/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR k l x y z) (RULES +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ) Problem 1: Dependency Pairs Processor: -> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) +#(s(x),s(y)) -> +#(x,y) +#(s(x),s(y)) -> S(+(x,y)) +#(s(x),s(y)) -> S(s(+(x,y))) A(s(l),h,h,z) -> A(l,z,h,z) A(h,h,h,x) -> S(x) A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) A(l,x,s(y),h) -> S(h) APP(cons(x,l),k) -> APP(l,k) SUM(cons(x,cons(y,l))) -> A(x,y,h,h) SUM(cons(x,cons(y,l))) -> SUM(cons(a(x,y,h,h),l)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) Problem 1: SCC Processor: -> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) +#(s(x),s(y)) -> +#(x,y) +#(s(x),s(y)) -> S(+(x,y)) +#(s(x),s(y)) -> S(s(+(x,y))) A(s(l),h,h,z) -> A(l,z,h,z) A(h,h,h,x) -> S(x) A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) A(l,x,s(y),h) -> S(h) APP(cons(x,l),k) -> APP(l,k) SUM(cons(x,cons(y,l))) -> A(x,y,h,h) SUM(cons(x,cons(y,l))) -> SUM(cons(a(x,y,h,h),l)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(cons(x,l),k) -> APP(l,k) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->->Cycle: ->->-> Pairs: A(s(l),h,h,z) -> A(l,z,h,z) A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->->Cycle: ->->-> Pairs: SUM(cons(x,cons(y,l))) -> SUM(cons(a(x,y,h,h),l)) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->->Cycle: ->->-> Pairs: +#(+(x,y),z) -> +#(x,+(y,z)) +#(+(x,y),z) -> +#(y,z) +#(s(x),s(y)) -> +#(x,y) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(cons(x,l),k) -> APP(l,k) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(APP) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: A(s(l),h,h,z) -> A(l,z,h,z) A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(A) = 1 Problem 1.2: SCC Processor: -> Pairs: A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) Problem 1.2: Subterm Processor: -> Pairs: A(l,s(x),h,z) -> A(l,x,z,z) A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(A) = 2 Problem 1.2: SCC Processor: -> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) Problem 1.2: Subterm Processor: -> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) A(l,x,s(y),s(z)) -> A(l,x,y,a(l,x,s(y),z)) A(l,x,s(y),h) -> A(l,x,y,s(h)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(A) = 3 Problem 1.2: SCC Processor: -> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) ->->-> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) Problem 1.2: Subterm Processor: -> Pairs: A(l,x,s(y),s(z)) -> A(l,x,s(y),z) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(A) = 4 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pair Processor: -> Pairs: SUM(cons(x,cons(y,l))) -> SUM(cons(a(x,y,h,h),l)) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) -> Usable rules: a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) s(h) -> 1 ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X1,X2,X3,X4) = 2.X4 [s](X) = 2.X [1] = 0 [cons](X1,X2) = X1 + X2 + 2 [h] = 0 [SUM](X) = X Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->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) +#(s(x),s(y)) -> +#(x,y) -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Projection: pi(+#) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: +(+(x,y),z) -> +(x,+(y,z)) +(s(x),s(y)) -> s(s(+(x,y))) +(h,x) -> x +(x,h) -> x a(s(l),h,h,z) -> a(l,z,h,z) a(h,h,h,x) -> s(x) a(l,s(x),h,z) -> a(l,x,z,z) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,x,s(y),h) -> a(l,x,y,s(h)) app(cons(x,l),k) -> cons(x,app(l,k)) app(nil,k) -> k app(l,nil) -> l s(h) -> 1 sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h,h),l)) sum(cons(x,nil)) -> cons(x,nil) ->Strongly Connected Components: There is no strongly connected component The problem is finite.