/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR I P X X1 X2 Y Z) (RULES __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ) Problem 1: Dependency Pairs Processor: -> Pairs: __#(mark(X1),X2) -> __#(X1,X2) __#(ok(X1),ok(X2)) -> __#(X1,X2) __#(X1,mark(X2)) -> __#(X1,X2) ACTIVE(__(__(X,Y),Z)) -> __#(X,__(Y,Z)) ACTIVE(__(__(X,Y),Z)) -> __#(Y,Z) ACTIVE(__(X1,X2)) -> __#(active(X1),X2) ACTIVE(__(X1,X2)) -> __#(X1,active(X2)) ACTIVE(__(X1,X2)) -> ACTIVE(X1) ACTIVE(__(X1,X2)) -> ACTIVE(X2) ACTIVE(and(X1,X2)) -> ACTIVE(X1) ACTIVE(and(X1,X2)) -> AND(active(X1),X2) ACTIVE(isNePal(X)) -> ACTIVE(X) ACTIVE(isNePal(X)) -> ISNEPAL(active(X)) AND(mark(X1),X2) -> AND(X1,X2) AND(ok(X1),ok(X2)) -> AND(X1,X2) ISNEPAL(mark(X)) -> ISNEPAL(X) ISNEPAL(ok(X)) -> ISNEPAL(X) PROPER(__(X1,X2)) -> __#(proper(X1),proper(X2)) PROPER(__(X1,X2)) -> PROPER(X1) PROPER(__(X1,X2)) -> PROPER(X2) PROPER(and(X1,X2)) -> AND(proper(X1),proper(X2)) PROPER(and(X1,X2)) -> PROPER(X1) PROPER(and(X1,X2)) -> PROPER(X2) PROPER(isNePal(X)) -> ISNEPAL(proper(X)) PROPER(isNePal(X)) -> PROPER(X) TOP(mark(X)) -> PROPER(X) TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> ACTIVE(X) TOP(ok(X)) -> TOP(active(X)) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1: SCC Processor: -> Pairs: __#(mark(X1),X2) -> __#(X1,X2) __#(ok(X1),ok(X2)) -> __#(X1,X2) __#(X1,mark(X2)) -> __#(X1,X2) ACTIVE(__(__(X,Y),Z)) -> __#(X,__(Y,Z)) ACTIVE(__(__(X,Y),Z)) -> __#(Y,Z) ACTIVE(__(X1,X2)) -> __#(active(X1),X2) ACTIVE(__(X1,X2)) -> __#(X1,active(X2)) ACTIVE(__(X1,X2)) -> ACTIVE(X1) ACTIVE(__(X1,X2)) -> ACTIVE(X2) ACTIVE(and(X1,X2)) -> ACTIVE(X1) ACTIVE(and(X1,X2)) -> AND(active(X1),X2) ACTIVE(isNePal(X)) -> ACTIVE(X) ACTIVE(isNePal(X)) -> ISNEPAL(active(X)) AND(mark(X1),X2) -> AND(X1,X2) AND(ok(X1),ok(X2)) -> AND(X1,X2) ISNEPAL(mark(X)) -> ISNEPAL(X) ISNEPAL(ok(X)) -> ISNEPAL(X) PROPER(__(X1,X2)) -> __#(proper(X1),proper(X2)) PROPER(__(X1,X2)) -> PROPER(X1) PROPER(__(X1,X2)) -> PROPER(X2) PROPER(and(X1,X2)) -> AND(proper(X1),proper(X2)) PROPER(and(X1,X2)) -> PROPER(X1) PROPER(and(X1,X2)) -> PROPER(X2) PROPER(isNePal(X)) -> ISNEPAL(proper(X)) PROPER(isNePal(X)) -> PROPER(X) TOP(mark(X)) -> PROPER(X) TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> ACTIVE(X) TOP(ok(X)) -> TOP(active(X)) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNEPAL(mark(X)) -> ISNEPAL(X) ISNEPAL(ok(X)) -> ISNEPAL(X) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: AND(mark(X1),X2) -> AND(X1,X2) AND(ok(X1),ok(X2)) -> AND(X1,X2) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: __#(mark(X1),X2) -> __#(X1,X2) __#(ok(X1),ok(X2)) -> __#(X1,X2) __#(X1,mark(X2)) -> __#(X1,X2) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: ACTIVE(__(X1,X2)) -> ACTIVE(X1) ACTIVE(__(X1,X2)) -> ACTIVE(X2) ACTIVE(and(X1,X2)) -> ACTIVE(X1) ACTIVE(isNePal(X)) -> ACTIVE(X) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: PROPER(__(X1,X2)) -> PROPER(X1) PROPER(__(X1,X2)) -> PROPER(X2) PROPER(and(X1,X2)) -> PROPER(X1) PROPER(and(X1,X2)) -> PROPER(X2) PROPER(isNePal(X)) -> PROPER(X) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->->Cycle: ->->-> Pairs: TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> TOP(active(X)) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: ISNEPAL(mark(X)) -> ISNEPAL(X) ISNEPAL(ok(X)) -> ISNEPAL(X) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(ISNEPAL) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: AND(mark(X1),X2) -> AND(X1,X2) AND(ok(X1),ok(X2)) -> AND(X1,X2) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(AND) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: __#(mark(X1),X2) -> __#(X1,X2) __#(ok(X1),ok(X2)) -> __#(X1,X2) __#(X1,mark(X2)) -> __#(X1,X2) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(__#) = 1 Problem 1.3: SCC Processor: -> Pairs: __#(X1,mark(X2)) -> __#(X1,X2) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: __#(X1,mark(X2)) -> __#(X1,X2) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1.3: Subterm Processor: -> Pairs: __#(X1,mark(X2)) -> __#(X1,X2) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(__#) = 2 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ACTIVE(__(X1,X2)) -> ACTIVE(X1) ACTIVE(__(X1,X2)) -> ACTIVE(X2) ACTIVE(and(X1,X2)) -> ACTIVE(X1) ACTIVE(isNePal(X)) -> ACTIVE(X) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(ACTIVE) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: PROPER(__(X1,X2)) -> PROPER(X1) PROPER(__(X1,X2)) -> PROPER(X2) PROPER(and(X1,X2)) -> PROPER(X1) PROPER(and(X1,X2)) -> PROPER(X2) PROPER(isNePal(X)) -> PROPER(X) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Projection: pi(PROPER) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pair Processor: -> Pairs: TOP(mark(X)) -> TOP(proper(X)) TOP(ok(X)) -> TOP(active(X)) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -> Usable rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X1 + X2 + 2 [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 + 2 [isNePal](X) = 2.X + 2 [proper](X) = X [mark](X) = X + 2 [nil] = 0 [ok](X) = X [tt] = 2 [TOP](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: TOP(ok(X)) -> TOP(active(X)) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TOP(ok(X)) -> TOP(active(X)) ->->-> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Problem 1.6: Reduction Pair Processor: -> Pairs: TOP(ok(X)) -> TOP(active(X)) -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -> Usable rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [__](X1,X2) = 2.X2 + 1 [active](X) = 2.X [and](X1,X2) = 2.X1 + X2 + 1 [isNePal](X) = 2.X + 2 [mark](X) = 1 [nil] = 2 [ok](X) = 2.X + 2 [tt] = 0 [TOP](X) = X Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: __(mark(X1),X2) -> mark(__(X1,X2)) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(nil,X)) -> mark(X) active(__(X,nil)) -> mark(X) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(tt,X)) -> mark(X) active(and(X1,X2)) -> and(active(X1),X2) active(isNePal(__(I,__(P,I)))) -> mark(tt) active(isNePal(X)) -> isNePal(active(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNePal(mark(X)) -> mark(isNePal(X)) isNePal(ok(X)) -> ok(isNePal(X)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNePal(X)) -> isNePal(proper(X)) proper(nil) -> ok(nil) proper(tt) -> ok(tt) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.