YES

Problem 1: 

(VAR v_NonEmpty:S n:S u:S v:S w:S x:S y:S z:S)
(RULES
app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
app(nil,y:S) -> y:S
concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
concat(leaf,y:S) -> y:S
less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
less_leaves(leaf,cons(w:S,z:S)) -> ttrue
less_leaves(x:S,leaf) -> ffalse
minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
minus(x:S,0) -> x:S
quot(0,s(y:S)) -> 0
quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
reverse(nil) -> nil
shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
shuffle(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(add(n:S,x:S),y:S) -> APP(x:S,y:S)
 CONCAT(cons(u:S,v:S),y:S) -> CONCAT(v:S,y:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> CONCAT(u:S,v:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> CONCAT(w:S,z:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> LESS_LEAVES(concat(u:S,v:S),concat(w:S,z:S))
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 REVERSE(add(n:S,x:S)) -> APP(reverse(x:S),add(n:S,nil))
 REVERSE(add(n:S,x:S)) -> REVERSE(x:S)
 SHUFFLE(add(n:S,x:S)) -> REVERSE(x:S)
 SHUFFLE(add(n:S,x:S)) -> SHUFFLE(reverse(x:S))
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 APP(add(n:S,x:S),y:S) -> APP(x:S,y:S)
 CONCAT(cons(u:S,v:S),y:S) -> CONCAT(v:S,y:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> CONCAT(u:S,v:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> CONCAT(w:S,z:S)
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> LESS_LEAVES(concat(u:S,v:S),concat(w:S,z:S))
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
 REVERSE(add(n:S,x:S)) -> APP(reverse(x:S),add(n:S,nil))
 REVERSE(add(n:S,x:S)) -> REVERSE(x:S)
 SHUFFLE(add(n:S,x:S)) -> REVERSE(x:S)
 SHUFFLE(add(n:S,x:S)) -> SHUFFLE(reverse(x:S))
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 CONCAT(cons(u:S,v:S),y:S) -> CONCAT(v:S,y:S)
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> LESS_LEAVES(concat(u:S,v:S),concat(w:S,z:S))
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 APP(add(n:S,x:S),y:S) -> APP(x:S,y:S)
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 REVERSE(add(n:S,x:S)) -> REVERSE(x:S)
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->->Cycle:
->->-> Pairs:
 SHUFFLE(add(n:S,x:S)) -> SHUFFLE(reverse(x:S))
->->-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil


The problem is decomposed in 7 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 QUOT(s(x:S),s(y:S)) -> QUOT(minus(x:S,y:S),s(y:S))
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
-> Usable rules:
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 0
[concat](X1,X2) = 0
[less_leaves](X1,X2) = 0
[minus](X1,X2) = 2.X1
[quot](X1,X2) = 0
[reverse](X) = 0
[shuffle](X) = 0
[0] = 0
[add](X1,X2) = 0
[cons](X1,X2) = 0
[fSNonEmpty] = 0
[false] = 0
[leaf] = 0
[nil] = 0
[s](X) = 2.X + 2
[true] = 0
[APP](X1,X2) = 0
[CONCAT](X1,X2) = 0
[LESS_LEAVES](X1,X2) = 0
[MINUS](X1,X2) = 0
[QUOT](X1,X2) = 2.X1
[REVERSE](X) = 0
[SHUFFLE](X) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 CONCAT(cons(u:S,v:S),y:S) -> CONCAT(v:S,y:S)
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Projection:
 pi(CONCAT) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> Pairs:
 LESS_LEAVES(cons(u:S,v:S),cons(w:S,z:S)) -> LESS_LEAVES(concat(u:S,v:S),concat(w:S,z:S))
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
-> Usable rules:
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 0
[concat](X1,X2) = 2.X1 + X2
[less_leaves](X1,X2) = 0
[minus](X1,X2) = 0
[quot](X1,X2) = 0
[reverse](X) = 0
[shuffle](X) = 0
[0] = 0
[add](X1,X2) = 0
[cons](X1,X2) = 2.X1 + X2 + 2
[fSNonEmpty] = 0
[false] = 0
[leaf] = 0
[nil] = 0
[s](X) = 0
[true] = 0
[APP](X1,X2) = 0
[CONCAT](X1,X2) = 0
[LESS_LEAVES](X1,X2) = 2.X1 + X2
[MINUS](X1,X2) = 0
[QUOT](X1,X2) = 0
[REVERSE](X) = 0
[SHUFFLE](X) = 0

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 APP(add(n:S,x:S),y:S) -> APP(x:S,y:S)
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Projection:
 pi(APP) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 REVERSE(add(n:S,x:S)) -> REVERSE(x:S)
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Projection:
 pi(REVERSE) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 SHUFFLE(add(n:S,x:S)) -> SHUFFLE(reverse(x:S))
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
-> Usable rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = X1 + X2
[concat](X1,X2) = 0
[less_leaves](X1,X2) = 0
[minus](X1,X2) = 0
[quot](X1,X2) = 0
[reverse](X) = X
[shuffle](X) = 0
[0] = 0
[add](X1,X2) = X1 + X2 + 2
[cons](X1,X2) = 0
[fSNonEmpty] = 0
[false] = 0
[leaf] = 0
[nil] = 0
[s](X) = 0
[true] = 0
[APP](X1,X2) = 0
[CONCAT](X1,X2) = 0
[LESS_LEAVES](X1,X2) = 0
[MINUS](X1,X2) = 0
[QUOT](X1,X2) = 0
[REVERSE](X) = 0
[SHUFFLE](X) = X

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S))
 app(nil,y:S) -> y:S
 concat(cons(u:S,v:S),y:S) -> cons(u:S,concat(v:S,y:S))
 concat(leaf,y:S) -> y:S
 less_leaves(cons(u:S,v:S),cons(w:S,z:S)) -> less_leaves(concat(u:S,v:S),concat(w:S,z:S))
 less_leaves(leaf,cons(w:S,z:S)) -> ttrue
 less_leaves(x:S,leaf) -> ffalse
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 quot(0,s(y:S)) -> 0
 quot(s(x:S),s(y:S)) -> s(quot(minus(x:S,y:S),s(y:S)))
 reverse(add(n:S,x:S)) -> app(reverse(x:S),add(n:S,nil))
 reverse(nil) -> nil
 shuffle(add(n:S,x:S)) -> add(n:S,shuffle(reverse(x:S)))
 shuffle(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.