YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 5 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.4: 

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

Problem 1.5: 

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

The problem is finite.