YES

Problem 1: 

(VAR X Y Z x y)
(THEORY 
(AC app)
(C max'))
(RULES
1 -> s(0)
2 -> s(1)
3 -> s(2)
4 -> s(3)
5 -> s(4)
6 -> s(5)
7 -> s(6)
8 -> s(7)
9 -> s(8)
app(empty,X) -> X
max(app(singl(x),Y)) -> max2(x,Y)
max(singl(x)) -> x
max'(0,x) -> x
max'(s(x),s(y)) -> s(max'(x,y))
max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
max2(x,empty) -> x
max2(x,singl(y)) -> max'(x,y)
)

Problem 1: 

Reduction Order Processor:
-> Rules:
 1 -> s(0)
 2 -> s(1)
 3 -> s(2)
 4 -> s(3)
 5 -> s(4)
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 2
[2] = 2
[3] = 2
[4] = 2
[5] = 2
[6] = 2
[7] = 2
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2
[max](X) = 2.X + 1
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 1
[empty] = 0
[s](X) = X
[singl](X) = X + 2

Problem 1: 

Reduction Order Processor:
-> Rules:
 2 -> s(1)
 3 -> s(2)
 4 -> s(3)
 5 -> s(4)
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 1
[3] = 1
[4] = 1
[5] = 1
[6] = 1
[7] = 1
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2 + 2
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2 + 1
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 3 -> s(2)
 4 -> s(3)
 5 -> s(4)
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 1
[3] = 2
[4] = 2
[5] = 2
[6] = 2
[7] = 2
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2 + 1
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = 2.X + 1

Problem 1: 

Reduction Order Processor:
-> Rules:
 4 -> s(3)
 5 -> s(4)
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 1
[4] = 2
[5] = 2
[6] = 2
[7] = 2
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2 + 2
[max](X) = X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = X1 + X2 + 2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = X

Problem 1: 

Reduction Order Processor:
-> Rules:
 5 -> s(4)
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 1
[6] = 1
[7] = 1
[8] = 1
[9] = 1
[app](X1,X2) = X1 + X2 + 1
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = 2.X + 2

Problem 1: 

Reduction Order Processor:
-> Rules:
 6 -> s(5)
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 1
[7] = 2
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2 + 2
[max](X) = X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = X1 + X2 + 2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = X

Problem 1: 

Reduction Order Processor:
-> Rules:
 7 -> s(6)
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 1
[7] = 2
[8] = 2
[9] = 2
[app](X1,X2) = X1 + X2 + 1
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2 + 1
[max2](X1,X2) = 2.X1 + 2.X2
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = X + 1

Problem 1: 

Reduction Order Processor:
-> Rules:
 8 -> s(7)
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 1
[9] = 2
[app](X1,X2) = X1 + X2 + 2
[max](X) = X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = X1 + X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X
[singl](X) = X

Problem 1: 

Reduction Order Processor:
-> Rules:
 9 -> s(8)
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 2
[app](X1,X2) = X1 + X2 + 2
[max](X) = X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = X1 + X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X
[singl](X) = X

Problem 1: 

Reduction Order Processor:
-> Rules:
 app(empty,X) -> X
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 2
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2 + 2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 2
[s](X) = 2.X + 2
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max(app(singl(x),Y)) -> max2(x,Y)
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 1
[max](X) = 2.X + 2
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 0
[empty] = 0
[s](X) = 2.X + 2
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max(singl(x)) -> x
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 2
[max](X) = X + 2
[max'](X1,X2) = X1 + X2 + 2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X + 2
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max'(0,x) -> x
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 2
[max](X) = 2.X
[max'](X1,X2) = X1 + X2 + 2
[max2](X1,X2) = X1 + 2.X2 + 2
[0] = 1
[empty] = 0
[s](X) = 2.X + 2
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max'(s(x),s(y)) -> s(max'(x,y))
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2
[max](X) = 2.X
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X + 2
[singl](X) = 2.X + 2

Problem 1: 

Reduction Order Processor:
-> Rules:
 max2(x,app(singl(y),Z)) -> max2(max'(x,y),Z)
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 2
[max](X) = 2.X
[max'](X1,X2) = X1 + X2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X
[singl](X) = 2.X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max2(x,empty) -> x
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 2
[max](X) = 2.X
[max'](X1,X2) = 2.X1 + 2.X2 + 2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X
[singl](X) = X

Problem 1: 

Reduction Order Processor:
-> Rules:
 max2(x,singl(y)) -> max'(x,y)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[1] = 0
[2] = 0
[3] = 0
[4] = 0
[5] = 0
[6] = 0
[7] = 0
[8] = 0
[9] = 0
[app](X1,X2) = X1 + X2 + 1
[max](X) = 2.X
[max'](X1,X2) = 2.X1 + 2.X2 + 2
[max2](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[empty] = 0
[s](X) = 2.X
[singl](X) = 2.X + 2

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 APP(app(x5,x6),x7) = APP(x5,app(x6,x7))
 APP(x5,x6) = APP(x6,x5)
 MAX'(x5,x6) = MAX'(x6,x5)
-> Pairs:
 Empty
-> EAxioms:
 app(app(x5,x6),x7) = app(x5,app(x6,x7))
 app(x5,x6) = app(x6,x5)
 max'(x5,x6) = max'(x6,x5)
-> Rules:
 Empty
-> SRules:
 APP(app(x5,x6),x7) -> APP(x5,x6)
 APP(x5,app(x6,x7)) -> APP(x6,x7)

Problem 1: 

SCC Processor:
-> FAxioms:
 APP(app(x5,x6),x7) = APP(x5,app(x6,x7))
 APP(x5,x6) = APP(x6,x5)
 MAX'(x5,x6) = MAX'(x6,x5)
-> Pairs:
 Empty
-> EAxioms:
 app(app(x5,x6),x7) = app(x5,app(x6,x7))
 app(x5,x6) = app(x6,x5)
 max'(x5,x6) = max'(x6,x5)
-> Rules:
 Empty
-> SRules:
 APP(app(x5,x6),x7) -> APP(x5,x6)
 APP(x5,app(x6,x7)) -> APP(x6,x7)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.