By N. Vilenkin, George Yankovsky (translator)

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This leads to ≡ ≡ P 0(y := 1) {substitution} x2 + 1 = k · p {} x2 =p −1 ∧ k = (x2 + 1)/p . Hence k := (x2 + 1)/p establishes P 0(y := 1) provided x is chosen such that x2 =p −1. This value of k is obviously positive, so P 1 is established as well. As to P 2, we have ≡ ⇐ (k < p)(k := (x2 + 1)/p) {substitution} x 2 + 1 < p2 {(p − 1)2 + 1 < p2 since p ≥ 2} 1≤x

2. Every vertex of G is also a vertex of T (T spans G). Let us assume that G is given as an adjacence relation, that is an irreflexive and symmetric relation on a set V (the set of vertices), and that the output T is required in the same format. Representing graphs as relations facilitates the implementation of algorithms, because relations can be encoded efficiently as Boolean matrices, linked lists, or binary decision diagrams. It also facilitates the design of algorithms, because relations are the objects of a concise algebraic calculus (which was formalized in 1941 by Tarski, see also [11,4]).

They can be thought of as denoting sets of edges. Lemma 21 Γ (p) = (f ∩ p)∪ g ∪ (g ∩ p)∪ f = f ∪ (I ∩ p)g ∪ g∪ (I ∩ p)f . Proof. Γ (p) = = = = = {Definition of Γ , vector associativity} I ∩ (M ∩ p)∪ M {M = f ∪ g, distributivity} I ∩ ((f ∩ p)∪ f ∪ (f ∩ p)∪ g ∪ (g ∩ p)∪ f ∪ (g ∩ p)∪ g) {f ∪ f ⊆ I and g∪ g ⊆ I since f and g are a functions} I ∩ ((f ∩ p)∪ g ∪ (g ∩ p)∪ f) {f ∪ g ⊆ I and g∪ f ⊆ I since f, g are disjoint (Schr¨ oder)} ∪ (f ∩ p) g ∪ (g ∩ p)∪ f {Vector identity rule} f ∪ (I ∩ p)g ∪ g∪ (I ∩ p)f .