0.00/0.07 YES 0.00/0.07 0.00/0.07 Problem 1: 0.00/0.07 0.00/0.07 (VAR v_NonEmpty:S x:S y:S) 0.00/0.07 (RULES 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ) 0.00/0.07 (STRATEGY INNERMOST) 0.00/0.07 0.00/0.07 Problem 1: 0.00/0.07 0.00/0.07 Dependency Pairs Processor: 0.00/0.07 -> Pairs: 0.00/0.07 F(s(x:S)) -> F(x:S) 0.00/0.07 G(s(x:S),s(y:S)) -> F(x:S) 0.00/0.07 G(s(x:S),s(y:S)) -> IF(f(x:S),s(x:S),s(y:S)) 0.00/0.07 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S) 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 0.00/0.07 Problem 1: 0.00/0.07 0.00/0.07 SCC Processor: 0.00/0.07 -> Pairs: 0.00/0.07 F(s(x:S)) -> F(x:S) 0.00/0.07 G(s(x:S),s(y:S)) -> F(x:S) 0.00/0.07 G(s(x:S),s(y:S)) -> IF(f(x:S),s(x:S),s(y:S)) 0.00/0.07 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S) 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Strongly Connected Components: 0.00/0.07 ->->Cycle: 0.00/0.07 ->->-> Pairs: 0.00/0.07 F(s(x:S)) -> F(x:S) 0.00/0.07 ->->-> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->->Cycle: 0.00/0.07 ->->-> Pairs: 0.00/0.07 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S) 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 ->->-> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 0.00/0.07 0.00/0.07 The problem is decomposed in 2 subproblems. 0.00/0.07 0.00/0.07 Problem 1.1: 0.00/0.07 0.00/0.07 Subterm Processor: 0.00/0.07 -> Pairs: 0.00/0.07 F(s(x:S)) -> F(x:S) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Projection: 0.00/0.07 pi(F) = 1 0.00/0.07 0.00/0.07 Problem 1.1: 0.00/0.07 0.00/0.07 SCC Processor: 0.00/0.07 -> Pairs: 0.00/0.07 Empty 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Strongly Connected Components: 0.00/0.07 There is no strongly connected component 0.00/0.07 0.00/0.07 The problem is finite. 0.00/0.07 0.00/0.07 Problem 1.2: 0.00/0.07 0.00/0.07 Reduction Pairs Processor: 0.00/0.07 -> Pairs: 0.00/0.07 G(x:S,c(y:S)) -> G(s(c(y:S)),y:S) 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 -> Usable rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Interpretation type: 0.00/0.07 Linear 0.00/0.07 ->Coefficients: 0.00/0.07 Natural Numbers 0.00/0.07 ->Dimension: 0.00/0.07 1 0.00/0.07 ->Bound: 0.00/0.07 2 0.00/0.07 ->Interpretation: 0.00/0.07 0.00/0.07 [f](X) = 0 0.00/0.07 [g](X1,X2) = 2.X2 + 2 0.00/0.07 [if](X1,X2,X3) = X2 + 2.X3 0.00/0.07 [0] = 1 0.00/0.07 [1] = 0 0.00/0.07 [c](X) = 2.X + 2 0.00/0.07 [fSNonEmpty] = 0 0.00/0.07 [false] = 0 0.00/0.07 [s](X) = 2 0.00/0.07 [true] = 0 0.00/0.07 [F](X) = 0 0.00/0.07 [G](X1,X2) = 2.X2 0.00/0.07 [IF](X1,X2,X3) = 0 0.00/0.07 0.00/0.07 Problem 1.2: 0.00/0.07 0.00/0.07 SCC Processor: 0.00/0.07 -> Pairs: 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Strongly Connected Components: 0.00/0.07 ->->Cycle: 0.00/0.07 ->->-> Pairs: 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 ->->-> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 0.00/0.07 Problem 1.2: 0.00/0.07 0.00/0.07 Reduction Pairs Processor: 0.00/0.07 -> Pairs: 0.00/0.07 G(x:S,c(y:S)) -> G(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 -> Usable rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Interpretation type: 0.00/0.07 Linear 0.00/0.07 ->Coefficients: 0.00/0.07 Natural Numbers 0.00/0.07 ->Dimension: 0.00/0.07 1 0.00/0.07 ->Bound: 0.00/0.07 2 0.00/0.07 ->Interpretation: 0.00/0.07 0.00/0.07 [f](X) = 2 0.00/0.07 [g](X1,X2) = X1 + 2.X2 0.00/0.07 [if](X1,X2,X3) = 2.X2 + X3 0.00/0.07 [0] = 2 0.00/0.07 [1] = 1 0.00/0.07 [c](X) = 2.X + 2 0.00/0.07 [fSNonEmpty] = 0 0.00/0.07 [false] = 1 0.00/0.07 [s](X) = 1 0.00/0.07 [true] = 1 0.00/0.07 [F](X) = 0 0.00/0.07 [G](X1,X2) = 2.X2 0.00/0.07 [IF](X1,X2,X3) = 0 0.00/0.07 0.00/0.07 Problem 1.2: 0.00/0.07 0.00/0.07 SCC Processor: 0.00/0.07 -> Pairs: 0.00/0.07 Empty 0.00/0.07 -> Rules: 0.00/0.07 f(0) -> ttrue 0.00/0.07 f(1) -> ffalse 0.00/0.07 f(s(x:S)) -> f(x:S) 0.00/0.07 g(s(x:S),s(y:S)) -> if(f(x:S),s(x:S),s(y:S)) 0.00/0.07 g(x:S,c(y:S)) -> g(x:S,g(s(c(y:S)),y:S)) 0.00/0.07 if(ffalse,x:S,y:S) -> y:S 0.00/0.07 if(ttrue,x:S,y:S) -> x:S 0.00/0.07 ->Strongly Connected Components: 0.00/0.07 There is no strongly connected component 0.00/0.07 0.00/0.07 The problem is finite. 0.00/0.07 EOF