YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
a(lambda(x:S),y:S) -> lambda(a(x:S,1))
a(x:S,y:S) -> x:S
a(x:S,y:S) -> y:S
lambda(x:S) -> x:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,a(y:S,t))
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
 A(lambda(x:S),y:S) -> LAMBDA(a(x:S,a(y:S,t)))
 A(lambda(x:S),y:S) -> LAMBDA(a(x:S,1))
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,a(y:S,t))
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
 A(lambda(x:S),y:S) -> LAMBDA(a(x:S,a(y:S,t)))
 A(lambda(x:S),y:S) -> LAMBDA(a(x:S,1))
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,a(y:S,t))
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
->->-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,a(y:S,t))
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
-> Usable rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X1,X2) = X1 + X2
[lambda](X) = X + 2
[1] = 0
[t] = 0
[A](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
->->-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(x:S,1)
 A(lambda(x:S),y:S) -> A(y:S,t)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
-> Usable rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X1,X2) = X1 + X2
[lambda](X) = X + 2
[1] = 0
[t] = 0
[A](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(y:S,t)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(y:S,t)
->->-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
 A(lambda(x:S),y:S) -> A(y:S,t)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
-> Usable rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X1,X2) = X1 + X2
[lambda](X) = X + 2
[1] = 0
[t] = 0
[A](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
->->-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S

Problem 1: 

Subterm Processor:
-> Pairs:
 A(a(x:S,y:S),z:S) -> A(x:S,a(y:S,z:S))
 A(a(x:S,y:S),z:S) -> A(y:S,z:S)
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Projection:
 pi(A) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(a(x:S,y:S),z:S) -> a(x:S,a(y:S,z:S))
 a(lambda(x:S),y:S) -> lambda(a(x:S,a(y:S,t)))
 a(lambda(x:S),y:S) -> lambda(a(x:S,1))
 a(x:S,y:S) -> x:S
 a(x:S,y:S) -> y:S
 lambda(x:S) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.