/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 v_NonEmpty:S x:S y:S z:S)
(RULES
f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
f(x:S,nil) -> g(nil,x:S)
norm(g(x:S,y:S)) -> s(norm(x:S))
norm(nil) -> 0
rem(g(x:S,y:S),0) -> g(x:S,y:S)
rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
rem(nil,y:S) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(x:S,g(y:S,z:S)) -> F(x:S,y:S)
 NORM(g(x:S,y:S)) -> NORM(x:S)
 REM(g(x:S,y:S),s(z:S)) -> REM(x:S,z:S)
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 F(x:S,g(y:S,z:S)) -> F(x:S,y:S)
 NORM(g(x:S,y:S)) -> NORM(x:S)
 REM(g(x:S,y:S),s(z:S)) -> REM(x:S,z:S)
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REM(g(x:S,y:S),s(z:S)) -> REM(x:S,z:S)
->->-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->->Cycle:
->->-> Pairs:
 NORM(g(x:S,y:S)) -> NORM(x:S)
->->-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->->Cycle:
->->-> Pairs:
 F(x:S,g(y:S,z:S)) -> F(x:S,y:S)
->->-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 REM(g(x:S,y:S),s(z:S)) -> REM(x:S,z:S)
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Projection:
 pi(REM) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 NORM(g(x:S,y:S)) -> NORM(x:S)
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Projection:
 pi(NORM) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 F(x:S,g(y:S,z:S)) -> F(x:S,y:S)
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Projection:
 pi(F) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(x:S,g(y:S,z:S)) -> g(f(x:S,y:S),z:S)
 f(x:S,nil) -> g(nil,x:S)
 norm(g(x:S,y:S)) -> s(norm(x:S))
 norm(nil) -> 0
 rem(g(x:S,y:S),0) -> g(x:S,y:S)
 rem(g(x:S,y:S),s(z:S)) -> rem(x:S,z:S)
 rem(nil,y:S) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.