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Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270307
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
n134.star.cs.uiowa.edu
space
Mixed_HO_10
run statistics
property
value
solver
Wanda 2.1c
configuration
default
runtime (wallclock)
0.281437 seconds
cpu usage
0.281495
user time
0.239311
system time
0.042184
max virtual memory
113176.0
max residence set size
7672.0
stage attributes
key
value
starexec-result
YES
output
0.00/0.26 YES 0.00/0.28 We consider the system theBenchmark. 0.00/0.28 0.00/0.28 Alphabet: 0.00/0.28 0.00/0.28 0 : [] --> nat 0.00/0.28 add : [nat] --> nat -> nat 0.00/0.28 eq : [nat] --> nat -> bool 0.00/0.28 err : [] --> nat 0.00/0.28 false : [] --> bool 0.00/0.28 id : [] --> nat -> nat 0.00/0.28 nul : [] --> nat -> bool 0.00/0.28 pred : [nat] --> nat 0.00/0.28 s : [nat] --> nat 0.00/0.28 true : [] --> bool 0.00/0.28 0.00/0.28 Rules: 0.00/0.28 0.00/0.28 nul 0 => true 0.00/0.28 nul s(x) => false 0.00/0.28 nul err => false 0.00/0.28 pred(0) => err 0.00/0.28 pred(s(x)) => x 0.00/0.28 id x => x 0.00/0.28 eq(0) => nul 0.00/0.28 eq(s(x)) => /\y.eq(x) pred(y) 0.00/0.28 add(0) => id 0.00/0.28 add(s(x)) => /\y.add(x) s(y) 0.00/0.28 0.00/0.28 This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). 0.00/0.28 0.00/0.28 We use rule removal, following [Kop12, Theorem 2.23]. 0.00/0.28 0.00/0.28 This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): 0.00/0.28 0.00/0.28 nul 0 >? true 0.00/0.28 nul s(X) >? false 0.00/0.28 nul err >? false 0.00/0.28 pred(0) >? err 0.00/0.28 pred(s(X)) >? X 0.00/0.28 id X >? X 0.00/0.28 eq(0) >? nul 0.00/0.28 eq(s(X)) >? /\x.eq(X) pred(x) 0.00/0.28 add(0) >? id 0.00/0.28 add(s(X)) >? /\x.add(X) s(x) 0.00/0.28 0.00/0.28 We use a recursive path ordering as defined in [Kop12, Chapter 5]. 0.00/0.28 0.00/0.28 Argument functions: 0.00/0.28 0.00/0.28 [[false]] = _|_ 0.00/0.28 [[id]] = _|_ 0.00/0.28 [[nul]] = _|_ 0.00/0.28 [[true]] = _|_ 0.00/0.28 0.00/0.28 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 0.00/0.28 0.00/0.28 Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: 0.00/0.28 0.00/0.28 @_{o -> o}(_|_, 0) >= _|_ 0.00/0.28 @_{o -> o}(_|_, s(X)) >= _|_ 0.00/0.28 @_{o -> o}(_|_, err) >= _|_ 0.00/0.28 pred(0) >= err 0.00/0.28 pred(s(X)) > X 0.00/0.28 @_{o -> o}(_|_, X) >= X 0.00/0.28 eq(0) >= _|_ 0.00/0.28 eq(s(X)) >= /\x.@_{o -> o}(eq(X), pred(x)) 0.00/0.28 add(0) >= _|_ 0.00/0.28 add(s(X)) >= /\x.@_{o -> o}(add(X), s(x)) 0.00/0.28 0.00/0.28 With these choices, we have: 0.00/0.28 0.00/0.28 1] @_{o -> o}(_|_, 0) >= _|_ by (Bot) 0.00/0.28 0.00/0.28 2] @_{o -> o}(_|_, s(X)) >= _|_ by (Bot) 0.00/0.28 0.00/0.28 3] @_{o -> o}(_|_, err) >= _|_ by (Bot) 0.00/0.28 0.00/0.28 4] pred(0) >= err because [5], by (Star) 0.00/0.28 5] pred*(0) >= err because pred > err, by (Copy) 0.00/0.28 0.00/0.28 6] pred(s(X)) > X because [7], by definition 0.00/0.28 7] pred*(s(X)) >= X because [8], by (Select) 0.00/0.28 8] s(X) >= X because [9], by (Star) 0.00/0.28 9] s*(X) >= X because [10], by (Select) 0.00/0.28 10] X >= X by (Meta) 0.00/0.28 0.00/0.28 11] @_{o -> o}(_|_, X) >= X because [12], by (Star) 0.00/0.28 12] @_{o -> o}*(_|_, X) >= X because [13], by (Select) 0.00/0.28 13] X >= X by (Meta) 0.00/0.28 0.00/0.28 14] eq(0) >= _|_ by (Bot) 0.00/0.28 0.00/0.28 15] eq(s(X)) >= /\x.@_{o -> o}(eq(X), pred(x)) because [16], by (Star) 0.00/0.28 16] eq*(s(X)) >= /\y.@_{o -> o}(eq(X), pred(y)) because [17], by (F-Abs) 0.00/0.28 17] eq*(s(X), x) >= @_{o -> o}(eq(X), pred(x)) because eq > @_{o -> o}, [18] and [22], by (Copy) 0.00/0.28 18] eq*(s(X), x) >= eq(X) because eq in Mul and [19], by (Stat) 0.00/0.28 19] s(X) > X because [20], by definition 0.00/0.28 20] s*(X) >= X because [21], by (Select) 0.00/0.28 21] X >= X by (Meta)
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