0.00/0.21 YES 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S) 0.00/0.21 (RULES 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ) 0.00/0.21 (STRATEGY INNERMOST) 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 Dependency Pairs Processor: 0.00/0.21 -> Pairs: 0.00/0.21 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 SCC Processor: 0.00/0.21 -> Pairs: 0.00/0.21 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Strongly Connected Components: 0.00/0.21 ->->Cycle: 0.00/0.21 ->->-> Pairs: 0.00/0.21 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 ->->-> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 Reduction Pairs Processor: 0.00/0.21 -> Pairs: 0.00/0.21 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 -> Usable rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Interpretation type: 0.00/0.21 Linear 0.00/0.21 ->Coefficients: 0.00/0.21 Natural Numbers 0.00/0.21 ->Dimension: 0.00/0.21 1 0.00/0.21 ->Bound: 0.00/0.21 2 0.00/0.21 ->Interpretation: 0.00/0.21 0.00/0.21 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.21 [a__from](X) = 2.X + 2 0.00/0.21 [mark](X) = 2.X 0.00/0.21 [0] = 2 0.00/0.21 [cons](X1,X2) = X1 + 2 0.00/0.21 [fSNonEmpty] = 0 0.00/0.21 [first](X1,X2) = 2.X1 + 2.X2 + 1 0.00/0.21 [from](X) = 2.X + 2 0.00/0.21 [nil] = 0 0.00/0.21 [s](X) = X + 2 0.00/0.21 [A__FIRST](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.21 [A__FROM](X) = 2.X + 2 0.00/0.21 [MARK](X) = 2.X + 2 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 SCC Processor: 0.00/0.21 -> Pairs: 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Strongly Connected Components: 0.00/0.21 ->->Cycle: 0.00/0.21 ->->-> Pairs: 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 ->->-> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 Reduction Pairs Processor: 0.00/0.21 -> Pairs: 0.00/0.21 A__FROM(X:S) -> MARK(X:S) 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 -> Usable rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Interpretation type: 0.00/0.21 Linear 0.00/0.21 ->Coefficients: 0.00/0.21 Natural Numbers 0.00/0.21 ->Dimension: 0.00/0.21 1 0.00/0.21 ->Bound: 0.00/0.21 2 0.00/0.21 ->Interpretation: 0.00/0.21 0.00/0.21 [a__first](X1,X2) = X1 + 2.X2 + 2 0.00/0.21 [a__from](X) = 2.X + 2 0.00/0.21 [mark](X) = 2.X + 1 0.00/0.21 [0] = 2 0.00/0.21 [cons](X1,X2) = X1 0.00/0.21 [fSNonEmpty] = 0 0.00/0.21 [first](X1,X2) = X1 + 2.X2 + 2 0.00/0.21 [from](X) = 2.X + 2 0.00/0.21 [nil] = 2 0.00/0.21 [s](X) = X 0.00/0.21 [A__FIRST](X1,X2) = 0 0.00/0.21 [A__FROM](X) = 2.X + 2 0.00/0.21 [MARK](X) = 2.X + 1 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 SCC Processor: 0.00/0.21 -> Pairs: 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> A__FROM(mark(X:S)) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Strongly Connected Components: 0.00/0.21 ->->Cycle: 0.00/0.21 ->->-> Pairs: 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 ->->-> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 Subterm Processor: 0.00/0.21 -> Pairs: 0.00/0.21 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.21 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.21 MARK(from(X:S)) -> MARK(X:S) 0.00/0.21 MARK(s(X:S)) -> MARK(X:S) 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Projection: 0.00/0.21 pi(MARK) = 1 0.00/0.21 0.00/0.21 Problem 1: 0.00/0.21 0.00/0.21 SCC Processor: 0.00/0.21 -> Pairs: 0.00/0.21 Empty 0.00/0.21 -> Rules: 0.00/0.21 a__first(0,X:S) -> nil 0.00/0.21 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.21 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.21 a__from(X:S) -> cons(mark(X:S),from(s(X:S))) 0.00/0.21 a__from(X:S) -> from(X:S) 0.00/0.21 mark(0) -> 0 0.00/0.21 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.21 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.21 mark(from(X:S)) -> a__from(mark(X:S)) 0.00/0.21 mark(nil) -> nil 0.00/0.21 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.21 ->Strongly Connected Components: 0.00/0.21 There is no strongly connected component 0.00/0.21 0.00/0.21 The problem is finite. 0.00/0.21 EOF