0.00/0.27 YES 0.00/0.27 0.00/0.27 Problem 1: 0.00/0.27 0.00/0.27 (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) 0.00/0.27 (RULES 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ) 0.00/0.27 (STRATEGY INNERMOST) 0.00/0.27 0.00/0.27 Problem 1: 0.00/0.27 0.00/0.27 Dependency Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S)) 0.00/0.27 A__U11(tt,V2:S) -> A__ISNAT(V2:S) 0.00/0.27 A__U31(tt,N:S) -> MARK(N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__ISNAT(N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) 0.00/0.27 A__PLUS(N:S,0) -> A__ISNAT(N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> A__U12(mark(X:S)) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> A__U21(mark(X:S)) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 0.00/0.27 Problem 1: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U11(tt,V2:S) -> A__U12(a__isNat(V2:S)) 0.00/0.27 A__U11(tt,V2:S) -> A__ISNAT(V2:S) 0.00/0.27 A__U31(tt,N:S) -> MARK(N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__ISNAT(N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__U21(a__isNat(V1:S)) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) 0.00/0.27 A__PLUS(N:S,0) -> A__ISNAT(N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> A__U12(mark(X:S)) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> A__U21(mark(X:S)) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(isNat(X:S)) -> A__ISNAT(X:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 A__U11(tt,V2:S) -> A__ISNAT(V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 A__U31(tt,N:S) -> MARK(N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 0.00/0.27 0.00/0.27 The problem is decomposed in 2 subproblems. 0.00/0.27 0.00/0.27 Problem 1.1: 0.00/0.27 0.00/0.27 Subterm Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U11(tt,V2:S) -> A__ISNAT(V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNat(V1:S),V2:S) 0.00/0.27 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNAT(V1:S) 0.00/0.27 A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Projection: 0.00/0.27 pi(A__U11) = 2 0.00/0.27 pi(A__ISNAT) = 1 0.00/0.27 0.00/0.27 Problem 1.1: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U11(tt,V2:S) -> A__ISNAT(V2:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 There is no strongly connected component 0.00/0.27 0.00/0.27 The problem is finite. 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 Reduction Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U31(tt,N:S) -> MARK(N:S) 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 -> Usable rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Interpretation type: 0.00/0.27 Linear 0.00/0.27 ->Coefficients: 0.00/0.27 Natural Numbers 0.00/0.27 ->Dimension: 0.00/0.27 1 0.00/0.27 ->Bound: 0.00/0.27 2 0.00/0.27 ->Interpretation: 0.00/0.27 0.00/0.27 [a__U11](X1,X2) = 2.X1 0.00/0.27 [a__U12](X) = 2.X 0.00/0.27 [a__U21](X) = 2.X 0.00/0.27 [a__U31](X1,X2) = 2.X1 + 2.X2 + 1 0.00/0.27 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [a__U42](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [a__isNat](X) = 0 0.00/0.27 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [mark](X) = X 0.00/0.27 [0] = 2 0.00/0.27 [U11](X1,X2) = 2.X1 0.00/0.27 [U12](X) = 2.X 0.00/0.27 [U21](X) = 2.X 0.00/0.27 [U31](X1,X2) = 2.X1 + 2.X2 + 1 0.00/0.27 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [U42](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [fSNonEmpty] = 0 0.00/0.27 [isNat](X) = 0 0.00/0.27 [plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [s](X) = X 0.00/0.27 [tt] = 0 0.00/0.27 [A__U11](X1,X2) = 0 0.00/0.27 [A__U12](X) = 0 0.00/0.27 [A__U21](X) = 0 0.00/0.27 [A__U31](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.27 [A__U42](X1,X2,X3) = 2.X2 + 2.X3 0.00/0.27 [A__ISNAT](X) = 0 0.00/0.27 [A__PLUS](X1,X2) = 2.X1 + 2.X2 0.00/0.27 [MARK](X) = 2.X 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,0) -> A__U31(a__isNat(N:S),N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 Reduction Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U41(tt,M:S,N:S) -> A__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 -> Usable rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Interpretation type: 0.00/0.27 Linear 0.00/0.27 ->Coefficients: 0.00/0.27 Natural Numbers 0.00/0.27 ->Dimension: 0.00/0.27 1 0.00/0.27 ->Bound: 0.00/0.27 2 0.00/0.27 ->Interpretation: 0.00/0.27 0.00/0.27 [a__U11](X1,X2) = X1 0.00/0.27 [a__U12](X) = X 0.00/0.27 [a__U21](X) = X 0.00/0.27 [a__U31](X1,X2) = X1 + X2 + 2 0.00/0.27 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [a__U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [a__isNat](X) = 2 0.00/0.27 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [mark](X) = X 0.00/0.27 [0] = 2 0.00/0.27 [U11](X1,X2) = X1 0.00/0.27 [U12](X) = X 0.00/0.27 [U21](X) = X 0.00/0.27 [U31](X1,X2) = X1 + X2 + 2 0.00/0.27 [U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [fSNonEmpty] = 0 0.00/0.27 [isNat](X) = 2 0.00/0.27 [plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [s](X) = X + 2 0.00/0.27 [tt] = 2 0.00/0.27 [A__U11](X1,X2) = 0 0.00/0.27 [A__U12](X) = 0 0.00/0.27 [A__U21](X) = 0 0.00/0.27 [A__U31](X1,X2) = 0 0.00/0.27 [A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.27 [A__U42](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 0.00/0.27 [A__ISNAT](X) = 0 0.00/0.27 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.27 [MARK](X) = 2.X + 2 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 SCC Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U42(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 A__PLUS(N:S,s(M:S)) -> A__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> A__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Strongly Connected Components: 0.00/0.27 ->->Cycle: 0.00/0.27 ->->-> Pairs: 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 ->->-> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 0.00/0.27 Problem 1.2: 0.00/0.27 0.00/0.27 Reduction Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(M:S) 0.00/0.27 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.27 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.27 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.27 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.27 MARK(s(X:S)) -> MARK(X:S) 0.00/0.27 -> Rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 -> Usable rules: 0.00/0.27 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.27 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.27 a__U12(tt) -> tt 0.00/0.27 a__U12(X:S) -> U12(X:S) 0.00/0.27 a__U21(tt) -> tt 0.00/0.27 a__U21(X:S) -> U21(X:S) 0.00/0.27 a__U31(tt,N:S) -> mark(N:S) 0.00/0.27 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.27 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.27 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.27 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.27 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.27 a__isNat(0) -> tt 0.00/0.27 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.27 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.27 a__isNat(X:S) -> isNat(X:S) 0.00/0.27 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.27 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.27 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.27 mark(0) -> 0 0.00/0.27 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.27 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.27 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.27 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.27 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.27 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.27 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.27 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.27 mark(tt) -> tt 0.00/0.27 ->Interpretation type: 0.00/0.27 Linear 0.00/0.27 ->Coefficients: 0.00/0.27 Natural Numbers 0.00/0.27 ->Dimension: 0.00/0.27 1 0.00/0.27 ->Bound: 0.00/0.27 2 0.00/0.27 ->Interpretation: 0.00/0.27 0.00/0.27 [a__U11](X1,X2) = X1 0.00/0.27 [a__U12](X) = X 0.00/0.27 [a__U21](X) = X 0.00/0.27 [a__U31](X1,X2) = X1 + X2 + 2 0.00/0.27 [a__U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 0.00/0.27 [a__U42](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 0.00/0.27 [a__isNat](X) = 2 0.00/0.27 [a__plus](X1,X2) = X1 + 2.X2 + 2 0.00/0.27 [mark](X) = X 0.00/0.27 [0] = 1 0.00/0.27 [U11](X1,X2) = X1 0.00/0.27 [U12](X) = X 0.00/0.27 [U21](X) = X 0.00/0.27 [U31](X1,X2) = X1 + X2 + 2 0.00/0.27 [U41](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2 0.00/0.27 [U42](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 0.00/0.27 [fSNonEmpty] = 0 0.00/0.27 [isNat](X) = 2 0.00/0.27 [plus](X1,X2) = X1 + 2.X2 + 2 0.00/0.27 [s](X) = X + 2 0.00/0.27 [tt] = 2 0.00/0.27 [A__U11](X1,X2) = 0 0.00/0.27 [A__U12](X) = 0 0.00/0.27 [A__U21](X) = 0 0.00/0.28 [A__U31](X1,X2) = 0 0.00/0.28 [A__U41](X1,X2,X3) = 0 0.00/0.28 [A__U42](X1,X2,X3) = 2.X2 + 2.X3 + 1 0.00/0.28 [A__ISNAT](X) = 0 0.00/0.28 [A__PLUS](X1,X2) = 0 0.00/0.28 [MARK](X) = 2.X 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.28 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.28 MARK(s(X:S)) -> MARK(X:S) 0.00/0.28 -> Rules: 0.00/0.28 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.28 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.28 a__U12(tt) -> tt 0.00/0.28 a__U12(X:S) -> U12(X:S) 0.00/0.28 a__U21(tt) -> tt 0.00/0.28 a__U21(X:S) -> U21(X:S) 0.00/0.28 a__U31(tt,N:S) -> mark(N:S) 0.00/0.28 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.28 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.28 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.28 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.28 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.28 a__isNat(0) -> tt 0.00/0.28 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.28 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.28 a__isNat(X:S) -> isNat(X:S) 0.00/0.28 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.28 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.28 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.28 mark(0) -> 0 0.00/0.28 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.28 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.28 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.28 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.28 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.28 