0.00/0.08 YES 0.00/0.08 0.00/0.08 Problem 1: 0.00/0.08 0.00/0.08 (VAR v_NonEmpty:S m:S n:S x:S y:S) 0.00/0.08 (RULES 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ) 0.00/0.08 (STRATEGY INNERMOST) 0.00/0.08 0.00/0.08 Problem 1: 0.00/0.08 0.00/0.08 Dependency Pairs Processor: 0.00/0.08 -> Pairs: 0.00/0.08 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S) 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> SUM(cons(n:S,cons(m:S,x:S)),cons(0,x:S)) 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 0.00/0.08 Problem 1: 0.00/0.08 0.00/0.08 SCC Processor: 0.00/0.08 -> Pairs: 0.00/0.08 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S) 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> SUM(cons(n:S,cons(m:S,x:S)),cons(0,x:S)) 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ->Strongly Connected Components: 0.00/0.08 ->->Cycle: 0.00/0.08 ->->-> Pairs: 0.00/0.08 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S) 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 ->->-> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ->->Cycle: 0.00/0.08 ->->-> Pairs: 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 ->->-> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 0.00/0.08 0.00/0.08 The problem is decomposed in 2 subproblems. 0.00/0.08 0.00/0.08 Problem 1.1: 0.00/0.08 0.00/0.08 Reduction Pairs Processor: 0.00/0.08 -> Pairs: 0.00/0.08 SUM(cons(0,x:S),y:S) -> SUM(x:S,y:S) 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 -> Usable rules: 0.00/0.08 Empty 0.00/0.08 ->Interpretation type: 0.00/0.08 Linear 0.00/0.08 ->Coefficients: 0.00/0.08 Natural Numbers 0.00/0.08 ->Dimension: 0.00/0.08 1 0.00/0.08 ->Bound: 0.00/0.08 2 0.00/0.08 ->Interpretation: 0.00/0.08 0.00/0.08 [sum](X1,X2) = 0 0.00/0.08 [weight](X) = 0 0.00/0.08 [0] = 2 0.00/0.08 [cons](X1,X2) = 2.X2 + 2 0.00/0.08 [fSNonEmpty] = 0 0.00/0.08 [nil] = 0 0.00/0.08 [s](X) = 2.X + 2 0.00/0.08 [SUM](X1,X2) = 2.X1 0.00/0.08 [WEIGHT](X) = 0 0.00/0.08 0.00/0.08 Problem 1.1: 0.00/0.08 0.00/0.08 SCC Processor: 0.00/0.08 -> Pairs: 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ->Strongly Connected Components: 0.00/0.08 ->->Cycle: 0.00/0.08 ->->-> Pairs: 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 ->->-> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 0.00/0.08 Problem 1.1: 0.00/0.08 0.00/0.08 Reduction Pairs Processor: 0.00/0.08 -> Pairs: 0.00/0.08 SUM(cons(s(n:S),x:S),cons(m:S,y:S)) -> SUM(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 -> Usable rules: 0.00/0.08 Empty 0.00/0.08 ->Interpretation type: 0.00/0.08 Linear 0.00/0.08 ->Coefficients: 0.00/0.08 Natural Numbers 0.00/0.08 ->Dimension: 0.00/0.08 1 0.00/0.08 ->Bound: 0.00/0.08 2 0.00/0.08 ->Interpretation: 0.00/0.08 0.00/0.08 [sum](X1,X2) = 0 0.00/0.08 [weight](X) = 0 0.00/0.08 [0] = 0 0.00/0.08 [cons](X1,X2) = 2.X1 0.00/0.08 [fSNonEmpty] = 0 0.00/0.08 [nil] = 0 0.00/0.08 [s](X) = 2.X + 2 0.00/0.08 [SUM](X1,X2) = 2.X1 0.00/0.08 [WEIGHT](X) = 0 0.00/0.08 0.00/0.08 Problem 1.1: 0.00/0.08 0.00/0.08 SCC Processor: 0.00/0.08 -> Pairs: 0.00/0.08 Empty 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ->Strongly Connected Components: 0.00/0.08 There is no strongly connected component 0.00/0.08 0.00/0.08 The problem is finite. 0.00/0.08 0.00/0.08 Problem 1.2: 0.00/0.08 0.00/0.08 Reduction Pairs Processor: 0.00/0.08 -> Pairs: 0.00/0.08 WEIGHT(cons(n:S,cons(m:S,x:S))) -> WEIGHT(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 -> Usable rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 ->Interpretation type: 0.00/0.08 Linear 0.00/0.08 ->Coefficients: 0.00/0.08 Natural Numbers 0.00/0.08 ->Dimension: 0.00/0.08 1 0.00/0.08 ->Bound: 0.00/0.08 2 0.00/0.08 ->Interpretation: 0.00/0.08 0.00/0.08 [sum](X1,X2) = X2 + 1 0.00/0.08 [weight](X) = 0 0.00/0.08 [0] = 1 0.00/0.08 [cons](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.08 [fSNonEmpty] = 0 0.00/0.08 [nil] = 0 0.00/0.08 [s](X) = 0 0.00/0.08 [SUM](X1,X2) = 0 0.00/0.08 [WEIGHT](X) = 2.X 0.00/0.08 0.00/0.08 Problem 1.2: 0.00/0.08 0.00/0.08 SCC Processor: 0.00/0.08 -> Pairs: 0.00/0.08 Empty 0.00/0.08 -> Rules: 0.00/0.08 sum(cons(0,x:S),y:S) -> sum(x:S,y:S) 0.00/0.08 sum(cons(s(n:S),x:S),cons(m:S,y:S)) -> sum(cons(n:S,x:S),cons(s(m:S),y:S)) 0.00/0.08 sum(nil,y:S) -> y:S 0.00/0.08 weight(cons(n:S,cons(m:S,x:S))) -> weight(sum(cons(n:S,cons(m:S,x:S)),cons(0,x:S))) 0.00/0.08 weight(cons(n:S,nil)) -> n:S 0.00/0.08 ->Strongly Connected Components: 0.00/0.08 There is no strongly connected component 0.00/0.08 0.00/0.08 The problem is finite. 0.00/0.08 EOF