0.00/0.03 YES 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 (VAR v_NonEmpty:S fun:S x:S xs:S y:S z:S) 0.00/0.03 (RULES 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ) 0.00/0.03 (STRATEGY INNERMOST) 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 Dependency Pairs Processor: 0.00/0.03 -> Pairs: 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(f,app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(h,x:S) 0.00/0.03 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(f,app(app(g,x:S),y:S)) 0.00/0.03 APP(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> APP(h,x:S) 0.00/0.03 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 ->->Cycle: 0.00/0.03 ->->-> Pairs: 0.00/0.03 APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) 0.00/0.03 ->->-> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->->Cycle: 0.00/0.03 ->->-> Pairs: 0.00/0.03 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 ->->-> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 0.00/0.03 0.00/0.03 The problem is decomposed in 2 subproblems. 0.00/0.03 0.00/0.03 Problem 1.1: 0.00/0.03 0.00/0.03 Subterm Processor: 0.00/0.03 -> Pairs: 0.00/0.03 APP(h,app(app(g,x:S),y:S)) -> APP(h,x:S) 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->Projection: 0.00/0.03 pi(APP) = 2 0.00/0.03 0.00/0.03 Problem 1.1: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 Empty 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 There is no strongly connected component 0.00/0.03 0.00/0.03 The problem is finite. 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 Subterm Processor: 0.00/0.03 -> Pairs: 0.00/0.03 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.00/0.03 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->Projection: 0.00/0.03 pi(APP) = 2 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.00/0.03 -> Rules: 0.00/0.03 app(app(app(app(f,0),1),app(app(g,x:S),y:S)),z:S) -> app(app(app(app(f,app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(app(g,x:S),y:S)),app(h,x:S)) 0.00/0.03 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.00/0.03 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.00/0.03 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.00/0.03 app(app(filter,fun:S),nil) -> nil 0.00/0.03 app(app(g,0),1) -> 0 0.00/0.03 app(app(g,0),1) -> 1 0.00/0.03 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.00/0.03 app(app(map,fun:S),nil) -> nil 0.00/0.03 app(h,app(app(g,x:S),y:S)) -> app(h,x:S) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 There is no strongly connected component 0.00/0.03 0.00/0.03 The problem is finite. 0.00/0.03 EOF