/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S x:S xs:S y:S ys:S) (RULES +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ) Problem 1: Dependency Pairs Processor: -> Pairs: +#(s(x:S),y:S) -> +#(x:S,y:S) ++#(:(x:S,xs:S),ys:S) -> ++#(xs:S,ys:S) -#(s(x:S),s(y:S)) -> -#(x:S,y:S) AVG(xs:S) -> HD(sum(xs:S)) AVG(xs:S) -> LENGTH(xs:S) AVG(xs:S) -> QUOT(hd(sum(xs:S)),length(xs:S)) AVG(xs:S) -> SUM(xs:S) LENGTH(:(x:S,xs:S)) -> LENGTH(xs:S) QUOT(s(x:S),s(y:S)) -> -#(x:S,y:S) QUOT(s(x:S),s(y:S)) -> QUOT(-(x:S,y:S),s(y:S)) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> ++#(xs:S,sum(:(x:S,:(y:S,ys:S)))) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(:(x:S,:(y:S,ys:S))) SUM(:(x:S,:(y:S,xs:S))) -> +#(x:S,y:S) SUM(:(x:S,:(y:S,xs:S))) -> SUM(:(+(x:S,y:S),xs:S)) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) Problem 1: SCC Processor: -> Pairs: +#(s(x:S),y:S) -> +#(x:S,y:S) ++#(:(x:S,xs:S),ys:S) -> ++#(xs:S,ys:S) -#(s(x:S),s(y:S)) -> -#(x:S,y:S) AVG(xs:S) -> HD(sum(xs:S)) AVG(xs:S) -> LENGTH(xs:S) AVG(xs:S) -> QUOT(hd(sum(xs:S)),length(xs:S)) AVG(xs:S) -> SUM(xs:S) LENGTH(:(x:S,xs:S)) -> LENGTH(xs:S) QUOT(s(x:S),s(y:S)) -> -#(x:S,y:S) QUOT(s(x:S),s(y:S)) -> QUOT(-(x:S,y:S),s(y:S)) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> ++#(xs:S,sum(:(x:S,:(y:S,ys:S)))) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(:(x:S,:(y:S,ys:S))) SUM(:(x:S,:(y:S,xs:S))) -> +#(x:S,y:S) SUM(:(x:S,:(y:S,xs:S))) -> SUM(:(+(x:S,y:S),xs:S)) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: LENGTH(:(x:S,xs:S)) -> LENGTH(xs:S) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: -#(s(x:S),s(y:S)) -> -#(x:S,y:S) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: QUOT(s(x:S),s(y:S)) -> QUOT(-(x:S,y:S),s(y:S)) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: ++#(:(x:S,xs:S),ys:S) -> ++#(xs:S,ys:S) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: +#(s(x:S),y:S) -> +#(x:S,y:S) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: SUM(:(x:S,:(y:S,xs:S))) -> SUM(:(+(x:S,y:S),xs:S)) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->->Cycle: ->->-> Pairs: SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) ->->-> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) The problem is decomposed in 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: LENGTH(:(x:S,xs:S)) -> LENGTH(xs:S) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Projection: pi(LENGTH) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: -#(s(x:S),s(y:S)) -> -#(x:S,y:S) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Projection: pi(-#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pair Processor: -> Pairs: QUOT(s(x:S),s(y:S)) -> QUOT(-(x:S,y:S),s(y:S)) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) -> Usable rules: -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [-](X1,X2) = X1 + 1 [0] = 1 [s](X) = X + 2 [QUOT](X1,X2) = 2.X1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ++#(:(x:S,xs:S),ys:S) -> ++#(xs:S,ys:S) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Projection: pi(++#) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: +#(s(x:S),y:S) -> +#(x:S,y:S) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Projection: pi(+#) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pair Processor: -> Pairs: SUM(:(x:S,:(y:S,xs:S))) -> SUM(:(+(x:S,y:S),xs:S)) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) -> Usable rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = 2.X1 + 2.X2 [0] = 0 [:](X1,X2) = X2 + 2 [s](X) = X + 2 [SUM](X) = X Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Reduction Pair Processor: -> Pairs: SUM(++(xs:S,:(x:S,:(y:S,ys:S)))) -> SUM(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) -> Usable rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [+](X1,X2) = X2 + 1 [++](X1,X2) = 2.X1 + 2.X2 + 2 [sum](X) = 2 [0] = 0 [:](X1,X2) = X2 + 2 [nil] = 0 [s](X) = 0 [SUM](X) = 2.X Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: +(0,y:S) -> y:S +(s(x:S),y:S) -> s(+(x:S,y:S)) ++(:(x:S,xs:S),ys:S) -> :(x:S,++(xs:S,ys:S)) ++(nil,ys:S) -> ys:S -(0,s(y:S)) -> 0 -(s(x:S),s(y:S)) -> -(x:S,y:S) -(x:S,0) -> x:S avg(xs:S) -> quot(hd(sum(xs:S)),length(xs:S)) hd(:(x:S,xs:S)) -> x:S length(:(x:S,xs:S)) -> s(length(xs:S)) length(nil) -> 0 quot(0,s(y:S)) -> 0 quot(s(x:S),s(y:S)) -> s(quot(-(x:S,y:S),s(y:S))) sum(++(xs:S,:(x:S,:(y:S,ys:S)))) -> sum(++(xs:S,sum(:(x:S,:(y:S,ys:S))))) sum(:(x:S,:(y:S,xs:S))) -> sum(:(+(x:S,y:S),xs:S)) sum(:(x:S,nil)) -> :(x:S,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite.