0.00/0.06 YES 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 (VAR v_NonEmpty:S x:S y:S) 0.00/0.06 (RULES 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ) 0.00/0.06 (STRATEGY INNERMOST) 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 Dependency Pairs Processor: 0.00/0.06 -> Pairs: 0.00/0.06 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.06 MINUS(x:S,s(y:S)) -> IF(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 MINUS(x:S,s(y:S)) -> LE(x:S,s(y:S)) 0.00/0.06 MINUS(x:S,s(y:S)) -> MINUS(x:S,p(s(y:S))) 0.00/0.06 MINUS(x:S,s(y:S)) -> P(minus(x:S,p(s(y:S)))) 0.00/0.06 MINUS(x:S,s(y:S)) -> P(s(y:S)) 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 0.00/0.06 Problem 1: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.06 MINUS(x:S,s(y:S)) -> IF(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 MINUS(x:S,s(y:S)) -> LE(x:S,s(y:S)) 0.00/0.06 MINUS(x:S,s(y:S)) -> MINUS(x:S,p(s(y:S))) 0.00/0.06 MINUS(x:S,s(y:S)) -> P(minus(x:S,p(s(y:S)))) 0.00/0.06 MINUS(x:S,s(y:S)) -> P(s(y:S)) 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.06 ->->-> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 MINUS(x:S,s(y:S)) -> MINUS(x:S,p(s(y:S))) 0.00/0.06 ->->-> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 0.00/0.06 0.00/0.06 The problem is decomposed in 2 subproblems. 0.00/0.06 0.00/0.06 Problem 1.1: 0.00/0.06 0.00/0.06 Subterm Processor: 0.00/0.06 -> Pairs: 0.00/0.06 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->Projection: 0.00/0.06 pi(LE) = 1 0.00/0.06 0.00/0.06 Problem 1.1: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 There is no strongly connected component 0.00/0.06 0.00/0.06 The problem is finite. 0.00/0.06 0.00/0.06 Problem 1.2: 0.00/0.06 0.00/0.06 Reduction Pairs Processor: 0.00/0.06 -> Pairs: 0.00/0.06 MINUS(x:S,s(y:S)) -> MINUS(x:S,p(s(y:S))) 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 -> Usable rules: 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->Interpretation type: 0.00/0.06 Linear 0.00/0.06 ->Coefficients: 0.00/0.06 All rationals 0.00/0.06 ->Dimension: 0.00/0.06 1 0.00/0.06 ->Bound: 0.00/0.06 2 0.00/0.06 ->Interpretation: 0.00/0.06 0.00/0.06 [if](X1,X2,X3) = 0 0.00/0.06 [le](X1,X2) = 0 0.00/0.06 [minus](X1,X2) = 0 0.00/0.06 [p](X) = 1/2.X + 1/2 0.00/0.06 [0] = 1/2 0.00/0.06 [fSNonEmpty] = 0 0.00/0.06 [false] = 0 0.00/0.06 [s](X) = 2.X + 2 0.00/0.06 [true] = 0 0.00/0.06 [IF](X1,X2,X3) = 0 0.00/0.06 [LE](X1,X2) = 0 0.00/0.06 [MINUS](X1,X2) = 2.X2 0.00/0.06 [P](X) = 0 0.00/0.06 0.00/0.06 Problem 1.2: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 if(ffalse,x:S,y:S) -> y:S 0.00/0.06 if(ttrue,x:S,y:S) -> x:S 0.00/0.06 le(0,y:S) -> ttrue 0.00/0.06 le(s(x:S),0) -> ffalse 0.00/0.06 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 minus(x:S,s(y:S)) -> if(le(x:S,s(y:S)),0,p(minus(x:S,p(s(y:S))))) 0.00/0.06 p(0) -> 0 0.00/0.06 p(s(x:S)) -> x:S 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 There is no strongly connected component 0.00/0.06 0.00/0.06 The problem is finite. 0.00/0.06 EOF