0.53/0.60 YES 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 (VAR x y) 0.53/0.60 (THEORY 0.53/0.60 (AC plus)) 0.53/0.60 (RULES 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 ) 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 Dependency Pairs Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 SCC Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Strongly Connected Components: 0.53/0.60 ->->Cycle: 0.53/0.60 ->->-> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> FAxioms: 0.53/0.60 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) -> plus(x3,x2) 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 ->->-> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 Reduction Pairs Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Usable Equations: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> Usable Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Interpretation type: 0.53/0.60 Linear 0.53/0.60 ->Coefficients: 0.53/0.60 Natural Numbers 0.53/0.60 ->Dimension: 0.53/0.60 1 0.53/0.60 ->Bound: 0.53/0.60 2 0.53/0.60 ->Interpretation: 0.53/0.60 0.53/0.60 [plus](X1,X2) = X1 + X2 0.53/0.60 [0] = 0 0.53/0.60 [s](X) = X + 2 0.53/0.60 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 SCC Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Strongly Connected Components: 0.53/0.60 ->->Cycle: 0.53/0.60 ->->-> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> FAxioms: 0.53/0.60 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) -> plus(x3,x2) 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 ->->-> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 Reduction Pairs Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(y,0) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Usable Equations: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> Usable Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Interpretation type: 0.53/0.60 Linear 0.53/0.60 ->Coefficients: 0.53/0.60 Natural Numbers 0.53/0.60 ->Dimension: 0.53/0.60 1 0.53/0.60 ->Bound: 0.53/0.60 2 0.53/0.60 ->Interpretation: 0.53/0.60 0.53/0.60 [plus](X1,X2) = X1 + X2 0.53/0.60 [0] = 0 0.53/0.60 [s](X) = X + 2 0.53/0.60 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 SCC Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Strongly Connected Components: 0.53/0.60 ->->Cycle: 0.53/0.60 ->->-> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> FAxioms: 0.53/0.60 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) -> plus(x3,x2) 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 ->->-> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 Reduction Pairs Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.60 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(s(x),plus(y,0)) 0.53/0.60 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.60 -> EAxioms: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Usable Equations: 0.53/0.60 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.60 plus(x2,x3) = plus(x3,x2) 0.53/0.60 -> Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> Usable Rules: 0.53/0.60 plus(0,y) -> y 0.53/0.60 plus(s(x),0) -> s(x) 0.53/0.60 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.60 -> SRules: 0.53/0.60 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.60 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.60 ->Interpretation type: 0.53/0.60 Linear 0.53/0.60 ->Coefficients: 0.53/0.60 Natural Numbers 0.53/0.60 ->Dimension: 0.53/0.60 1 0.53/0.60 ->Bound: 0.53/0.60 2 0.53/0.60 ->Interpretation: 0.53/0.60 0.53/0.60 [plus](X1,X2) = X1 + X2 0.53/0.60 [0] = 0 0.53/0.60 [s](X) = X + 2 0.53/0.60 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.60 0.53/0.60 Problem 1: 0.53/0.60 0.53/0.60 SCC Processor: 0.53/0.60 -> FAxioms: 0.53/0.60 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.60 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.60 -> Pairs: 0.53/0.60 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.60 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Strongly Connected Components: 0.53/0.61 ->->Cycle: 0.53/0.61 ->->-> Pairs: 0.53/0.61 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.61 -> FAxioms: 0.53/0.61 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) -> plus(x3,x2) 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 ->->-> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 Reduction Pairs Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 