/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR x y)
(RULES
le(0,s(x)) -> true
le(s(x),s(y)) -> le(x,y)
le(x,0) -> false
m -> s(s(s(s(0))))
pop(empty) -> empty
pop(push(x,y)) -> x | le(size(x),m) -> true
size(empty) -> 0
size(push(x,y)) -> s(size(x))
top(empty) -> eentry
top(push(x,y)) -> y | le(size(x),m) -> true
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
 SIZE(push(x,y)) -> SIZE(x)
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true

Conditional Termination Problem 2:
-> Pairs:
 POP(push(x,y)) -> LE(size(x),m)
 POP(push(x,y)) -> M
 POP(push(x,y)) -> SIZE(x)
 TOP(push(x,y)) -> LE(size(x),m)
 TOP(push(x,y)) -> M
 TOP(push(x,y)) -> SIZE(x)
-> QPairs:
 LE(s(x),s(y)) -> LE(x,y)
 SIZE(push(x,y)) -> SIZE(x)
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
 SIZE(push(x,y)) -> SIZE(x)
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 SIZE(push(x,y)) -> SIZE(x)
->->-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->->Cycle:
->->-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
->->-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true


The problem is decomposed in 2 subproblems.

Problem 1.1.1: 

Conditional Subterm Processor:
-> Pairs:
 SIZE(push(x,y)) -> SIZE(x)
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Projection:
 pi(SIZE) = 1

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.1.2: 

Conditional Subterm Processor:
-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Projection:
 pi(LE) = 1

Problem 1.1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 POP(push(x,y)) -> LE(size(x),m)
 POP(push(x,y)) -> M
 POP(push(x,y)) -> SIZE(x)
 TOP(push(x,y)) -> LE(size(x),m)
 TOP(push(x,y)) -> M
 TOP(push(x,y)) -> SIZE(x)
-> QPairs:
 LE(s(x),s(y)) -> LE(x,y)
 SIZE(push(x,y)) -> SIZE(x)
-> Rules:
 le(0,s(x)) -> true
 le(s(x),s(y)) -> le(x,y)
 le(x,0) -> false
 m -> s(s(s(s(0))))
 pop(empty) -> empty
 pop(push(x,y)) -> x | le(size(x),m) -> true
 size(empty) -> 0
 size(push(x,y)) -> s(size(x))
 top(empty) -> eentry
 top(push(x,y)) -> y | le(size(x),m) -> true
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.