27.80/29.94 YES 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S) 27.80/29.94 (RULES 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ) 27.80/29.94 (STRATEGY INNERMOST) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Dependency Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__AND(tt,X:S) -> MARK(X:S) 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 SCC Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__AND(tt,X:S) -> MARK(X:S) 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Strongly Connected Components: 27.80/29.94 ->->Cycle: 27.80/29.94 ->->-> Pairs: 27.80/29.94 A__AND(tt,X:S) -> MARK(X:S) 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 ->->-> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Reduction Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__AND(tt,X:S) -> MARK(X:S) 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 -> Usable rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Interpretation type: 27.80/29.94 Simple mixed 27.80/29.94 ->Coefficients: 27.80/29.94 Natural Numbers 27.80/29.94 ->Dimension: 27.80/29.94 1 27.80/29.94 ->Bound: 27.80/29.94 2 27.80/29.94 ->Interpretation: 27.80/29.94 27.80/29.94 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 27.80/29.94 [mark](X) = X 27.80/29.94 [0] = 2 27.80/29.94 [and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [fSNonEmpty] = 0 27.80/29.94 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [s](X) = X + 2 27.80/29.94 [tt] = 1 27.80/29.94 [x](X1,X2) = X1.X2 + 2.X1 + X2 27.80/29.94 [A__AND](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 1 27.80/29.94 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 27.80/29.94 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 27.80/29.94 [MARK](X) = 2.X + 2 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 SCC Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Strongly Connected Components: 27.80/29.94 ->->Cycle: 27.80/29.94 ->->-> Pairs: 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 ->->-> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Reduction Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,0) -> MARK(N:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 -> Usable rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Interpretation type: 27.80/29.94 Simple mixed 27.80/29.94 ->Coefficients: 27.80/29.94 Natural Numbers 27.80/29.94 ->Dimension: 27.80/29.94 1 27.80/29.94 ->Bound: 27.80/29.94 2 27.80/29.94 ->Interpretation: 27.80/29.94 27.80/29.94 [a__and](X1,X2) = X1 + X2 27.80/29.94 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [mark](X) = X 27.80/29.94 [0] = 2 27.80/29.94 [and](X1,X2) = X1 + X2 27.80/29.94 [fSNonEmpty] = 0 27.80/29.94 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [s](X) = X + 2 27.80/29.94 [tt] = 0 27.80/29.94 [x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [A__AND](X1,X2) = 0 27.80/29.94 [A__PLUS](X1,X2) = 2.X1 + 2.X2 27.80/29.94 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 27.80/29.94 [MARK](X) = 2.X + 2 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 SCC Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Strongly Connected Components: 27.80/29.94 ->->Cycle: 27.80/29.94 ->->-> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 ->->-> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Reduction Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> A__PLUS(mark(N:S),mark(M:S)) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 -> Usable rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Interpretation type: 27.80/29.94 Simple mixed 27.80/29.94 ->Coefficients: 27.80/29.94 Natural Numbers 27.80/29.94 ->Dimension: 27.80/29.94 1 27.80/29.94 ->Bound: 27.80/29.94 2 27.80/29.94 ->Interpretation: 27.80/29.94 27.80/29.94 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [mark](X) = X 27.80/29.94 [0] = 1 27.80/29.94 [and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [fSNonEmpty] = 0 27.80/29.94 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.94 [s](X) = X + 2 27.80/29.94 [tt] = 2 27.80/29.94 [x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [A__AND](X1,X2) = 0 27.80/29.94 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 27.80/29.94 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 27.80/29.94 [MARK](X) = 2.X 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 SCC Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Strongly Connected Components: 27.80/29.94 ->->Cycle: 27.80/29.94 ->->-> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 ->->-> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Reduction Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 -> Usable rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Interpretation type: 27.80/29.94 Simple mixed 27.80/29.94 ->Coefficients: 27.80/29.94 Natural Numbers 27.80/29.94 ->Dimension: 27.80/29.94 1 27.80/29.94 ->Bound: 27.80/29.94 2 27.80/29.94 ->Interpretation: 27.80/29.94 27.80/29.94 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [a__plus](X1,X2) = X1 + 2.X2 + 1 27.80/29.94 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 27.80/29.94 [mark](X) = X 27.80/29.94 [0] = 1 27.80/29.94 [and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.94 [fSNonEmpty] = 0 27.80/29.94 [plus](X1,X2) = X1 + 2.X2 + 1 27.80/29.94 [s](X) = X + 2 27.80/29.94 [tt] = 2 27.80/29.94 [x](X1,X2) = X1.X2 + 2.X1 + X2 27.80/29.94 [A__AND](X1,X2) = 0 27.80/29.94 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 1 27.80/29.94 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 27.80/29.94 [MARK](X) = 2.X 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 SCC Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Strongly Connected Components: 27.80/29.94 ->->Cycle: 27.80/29.94 ->->-> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 ->->-> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 27.80/29.94 Problem 1: 27.80/29.94 27.80/29.94 Reduction Pairs Processor: 27.80/29.94 -> Pairs: 27.80/29.94 A__PLUS(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.94 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.94 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 MARK(s(X:S)) -> MARK(X:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.94 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.94 -> Rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 -> Usable rules: 27.80/29.94 a__and(tt,X:S) -> mark(X:S) 27.80/29.94 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.94 a__plus(N:S,0) -> mark(N:S) 27.80/29.94 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.94 a__x(N:S,0) -> 0 27.80/29.94 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.94 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.94 mark(0) -> 0 27.80/29.94 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.94 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.94 mark(tt) -> tt 27.80/29.94 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.94 ->Interpretation type: 27.80/29.95 Simple mixed 27.80/29.95 ->Coefficients: 27.80/29.95 Natural Numbers 27.80/29.95 ->Dimension: 27.80/29.95 1 27.80/29.95 ->Bound: 27.80/29.95 2 27.80/29.95 ->Interpretation: 27.80/29.95 27.80/29.95 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [a__plus](X1,X2) = X1 + X2 + 2 27.80/29.95 [a__x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [mark](X) = X 27.80/29.95 [0] = 2 27.80/29.95 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [fSNonEmpty] = 0 27.80/29.95 [plus](X1,X2) = X1 + X2 + 2 27.80/29.95 [s](X) = X + 2 27.80/29.95 [tt] = 1 27.80/29.95 [x](X1,X2) = X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [A__AND](X1,X2) = 0 27.80/29.95 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 27.80/29.95 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [MARK](X) = 2.X + 2 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 SCC Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> A__PLUS(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Strongly Connected Components: 27.80/29.95 ->->Cycle: 27.80/29.95 ->->-> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 ->->-> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 Reduction Pairs Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> A__X(mark(N:S),mark(M:S)) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 -> Usable rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Interpretation type: 27.80/29.95 Simple mixed 27.80/29.95 ->Coefficients: 27.80/29.95 Natural Numbers 27.80/29.95 ->Dimension: 27.80/29.95 1 27.80/29.95 ->Bound: 27.80/29.95 2 27.80/29.95 ->Interpretation: 27.80/29.95 27.80/29.95 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [a__x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [mark](X) = X 27.80/29.95 [0] = 2 27.80/29.95 [and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [fSNonEmpty] = 0 27.80/29.95 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [s](X) = X + 2 27.80/29.95 [tt] = 2 27.80/29.95 [x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [A__AND](X1,X2) = 