TRS Innermost pair #487092940

details

property	value
status	complete
benchmark	#4.37.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n149.star.cs.uiowa.edu
space	AG01_innermost

run statistics

property	value
solver	muterm 6.0.3
configuration	default
runtime (wallclock)	0.0771191 seconds
cpu usage	0.050272
user time	0.026663
system time	0.023609
max virtual memory	113188.0
max residence set size	4960.0

stage attributes

key	value
starexec-result	YES

output

YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(c(s(x:S),y:S)) -> F(c(x:S,s(y:S)))
 G(c(x:S,s(y:S))) -> G(c(s(x:S),y:S))
-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 F(c(s(x:S),y:S)) -> F(c(x:S,s(y:S)))
 G(c(x:S,s(y:S))) -> G(c(s(x:S),y:S))
-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(c(x:S,s(y:S))) -> G(c(s(x:S),y:S))
->->-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
->->Cycle:
->->-> Pairs:
 F(c(s(x:S),y:S)) -> F(c(x:S,s(y:S)))
->->-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 G(c(x:S,s(y:S))) -> G(c(s(x:S),y:S))
-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[f](X) = 0
[g](X) = 0
[c](X1,X2) = 2.X2
[fSNonEmpty] = 0
[s](X) = 2.X + 2
[F](X) = 0
[G](X) = 2.X

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 F(c(s(x:S),y:S)) -> F(c(x:S,s(y:S)))
-> Rules:
 f(c(s(x:S),y:S)) -> f(c(x:S,s(y:S)))
 g(c(x:S,s(y:S))) -> g(c(s(x:S),y:S))
-> Usable rules:
 Empty
->Interpretation type:
 Linear

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