/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S) (RULES 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U21#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S))) ACTIVATE(n__0) -> 0# ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) ACTIVATE(n__s(X:S)) -> S(X:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U11#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: SCC Processor: -> Pairs: U11#(tt,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U21#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S))) ACTIVATE(n__0) -> 0# ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) ACTIVATE(n__s(X:S)) -> S(X:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U11#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U11#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: U11#(tt,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U11#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 0 [U11](X1,X2) = 2.X2 + 2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = 2.X2 + 2 [isNat](X) = 2.X + 1 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [n__0] = 0 [n__isNat](X) = 2.X + 1 [n__plus](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X [tt] = 0 [U11#](X1,X2) = 2.X2 + 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 + 1 [ISNAT](X) = 2.X [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> U11#(isNat(N:S),N:S) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 1 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = 2.X2 + 2 [isNat](X) = 2.X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [n__0] = 2 [n__isNat](X) = 2.X [n__plus](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X + 1 [tt] = 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X + 1 [AND](X1,X2) = X1 + X2 + 1 [ISNAT](X) = X [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X1 + X2 + 1 [isNat](X) = 2.X + 1 [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [n__0] = 2 [n__isNat](X) = 2.X + 1 [n__plus](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X + 1 [tt] = 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 + 2 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: U21#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 1 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 2 [isNat](X) = 2.X + 2 [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [n__0] = 2 [n__isNat](X) = 2.X + 2 [n__plus](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X + 1 [tt] = 2 [U21#](X1,X2,X3) = 2.X2 + 2.X3 + 2 [ACTIVATE](X) = X [AND](X1,X2) = X2 [ISNAT](X) = 2.X + 1 [PLUS](X1,X2) = 2.X1 + 2.X2 + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> U21#(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__isNat(X:S)) -> ISNAT(X:S) ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 1 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = 2.X2 [isNat](X) = 2.X + 2 [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [n__0] = 2 [n__isNat](X) = 2.X + 2 [n__plus](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X + 1 [tt] = 0 [ACTIVATE](X) = X + 1 [AND](X1,X2) = X2 + 2 [ISNAT](X) = 2.X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(X1:S,X2:S) AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = X2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = 2.X1 + X2 + 2 [isNat](X) = 2.X + 1 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [n__0] = 2 [n__isNat](X) = 2.X + 1 [n__plus](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X [tt] = 2 [ACTIVATE](X) = X + 1 [AND](X1,X2) = X1 + X2 [ISNAT](X) = X + 2 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1: SCC Processor: -> Pairs: AND(tt,X:S) -> ACTIVATE(X:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S) ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S) ISNAT(n__plus(V1:S,V2:S)) -> AND(isNat(activate(V1:S)),n__isNat(activate(V2:S))) ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) PLUS(N:S,0) -> ISNAT(N:S) PLUS(N:S,s(M:S)) -> AND(isNat(M:S),n__isNat(N:S)) PLUS(N:S,s(M:S)) -> ISNAT(M:S) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__plus(V1:S,V2:S)) -> ISNAT(activate(V1:S)) ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X1 + X2 [U21](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 [isNat](X) = 2 [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [n__0] = 2 [n__isNat](X) = 2 [n__plus](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X + 2 [tt] = 1 [ISNAT](X) = 2.X Problem 1: SCC Processor: -> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) ->->-> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ISNAT(n__s(V1:S)) -> ISNAT(activate(V1:S)) -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) -> Usable rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0] = 2 [U11](X1,X2) = 2.X2 + 2 [U21](X1,X2,X3) = 2.X2 + 2.X3 + 2 [activate](X) = X [and](X1,X2) = X2 + 2 [isNat](X) = 2.X + 2 [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [n__0] = 2 [n__isNat](X) = 2.X + 2 [n__plus](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X + 1 [tt] = 0 [ISNAT](X) = 2.X Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 0 -> n__0 U11(tt,N:S) -> activate(N:S) U21(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) activate(n__0) -> 0 activate(n__isNat(X:S)) -> isNat(X:S) activate(n__plus(X1:S,X2:S)) -> plus(X1:S,X2:S) activate(n__s(X:S)) -> s(X:S) activate(X:S) -> X:S and(tt,X:S) -> activate(X:S) isNat(n__0) -> tt isNat(n__plus(V1:S,V2:S)) -> and(isNat(activate(V1:S)),n__isNat(activate(V2:S))) isNat(n__s(V1:S)) -> isNat(activate(V1:S)) isNat(X:S) -> n__isNat(X:S) plus(N:S,0) -> U11(isNat(N:S),N:S) plus(N:S,s(M:S)) -> U21(and(isNat(M:S),n__isNat(N:S)),M:S,N:S) plus(X1:S,X2:S) -> n__plus(X1:S,X2:S) s(X:S) -> n__s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.