/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S M:S N:S X:S) (RULES U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) ) Problem 1: Innermost Equivalent Processor: -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: U11#(tt,M:S,N:S) -> U12#(tt,activate(M:S),activate(N:S)) U11#(tt,M:S,N:S) -> ACTIVATE(M:S) U11#(tt,M:S,N:S) -> ACTIVATE(N:S) U12#(tt,M:S,N:S) -> ACTIVATE(M:S) U12#(tt,M:S,N:S) -> ACTIVATE(N:S) U12#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U21#(tt,M:S,N:S) -> U22#(tt,activate(M:S),activate(N:S)) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U22#(tt,M:S,N:S) -> ACTIVATE(M:S) U22#(tt,M:S,N:S) -> ACTIVATE(N:S) U22#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U22#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) PLUS(N:S,s(M:S)) -> U11#(tt,M:S,N:S) X(N:S,s(M:S)) -> U21#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) Problem 1: SCC Processor: -> Pairs: U11#(tt,M:S,N:S) -> U12#(tt,activate(M:S),activate(N:S)) U11#(tt,M:S,N:S) -> ACTIVATE(M:S) U11#(tt,M:S,N:S) -> ACTIVATE(N:S) U12#(tt,M:S,N:S) -> ACTIVATE(M:S) U12#(tt,M:S,N:S) -> ACTIVATE(N:S) U12#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) U21#(tt,M:S,N:S) -> U22#(tt,activate(M:S),activate(N:S)) U21#(tt,M:S,N:S) -> ACTIVATE(M:S) U21#(tt,M:S,N:S) -> ACTIVATE(N:S) U22#(tt,M:S,N:S) -> ACTIVATE(M:S) U22#(tt,M:S,N:S) -> ACTIVATE(N:S) U22#(tt,M:S,N:S) -> PLUS(x(activate(N:S),activate(M:S)),activate(N:S)) U22#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) PLUS(N:S,s(M:S)) -> U11#(tt,M:S,N:S) X(N:S,s(M:S)) -> U21#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(tt,M:S,N:S) -> U12#(tt,activate(M:S),activate(N:S)) U12#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) PLUS(N:S,s(M:S)) -> U11#(tt,M:S,N:S) ->->-> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) ->->Cycle: ->->-> Pairs: U21#(tt,M:S,N:S) -> U22#(tt,activate(M:S),activate(N:S)) U22#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) X(N:S,s(M:S)) -> U21#(tt,M:S,N:S) ->->-> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) The problem is decomposed in 2 subproblems. Problem 1.1: Reduction Pairs Processor: -> Pairs: U11#(tt,M:S,N:S) -> U12#(tt,activate(M:S),activate(N:S)) U12#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) PLUS(N:S,s(M:S)) -> U11#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) -> Usable rules: activate(X:S) -> X:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 0 [U21](X1,X2,X3) = 0 [U22](X1,X2,X3) = 0 [activate](X) = X [plus](X1,X2) = 0 [x](X1,X2) = 0 [0] = 0 [fSNonEmpty] = 0 [s](X) = 2.X + 2 [tt] = 2 [U11#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U21#](X1,X2,X3) = 0 [U22#](X1,X2,X3) = 0 [ACTIVATE](X) = 0 [PLUS](X1,X2) = 2.X1 + 2.X2 + 2 [X](X1,X2) = 0 Problem 1.1: SCC Processor: -> Pairs: U12#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S)) PLUS(N:S,s(M:S)) -> U11#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pairs Processor: -> Pairs: U21#(tt,M:S,N:S) -> U22#(tt,activate(M:S),activate(N:S)) U22#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) X(N:S,s(M:S)) -> U21#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) -> Usable rules: activate(X:S) -> X:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 0 [U21](X1,X2,X3) = 0 [U22](X1,X2,X3) = 0 [activate](X) = X [plus](X1,X2) = 0 [x](X1,X2) = 0 [0] = 0 [fSNonEmpty] = 0 [s](X) = 2.X + 2 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U21#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U22#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [ACTIVATE](X) = 0 [PLUS](X1,X2) = 0 [X](X1,X2) = 2.X1 + 2.X2 + 2 Problem 1.2: SCC Processor: -> Pairs: U22#(tt,M:S,N:S) -> X(activate(N:S),activate(M:S)) X(N:S,s(M:S)) -> U21#(tt,M:S,N:S) -> Rules: U11(tt,M:S,N:S) -> U12(tt,activate(M:S),activate(N:S)) U12(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S))) U21(tt,M:S,N:S) -> U22(tt,activate(M:S),activate(N:S)) U22(tt,M:S,N:S) -> plus(x(activate(N:S),activate(M:S)),activate(N:S)) activate(X:S) -> X:S plus(N:S,0) -> N:S plus(N:S,s(M:S)) -> U11(tt,M:S,N:S) x(N:S,0) -> 0 x(N:S,s(M:S)) -> U21(tt,M:S,N:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.