/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR f x xs y ys) (RULES app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ) Problem 1: Innermost Equivalent Processor: -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: APP(app(le,app(s,x)),app(s,y)) -> APP(app(le,x),y) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs)) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x)) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(le,x),y) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(maxlist,y),ys) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(if,app(app(le,x),y)) APP(height,app(app(node,x),xs)) -> APP(app(map,height),xs) APP(height,app(app(node,x),xs)) -> APP(app(maxlist,0),app(app(map,height),xs)) APP(height,app(app(node,x),xs)) -> APP(s,app(app(maxlist,0),app(app(map,height),xs))) -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) Problem 1: SCC Processor: -> Pairs: APP(app(le,app(s,x)),app(s,y)) -> APP(app(le,x),y) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(cons,app(f,x)),app(app(map,f),xs)) APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(cons,app(f,x)) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(le,x),y) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(maxlist,y),ys) APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(if,app(app(le,x),y)) APP(height,app(app(node,x),xs)) -> APP(app(map,height),xs) APP(height,app(app(node,x),xs)) -> APP(app(maxlist,0),app(app(map,height),xs)) APP(height,app(app(node,x),xs)) -> APP(s,app(app(maxlist,0),app(app(map,height),xs))) -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(le,app(s,x)),app(s,y)) -> APP(app(le,x),y) ->->-> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->->Cycle: ->->-> Pairs: APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(maxlist,y),ys) ->->-> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->->Cycle: ->->-> Pairs: APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(height,app(app(node,x),xs)) -> APP(app(map,height),xs) ->->-> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(app(le,app(s,x)),app(s,y)) -> APP(app(le,x),y) -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: APP(app(maxlist,x),app(app(cons,y),ys)) -> APP(app(maxlist,y),ys) -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Projection: pi(APP) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: APP(app(map,f),app(app(cons,x),xs)) -> APP(app(map,f),xs) APP(app(map,f),app(app(cons,x),xs)) -> APP(f,x) APP(height,app(app(node,x),xs)) -> APP(app(map,height),xs) -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Projection: pi(APP) = 2 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: app(app(le,app(s,x)),app(s,y)) -> app(app(le,x),y) app(app(le,app(s,x)),0) -> false app(app(le,0),y) -> true app(app(map,f),app(app(cons,x),xs)) -> app(app(cons,app(f,x)),app(app(map,f),xs)) app(app(map,f),nil) -> nil app(app(maxlist,x),app(app(cons,y),ys)) -> app(app(if,app(app(le,x),y)),app(app(maxlist,y),ys)) app(app(maxlist,x),nil) -> x app(height,app(app(node,x),xs)) -> app(s,app(app(maxlist,0),app(app(map,height),xs))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.