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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(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))) -> 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(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(div(X1:S,X2:S)) -> DIV(active(X1:S),X2:S) ACTIVE(geq(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X:S)) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(ok(X1:S),ok(X2:S)) -> DIV(X1:S,X2:S) GEQ(ok(X1:S),ok(X2:S)) -> GEQ(X1:S,X2:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) MINUS(ok(X1:S),ok(X2:S)) -> MINUS(X1:S,X2:S) PROPER(div(X1:S,X2:S)) -> DIV(proper(X1:S),proper(X2:S)) PROPER(div(X1:S,X2:S)) -> PROPER(X1:S) PROPER(div(X1:S,X2:S)) -> PROPER(X2:S) PROPER(geq(X1:S,X2:S)) -> GEQ(proper(X1:S),proper(X2:S)) PROPER(geq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> IF(proper(X1:S),proper(X2:S),proper(X3:S)) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) PROPER(minus(X1:S,X2:S)) -> MINUS(proper(X1:S),proper(X2:S)) PROPER(minus(X1:S,X2:S)) -> PROPER(X1:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X2:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(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))) -> 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(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(div(X1:S,X2:S)) -> DIV(active(X1:S),X2:S) ACTIVE(geq(s(X:S),s(Y:S))) -> GEQ(X:S,Y:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) ACTIVE(minus(s(X:S),s(Y:S))) -> MINUS(X:S,Y:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X:S)) DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(ok(X1:S),ok(X2:S)) -> DIV(X1:S,X2:S) GEQ(ok(X1:S),ok(X2:S)) -> GEQ(X1:S,X2:S) IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) MINUS(ok(X1:S),ok(X2:S)) -> MINUS(X1:S,X2:S) PROPER(div(X1:S,X2:S)) -> DIV(proper(X1:S),proper(X2:S)) PROPER(div(X1:S,X2:S)) -> PROPER(X1:S) PROPER(div(X1:S,X2:S)) -> PROPER(X2:S) PROPER(geq(X1:S,X2:S)) -> GEQ(proper(X1:S),proper(X2:S)) PROPER(geq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> IF(proper(X1:S),proper(X2:S),proper(X3:S)) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) PROPER(minus(X1:S,X2:S)) -> MINUS(proper(X1:S),proper(X2:S)) PROPER(minus(X1:S,X2:S)) -> PROPER(X1:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X2:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TOP(mark(X:S)) -> PROPER(X:S) TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> ACTIVE(X:S) TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(mark(X:S)) -> S(X:S) S(ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: MINUS(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(ok(X1:S),ok(X2:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: GEQ(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: PROPER(div(X1:S,X2:S)) -> PROPER(X1:S) PROPER(div(X1:S,X2:S)) -> PROPER(X2:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X1:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X2:S) PROPER(s(X:S)) -> PROPER(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ACTIVE(div(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(s(X:S)) -> ACTIVE(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) The problem is decomposed in 8 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(mark(X:S)) -> S(X:S) S(ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: MINUS(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(MINUS) = 1 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) IF(ok(X1:S),ok(X2:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(IF) = 1 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: GEQ(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(GEQ) = 1 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: DIV(mark(X1:S),X2:S) -> DIV(X1:S,X2:S) DIV(ok(X1:S),ok(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(DIV) = 1 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: PROPER(div(X1:S,X2:S)) -> PROPER(X1:S) PROPER(div(X1:S,X2:S)) -> PROPER(X2:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(geq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X1:S) PROPER(minus(X1:S,X2:S)) -> PROPER(X2:S) PROPER(s(X:S)) -> PROPER(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(PROPER) = 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: ACTIVE(div(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(s(X:S)) -> ACTIVE(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.7: 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [active](X) = X [div](X1,X2) = 3.X1 + X2 + 3 [geq](X1,X2) = 1/2.X1 + 4/3 [if](X1,X2,X3) = X1 + X2 + X3 + 4/3 [minus](X1,X2) = 1/3.X1 + 4/3 [proper](X) = X [s](X) = X + 4 [top](X) = 0 [0] = 0 [fSNonEmpty] = 0 [false] = 0 [mark](X) = X + 4/3 [ok](X) = X [true] = 0 [ACTIVE](X) = 0 [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MINUS](X1,X2) = 0 [PROPER](X) = 0 [S](X) = 0 [TOP](X) = 1/3.X Problem 1.8: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.8: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 1 [div](X1,X2) = X1 [geq](X1,X2) = 2.X1 [if](X1,X2,X3) = X1 [minus](X1,X2) = 2.X1 + 2.X2 [proper](X) = 0 [s](X) = X [top](X) = 0 [0] = 2 [fSNonEmpty] = 0 [false] = 2 [mark](X) = 1 [ok](X) = 2 [true] = 1 [ACTIVE](X) = 0 [DIV](X1,X2) = 0 [GEQ](X1,X2) = 0 [IF](X1,X2,X3) = 0 [MINUS](X1,X2) = 0 [PROPER](X) = 0 [S](X) = 0 [TOP](X) = 2.X Problem 1.8: 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(div(X1:S,X2:S)) -> div(active(X1:S),X2: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(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) active(minus(s(X:S),s(Y:S))) -> mark(minus(X:S,Y:S)) active(minus(0,Y:S)) -> mark(0) active(s(X:S)) -> s(active(X:S)) div(mark(X1:S),X2:S) -> mark(div(X1:S,X2:S)) div(ok(X1:S),ok(X2:S)) -> ok(div(X1:S,X2:S)) geq(ok(X1:S),ok(X2:S)) -> ok(geq(X1:S,X2:S)) if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) minus(ok(X1:S),ok(X2:S)) -> ok(minus(X1:S,X2:S)) proper(div(X1:S,X2:S)) -> div(proper(X1:S),proper(X2:S)) proper(geq(X1:S,X2:S)) -> geq(proper(X1:S),proper(X2:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(minus(X1:S,X2:S)) -> minus(proper(X1:S),proper(X2:S)) proper(s(X:S)) -> s(proper(X:S)) proper(0) -> ok(0) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.