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Higher Order Rewriting Union Beta pair #487093835
details
property
value
status
complete
benchmark
onearg.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n151.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.300307 seconds
cpu usage
0.293992
user time
0.246316
system time
0.047676
max virtual memory
113188.0
max residence set size
7640.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat add : [nat] --> nat -> nat eq : [nat] --> nat -> bool err : [] --> nat false : [] --> bool id : [] --> nat -> nat nul : [] --> nat -> bool pred : [nat] --> nat s : [nat] --> nat true : [] --> bool Rules: nul 0 => true nul s(x) => false nul err => false pred(0) => err pred(s(x)) => x id x => x eq(0) => nul eq(s(x)) => /\y.eq(x) pred(y) add(0) => id add(s(x)) => /\y.add(x) s(y) 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]): nul 0 >? true nul s(X) >? false nul err >? false pred(0) >? err pred(s(X)) >? X id X >? X eq(0) >? nul eq(s(X)) >? /\x.eq(X) pred(x) add(0) >? id add(s(X)) >? /\x.add(X) s(x) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[false]] = _|_ [[id]] = _|_ [[nul]] = _|_ [[true]] = _|_ We choose Lex = {} and Mul = {0, @_{o -> o}, add, eq, err, pred, s}, and the following precedence: add > 0 > s > eq > pred > @_{o -> o} > err Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: @_{o -> o}(_|_, 0) >= _|_ @_{o -> o}(_|_, s(X)) >= _|_ @_{o -> o}(_|_, err) >= _|_ pred(0) >= err pred(s(X)) > X @_{o -> o}(_|_, X) >= X eq(0) >= _|_ eq(s(X)) >= /\x.@_{o -> o}(eq(X), pred(x)) add(0) >= _|_ add(s(X)) >= /\x.@_{o -> o}(add(X), s(x)) With these choices, we have: 1] @_{o -> o}(_|_, 0) >= _|_ by (Bot) 2] @_{o -> o}(_|_, s(X)) >= _|_ by (Bot) 3] @_{o -> o}(_|_, err) >= _|_ by (Bot) 4] pred(0) >= err because [5], by (Star) 5] pred*(0) >= err because pred > err, by (Copy) 6] pred(s(X)) > X because [7], by definition 7] pred*(s(X)) >= X because [8], by (Select) 8] s(X) >= X because [9], by (Star) 9] s*(X) >= X because [10], by (Select) 10] X >= X by (Meta) 11] @_{o -> o}(_|_, X) >= X because [12], by (Star) 12] @_{o -> o}*(_|_, X) >= X because [13], by (Select) 13] X >= X by (Meta) 14] eq(0) >= _|_ by (Bot) 15] eq(s(X)) >= /\x.@_{o -> o}(eq(X), pred(x)) because [16], by (Star) 16] eq*(s(X)) >= /\y.@_{o -> o}(eq(X), pred(y)) because [17], by (F-Abs) 17] eq*(s(X), x) >= @_{o -> o}(eq(X), pred(x)) because eq > @_{o -> o}, [18] and [22], by (Copy) 18] eq*(s(X), x) >= eq(X) because eq in Mul and [19], by (Stat) 19] s(X) > X because [20], by definition 20] s*(X) >= X because [21], by (Select) 21] X >= X by (Meta)
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