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Higher Order Rewriting Union Beta pair #487093877
details
property
value
status
complete
benchmark
06plusmult.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n150.star.cs.uiowa.edu
space
Blanqui_15
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.455026 seconds
cpu usage
0.439388
user time
0.336817
system time
0.102571
max virtual memory
113188.0
max residence set size
8096.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: mult : [N * N] --> N plus : [N * N] --> N s : [N] --> N z : [] --> N Rules: plus(z, x) => x plus(s(x), y) => plus(x, s(y)) plus(plus(x, y), u) => plus(x, plus(y, u)) mult(z, x) => z mult(s(x), y) => plus(mult(x, y), y) mult(plus(x, y), u) => plus(mult(x, u), mult(y, u)) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): plus(z, X) >? X plus(s(X), Y) >? plus(X, s(Y)) plus(plus(X, Y), Z) >? plus(X, plus(Y, Z)) mult(z, X) >? z mult(s(X), Y) >? plus(mult(X, Y), Y) mult(plus(X, Y), Z) >? plus(mult(X, Z), mult(Y, Z)) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[z]] = _|_ We choose Lex = {plus} and Mul = {mult, s}, and the following precedence: mult > plus > s Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: plus(_|_, X) >= X plus(s(X), Y) >= plus(X, s(Y)) plus(plus(X, Y), Z) > plus(X, plus(Y, Z)) mult(_|_, X) >= _|_ mult(s(X), Y) >= plus(mult(X, Y), Y) mult(plus(X, Y), Z) > plus(mult(X, Z), mult(Y, Z)) With these choices, we have: 1] plus(_|_, X) >= X because [2], by (Star) 2] plus*(_|_, X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] plus(s(X), Y) >= plus(X, s(Y)) because [5], by (Star) 5] plus*(s(X), Y) >= plus(X, s(Y)) because [6], [9] and [11], by (Stat) 6] s(X) > X because [7], by definition 7] s*(X) >= X because [8], by (Select) 8] X >= X by (Meta) 9] plus*(s(X), Y) >= X because [10], by (Select) 10] s(X) >= X because [7], by (Star) 11] plus*(s(X), Y) >= s(Y) because plus > s and [12], by (Copy) 12] plus*(s(X), Y) >= Y because [13], by (Select) 13] Y >= Y by (Meta) 14] plus(plus(X, Y), Z) > plus(X, plus(Y, Z)) because [15], by definition 15] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z)) because [16], [19] and [21], by (Stat) 16] plus(X, Y) > X because [17], by definition 17] plus*(X, Y) >= X because [18], by (Select) 18] X >= X by (Meta) 19] plus*(plus(X, Y), Z) >= X because [20], by (Select) 20] plus(X, Y) >= X because [17], by (Star) 21] plus*(plus(X, Y), Z) >= plus(Y, Z) because [22], [25] and [27], by (Stat) 22] plus(X, Y) > Y because [23], by definition 23] plus*(X, Y) >= Y because [24], by (Select) 24] Y >= Y by (Meta) 25] plus*(plus(X, Y), Z) >= Y because [26], by (Select) 26] plus(X, Y) >= Y because [23], by (Star) 27] plus*(plus(X, Y), Z) >= Z because [28], by (Select) 28] Z >= Z by (Meta) 29] mult(_|_, X) >= _|_ by (Bot) 30] mult(s(X), Y) >= plus(mult(X, Y), Y) because [31], by (Star) 31] mult*(s(X), Y) >= plus(mult(X, Y), Y) because mult > plus, [32] and [37], by (Copy) 32] mult*(s(X), Y) >= mult(X, Y) because mult in Mul, [33] and [36], by (Stat) 33] s(X) > X because [34], by definition 34] s*(X) >= X because [35], by (Select) 35] X >= X by (Meta) 36] Y >= Y by (Meta) 37] mult*(s(X), Y) >= Y because [36], by (Select) 38] mult(plus(X, Y), Z) > plus(mult(X, Z), mult(Y, Z)) because [39], by definition 39] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) because mult > plus, [40] and [45], by (Copy) 40] mult*(plus(X, Y), Z) >= mult(X, Z) because mult in Mul, [41] and [44], by (Stat) 41] plus(X, Y) > X because [42], by definition 42] plus*(X, Y) >= X because [43], by (Select) 43] X >= X by (Meta) 44] Z >= Z by (Meta)
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