0 27.80/29.95 [A__PLUS](X1,X2) = 0 27.80/29.95 [A__X](X1,X2) = 2.X1 + 2.X2 + 2 27.80/29.95 [MARK](X) = 2.X 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 SCC Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Strongly Connected Components: 27.80/29.95 ->->Cycle: 27.80/29.95 ->->-> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 ->->-> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 Reduction Pairs Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(M:S) 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 -> Usable rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Interpretation type: 27.80/29.95 Simple mixed 27.80/29.95 ->Coefficients: 27.80/29.95 Natural Numbers 27.80/29.95 ->Dimension: 27.80/29.95 1 27.80/29.95 ->Bound: 27.80/29.95 2 27.80/29.95 ->Interpretation: 27.80/29.95 27.80/29.95 [a__and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 27.80/29.95 [mark](X) = X 27.80/29.95 [0] = 2 27.80/29.95 [and](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 27.80/29.95 [fSNonEmpty] = 0 27.80/29.95 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [s](X) = X + 2 27.80/29.95 [tt] = 0 27.80/29.95 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 27.80/29.95 [A__AND](X1,X2) = 0 27.80/29.95 [A__PLUS](X1,X2) = 0 27.80/29.95 [A__X](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [MARK](X) = 2.X + 2 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 SCC Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Strongly Connected Components: 27.80/29.95 ->->Cycle: 27.80/29.95 ->->-> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 ->->-> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 Reduction Pairs Processor: 27.80/29.95 -> Pairs: 27.80/29.95 A__X(N:S,s(M:S)) -> MARK(N:S) 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 -> Usable rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Interpretation type: 27.80/29.95 Simple mixed 27.80/29.95 ->Coefficients: 27.80/29.95 Natural Numbers 27.80/29.95 ->Dimension: 27.80/29.95 1 27.80/29.95 ->Bound: 27.80/29.95 2 27.80/29.95 ->Interpretation: 27.80/29.95 27.80/29.95 [a__and](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [a__plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [a__x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [mark](X) = X 27.80/29.95 [0] = 2 27.80/29.95 [and](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [fSNonEmpty] = 0 27.80/29.95 [plus](X1,X2) = X1 + 2.X2 + 2 27.80/29.95 [s](X) = X + 2 27.80/29.95 [tt] = 2 27.80/29.95 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 27.80/29.95 [A__AND](X1,X2) = 0 27.80/29.95 [A__PLUS](X1,X2) = 0 27.80/29.95 [A__X](X1,X2) = 2.X1.X2 + X2 + 1 27.80/29.95 [MARK](X) = 2.X + 2 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 SCC Processor: 27.80/29.95 -> Pairs: 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> A__X(mark(X1:S),mark(X2:S)) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Strongly Connected Components: 27.80/29.95 ->->Cycle: 27.80/29.95 ->->-> Pairs: 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 ->->-> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 Subterm Processor: 27.80/29.95 -> Pairs: 27.80/29.95 MARK(and(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 MARK(s(X:S)) -> MARK(X:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X1:S) 27.80/29.95 MARK(x(X1:S,X2:S)) -> MARK(X2:S) 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Projection: 27.80/29.95 pi(MARK) = 1 27.80/29.95 27.80/29.95 Problem 1: 27.80/29.95 27.80/29.95 SCC Processor: 27.80/29.95 -> Pairs: 27.80/29.95 Empty 27.80/29.95 -> Rules: 27.80/29.95 a__and(tt,X:S) -> mark(X:S) 27.80/29.95 a__and(X1:S,X2:S) -> and(X1:S,X2:S) 27.80/29.95 a__plus(N:S,0) -> mark(N:S) 27.80/29.95 a__plus(N:S,s(M:S)) -> s(a__plus(mark(N:S),mark(M:S))) 27.80/29.95 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 27.80/29.95 a__x(N:S,0) -> 0 27.80/29.95 a__x(N:S,s(M:S)) -> a__plus(a__x(mark(N:S),mark(M:S)),mark(N:S)) 27.80/29.95 a__x(X1:S,X2:S) -> x(X1:S,X2:S) 27.80/29.95 mark(0) -> 0 27.80/29.95 mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) 27.80/29.95 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 27.80/29.95 mark(s(X:S)) -> s(mark(X:S)) 27.80/29.95 mark(tt) -> tt 27.80/29.95 mark(x(X1:S,X2:S)) -> a__x(mark(X1:S),mark(X2:S)) 27.80/29.95 ->Strongly Connected Components: 27.80/29.95 There is no strongly connected component 27.80/29.95 27.80/29.95 The problem is finite. 27.80/29.95 EOF