YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S) (RULES active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(div(s(X:S),s(Y:S))) -> DIV(minus(X:S,Y:S),s(Y:S)) ACTIVE(div(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(div(s(X:S),s(Y:S))) -> IF(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0) ACTIVE(div(s(X:S),s(Y:S))) -> MARK(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) ACTIVE(div(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(div(s(X:S),s(Y:S))) -> S(div(minus(X:S,Y:S),s(Y:S))) ACTIVE(div(0,s(Y:S))) -> MARK(0) ACTIVE(geq(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(geq(0,s(Y:S))) -> MARK(ffalse) ACTIVE(geq(X:S,0)) -> MARK(ttrue) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) ACTIVE(minus(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(minus(0,Y:S)) -> MARK(0) DIV(active(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) GEQ(active(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(mark(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) IF(active(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> DIV(mark(X1:S),X2:S) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) MINUS(active(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(mark(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(div(s(X:S),s(Y:S))) -> DIV(minus(X:S,Y:S),s(Y:S)) ACTIVE(div(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(div(s(X:S),s(Y:S))) -> IF(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0) ACTIVE(div(s(X:S),s(Y:S))) -> MARK(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) ACTIVE(div(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(div(s(X:S),s(Y:S))) -> S(div(minus(X:S,Y:S),s(Y:S))) ACTIVE(div(0,s(Y:S))) -> MARK(0) ACTIVE(geq(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(geq(0,s(Y:S))) -> MARK(ffalse) ACTIVE(geq(X:S,0)) -> MARK(ttrue) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) ACTIVE(minus(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(minus(0,Y:S)) -> MARK(0) DIV(active(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) GEQ(active(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(mark(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) IF(active(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> DIV(mark(X1:S),X2:S) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) MINUS(active(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(mark(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: MINUS(active(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(mark(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: IF(active(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: GEQ(active(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(mark(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: DIV(active(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(div(s(X:S),s(Y:S))) -> MARK(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: MINUS(active(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(mark(X1:S),X2:S) -> MINUS(X1:S,X2:S) MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MINUS) = 1 Problem 1.2: SCC Processor: -> Pairs: MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.2: Subterm Processor: -> Pairs: MINUS(X1:S,active(X2:S)) -> MINUS(X1:S,X2:S) MINUS(X1:S,mark(X2:S)) -> MINUS(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MINUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: IF(active(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(IF) = 1 Problem 1.3: SCC Processor: -> Pairs: IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.3: Subterm Processor: -> Pairs: IF(X1:S,active(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(IF) = 2 Problem 1.3: SCC Processor: -> Pairs: IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.3: Subterm Processor: -> Pairs: IF(X1:S,X2:S,active(X3:S)) -> IF(X1:S,X2:S,X3:S) IF(X1:S,X2:S,mark(X3:S)) -> IF(X1:S,X2:S,X3:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(IF) = 3 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: GEQ(active(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(mark(X1:S),X2:S) -> GEQ(X1:S,X2:S) GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(GEQ) = 1 Problem 1.4: SCC Processor: -> Pairs: GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.4: Subterm Processor: -> Pairs: GEQ(X1:S,active(X2:S)) -> GEQ(X1:S,X2:S) GEQ(X1:S,mark(X2:S)) -> GEQ(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(GEQ) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: