### Table 5. Laws for Commuting and Distributing Update Connectives

2006

"... In PAGE 55: ...Schema Variables Table5 . Modi ers for Schema Variables Modi er Applicable to rigid \term A \formula Terms or formulae that can syntactically be identi ed as rigid strict \term A Terms of type A (and not of proper subtypes of A) list \program t Sequences of program entities.... In PAGE 106: ...xample 2. We continue Example 1 and assume the same vocabulary/algebra. a := 1 ; f(a) := 2 a := 1 j f(1) := 2 valS; (a := 1 ; f(a) := 2) = fhai 7! 1; hf; (1)i 7! 2g valS; (a := 1 ; (a := 3 j f(a) := 2)) = fhai 7! 3; hf; (1)i 7! 2g We normalise the update in the second line using the given rewriting rules: a := 1 ; (a := 3 j f(a) := 2) (R45) ! a := 1 j fa := 1g (a := 3 j f(a) := 2) (R48) ! a := 1 j (fa := 1g a := 3 j fa := 1g f(a) := 2) (R47) ! a := 1 j (a := fa := 1g 3 j f(fa := 1g a) := fa := 1g 2) (R2); (R12) ! a := 1 j (a := 3 j f(non-rec(a := 1; a; ())) := 2) (R11) ! a := 1 j (a := 3 j f(if true then 1 else a) := 2) The last expression can be simpli ed further using rules for conditional terms, which are, however, beyond the scope of this paper. Further, using (R54) in Table5 , it is possible to eliminate the assignment a := 1, which is overridden by a := 3. 8 Soundness and Completeness of Update Application The following two lemmas state that the rewriting rules from Sect.... In PAGE 111: ...ewriting rules for update application (than the ones given in Sect. 5). This has been done for the implementation of updates in KeY. Table5 gives, besides others, identities that enable to establish form (1) by turning sequential composition into parallel composition, distributing if and for through parallel composition and commuting if and for. Another impor-... ..."

### TABLE I COMMUTATIVE GROUPS AND THEIR DUALS

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Cited by 11

### Table 1: First functions and their orders for the method of iterated commutators

"... In PAGE 32: ...Table1 0: A Matlab program for the IC 4 GR for Toda #0Dows function L1 = TodaIC4_Radau#28L0, h, t0, tf#29; #25 function L1 = TodaIC4_Radau#28L0, h, t0, tf#29; #25 #25 Solves the Toda flow with the Lie group invariant IC4GR #25 B_IC4R#28L0, t1,t2,t3,t4,t5,t6#29 evaluates the 3-rd order interpolant #25 to B at t=t1,.... In PAGE 33: ...Table1 1: A Matlab program for the Magnus method with r = 2 and order 4 for Toda #0Dows function L1 = TodaMagnus4#28L0, h, t0, tf#29; #25 function L1 = TodaMagnus4#28L0, h, t0, tf#29; #25 #25 Solves the Toda flow with the Lie group invariant method Magnus4 #25 #28cf. Iserles and Norsett#29 #25 c1 = 1#2F2 - sqrt#283#29#2F6; c2 = 1#2F2 + sqrt#283#29#2F6; w11 = 1#2F2; w12 = 1#2F2; w21 = sqrt#283#29#2F6; for tn = t0 : h : tf-eps #25 #25 integrate the orthogonal flow #25 #5BX1, X2, Xh#5D = B_M4#28L0, c1*h, c2*h, h#29; Psi_h = h * #28w11*X1 + w12*X2#29; sigma0_h = Psi_h + 0.... In PAGE 34: ...Table1 2: A Matlab program for a 4-th order Lie-group method of Munthe-Kaas for Toda #0Dows function L1 = TodaMK4#28L0, h, t0, tf#29; #25 function L1 = TodaMK4#28L0, h, t0, tf#29 #25 #25 Solves the Toda flow with the Lie group invariant RK method RK4r.m c = #5B0 1#2F2 1#2F2 1#5D; a=#5B0 0 0 0 1#2F2 0 0 0 0 1#2F2 0 0 0 0 1 0 #5D; b = #5B1#2F6 1#2F3 1#2F3 1#2F6#5D; ord = 4; #5Bn, m#5D = size#28L0#29; I = eye#28n#29; for tn = t0 : h : tf-eps #25 #25 integrate the orthogonal flow #25 P1 = I; K1 = QRflow#28L0#29; U2 = h * a#282,1#29 * K1; P2 = ER_expm#28U2#29*P1; K2 = B_RK4#28P2, L0#29; #25 evaluates B#28P2*L0*P2^T#29 K2 = dexpinvr#28U2,K2,ord#29; U3 = h * #28a#283,1#29 * K1 + a#283,2#29 * K2#29; P3 = ER_expm#28U3#29*P1; K3 = B_RK4#28P3, L0#29; K3 = dexpinvr#28U3,K3,ord#29; U4 = h * #28a#284,1#29 * K1 + a#284,2#29 * K2 + a#284,3#29*K3#29; P4 = ER_expm#28U4#29*P1; K4 = B_RK4#28P4, L0#29; K4 = dexpinvr#28U4,K4,ord#29; V = h * #28b#281#29 * K1 + b#282#29 * K2 + b#283#29 * K3 + b#284#29*K4#29; Y1 = ER_expm#28V#29*P1; L1 = Y1 * L0 * Y1 apos;; #25 update the numerical solution L0 = L1; end... ..."

### Table I. Estimated fault coverage for benchmark programs. Branch

2002

Cited by 28

### Table 4: Precondition formulae for the true-branch part of atomic program conditions that manipulate linked lists.

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Cited by 70

### Table 4: Precondition formulae for the true-branch part of atomic program conditions that manipulate linked lists.

### Table 2.3: Multiplication table of the eight{dimensional commutative algebra

### Table 1 Running times, mathematical programming vs. direct branch and bound

"... In PAGE 17: ...ection 4.2.4 were used in the experiments. Table1 shows the absolute and relative performance of the two algorithms. Each cell of the table contains the mean and standard deviations of execution times as well as the number of experiments I.... ..."

### Table 1: Icosahedron Subdivision Parameters

"... In PAGE 3: ... The vertices (normalized to lie on the unit sphere) of the l-frequency subdivision of the icosahedron were used to de ne viewpoints around the object in its canonical position. Table1 enumer- ates the number of viewpoints and the angle between view directions for a range of values of the subdivi- sion parameter l. The choice of l allows the number of viewpoints (hence the angular spacing between view-... ..."

### Table 4: Environment Part I

"... In PAGE 5: ... We assume that the reader is familiar with the class file format as described in the official specification of JVML [1]. The environment as described in Table4 and Table 5 models the different program declarations and is represented as a map that associates a set of classes to a set of reference types. A class is a record containing a constant pool, a super-class, a set of interfaces, a list of fields, a map that associates values to static fields, a list of methods, two flags that indicate whether the class is initialized or not and if the VOL 6, NO.... ..."

Cited by 1