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