0.00/0.01 YES 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 0.00/0.01 (VAR v_NonEmpty:S x:S y:S z:S) 0.00/0.01 (RULES 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 ) 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 Valid CTRS Processor: 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 -> The system is a deterministic 3-CTRS. 0.00/0.01 0.00/0.01 Problem 1: 0.00/0.01 0.00/0.01 Dependency Pairs Processor: 0.00/0.01 0.00/0.01 Conditional Termination Problem 1: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 0.00/0.01 Conditional Termination Problem 2: 0.00/0.01 -> Pairs: 0.00/0.01 MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 0.00/0.01 0.00/0.01 The problem is decomposed in 2 subproblems. 0.00/0.01 0.00/0.01 Problem 1.1: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 0.00/0.01 Problem 1.2: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 ->->-> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 0.00/0.01 Problem 1.2: 0.00/0.01 0.00/0.01 Conditional Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 ->Projection: 0.00/0.01 pi(MINUS) = 2 0.00/0.01 0.00/0.01 Problem 1.2: 0.00/0.01 0.00/0.01 SCC Processor: 0.00/0.01 -> Pairs: 0.00/0.01 Empty 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 minus(x:S,0) -> x:S 0.00/0.01 minus(x:S,s(y:S)) -> z:S | minus(x:S,y:S) ->* s(z:S) 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 There is no strongly connected component 0.00/0.01 0.00/0.01 The problem is finite. 0.00/0.01 EOF