/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
*(O(x),y) -> O(*(x,y))
*(0,x) -> 0
*(I(x),y) -> +(O(*(x,y)),y)
*(x,0) -> 0
+(O(x),O(y)) -> O(+(x,y))
+(O(x),I(y)) -> I(+(x,y))
+(0,x) -> x
+(I(x),O(y)) -> I(+(x,y))
+(I(x),I(y)) -> O(+(+(x,y),I(0)))
+(x,0) -> x
-(O(x),O(y)) -> O(-(x,y))
-(O(x),I(y)) -> I(-(-(x,y),I(1)))
-(0,x) -> 0
-(I(x),O(y)) -> I(-(x,y))
-(I(x),I(y)) -> O(-(x,y))
-(x,0) -> x
O(0) -> 0
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *#(O(x),y) -> *#(x,y)
 *#(O(x),y) -> O#(*(x,y))
 *#(I(x),y) -> *#(x,y)
 *#(I(x),y) -> +#(O(*(x,y)),y)
 *#(I(x),y) -> O#(*(x,y))
 +#(O(x),O(y)) -> +#(x,y)
 +#(O(x),O(y)) -> O#(+(x,y))
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
 +#(I(x),I(y)) -> O#(+(+(x,y),I(0)))
 -#(O(x),O(y)) -> -#(x,y)
 -#(O(x),O(y)) -> O#(-(x,y))
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
 -#(I(x),I(y)) -> O#(-(x,y))
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1: 

SCC Processor:
-> Pairs:
 *#(O(x),y) -> *#(x,y)
 *#(O(x),y) -> O#(*(x,y))
 *#(I(x),y) -> *#(x,y)
 *#(I(x),y) -> +#(O(*(x,y)),y)
 *#(I(x),y) -> O#(*(x,y))
 +#(O(x),O(y)) -> +#(x,y)
 +#(O(x),O(y)) -> O#(+(x,y))
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
 +#(I(x),I(y)) -> O#(+(+(x,y),I(0)))
 -#(O(x),O(y)) -> -#(x,y)
 -#(O(x),O(y)) -> O#(-(x,y))
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
 -#(I(x),I(y)) -> O#(-(x,y))
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(O(x),O(y)) -> -#(x,y)
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->->Cycle:
->->-> Pairs:
 +#(O(x),O(y)) -> +#(x,y)
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->->Cycle:
->->-> Pairs:
 *#(O(x),y) -> *#(x,y)
 *#(I(x),y) -> *#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 -#(O(x),O(y)) -> -#(x,y)
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1 + 2.X2
[O](X) = X + 2
[0] = 2
[1] = 0
[I](X) = X + 2
[-#](X1,X2) = X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 -#(O(x),I(y)) -> -#(-(x,y),I(1))
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[-](X1,X2) = X1 + X2 + 1
[O](X) = X + 2
[0] = 1
[1] = 0
[I](X) = X + 1
[-#](X1,X2) = 2.X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.1: 

Subterm Processor:
-> Pairs:
 -#(O(x),I(y)) -> -#(x,y)
 -#(I(x),O(y)) -> -#(x,y)
 -#(I(x),I(y)) -> -#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Projection:
 pi(-#) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(O(x),O(y)) -> +#(x,y)
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + 2.X2 + 1
[O](X) = X + 1
[0] = 0
[I](X) = X + 2
[+#](X1,X2) = X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(O(x),I(y)) -> +#(x,y)
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[O](X) = X
[0] = 0
[I](X) = X + 2
[+#](X1,X2) = X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(I(x),O(y)) -> +#(x,y)
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[O](X) = X + 1
[0] = 0
[I](X) = X + 1
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(I(x),I(y)) -> +#(+(x,y),I(0))
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
-> Usable rules:
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 O(0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[O](X) = 2.X
[0] = 0
[I](X) = 2.X + 2
[+#](X1,X2) = 2.X1 + X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(I(x),I(y)) -> +#(x,y)
->->-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0

Problem 1.2: 

Subterm Processor:
-> Pairs:
 +#(I(x),I(y)) -> +#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Projection:
 pi(+#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 *#(O(x),y) -> *#(x,y)
 *#(I(x),y) -> *#(x,y)
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Projection:
 pi(*#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(O(x),y) -> O(*(x,y))
 *(0,x) -> 0
 *(I(x),y) -> +(O(*(x,y)),y)
 *(x,0) -> 0
 +(O(x),O(y)) -> O(+(x,y))
 +(O(x),I(y)) -> I(+(x,y))
 +(0,x) -> x
 +(I(x),O(y)) -> I(+(x,y))
 +(I(x),I(y)) -> O(+(+(x,y),I(0)))
 +(x,0) -> x
 -(O(x),O(y)) -> O(-(x,y))
 -(O(x),I(y)) -> I(-(-(x,y),I(1)))
 -(0,x) -> 0
 -(I(x),O(y)) -> I(-(x,y))
 -(I(x),I(y)) -> O(-(x,y))
 -(x,0) -> x
 O(0) -> 0
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.