy that simply routs its inputs and outputs through a simulator S1 that executes the protocol.