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


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR x y)
(RULES
-(s(x),s(y)) -> -(x,y)
-(x,0) -> x
if(false,x,y) -> y
if(true,x,y) -> x
leq(0,y) -> true
leq(s(x),0) -> false
leq(s(x),s(y)) -> leq(x,y)
mod(0,y) -> 0
mod(s(x),0) -> 0
mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 LEQ(s(x),s(y)) -> LEQ(x,y)
 MOD(s(x),s(y)) -> -#(s(x),s(y))
 MOD(s(x),s(y)) -> IF(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
 MOD(s(x),s(y)) -> LEQ(y,x)
 MOD(s(x),s(y)) -> MOD(-(s(x),s(y)),s(y))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))

Problem 1: 

SCC Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
 LEQ(s(x),s(y)) -> LEQ(x,y)
 MOD(s(x),s(y)) -> -#(s(x),s(y))
 MOD(s(x),s(y)) -> IF(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
 MOD(s(x),s(y)) -> LEQ(y,x)
 MOD(s(x),s(y)) -> MOD(-(s(x),s(y)),s(y))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LEQ(s(x),s(y)) -> LEQ(x,y)
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->->Cycle:
->->-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->->Cycle:
->->-> Pairs:
 MOD(s(x),s(y)) -> MOD(-(s(x),s(y)),s(y))
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LEQ(s(x),s(y)) -> LEQ(x,y)
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Projection:
 pi(LEQ) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 -#(s(x),s(y)) -> -#(x,y)
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Projection:
 pi(-#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 MOD(s(x),s(y)) -> MOD(-(s(x),s(y)),s(y))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Narrowed Pairs:
->->Original Pair:
 MOD(s(x),s(y)) -> MOD(-(s(x),s(y)),s(y))
->-> Narrowed pairs:
 MOD(s(x),s(y)) -> MOD(-(x,y),s(y))

Problem 1.3: 

SCC Processor:
-> Pairs:
 MOD(s(x),s(y)) -> MOD(-(x,y),s(y))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MOD(s(x),s(y)) -> MOD(-(x,y),s(y))
->->-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 MOD(s(x),s(y)) -> MOD(-(x,y),s(y))
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
-> Usable rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = 2.X1 + 1
[0] = 0
[s](X) = 2.X + 2
[MOD](X1,X2) = 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 -(s(x),s(y)) -> -(x,y)
 -(x,0) -> x
 if(false,x,y) -> y
 if(true,x,y) -> x
 leq(0,y) -> true
 leq(s(x),0) -> false
 leq(s(x),s(y)) -> leq(x,y)
 mod(0,y) -> 0
 mod(s(x),0) -> 0
 mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.