/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N X X1 X2 X3) (RULES U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ACTIVE(U11(tt,M,N)) -> U12#(tt,M,N) ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N)) ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U12(tt,M,N)) -> PLUS(N,M) ACTIVE(U12(tt,M,N)) -> S(plus(N,M)) ACTIVE(plus(N,s(M))) -> U11#(tt,M,N) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1: SCC Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ACTIVE(U11(tt,M,N)) -> U12#(tt,M,N) ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N)) ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U12(tt,M,N)) -> PLUS(N,M) ACTIVE(U12(tt,M,N)) -> S(plus(N,M)) ACTIVE(plus(N,s(M))) -> U11#(tt,M,N) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N)) ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 5 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 1 Problem 1.2: SCC Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.2: Subterm Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 1 Problem 1.3: SCC Processor: -> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.3: Subterm Processor: -> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 2 Problem 1.3: SCC Processor: -> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.3: Subterm Processor: -> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 3 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 1 Problem 1.4: SCC Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.4: Subterm Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 2 Problem 1.4: SCC Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.4: Subterm Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 3 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,M,N)) -> MARK(U12(tt,M,N)) ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [active](X) = X [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(U12(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(plus(N,s(M))) -> MARK(U11(tt,M,N)) ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [active](X) = X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Reduction Pair Processor: -> Pairs: ACTIVE(plus(N,0)) -> MARK(N) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 1 [tt] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.5: SCC Processor: -> Pairs: MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Subterm Processor: -> Pairs: MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(MARK) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) active(U11(tt,M,N)) -> mark(U12(tt,M,N)) active(U12(tt,M,N)) -> mark(s(plus(N,M))) active(plus(N,s(M))) -> mark(U11(tt,M,N)) active(plus(N,0)) -> mark(N) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.