0.00/0.36 YES 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 (VAR k l x y) 0.00/0.36 (THEORY 0.00/0.36 (AC plus)) 0.00/0.36 (RULES 0.00/0.36 app(cons(x,l),k) -> cons(x,app(l,k)) 0.00/0.36 app(nil,k) -> k 0.00/0.36 app(l,nil) -> l 0.00/0.36 plus(0,y) -> y 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ) 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Reduction Order Processor: 0.00/0.36 -> Rules: 0.00/0.36 app(cons(x,l),k) -> cons(x,app(l,k)) 0.00/0.36 app(nil,k) -> k 0.00/0.36 app(l,nil) -> l 0.00/0.36 plus(0,y) -> y 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 2.X1 + 2.X2 + 1 0.00/0.36 [plus](X1,X2) = X1 + X2 + 2 0.00/0.36 [sum](X) = X 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = X1 + X2 + 2 0.00/0.36 [nil] = 2 0.00/0.36 [s](X) = X + 2 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Reduction Order Processor: 0.00/0.36 -> Rules: 0.00/0.36 app(nil,k) -> k 0.00/0.36 app(l,nil) -> l 0.00/0.36 plus(0,y) -> y 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = X1 + 2.X2 0.00/0.36 [plus](X1,X2) = X1 + X2 + 2 0.00/0.36 [sum](X) = X 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.36 [nil] = 2 0.00/0.36 [s](X) = X 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Reduction Order Processor: 0.00/0.36 -> Rules: 0.00/0.36 app(l,nil) -> l 0.00/0.36 plus(0,y) -> y 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.36 [plus](X1,X2) = X1 + X2 + 1 0.00/0.36 [sum](X) = X 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.36 [nil] = 2 0.00/0.36 [s](X) = X + 2 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Reduction Order Processor: 0.00/0.36 -> Rules: 0.00/0.36 plus(0,y) -> y 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 2.X1 + X2 0.00/0.36 [plus](X1,X2) = X1 + X2 + 2 0.00/0.36 [sum](X) = X 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = X1 + 2.X2 + 2 0.00/0.36 [nil] = 2 0.00/0.36 [s](X) = X + 2 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Reduction Order Processor: 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 2.X1 + 2.X2 0.00/0.36 [plus](X1,X2) = X1 + X2 0.00/0.36 [sum](X) = X 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.36 [nil] = 2 0.00/0.36 [s](X) = X 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 Dependency Pairs Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.36 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.36 -> Pairs: 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(x,y) 0.00/0.36 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k)))) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k))) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.36 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.36 0.00/0.36 Problem 1: 0.00/0.36 0.00/0.36 SCC Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.36 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.36 -> Pairs: 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(x,y) 0.00/0.36 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k)))) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k))) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.36 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.36 ->Strongly Connected Components: 0.00/0.36 ->->Cycle: 0.00/0.36 ->->-> Pairs: 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 -> FAxioms: 0.00/0.36 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) -> plus(x5,x4) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 ->->-> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 Empty 0.00/0.36 ->->Cycle: 0.00/0.36 ->->-> Pairs: 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(x,y) 0.00/0.36 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.36 -> FAxioms: 0.00/0.36 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) -> plus(x5,x4) 0.00/0.36 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6)) 0.00/0.36 PLUS(x4,x5) -> PLUS(x5,x4) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 ->->-> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.36 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.36 0.00/0.36 0.00/0.36 The problem is decomposed in 2 subproblems. 0.00/0.36 0.00/0.36 Problem 1.1: 0.00/0.36 0.00/0.36 Reduction Pairs Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 Empty 0.00/0.36 -> Pairs: 0.00/0.36 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Usable Equations: 0.00/0.36 Empty 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> Usable Rules: 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 Empty 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 2.X2 0.00/0.36 [plus](X1,X2) = 0 0.00/0.36 [sum](X) = 2 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = 2.X2 + 2 0.00/0.36 [nil] = 0 0.00/0.36 [s](X) = 0 0.00/0.36 [APP](X1,X2) = 0 0.00/0.36 [PLUS](X1,X2) = 0 0.00/0.36 [SUM](X) = 2.X 0.00/0.36 0.00/0.36 Problem 1.1: 0.00/0.36 0.00/0.36 SCC Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 Empty 0.00/0.36 -> Pairs: 0.00/0.36 Empty 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> SRules: 0.00/0.36 Empty 0.00/0.36 ->Strongly Connected Components: 0.00/0.36 There is no strongly connected component 0.00/0.36 0.00/0.36 The problem is finite. 