YES Problem 1: (VAR v_NonEmpty:S ALPHA:S BETA:S X:S) (RULES dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one ) Problem 1: Dependency Pairs Processor: -> Pairs: DX(div(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(div(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(ln(ALPHA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(neg(ALPHA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(times(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(times(ALPHA:S,BETA:S)) -> DX(BETA:S) -> Rules: dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one Problem 1: SCC Processor: -> Pairs: DX(div(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(div(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(ln(ALPHA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(neg(ALPHA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(times(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(times(ALPHA:S,BETA:S)) -> DX(BETA:S) -> Rules: dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DX(div(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(div(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(ln(ALPHA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(neg(ALPHA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(times(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(times(ALPHA:S,BETA:S)) -> DX(BETA:S) ->->-> Rules: dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one Problem 1: Subterm Processor: -> Pairs: DX(div(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(div(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(exp(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(ln(ALPHA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(minus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(neg(ALPHA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(plus(ALPHA:S,BETA:S)) -> DX(BETA:S) DX(times(ALPHA:S,BETA:S)) -> DX(ALPHA:S) DX(times(ALPHA:S,BETA:S)) -> DX(BETA:S) -> Rules: dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one ->Projection: pi(DX) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: dx(a) -> zero dx(div(ALPHA:S,BETA:S)) -> minus(div(dx(ALPHA:S),BETA:S),times(ALPHA:S,div(dx(BETA:S),exp(BETA:S,two)))) dx(exp(ALPHA:S,BETA:S)) -> plus(times(BETA:S,times(exp(ALPHA:S,minus(BETA:S,one)),dx(ALPHA:S))),times(exp(ALPHA:S,BETA:S),times(ln(ALPHA:S),dx(BETA:S)))) dx(ln(ALPHA:S)) -> div(dx(ALPHA:S),ALPHA:S) dx(minus(ALPHA:S,BETA:S)) -> minus(dx(ALPHA:S),dx(BETA:S)) dx(neg(ALPHA:S)) -> neg(dx(ALPHA:S)) dx(plus(ALPHA:S,BETA:S)) -> plus(dx(ALPHA:S),dx(BETA:S)) dx(times(ALPHA:S,BETA:S)) -> plus(times(BETA:S,dx(ALPHA:S)),times(ALPHA:S,dx(BETA:S))) dx(X:S) -> one ->Strongly Connected Components: There is no strongly connected component The problem is finite.