YES Problem 1: (VAR v_NonEmpty:S x:S y:S) (RULES *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ) Problem 1: Dependency Pairs Processor: -> Pairs: *#(x:S,s(y:S)) -> *#(x:S,y:S) *#(x:S,s(y:S)) -> +#(*(x:S,y:S),x:S) +#(x:S,s(y:S)) -> +#(x:S,y:S) FAC(0) -> 1# FAC(s(x:S)) -> *#(s(x:S),fac(x:S)) FAC(s(x:S)) -> FAC(x:S) FLOOP(s(x:S),y:S) -> *#(s(x:S),y:S) FLOOP(s(x:S),y:S) -> FLOOP(x:S,*(s(x:S),y:S)) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) Problem 1: SCC Processor: -> Pairs: *#(x:S,s(y:S)) -> *#(x:S,y:S) *#(x:S,s(y:S)) -> +#(*(x:S,y:S),x:S) +#(x:S,s(y:S)) -> +#(x:S,y:S) FAC(0) -> 1# FAC(s(x:S)) -> *#(s(x:S),fac(x:S)) FAC(s(x:S)) -> FAC(x:S) FLOOP(s(x:S),y:S) -> *#(s(x:S),y:S) FLOOP(s(x:S),y:S) -> FLOOP(x:S,*(s(x:S),y:S)) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: +#(x:S,s(y:S)) -> +#(x:S,y:S) ->->-> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->->Cycle: ->->-> Pairs: *#(x:S,s(y:S)) -> *#(x:S,y:S) ->->-> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->->Cycle: ->->-> Pairs: FAC(s(x:S)) -> FAC(x:S) ->->-> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->->Cycle: ->->-> Pairs: FLOOP(s(x:S),y:S) -> FLOOP(x:S,*(s(x:S),y:S)) ->->-> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: +#(x:S,s(y:S)) -> +#(x:S,y:S) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Projection: pi(+#) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: *#(x:S,s(y:S)) -> *#(x:S,y:S) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Projection: pi(*#) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: FAC(s(x:S)) -> FAC(x:S) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Projection: pi(FAC) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: FLOOP(s(x:S),y:S) -> FLOOP(x:S,*(s(x:S),y:S)) -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Projection: pi(FLOOP) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: *(x:S,0) -> 0 *(x:S,s(y:S)) -> +(*(x:S,y:S),x:S) +(x:S,0) -> x:S +(x:S,s(y:S)) -> s(+(x:S,y:S)) 1 -> s(0) fac(0) -> 1 fac(0) -> s(0) fac(s(x:S)) -> *(s(x:S),fac(x:S)) floop(0,y:S) -> y:S floop(s(x:S),y:S) -> floop(x:S,*(s(x:S),y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.