/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S l:S l1:S l2:S l3:S x:S y:S)
(RULES
app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
app(nil,l:S) -> l:S
eq(0,0) -> ttrue
eq(0,s(x:S)) -> ffalse
eq(s(x:S),0) -> ffalse
eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
if(ffalse,x:S,y:S) -> y:S
if(ttrue,x:S,y:S) -> x:S
ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
ifmem(ttrue,x:S,l:S) -> ttrue
inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
inter(nil,x:S) -> nil
inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
inter(x:S,nil) -> nil
mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
mem(x:S,nil) -> ffalse
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(l1:S,l2:S),l3:S) -> APP(l1:S,app(l2:S,l3:S))
 APP(app(l1:S,l2:S),l3:S) -> APP(l2:S,l3:S)
 APP(cons(x:S,l1:S),l2:S) -> APP(l1:S,l2:S)
 EQ(s(x:S),s(y:S)) -> EQ(x:S,y:S)
 IFINTER(ffalse,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFMEM(ffalse,x:S,l:S) -> MEM(x:S,l:S)
 INTER(app(l1:S,l2:S),l3:S) -> APP(inter(l1:S,l3:S),inter(l2:S,l3:S))
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(cons(x:S,l1:S),l2:S) -> MEM(x:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> APP(inter(l1:S,l2:S),inter(l1:S,l3:S))
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
 INTER(l1:S,cons(x:S,l2:S)) -> MEM(x:S,l1:S)
 MEM(x:S,cons(y:S,l:S)) -> EQ(x:S,y:S)
 MEM(x:S,cons(y:S,l:S)) -> IFMEM(eq(x:S,y:S),x:S,l:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(l1:S,l2:S),l3:S) -> APP(l1:S,app(l2:S,l3:S))
 APP(app(l1:S,l2:S),l3:S) -> APP(l2:S,l3:S)
 APP(cons(x:S,l1:S),l2:S) -> APP(l1:S,l2:S)
 EQ(s(x:S),s(y:S)) -> EQ(x:S,y:S)
 IFINTER(ffalse,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFMEM(ffalse,x:S,l:S) -> MEM(x:S,l:S)
 INTER(app(l1:S,l2:S),l3:S) -> APP(inter(l1:S,l3:S),inter(l2:S,l3:S))
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(cons(x:S,l1:S),l2:S) -> MEM(x:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> APP(inter(l1:S,l2:S),inter(l1:S,l3:S))
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
 INTER(l1:S,cons(x:S,l2:S)) -> MEM(x:S,l1:S)
 MEM(x:S,cons(y:S,l:S)) -> EQ(x:S,y:S)
 MEM(x:S,cons(y:S,l:S)) -> IFMEM(eq(x:S,y:S),x:S,l:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 EQ(s(x:S),s(y:S)) -> EQ(x:S,y:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->->Cycle:
->->-> Pairs:
 IFMEM(ffalse,x:S,l:S) -> MEM(x:S,l:S)
 MEM(x:S,cons(y:S,l:S)) -> IFMEM(eq(x:S,y:S),x:S,l:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->->Cycle:
->->-> Pairs:
 APP(app(l1:S,l2:S),l3:S) -> APP(l1:S,app(l2:S,l3:S))
 APP(app(l1:S,l2:S),l3:S) -> APP(l2:S,l3:S)
 APP(cons(x:S,l1:S),l2:S) -> APP(l1:S,l2:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->->Cycle:
->->-> Pairs:
 IFINTER(ffalse,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 EQ(s(x:S),s(y:S)) -> EQ(x:S,y:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Projection:
 pi(EQ) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 IFMEM(ffalse,x:S,l:S) -> MEM(x:S,l:S)
 MEM(x:S,cons(y:S,l:S)) -> IFMEM(eq(x:S,y:S),x:S,l:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Projection:
 pi(IFMEM) = 3
 pi(MEM) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 IFMEM(ffalse,x:S,l:S) -> MEM(x:S,l:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 APP(app(l1:S,l2:S),l3:S) -> APP(l1:S,app(l2:S,l3:S))
 APP(app(l1:S,l2:S),l3:S) -> APP(l2:S,l3:S)
 APP(cons(x:S,l1:S),l2:S) -> APP(l1:S,l2:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Projection:
 pi(APP) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 IFINTER(ffalse,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
-> Usable rules:
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 2.X1 + 2.X2
[eq](X1,X2) = 0
[ifmem](X1,X2,X3) = 1
[mem](X1,X2) = 1
[0] = 0
[cons](X1,X2) = 2.X1 + 2.X2 + 1
[false] = 0
[nil] = 0
[s](X) = 2.X
[true] = 0
[IFINTER](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 + 2
[INTER](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 IFINTER(ttrue,x:S,l1:S,l2:S) -> INTER(l1:S,l2:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
-> Usable rules:
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 2.X1 + 2.X2
[eq](X1,X2) = 2.X2
[ifmem](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[mem](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 2
[cons](X1,X2) = 2.X1 + 2.X2 + 1
[false] = 1
[nil] = 2
[s](X) = 2.X + 2
[true] = 2
[IFINTER](X1,X2,X3,X4) = 2.X3 + 2.X4 + 2
[INTER](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(cons(x:S,l1:S),l2:S) -> IFINTER(mem(x:S,l2:S),x:S,l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
 INTER(l1:S,cons(x:S,l2:S)) -> IFINTER(mem(x:S,l1:S),x:S,l2:S,l1:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse

Problem 1.4: 

Subterm Processor:
-> Pairs:
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l1:S,l3:S)
 INTER(app(l1:S,l2:S),l3:S) -> INTER(l2:S,l3:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Projection:
 pi(INTER) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
->->-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse

Problem 1.4: 

Subterm Processor:
-> Pairs:
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l2:S)
 INTER(l1:S,app(l2:S,l3:S)) -> INTER(l1:S,l3:S)
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Projection:
 pi(INTER) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(l1:S,l2:S),l3:S) -> app(l1:S,app(l2:S,l3:S))
 app(cons(x:S,l1:S),l2:S) -> cons(x:S,app(l1:S,l2:S))
 app(nil,l:S) -> l:S
 eq(0,0) -> ttrue
 eq(0,s(x:S)) -> ffalse
 eq(s(x:S),0) -> ffalse
 eq(s(x:S),s(y:S)) -> eq(x:S,y:S)
 if(ffalse,x:S,y:S) -> y:S
 if(ttrue,x:S,y:S) -> x:S
 ifinter(ffalse,x:S,l1:S,l2:S) -> inter(l1:S,l2:S)
 ifinter(ttrue,x:S,l1:S,l2:S) -> cons(x:S,inter(l1:S,l2:S))
 ifmem(ffalse,x:S,l:S) -> mem(x:S,l:S)
 ifmem(ttrue,x:S,l:S) -> ttrue
 inter(app(l1:S,l2:S),l3:S) -> app(inter(l1:S,l3:S),inter(l2:S,l3:S))
 inter(cons(x:S,l1:S),l2:S) -> ifinter(mem(x:S,l2:S),x:S,l1:S,l2:S)
 inter(nil,x:S) -> nil
 inter(l1:S,app(l2:S,l3:S)) -> app(inter(l1:S,l2:S),inter(l1:S,l3:S))
 inter(l1:S,cons(x:S,l2:S)) -> ifinter(mem(x:S,l1:S),x:S,l2:S,l1:S)
 inter(x:S,nil) -> nil
 mem(x:S,cons(y:S,l:S)) -> ifmem(eq(x:S,y:S),x:S,l:S)
 mem(x:S,nil) -> ffalse
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.