0.00/0.36 0.00/0.36 Problem 1.2: 0.00/0.36 0.00/0.36 Reduction Pairs Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.36 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.36 -> Pairs: 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(x,y) 0.00/0.36 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Usable Equations: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.36 -> Usable Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 -> SRules: 0.00/0.36 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.36 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.36 ->Interpretation type: 0.00/0.36 Linear 0.00/0.36 ->Coefficients: 0.00/0.36 Natural Numbers 0.00/0.36 ->Dimension: 0.00/0.36 1 0.00/0.36 ->Bound: 0.00/0.36 2 0.00/0.36 ->Interpretation: 0.00/0.36 0.00/0.36 [app](X1,X2) = 0 0.00/0.36 [plus](X1,X2) = X1 + X2 + 2 0.00/0.36 [sum](X) = 0 0.00/0.36 [0] = 0 0.00/0.36 [cons](X1,X2) = 0 0.00/0.36 [nil] = 0 0.00/0.36 [s](X) = X + 2 0.00/0.36 [APP](X1,X2) = 0 0.00/0.36 [PLUS](X1,X2) = 2.X1 + 2.X2 0.00/0.36 [SUM](X) = 0 0.00/0.36 0.00/0.36 Problem 1.2: 0.00/0.36 0.00/0.36 SCC Processor: 0.00/0.36 -> FAxioms: 0.00/0.36 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.36 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.36 -> Pairs: 0.00/0.36 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.36 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.36 -> EAxioms: 0.00/0.36 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.36 plus(x4,x5) = plus(x5,x4) 0.00/0.36 -> Rules: 0.00/0.36 plus(s(x),y) -> s(plus(x,y)) 0.00/0.36 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.36 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Strongly Connected Components: 0.00/0.38 ->->Cycle: 0.00/0.38 ->->-> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.38 -> FAxioms: 0.00/0.38 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) -> plus(x5,x4) 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) -> PLUS(x5,x4) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 ->->-> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 Reduction Pairs Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Usable Equations: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> Usable Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5) 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Interpretation type: 0.00/0.38 Linear 0.00/0.38 ->Coefficients: 0.00/0.38 Natural Numbers 0.00/0.38 ->Dimension: 0.00/0.38 1 0.00/0.38 ->Bound: 0.00/0.38 2 0.00/0.38 ->Interpretation: 0.00/0.38 0.00/0.38 [app](X1,X2) = 0 0.00/0.38 [plus](X1,X2) = X1 + X2 + 2 0.00/0.38 [sum](X) = 0 0.00/0.38 [0] = 0 0.00/0.38 [cons](X1,X2) = 0 0.00/0.38 [nil] = 0 0.00/0.38 [s](X) = X + 2 0.00/0.38 [APP](X1,X2) = 0 0.00/0.38 [PLUS](X1,X2) = 2.X1 + 2.X2 0.00/0.38 [SUM](X) = 0 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 SCC Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Strongly Connected Components: 0.00/0.38 ->->Cycle: 0.00/0.38 ->->-> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.38 -> FAxioms: 0.00/0.38 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) -> plus(x5,x4) 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) -> PLUS(x5,x4) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 ->->-> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 Reduction Pairs Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 PLUS(s(x),y) -> PLUS(x,y) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Usable Equations: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> Usable Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Interpretation type: 0.00/0.38 Linear 0.00/0.38 ->Coefficients: 0.00/0.38 Natural Numbers 0.00/0.38 ->Dimension: 0.00/0.38 1 0.00/0.38 ->Bound: 0.00/0.38 2 0.00/0.38 ->Interpretation: 0.00/0.38 0.00/0.38 [app](X1,X2) = 0 0.00/0.38 [plus](X1,X2) = X1 + X2 + 2 0.00/0.38 [sum](X) = 0 0.00/0.38 [0] = 0 0.00/0.38 [cons](X1,X2) = 0 0.00/0.38 [nil] = 0 0.00/0.38 [s](X) = X + 2 0.00/0.38 [APP](X1,X2) = 0 0.00/0.38 [PLUS](X1,X2) = 2.X1 + 2.X2 0.00/0.38 [SUM](X) = 0 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 SCC Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Strongly Connected Components: 0.00/0.38 ->->Cycle: 0.00/0.38 ->->-> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 -> FAxioms: 0.00/0.38 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) -> plus(x5,x4) 0.00/0.38 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) -> PLUS(x5,x4) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 ->->-> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 Reduction Pairs Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 PLUS(plus(s(x),y),x4) -> PLUS(s(plus(x,y)),x4) 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Usable Equations: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> Usable Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Interpretation type: 0.00/0.38 Linear 0.00/0.38 ->Coefficients: 0.00/0.38 Natural Numbers 0.00/0.38 ->Dimension: 0.00/0.38 1 0.00/0.38 ->Bound: 0.00/0.38 2 0.00/0.38 ->Interpretation: 0.00/0.38 0.00/0.38 [app](X1,X2) = 0 0.00/0.38 [plus](X1,X2) = X1 + X2 + 2 0.00/0.38 [sum](X) = 0 0.00/0.38 [0] = 0 0.00/0.38 [cons](X1,X2) = 0 0.00/0.38 [nil] = 0 0.00/0.38 [s](X) = 2 0.00/0.38 [APP](X1,X2) = 0 0.00/0.38 [PLUS](X1,X2) = 2.X1 + 2.X2 0.00/0.38 [SUM](X) = 0 0.00/0.38 0.00/0.38 Problem 1.2: 0.00/0.38 0.00/0.38 SCC Processor: 0.00/0.38 -> FAxioms: 0.00/0.38 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6)) 0.00/0.38 PLUS(x4,x5) = PLUS(x5,x4) 0.00/0.38 -> Pairs: 0.00/0.38 Empty 0.00/0.38 -> EAxioms: 0.00/0.38 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6)) 0.00/0.38 plus(x4,x5) = plus(x5,x4) 0.00/0.38 -> Rules: 0.00/0.38 plus(s(x),y) -> s(plus(x,y)) 0.00/0.38 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) 0.00/0.38 sum(cons(x,nil)) -> cons(x,nil) 0.00/0.38 -> SRules: 0.00/0.38 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6) 0.00/0.38 ->Strongly Connected Components: 0.00/0.38 There is no strongly connected component 0.00/0.38 0.00/0.38 The problem is finite. 0.00/0.39 EOF