/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 x:S y:S) (RULES if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ) Problem 1: Innermost Equivalent Processor: -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: IF_MOD(ttrue,s(x:S),s(y:S)) -> MINUS(x:S,y:S) IF_MOD(ttrue,s(x:S),s(y:S)) -> MOD(minus(x:S,y:S),s(y:S)) LE(s(x:S),s(y:S)) -> LE(x:S,y:S) MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) MINUS(x:S,s(y:S)) -> PRED(minus(x:S,y:S)) MOD(s(x:S),s(y:S)) -> IF_MOD(le(y:S,x:S),s(x:S),s(y:S)) MOD(s(x:S),s(y:S)) -> LE(y:S,x:S) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S Problem 1: SCC Processor: -> Pairs: IF_MOD(ttrue,s(x:S),s(y:S)) -> MINUS(x:S,y:S) IF_MOD(ttrue,s(x:S),s(y:S)) -> MOD(minus(x:S,y:S),s(y:S)) LE(s(x:S),s(y:S)) -> LE(x:S,y:S) MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) MINUS(x:S,s(y:S)) -> PRED(minus(x:S,y:S)) MOD(s(x:S),s(y:S)) -> IF_MOD(le(y:S,x:S),s(x:S),s(y:S)) MOD(s(x:S),s(y:S)) -> LE(y:S,x:S) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) ->->-> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->->Cycle: ->->-> Pairs: LE(s(x:S),s(y:S)) -> LE(x:S,y:S) ->->-> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->->Cycle: ->->-> Pairs: IF_MOD(ttrue,s(x:S),s(y:S)) -> MOD(minus(x:S,y:S),s(y:S)) MOD(s(x:S),s(y:S)) -> IF_MOD(le(y:S,x:S),s(x:S),s(y:S)) ->->-> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: MINUS(x:S,s(y:S)) -> MINUS(x:S,y:S) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Projection: pi(MINUS) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: LE(s(x:S),s(y:S)) -> LE(x:S,y:S) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Projection: pi(LE) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pairs Processor: -> Pairs: IF_MOD(ttrue,s(x:S),s(y:S)) -> MOD(minus(x:S,y:S),s(y:S)) MOD(s(x:S),s(y:S)) -> IF_MOD(le(y:S,x:S),s(x:S),s(y:S)) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S -> Usable rules: le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) pred(s(x:S)) -> x:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [if_mod](X1,X2,X3) = 0 [le](X1,X2) = 1 [minus](X1,X2) = X1 [mod](X1,X2) = 0 [pred](X) = X [0] = 0 [fSNonEmpty] = 0 [false] = 0 [s](X) = X + 2 [true] = 1 [IF_MOD](X1,X2,X3) = 2.X1 + 2.X2 + X3 [LE](X1,X2) = 0 [MINUS](X1,X2) = 0 [MOD](X1,X2) = 2.X1 + X2 + 2 [PRED](X) = 0 Problem 1.3: SCC Processor: -> Pairs: MOD(s(x:S),s(y:S)) -> IF_MOD(le(y:S,x:S),s(x:S),s(y:S)) -> Rules: if_mod(ffalse,s(x:S),s(y:S)) -> s(x:S) if_mod(ttrue,s(x:S),s(y:S)) -> mod(minus(x:S,y:S),s(y:S)) le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) minus(x:S,0) -> x:S minus(x:S,s(y:S)) -> pred(minus(x:S,y:S)) mod(0,y:S) -> 0 mod(s(x:S),0) -> 0 mod(s(x:S),s(y:S)) -> if_mod(le(y:S,x:S),s(x:S),s(y:S)) pred(s(x:S)) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite.