0.00/0.05 YES 0.00/0.05 0.00/0.05 Problem 1: 0.00/0.05 0.00/0.05 (VAR v_NonEmpty:S x:S y:S) 0.00/0.05 (RULES 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ) 0.00/0.05 0.00/0.05 Problem 1: 0.00/0.05 Valid CTRS Processor: 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 -> The system is a deterministic 3-CTRS. 0.00/0.05 0.00/0.05 Problem 1: 0.00/0.05 0.00/0.05 Dependency Pairs Processor: 0.00/0.05 0.00/0.05 Conditional Termination Problem 1: 0.00/0.05 -> Pairs: 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> MINUS(x:S,y:S) | less(y:S,x:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> MINUS(y:S,x:S) | less(x:S,y:S) ->* ttrue 0.00/0.05 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 0.00/0.05 Conditional Termination Problem 2: 0.00/0.05 -> Pairs: 0.00/0.05 GCD(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 GCD(s(x:S),s(y:S)) -> LESS(y:S,x:S) 0.00/0.05 -> QPairs: 0.00/0.05 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 0.00/0.05 0.00/0.05 The problem is decomposed in 2 subproblems. 0.00/0.05 0.00/0.05 Problem 1.1: 0.00/0.05 0.00/0.05 SCC Processor: 0.00/0.05 -> Pairs: 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> MINUS(x:S,y:S) | less(y:S,x:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> MINUS(y:S,x:S) | less(x:S,y:S) ->* ttrue 0.00/0.05 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ->Strongly Connected Components: 0.00/0.05 ->->Cycle: 0.00/0.05 ->->-> Pairs: 0.00/0.05 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 ->->-> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ->->Cycle: 0.00/0.05 ->->-> Pairs: 0.00/0.05 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 ->->-> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ->->Cycle: 0.00/0.05 ->->-> Pairs: 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 ->->-> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 0.00/0.05 0.00/0.05 The problem is decomposed in 3 subproblems. 0.00/0.05 0.00/0.05 Problem 1.1.1: 0.00/0.05 0.00/0.05 Conditional Subterm Processor: 0.00/0.05 -> Pairs: 0.00/0.05 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ->Projection: 0.00/0.05 pi(MINUS) = 1 0.00/0.05 0.00/0.05 Problem 1.1.1: 0.00/0.05 0.00/0.05 SCC Processor: 0.00/0.05 -> Pairs: 0.00/0.05 Empty 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.05 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.05 gcd(x:S,x:S) -> x:S 0.00/0.05 less(0,s(x:S)) -> ttrue 0.00/0.05 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.05 less(x:S,0) -> ffalse 0.00/0.05 minus(0,s(y:S)) -> 0 0.00/0.05 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.05 minus(x:S,0) -> x:S 0.00/0.05 ->Strongly Connected Components: 0.00/0.05 There is no strongly connected component 0.00/0.05 0.00/0.05 The problem is finite. 0.00/0.05 0.00/0.05 Problem 1.1.2: 0.00/0.05 0.00/0.05 Conditional Subterm Processor: 0.00/0.05 -> Pairs: 0.00/0.05 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.05 -> QPairs: 0.00/0.05 Empty 0.00/0.05 -> Rules: 0.00/0.05 gcd(0,s(y:S)) -> s(y:S) 0.00/0.05 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 ->Projection: 0.00/0.06 pi(LESS) = 1 0.00/0.06 0.00/0.06 Problem 1.1.2: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> 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 1.1.3: 0.00/0.06 0.00/0.06 Reduction Triple Processor: 0.00/0.06 -> Pairs: 0.00/0.06 GCD(s(x:S),s(y:S)) -> GCD(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 -> Usable rules: 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 ->Interpretation type: 0.00/0.06 Linear 0.00/0.06 ->Coefficients: 0.00/0.06 Natural Numbers 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 [gcd](X1,X2) = 0 0.00/0.06 [less](X1,X2) = 2.X1 + 2 0.00/0.06 [minus](X1,X2) = X1 + 1 0.00/0.06 [0] = 0 0.00/0.06 [fSNonEmpty] = 0 0.00/0.06 [false] = 2 0.00/0.06 [s](X) = X + 2 0.00/0.06 [true] = 0 0.00/0.06 [GCD](X1,X2) = 2.X1 + 2.X2 0.00/0.06 [LESS](X1,X2) = 0 0.00/0.06 [MINUS](X1,X2) = 0 0.00/0.06 0.00/0.06 Problem 1.1.3: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 ->Strongly Connected Components: 0.00/0.06 ->->Cycle: 0.00/0.06 ->->-> Pairs: 0.00/0.06 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 ->->-> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 0.00/0.06 Problem 1.1.3: 0.00/0.06 0.00/0.06 Reduction Triple Processor: 0.00/0.06 -> Pairs: 0.00/0.06 GCD(s(x:S),s(y:S)) -> GCD(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 -> Usable rules: 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> x:S 0.00/0.06 ->Interpretation type: 0.00/0.06 Linear 0.00/0.06 ->Coefficients: 0.00/0.06 Natural Numbers 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 [gcd](X1,X2) = 0 0.00/0.06 [less](X1,X2) = 2 0.00/0.06 [minus](X1,X2) = X1 + 1 0.00/0.06 [0] = 1 0.00/0.06 [fSNonEmpty] = 0 0.00/0.06 [false] = 2 0.00/0.06 [s](X) = 2.X + 2 0.00/0.06 [true] = 1 0.00/0.06 [GCD](X1,X2) = 2.X2 0.00/0.06 [LESS](X1,X2) = 0 0.00/0.06 [MINUS](X1,X2) = 0 0.00/0.06 0.00/0.06 Problem 1.1.3: 0.00/0.06 0.00/0.06 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 Empty 0.00/0.06 -> QPairs: 0.00/0.06 Empty 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> 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 SCC Processor: 0.00/0.06 -> Pairs: 0.00/0.06 GCD(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.06 GCD(s(x:S),s(y:S)) -> LESS(y:S,x:S) 0.00/0.06 -> QPairs: 0.00/0.06 LESS(s(x:S),s(y:S)) -> LESS(x:S,y:S) 0.00/0.06 -> Rules: 0.00/0.06 gcd(0,s(y:S)) -> s(y:S) 0.00/0.06 gcd(s(x:S),0) -> s(x:S) 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(minus(x:S,y:S),s(y:S)) | less(y:S,x:S) ->* ttrue 0.00/0.06 gcd(s(x:S),s(y:S)) -> gcd(s(x:S),minus(y:S,x:S)) | less(x:S,y:S) ->* ttrue 0.00/0.06 gcd(x:S,x:S) -> x:S 0.00/0.06 less(0,s(x:S)) -> ttrue 0.00/0.06 less(s(x:S),s(y:S)) -> less(x:S,y:S) 0.00/0.06 less(x:S,0) -> ffalse 0.00/0.06 minus(0,s(y:S)) -> 0 0.00/0.06 minus(s(x:S),s(y:S)) -> minus(x:S,y:S) 0.00/0.06 minus(x:S,0) -> 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