0.00/0.03 YES 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 (VAR v_NonEmpty:S x:S y:S z:S) 0.00/0.03 (RULES 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ) 0.00/0.03 (STRATEGY INNERMOST) 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 Dependency Pairs Processor: 0.00/0.03 -> Pairs: 0.00/0.03 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) 0.00/0.03 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> PLUS(z:S,s(0)) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) 0.00/0.03 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> PLUS(z:S,s(0)) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 ->->Cycle: 0.00/0.03 ->->-> Pairs: 0.00/0.03 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) 0.00/0.03 ->->-> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->->Cycle: 0.00/0.03 ->->-> Pairs: 0.00/0.03 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 ->->-> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 0.00/0.03 0.00/0.03 The problem is decomposed in 2 subproblems. 0.00/0.03 0.00/0.03 Problem 1.1: 0.00/0.03 0.00/0.03 Subterm Processor: 0.00/0.03 -> Pairs: 0.00/0.03 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Projection: 0.00/0.03 pi(PLUS) = 1 0.00/0.03 0.00/0.03 Problem 1.1: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 Empty 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 There is no strongly connected component 0.00/0.03 0.00/0.03 The problem is finite. 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 Subterm Processor: 0.00/0.03 -> Pairs: 0.00/0.03 QUOT(s(x:S),s(y:S),z:S) -> QUOT(x:S,y:S,z:S) 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Projection: 0.00/0.03 pi(QUOT) = 1 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 ->->Cycle: 0.00/0.03 ->->-> Pairs: 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 ->->-> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 Reduction Pairs Processor: 0.00/0.03 -> Pairs: 0.00/0.03 QUOT(x:S,0,s(z:S)) -> QUOT(x:S,plus(z:S,s(0)),s(z:S)) 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 -> Usable rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 ->Interpretation type: 0.00/0.03 Linear 0.00/0.03 ->Coefficients: 0.00/0.03 Natural Numbers 0.00/0.03 ->Dimension: 0.00/0.03 1 0.00/0.03 ->Bound: 0.00/0.03 2 0.00/0.03 ->Interpretation: 0.00/0.03 0.00/0.03 [plus](X1,X2) = X2 + 1 0.00/0.03 [quot](X1,X2,X3) = 0 0.00/0.03 [0] = 2 0.00/0.03 [fSNonEmpty] = 0 0.00/0.03 [s](X) = 0 0.00/0.03 [PLUS](X1,X2) = 0 0.00/0.03 [QUOT](X1,X2,X3) = 2.X2 0.00/0.03 0.00/0.03 Problem 1.2: 0.00/0.03 0.00/0.03 SCC Processor: 0.00/0.03 -> Pairs: 0.00/0.03 Empty 0.00/0.03 -> Rules: 0.00/0.03 plus(0,y:S) -> y:S 0.00/0.03 plus(s(x:S),y:S) -> s(plus(x:S,y:S)) 0.00/0.03 quot(0,s(y:S),s(z:S)) -> 0 0.00/0.03 quot(s(x:S),s(y:S),z:S) -> quot(x:S,y:S,z:S) 0.00/0.03 quot(x:S,0,s(z:S)) -> s(quot(x:S,plus(z:S,s(0)),s(z:S))) 0.00/0.03 ->Strongly Connected Components: 0.00/0.03 There is no strongly connected component 0.00/0.03 0.00/0.03 The problem is finite. 0.00/0.04 EOF