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 l:S x:S y:S) 0.00/0.01 (RULES 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 0.00/0.01 Conditional Termination Problem 2: 0.00/0.01 -> Pairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> LE(x:S,min(l:S)) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) 0.00/0.01 -> QPairs: 0.00/0.01 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 ->->-> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 ->->-> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 1.1.1: 0.00/0.01 0.00/0.01 Conditional Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->Projection: 0.00/0.01 pi(MIN) = 1 0.00/0.01 0.00/0.01 Problem 1.1.1: 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 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 1.1.2: 0.00/0.01 0.00/0.01 Conditional Subterm Processor: 0.00/0.01 -> Pairs: 0.00/0.01 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 -> QPairs: 0.00/0.01 Empty 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->Projection: 0.00/0.01 pi(LE) = 1 0.00/0.01 0.00/0.01 Problem 1.1.2: 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 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 MIN(cons(x:S,l:S)) -> LE(x:S,min(l:S)) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) 0.00/0.01 -> QPairs: 0.00/0.01 LE(s(x:S),s(y:S)) -> LE(x:S,y:S) 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->Strongly Connected Components: 0.00/0.01 ->->Cycle: 0.00/0.01 ->->-> Pairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) 0.00/0.01 -> QPairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 ->->-> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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 MIN(cons(x:S,l:S)) -> MIN(l:S) 0.00/0.01 -> QPairs: 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S 0.00/0.01 -> Rules: 0.00/0.01 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 0.00/0.01 ->Projection: 0.00/0.01 pi(MIN) = 1 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 le(0,s(x:S)) -> ttrue 0.00/0.01 le(s(x:S),s(y:S)) -> le(x:S,y:S) 0.00/0.01 le(x:S,0) -> ffalse 0.00/0.01 min(cons(x:S,nil)) -> x:S 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse 0.00/0.01 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S 0.00/0.01 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue 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. 0.00/0.02 EOF