/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 Y)
(RULES
div(0,s(Y)) -> 0
div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
minus(s(X),s(Y)) -> p(minus(X,Y))
minus(X,0) -> X
p(s(X)) -> X
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
 DIV(s(X),s(Y)) -> MINUS(X,Y)
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
 MINUS(s(X),s(Y)) -> P(minus(X,Y))
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X

Problem 1: 

SCC Processor:
-> Pairs:
 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
 DIV(s(X),s(Y)) -> MINUS(X,Y)
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
 MINUS(s(X),s(Y)) -> P(minus(X,Y))
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
->->-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->->Cycle:
->->-> Pairs:
 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
->->-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
-> Usable rules:
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = X1 + 1
[p](X) = 2.X + 1
[0] = 0
[s](X) = 2.X + 2
[DIV](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 div(0,s(Y)) -> 0
 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
 minus(s(X),s(Y)) -> p(minus(X,Y))
 minus(X,0) -> X
 p(s(X)) -> X
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.