/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 l l1 l2 x y z)
(RULES
*(*(x,y),z) -> *(x,*(y,z))
*(0(x),y) -> 0(*(x,y))
*(#,x) -> #
*(1(x),y) -> +(0(*(x,y)),y)
*(x,+(y,z)) -> +(*(x,y),*(x,z))
+(+(x,y),z) -> +(x,+(y,z))
+(0(x),0(y)) -> 0(+(x,y))
+(0(x),1(y)) -> 1(+(x,y))
+(#,x) -> x
+(1(x),0(y)) -> 1(+(x,y))
+(1(x),1(y)) -> 0(+(+(x,y),1(#)))
+(x,#) -> x
0(#) -> #
app(cons(x,l1),l2) -> cons(x,app(l1,l2))
app(nil,l) -> l
prod(app(l1,l2)) -> *(prod(l1),prod(l2))
prod(cons(x,l)) -> *(x,prod(l))
prod(nil) -> 1(#)
sum(app(l1,l2)) -> +(sum(l1),sum(l2))
sum(cons(x,l)) -> +(x,sum(l))
sum(nil) -> 0(#)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 *#(*(x,y),z) -> *#(x,*(y,z))
 *#(*(x,y),z) -> *#(y,z)
 *#(0(x),y) -> *#(x,y)
 *#(0(x),y) -> 0#(*(x,y))
 *#(1(x),y) -> *#(x,y)
 *#(1(x),y) -> +#(0(*(x,y)),y)
 *#(1(x),y) -> 0#(*(x,y))
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
 *#(x,+(y,z)) -> +#(*(x,y),*(x,z))
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(+(x,y),z) -> +#(y,z)
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),0(y)) -> 0#(+(x,y))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(1(x),1(y)) -> 0#(+(+(x,y),1(#)))
 APP(cons(x,l1),l2) -> APP(l1,l2)
 PROD(app(l1,l2)) -> *#(prod(l1),prod(l2))
 PROD(app(l1,l2)) -> PROD(l1)
 PROD(app(l1,l2)) -> PROD(l2)
 PROD(cons(x,l)) -> *#(x,prod(l))
 PROD(cons(x,l)) -> PROD(l)
 SUM(app(l1,l2)) -> +#(sum(l1),sum(l2))
 SUM(app(l1,l2)) -> SUM(l1)
 SUM(app(l1,l2)) -> SUM(l2)
 SUM(cons(x,l)) -> +#(x,sum(l))
 SUM(cons(x,l)) -> SUM(l)
 SUM(nil) -> 0#(#)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1: 

SCC Processor:
-> Pairs:
 *#(*(x,y),z) -> *#(x,*(y,z))
 *#(*(x,y),z) -> *#(y,z)
 *#(0(x),y) -> *#(x,y)
 *#(0(x),y) -> 0#(*(x,y))
 *#(1(x),y) -> *#(x,y)
 *#(1(x),y) -> +#(0(*(x,y)),y)
 *#(1(x),y) -> 0#(*(x,y))
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
 *#(x,+(y,z)) -> +#(*(x,y),*(x,z))
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(+(x,y),z) -> +#(y,z)
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),0(y)) -> 0#(+(x,y))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
 +#(1(x),1(y)) -> 0#(+(+(x,y),1(#)))
 APP(cons(x,l1),l2) -> APP(l1,l2)
 PROD(app(l1,l2)) -> *#(prod(l1),prod(l2))
 PROD(app(l1,l2)) -> PROD(l1)
 PROD(app(l1,l2)) -> PROD(l2)
 PROD(cons(x,l)) -> *#(x,prod(l))
 PROD(cons(x,l)) -> PROD(l)
 SUM(app(l1,l2)) -> +#(sum(l1),sum(l2))
 SUM(app(l1,l2)) -> SUM(l1)
 SUM(app(l1,l2)) -> SUM(l2)
 SUM(cons(x,l)) -> +#(x,sum(l))
 SUM(cons(x,l)) -> SUM(l)
 SUM(nil) -> 0#(#)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(cons(x,l1),l2) -> APP(l1,l2)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(+(x,y),z) -> +#(y,z)
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->->Cycle:
->->-> Pairs:
 SUM(app(l1,l2)) -> SUM(l1)
 SUM(app(l1,l2)) -> SUM(l2)
 SUM(cons(x,l)) -> SUM(l)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->->Cycle:
->->-> Pairs:
 *#(*(x,y),z) -> *#(x,*(y,z))
 *#(*(x,y),z) -> *#(y,z)
 *#(0(x),y) -> *#(x,y)
 *#(1(x),y) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->->Cycle:
->->-> Pairs:
 PROD(app(l1,l2)) -> PROD(l1)
 PROD(app(l1,l2)) -> PROD(l2)
 PROD(cons(x,l)) -> PROD(l)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(cons(x,l1),l2) -> APP(l1,l2)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(APP) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(+(x,y),z) -> +#(y,z)
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[0](X) = X
[#] = 0
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),0(y)) -> +#(x,y)
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[0](X) = X + 1
[#] = 1
[1](X) = X + 2
[+#](X1,X2) = X1 + X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(0(x),1(y)) -> +#(x,y)
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 1
[0](X) = X + 1
[#] = 0
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),0(y)) -> +#(x,y)
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[0](X) = X
[#] = 2
[1](X) = X + 2
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(+(x,y),1(#))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
-> Usable rules:
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2
[0](X) = X + 1
[#] = 0
[1](X) = X + 1
[+#](X1,X2) = 2.X1 + 2.X2

Problem 1.2: 

SCC Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(x,y)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 +#(+(x,y),z) -> +#(x,+(y,z))
 +#(1(x),1(y)) -> +#(x,y)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(+#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 SUM(app(l1,l2)) -> SUM(l1)
 SUM(app(l1,l2)) -> SUM(l2)
 SUM(cons(x,l)) -> SUM(l)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(SUM) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 *#(*(x,y),z) -> *#(x,*(y,z))
 *#(*(x,y),z) -> *#(y,z)
 *#(0(x),y) -> *#(x,y)
 *#(1(x),y) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(*#) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
->->-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)

Problem 1.4: 

Subterm Processor:
-> Pairs:
 *#(x,+(y,z)) -> *#(x,y)
 *#(x,+(y,z)) -> *#(x,z)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(*#) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 PROD(app(l1,l2)) -> PROD(l1)
 PROD(app(l1,l2)) -> PROD(l2)
 PROD(cons(x,l)) -> PROD(l)
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Projection:
 pi(PROD) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 *(*(x,y),z) -> *(x,*(y,z))
 *(0(x),y) -> 0(*(x,y))
 *(#,x) -> #
 *(1(x),y) -> +(0(*(x,y)),y)
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 +(+(x,y),z) -> +(x,+(y,z))
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(#,x) -> x
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#)))
 +(x,#) -> x
 0(#) -> #
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 app(nil,l) -> l
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 prod(cons(x,l)) -> *(x,prod(l))
 prod(nil) -> 1(#)
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 sum(cons(x,l)) -> +(x,sum(l))
 sum(nil) -> 0(#)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.