YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S) (RULES active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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(f(X:S)) -> ACTIVE(X:S) ACTIVE(f(X:S)) -> F(active(X:S)) ACTIVE(f(X:S)) -> IF(X:S,c,f(ttrue)) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X2:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(X1:S,active(X2:S),X3:S) F(mark(X:S)) -> F(X:S) F(ok(X:S)) -> F(X: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) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) PROPER(f(X:S)) -> F(proper(X:S)) PROPER(f(X:S)) -> PROPER(X: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) 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(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1: SCC Processor: -> Pairs: ACTIVE(f(X:S)) -> ACTIVE(X:S) ACTIVE(f(X:S)) -> F(active(X:S)) ACTIVE(f(X:S)) -> IF(X:S,c,f(ttrue)) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X2:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(X1:S,active(X2:S),X3:S) F(mark(X:S)) -> F(X:S) F(ok(X:S)) -> F(X: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) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) PROPER(f(X:S)) -> F(proper(X:S)) PROPER(f(X:S)) -> PROPER(X: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) 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(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->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) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) ->->-> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: F(mark(X:S)) -> F(X:S) F(ok(X:S)) -> F(X:S) ->->-> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: PROPER(f(X:S)) -> PROPER(X: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) ->->-> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ACTIVE(f(X:S)) -> ACTIVE(X:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X2:S) ->->-> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) The problem is decomposed in 5 subproblems. Problem 1.1: 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) IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(IF) = 1 Problem 1.1: SCC Processor: -> Pairs: IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) ->->-> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.1: Subterm Processor: -> Pairs: IF(X1:S,mark(X2:S),X3:S) -> IF(X1:S,X2:S,X3:S) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(IF) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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: F(mark(X:S)) -> F(X:S) F(ok(X:S)) -> F(X:S) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(F) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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: PROPER(f(X:S)) -> PROPER(X: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) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(PROPER) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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: ACTIVE(f(X:S)) -> ACTIVE(X:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X2:S) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [f](X) = 2.X + 1 [if](X1,X2,X3) = 1/2.X1.X3 + X1 + X2 [proper](X) = X [top](X) = 0 [c] = 0 [fSNonEmpty] = 0 [false] = 2 [mark](X) = X + 1/2 [ok](X) = X [true] = 1/2 [ACTIVE](X) = 0 [F](X) = 0 [IF](X1,X2,X3) = 0 [PROPER](X) = 0 [TOP](X) = 1/2.X Problem 1.5: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.5: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X + 1 [f](X) = 2.X + 1 [if](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [proper](X) = 0 [top](X) = 0 [c] = 0 [fSNonEmpty] = 0 [false] = 0 [mark](X) = X [ok](X) = 2.X + 2 [true] = 0 [ACTIVE](X) = 0 [F](X) = 0 [IF](X1,X2,X3) = 0 [PROPER](X) = 0 [TOP](X) = 2.X Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(f(X:S)) -> f(active(X:S)) active(f(X:S)) -> mark(if(X:S,c,f(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(if(X1:S,X2:S,X3:S)) -> if(X1:S,active(X2:S),X3:S) f(mark(X:S)) -> mark(f(X:S)) f(ok(X:S)) -> ok(f(X: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)) if(X1:S,mark(X2:S),X3:S) -> mark(if(X1:S,X2:S,X3:S)) proper(f(X:S)) -> f(proper(X:S)) proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) proper(c) -> ok(c) proper(ffalse) -> ok(ffalse) proper(ttrue) -> ok(ttrue) 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.