YES Problem 1: (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) (RULES a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S)) A__U11(tt,V2:S) -> A__ISNAT(V2:S) A__U31(tt,N:S) -> MARK(N:S) A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U41(tt,M:S,N:S) -> A__ISNAT(N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) A__PLUS(N:S,0) -> A__ISNAT(N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> A__U12(mark(X:S)) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> A__U21(mark(X:S)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt Problem 1: SCC Processor: -> Pairs: A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S)) A__U11(tt,V2:S) -> A__ISNAT(V2:S) A__U31(tt,N:S) -> MARK(N:S) A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U41(tt,M:S,N:S) -> A__ISNAT(N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) A__PLUS(N:S,0) -> A__ISNAT(N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> A__U12(mark(X:S)) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> A__U21(mark(X:S)) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U11(tt,V2:S) -> A__ISNAT(V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) ->->-> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->->Cycle: ->->-> Pairs: A__U31(tt,N:S) -> MARK(N:S) A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__U11(tt,V2:S) -> A__ISNAT(V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Projection: pi(A__U11) = 2 pi(A__ISNAT) = 1 Problem 1.1: SCC Processor: -> Pairs: A__U11(tt,V2:S) -> A__ISNAT(V2:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: A__U31(tt,N:S) -> MARK(N:S) A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt -> Usable rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = 2.X1 [a__U12](X) = 2.X [a__U21](X) = 2.X [a__U31](X1,X2) = 2.X1 + 2.X2 + 1 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__isNat](X) = 0 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2) = 2.X1 [U12](X) = 2.X [U21](X) = 2.X [U31](X1,X2) = 2.X1 + 2.X2 + 1 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [fSNonEmpty] = 0 [isNat](X) = 0 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [tt] = 0 [A__U11](X1,X2) = 0 [A__U12](X) = 0 [A__U21](X) = 0 [A__U31](X1,X2) = 2.X1 + 2.X2 + 2 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [A__U42](X1,X2,X3) = 2.X2 + 2.X3 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 2.X1 + 2.X2 [MARK](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt Problem 1.2: Reduction Pairs Processor: -> Pairs: A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt -> Usable rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 [a__U12](X) = X [a__U21](X) = X [a__U31](X1,X2) = X1 + X2 + 2 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [a__U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [a__isNat](X) = 2 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 [mark](X) = X [0] = 2 [U11](X1,X2) = X1 [U12](X) = X [U21](X) = X [U31](X1,X2) = X1 + X2 + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [fSNonEmpty] = 0 [isNat](X) = 2 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [A__U11](X1,X2) = 0 [A__U12](X) = 0 [A__U21](X) = 0 [A__U31](X1,X2) = 0 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [A__U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt Problem 1.2: Reduction Pairs Processor: -> Pairs: A__U42(tt,M:S,N:S) -> MARK(M:S) A__U42(tt,M:S,N:S) -> MARK(N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt -> Usable rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 [a__U12](X) = X [a__U21](X) = X [a__U31](X1,X2) = X1 + X2 + 2 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [a__U42](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 [a__isNat](X) = 2 [a__plus](X1,X2) = X1 + 2.X2 + 2 [mark](X) = X [0] = 1 [U11](X1,X2) = X1 [U12](X) = X [U21](X) = X [U31](X1,X2) = X1 + X2 + 2 [U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 [U42](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 [fSNonEmpty] = 0 [isNat](X) = 2 [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [tt] = 2 [A__U11](X1,X2) = 0 [A__U12](X) = 0 [A__U21](X) = 0 [A__U31](X1,X2) = 0 [A__U41](X1,X2,X3) = 0 [A__U42](X1,X2,X3) = 2.X2 + 2.X3 + 1 [A__ISNAT](X) = 0 [A__PLUS](X1,X2) = 0 [MARK](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A__U42(tt,M:S,N:S) -> MARK(N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U42(tt,M:S,N:S) -> MARK(N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt Problem 1.2: Subterm Processor: -> Pairs: A__U42(tt,M:S,N:S) -> MARK(N:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(U12(X:S)) -> MARK(X:S) MARK(U21(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Projection: pi(A__U42) = 3 pi(MARK) = 1 Problem 1.2: SCC Processor: -> Pairs: A__U42(tt,M:S,N:S) -> MARK(N:S) -> Rules: a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__U12(tt) -> tt a__U12(X:S) -> U12(X:S) a__U21(tt) -> tt a__U21(X:S) -> U21(X:S) a__U31(tt,N:S) -> mark(N:S) a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) a__isNat(0) -> tt a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) a__isNat(X:S) -> isNat(X:S) a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(U12(X:S)) -> a__U12(mark(X:S)) mark(U21(X:S)) -> a__U21(mark(X:S)) mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt ->Strongly Connected Components: There is no strongly connected component The problem is finite.