/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR x x' y y' z)
(RULES
fib(0) -> pair(0,s(0))
fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
plus(x,y) -> y' | x -> 0, y -> y'
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 FIB(s(x)) -> PLUS(y,z) | fib(x) -> pair(y,z)
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'

Conditional Termination Problem 2:
-> Pairs:
 FIB(s(x)) -> FIB(x)
-> QPairs:
 FIB(s(x)) -> PLUS(y,z) | fib(x) -> pair(y,z)
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 FIB(s(x)) -> PLUS(y,z) | fib(x) -> pair(y,z)
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
->->-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-1+x_1 >= 0) /\ (0 >= 0))
->Interpretation:
 
[delta] = 1
[s](x_1) = 1+x_1
[PLUS](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 FIB(s(x)) -> FIB(x)
-> QPairs:
 FIB(s(x)) -> PLUS(y,z) | fib(x) -> pair(y,z)
 PLUS(x,y) -> PLUS(x',y') | x -> s(x'), y -> y'
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 FIB(s(x)) -> FIB(x)
->->-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'

Problem 1.2: 

Conditional Subterm Processor:
-> Pairs:
 FIB(s(x)) -> FIB(x)
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
->Projection:
 pi(FIB) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 fib(0) -> pair(0,s(0))
 fib(s(x)) -> pair(z,plus(y,z)) | fib(x) -> pair(y,z)
 plus(x,y) -> s(plus(x',y')) | x -> s(x'), y -> y'
 plus(x,y) -> y' | x -> 0, y -> y'
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.