DIV(active(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(DIV) = 1 Problem 1.5: SCC Processor: -> Pairs: DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: DIV(X1:S,active(X2:S)) -> DIV(X1:S,X2:S) DIV(X1:S,mark(X2:S)) -> DIV(X1:S,X2:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(DIV) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(div(s(X:S),s(Y:S))) -> MARK(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [div](X1,X2) = 2.X1 + 2.X2 + 2 [geq](X1,X2) = 0 [if](X1,X2,X3) = 2.X1 + X2 + X3 [mark](X) = X [minus](X1,X2) = X1 [s](X) = X + 1 [0] = 0 [fSNonEmpty] = 0 [false] = 0 [true] = 0 [ACTIVE](X) = 2.X + 2 [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MARK](X) = 2.X + 2 [MINUS](X1,X2) = 0 [S](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(geq(s(X:S),s(Y:S))) -> MARK(geq(X:S,Y:S)) ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [div](X1,X2) = 2.X1 [geq](X1,X2) = 2.X1 + 1 [if](X1,X2,X3) = X1 + X2 + X3 + 1 [mark](X) = X [minus](X1,X2) = 0 [s](X) = 2.X + 2 [0] = 0 [fSNonEmpty] = 0 [false] = 1 [true] = 0 [ACTIVE](X) = 2.X + 2 [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MARK](X) = 2.X + 2 [MINUS](X1,X2) = 0 [S](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(geq(X1:S,X2:S)) -> ACTIVE(geq(X1:S,X2:S)) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(if(ffalse,X:S,Y:S)) -> MARK(Y:S) ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [div](X1,X2) = 2.X1 + 2.X2 + 2 [geq](X1,X2) = 0 [if](X1,X2,X3) = 2.X1 + X2 + X3 + 1 [mark](X) = X [minus](X1,X2) = X1 [s](X) = X + 2 [0] = 1 [fSNonEmpty] = 0 [false] = 0 [true] = 0 [ACTIVE](X) = 2.X [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MARK](X) = 2.X [MINUS](X1,X2) = 0 [S](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(if(ttrue,X:S,Y:S)) -> MARK(X:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [div](X1,X2) = 2.X1 + 1 [geq](X1,X2) = 0 [if](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 1 [mark](X) = X [minus](X1,X2) = 0 [s](X) = 2.X + 2 [0] = 0 [fSNonEmpty] = 0 [false] = 0 [true] = 0 [ACTIVE](X) = 2.X [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MARK](X) = 2.X [MINUS](X1,X2) = 0 [S](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(minus(s(X:S),s(Y:S))) -> MARK(minus(X:S,Y:S)) MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [div](X1,X2) = 2.X1 [geq](X1,X2) = 2 [if](X1,X2,X3) = X1 + X2 + 2.X3 [mark](X) = X [minus](X1,X2) = X1 [s](X) = 2.X + 2 [0] = 0 [fSNonEmpty] = 0 [false] = 1 [true] = 2 [ACTIVE](X) = 2.X [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MARK](X) = 2.X + 2 [MINUS](X1,X2) = 0 [S](X) = 0 Problem 1.6: SCC Processor: -> Pairs: MARK(div(X1:S,X2:S)) -> ACTIVE(div(mark(X1:S),X2:S)) MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> ACTIVE(if(mark(X1:S),X2:S,X3:S)) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(minus(X1:S,X2:S)) -> ACTIVE(minus(X1:S,X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Subterm Processor: -> Pairs: MARK(div(X1:S,X2:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(div(s(X:S),s(Y:S))) -> mark(if(geq(X:S,Y:S),s(div(minus(X:S,Y:S),s(Y:S))),0)) active(div(0,s(Y:S))) -> mark(0) active(geq(s(X:S),s(Y:S))) -> mark(geq(X:S,Y:S)) active(geq(0,s(Y:S))) -> mark(ffalse) active(geq(X:S,0)) -> mark(ttrue) active(if(ffalse,X:S,Y:S)) -> mark(Y:S) active(if(ttrue,X:S,Y:S)) -> mark(X:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) div(active(X1:S),X2:S) -> div(X1:S,X2:S) div(mark(X1:S),X2:S) -> div(X1:S,X2:S) div(X1:S,active(X2:S)) -> div(X1:S,X2:S) div(X1:S,mark(X2:S)) -> div(X1:S,X2:S) geq(active(X1:S),X2:S) -> geq(X1:S,X2:S) geq(mark(X1:S),X2:S) -> geq(X1:S,X2:S) geq(X1:S,active(X2:S)) -> geq(X1:S,X2:S) geq(X1:S,mark(X2:S)) -> geq(X1:S,X2:S) if(active(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(mark(X1:S),X2:S,X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,active(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,mark(X2:S),X3:S) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,active(X3:S)) -> if(X1:S,X2:S,X3:S) if(X1:S,X2:S,mark(X3:S)) -> if(X1:S,X2:S,X3:S) mark(div(X1:S,X2:S)) -> active(div(mark(X1:S),X2:S)) mark(geq(X1:S,X2:S)) -> active(geq(X1:S,X2:S)) mark(if(X1:S,X2:S,X3:S)) -> active(if(mark(X1:S),X2:S,X3:S)) mark(minus(X1:S,X2:S)) -> active(minus(X1:S,X2:S)) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(ttrue) -> active(ttrue) minus(active(X1:S),X2:S) -> minus(X1:S,X2:S) minus(mark(X1:S),X2:S) -> minus(X1:S,X2:S) minus(X1:S,active(X2:S)) -> minus(X1:S,X2:S) minus(X1:S,mark(X2:S)) -> minus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.