YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
#(#(x1:S)) -> #(x1:S)
#(0(x1:S)) -> 0(#(x1:S))
#(1(x1:S)) -> 1(#(x1:S))
#($(x1:S)) -> *($(x1:S))
#(*(x1:S)) -> *(x1:S)
0(*(x1:S)) -> *(1(x1:S))
1(*(x1:S)) -> 0(#(x1:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ##(0(x1:S)) -> ##(x1:S)
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ##(0(x1:S)) -> ##(x1:S)
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ##(0(x1:S)) -> ##(x1:S)
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
->->-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ##(0(x1:S)) -> ##(x1:S)
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
-> Usable rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[#](X) = 2.X + 2
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[$](X) = 2.X + 2
[*](X) = 2.X + 2
[##](X) = 2.X + 2
[0#](X) = 2.X + 2
[1#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
->->-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ##(0(x1:S)) -> 0#(#(x1:S))
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
-> Usable rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[#](X) = 2.X + 2
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[$](X) = 2.X + 2
[*](X) = 2.X + 2
[##](X) = 2.X + 2
[0#](X) = X + 2
[1#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
->->-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ##(1(x1:S)) -> ##(x1:S)
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
-> Usable rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[#](X) = 2.X + 2
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[$](X) = 2.X + 2
[*](X) = 2.X + 2
[##](X) = 2.X
[0#](X) = 2.X
[1#](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
->->-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ##(1(x1:S)) -> 1#(#(x1:S))
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
-> Usable rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[#](X) = 2.X + 2
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[$](X) = 1
[*](X) = 2.X + 2
[##](X) = 2.X + 2
[0#](X) = 2.X + 1
[1#](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> ##(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
->->-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(*(x1:S)) -> 1#(x1:S)
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
-> Usable rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[#](X) = 2.X + 2
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[$](X) = X + 2
[*](X) = 2.X + 2
[0#](X) = X + 2
[1#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 1#(*(x1:S)) -> 0#(#(x1:S))
-> Rules:
 #(#(x1:S)) -> #(x1:S)
 #(0(x1:S)) -> 0(#(x1:S))
 #(1(x1:S)) -> 1(#(x1:S))
 #($(x1:S)) -> *($(x1:S))
 #(*(x1:S)) -> *(x1:S)
 0(*(x1:S)) -> *(1(x1:S))
 1(*(x1:S)) -> 0(#(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.