PLUS(s(x),s(y)) -> PLUS(y,0) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Usable Equations: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> Usable Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Interpretation type: 0.53/0.61 Linear 0.53/0.61 ->Coefficients: 0.53/0.61 Natural Numbers 0.53/0.61 ->Dimension: 0.53/0.61 1 0.53/0.61 ->Bound: 0.53/0.61 2 0.53/0.61 ->Interpretation: 0.53/0.61 0.53/0.61 [plus](X1,X2) = X1 + X2 0.53/0.61 [0] = 0 0.53/0.61 [s](X) = X + 2 0.53/0.61 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 SCC Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Strongly Connected Components: 0.53/0.61 ->->Cycle: 0.53/0.61 ->->-> Pairs: 0.53/0.61 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> FAxioms: 0.53/0.61 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) -> plus(x3,x2) 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 ->->-> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 Reduction Pairs Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(0,y),x2) -> PLUS(y,x2) 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Usable Equations: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> Usable Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Interpretation type: 0.53/0.61 Linear 0.53/0.61 ->Coefficients: 0.53/0.61 Natural Numbers 0.53/0.61 ->Dimension: 0.53/0.61 1 0.53/0.61 ->Bound: 0.53/0.61 2 0.53/0.61 ->Interpretation: 0.53/0.61 0.53/0.61 [plus](X1,X2) = X1 + X2 0.53/0.61 [0] = 2 0.53/0.61 [s](X) = 0 0.53/0.61 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 SCC Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Strongly Connected Components: 0.53/0.61 ->->Cycle: 0.53/0.61 ->->-> Pairs: 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> FAxioms: 0.53/0.61 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) -> plus(x3,x2) 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 ->->-> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 Reduction Pairs Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(s(x),0),x2) -> PLUS(s(x),x2) 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Usable Equations: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> Usable Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Interpretation type: 0.53/0.61 Linear 0.53/0.61 ->Coefficients: 0.53/0.61 Natural Numbers 0.53/0.61 ->Dimension: 0.53/0.61 1 0.53/0.61 ->Bound: 0.53/0.61 2 0.53/0.61 ->Interpretation: 0.53/0.61 0.53/0.61 [plus](X1,X2) = X1 + X2 + 2 0.53/0.61 [0] = 0 0.53/0.61 [s](X) = 0 0.53/0.61 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 SCC Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Strongly Connected Components: 0.53/0.61 ->->Cycle: 0.53/0.61 ->->-> Pairs: 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> FAxioms: 0.53/0.61 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) -> plus(x3,x2) 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) -> PLUS(x3,x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 ->->-> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 Reduction Pairs Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 PLUS(plus(s(x),s(y)),x2) -> PLUS(s(plus(s(x),plus(y,0))),x2) 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Usable Equations: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> Usable Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Interpretation type: 0.53/0.61 Linear 0.53/0.61 ->Coefficients: 0.53/0.61 Natural Numbers 0.53/0.61 ->Dimension: 0.53/0.61 1 0.53/0.61 ->Bound: 0.53/0.61 2 0.53/0.61 ->Interpretation: 0.53/0.61 0.53/0.61 [plus](X1,X2) = X1 + X2 0.53/0.61 [0] = 2 0.53/0.61 [s](X) = 2 0.53/0.61 [PLUS](X1,X2) = 2.X1 + 2.X2 0.53/0.61 0.53/0.61 Problem 1: 0.53/0.61 0.53/0.61 SCC Processor: 0.53/0.61 -> FAxioms: 0.53/0.61 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 0.53/0.61 PLUS(x2,x3) = PLUS(x3,x2) 0.53/0.61 -> Pairs: 0.53/0.61 Empty 0.53/0.61 -> EAxioms: 0.53/0.61 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 0.53/0.61 plus(x2,x3) = plus(x3,x2) 0.53/0.61 -> Rules: 0.53/0.61 plus(0,y) -> y 0.53/0.61 plus(s(x),0) -> s(x) 0.53/0.61 plus(s(x),s(y)) -> s(plus(s(x),plus(y,0))) 0.53/0.61 -> SRules: 0.53/0.61 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 0.53/0.61 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 0.53/0.61 ->Strongly Connected Components: 0.53/0.61 There is no strongly connected component 0.53/0.61 0.53/0.61 The problem is finite. 0.53/0.61 EOF