/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 u:S x:S y:S z:S) (RULES check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ) Problem 1: Innermost Equivalent Processor: -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: CHECK(s(s(s(x:S)))) -> CHECK(s(x:S)) HALF(s(s(x:S))) -> HALF(x:S) IF(even,x:S,y:S,z:S,u:S) -> HALF(s(z:S)) IF(even,x:S,y:S,z:S,u:S) -> HALF(x:S) IF(even,x:S,y:S,z:S,u:S) -> HALF(z:S) IF(even,x:S,y:S,z:S,u:S) -> PLUS(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(s(z:S))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> P(x:S) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) TIMES(x:S,y:S) -> TIMESITER(x:S,y:S,0) TIMESITER(x:S,y:S,z:S) -> CHECK(x:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) TIMESITER(x:S,y:S,z:S) -> PLUS(z:S,y:S) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) Problem 1: SCC Processor: -> Pairs: CHECK(s(s(s(x:S)))) -> CHECK(s(x:S)) HALF(s(s(x:S))) -> HALF(x:S) IF(even,x:S,y:S,z:S,u:S) -> HALF(s(z:S)) IF(even,x:S,y:S,z:S,u:S) -> HALF(x:S) IF(even,x:S,y:S,z:S,u:S) -> HALF(z:S) IF(even,x:S,y:S,z:S,u:S) -> PLUS(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(s(z:S))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> P(x:S) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) TIMES(x:S,y:S) -> TIMESITER(x:S,y:S,0) TIMESITER(x:S,y:S,z:S) -> CHECK(x:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) TIMESITER(x:S,y:S,z:S) -> PLUS(z:S,y:S) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->Cycle: ->->-> Pairs: HALF(s(s(x:S))) -> HALF(x:S) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->Cycle: ->->-> Pairs: CHECK(s(s(s(x:S)))) -> CHECK(s(x:S)) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->Cycle: ->->-> Pairs: IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(s(z:S))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) The problem is decomposed in 4 subproblems. Problem 1.1: Subterm Processor: -> Pairs: PLUS(s(x:S),y:S) -> PLUS(x:S,y:S) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Projection: pi(PLUS) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: HALF(s(s(x:S))) -> HALF(x:S) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Projection: pi(HALF) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: CHECK(s(s(s(x:S)))) -> CHECK(s(x:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Projection: pi(CHECK) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Reduction Pairs Processor: -> Pairs: IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(s(z:S))) IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Usable rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [check](X) = X + 1 [half](X) = 1/2.X [if](X1,X2,X3,X4,X5) = 0 [p](X) = 1/2.X [plus](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 1 [times](X1,X2) = 0 [timesIter](X1,X2,X3) = 0 [0] = 0 [even] = 2 [fSNonEmpty] = 0 [odd] = 1 [s](X) = 2.X + 2 [zero] = 1/2 [CHECK](X) = 0 [HALF](X) = 0 [IF](X1,X2,X3,X4,X5) = X1.X3 + X2.X3 + X1 + X2 + 1 [P](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 0 [TIMESITER](X1,X2,X3) = 2.X1.X2 + 2.X1 + X2 + 2 Problem 1.4: SCC Processor: -> Pairs: IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) Problem 1.4: Reduction Pairs Processor: -> Pairs: IF(even,x:S,y:S,z:S,u:S) -> TIMESITER(half(x:S),y:S,half(z:S)) IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Usable rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [check](X) = 1/2.X [half](X) = 1/2.X + 1/2 [if](X1,X2,X3,X4,X5) = 0 [p](X) = 1/2.X + 1/2 [plus](X1,X2) = 2.X1.X2 + 2.X1 + 2 [times](X1,X2) = 0 [timesIter](X1,X2,X3) = 0 [0] = 1 [even] = 2 [fSNonEmpty] = 0 [odd] = 2 [s](X) = 2.X + 2 [zero] = 1/2 [CHECK](X) = 0 [HALF](X) = 0 [IF](X1,X2,X3,X4,X5) = X1.X3 + X2.X3 + X1 + X2 + 2.X3 [P](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 0 [TIMESITER](X1,X2,X3) = 2.X1.X2 + 2.X1 + 2.X2 Problem 1.4: SCC Processor: -> Pairs: IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->->-> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) Problem 1.4: Reduction Pairs Processor: -> Pairs: IF(odd,x:S,y:S,z:S,u:S) -> TIMESITER(p(x:S),y:S,u:S) TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Usable rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [check](X) = 2.X [half](X) = 0 [if](X1,X2,X3,X4,X5) = 0 [p](X) = 1/2.X [plus](X1,X2) = 1/2.X1.X2 + X1 + X2 [times](X1,X2) = 0 [timesIter](X1,X2,X3) = 0 [0] = 0 [even] = 1 [fSNonEmpty] = 0 [odd] = 2 [s](X) = 2.X + 2 [zero] = 0 [CHECK](X) = 0 [HALF](X) = 0 [IF](X1,X2,X3,X4,X5) = 1/2.X2.X3 + 1/2.X1 + X2 [P](X) = 0 [PLUS](X1,X2) = 0 [TIMES](X1,X2) = 0 [TIMESITER](X1,X2,X3) = 1/2.X1.X2 + 2.X1 + 1/2 Problem 1.4: SCC Processor: -> Pairs: TIMESITER(x:S,y:S,z:S) -> IF(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) -> Rules: check(0) -> zero check(s(0)) -> odd check(s(s(0))) -> even check(s(s(s(x:S)))) -> check(s(x:S)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x:S))) -> s(half(x:S)) if(even,x:S,y:S,z:S,u:S) -> plus(timesIter(half(x:S),y:S,half(z:S)),timesIter(half(x:S),y:S,half(s(z:S)))) if(odd,x:S,y:S,z:S,u:S) -> timesIter(p(x:S),y:S,u:S) if(zero,x:S,y:S,z:S,u:S) -> z:S p(0) -> 0 p(s(x:S)) -> x:S plus(0,y:S) -> y:S plus(s(x:S),y:S) -> s(plus(x:S,y:S)) times(x:S,y:S) -> timesIter(x:S,y:S,0) timesIter(x:S,y:S,z:S) -> if(check(x:S),x:S,y:S,z:S,plus(z:S,y:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.