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.28 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.28 mark(tt) -> tt 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.28 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.28 MARK(s(X:S)) -> MARK(X:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.28 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.28 a__U12(tt) -> tt 0.00/0.28 a__U12(X:S) -> U12(X:S) 0.00/0.28 a__U21(tt) -> tt 0.00/0.28 a__U21(X:S) -> U21(X:S) 0.00/0.28 a__U31(tt,N:S) -> mark(N:S) 0.00/0.28 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.28 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.28 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.28 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.28 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.28 a__isNat(0) -> tt 0.00/0.28 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.28 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.28 a__isNat(X:S) -> isNat(X:S) 0.00/0.28 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.28 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.28 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.28 mark(0) -> 0 0.00/0.28 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.28 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.28 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.28 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.28 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.28 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.28 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.28 mark(tt) -> tt 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.28 MARK(U11(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U12(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U21(X:S)) -> MARK(X:S) 0.00/0.28 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> A__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 MARK(U42(X1:S,X2:S,X3:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.28 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.28 MARK(s(X:S)) -> MARK(X:S) 0.00/0.28 -> Rules: 0.00/0.28 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.28 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.28 a__U12(tt) -> tt 0.00/0.28 a__U12(X:S) -> U12(X:S) 0.00/0.28 a__U21(tt) -> tt 0.00/0.28 a__U21(X:S) -> U21(X:S) 0.00/0.28 a__U31(tt,N:S) -> mark(N:S) 0.00/0.28 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.28 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.28 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.28 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.28 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.28 a__isNat(0) -> tt 0.00/0.28 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.28 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.28 a__isNat(X:S) -> isNat(X:S) 0.00/0.28 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.28 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.28 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.28 mark(0) -> 0 0.00/0.28 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.28 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.28 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.28 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.28 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.28 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.28 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.28 mark(tt) -> tt 0.00/0.28 ->Projection: 0.00/0.28 pi(A__U42) = 3 0.00/0.28 pi(MARK) = 1 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 A__U42(tt,M:S,N:S) -> MARK(N:S) 0.00/0.28 -> Rules: 0.00/0.28 a__U11(tt,V2:S) -> a__U12(a__isNat(V2:S)) 0.00/0.28 a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) 0.00/0.28 a__U12(tt) -> tt 0.00/0.28 a__U12(X:S) -> U12(X:S) 0.00/0.28 a__U21(tt) -> tt 0.00/0.28 a__U21(X:S) -> U21(X:S) 0.00/0.28 a__U31(tt,N:S) -> mark(N:S) 0.00/0.28 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 0.00/0.28 a__U41(tt,M:S,N:S) -> a__U42(a__isNat(N:S),M:S,N:S) 0.00/0.28 a__U41(X1:S,X2:S,X3:S) -> U41(X1:S,X2:S,X3:S) 0.00/0.28 a__U42(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 0.00/0.28 a__U42(X1:S,X2:S,X3:S) -> U42(X1:S,X2:S,X3:S) 0.00/0.28 a__isNat(0) -> tt 0.00/0.28 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNat(V1:S),V2:S) 0.00/0.28 a__isNat(s(V1:S)) -> a__U21(a__isNat(V1:S)) 0.00/0.28 a__isNat(X:S) -> isNat(X:S) 0.00/0.28 a__plus(N:S,0) -> a__U31(a__isNat(N:S),N:S) 0.00/0.28 a__plus(N:S,s(M:S)) -> a__U41(a__isNat(M:S),M:S,N:S) 0.00/0.28 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 0.00/0.28 mark(0) -> 0 0.00/0.28 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) 0.00/0.28 mark(U12(X:S)) -> a__U12(mark(X:S)) 0.00/0.28 mark(U21(X:S)) -> a__U21(mark(X:S)) 0.00/0.28 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 0.00/0.28 mark(U41(X1:S,X2:S,X3:S)) -> a__U41(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(U42(X1:S,X2:S,X3:S)) -> a__U42(mark(X1:S),X2:S,X3:S) 0.00/0.28 mark(isNat(X:S)) -> a__isNat(X:S) 0.00/0.28 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 0.00/0.28 mark(s(X:S)) -> s(mark(X:S)) 0.00/0.28 mark(tt) -> tt 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 EOF