/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "div") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "false") :: [] -(0)-> "A"(0) F (TrsFun "geq") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "if") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) F (TrsFun "true") :: [] -(0)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) main(x1,x2) -> minus(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: geq(x,y){x -> s(x),y -> s(y)} = geq(s(x),s(y)) ->^+ geq(x,y) = C[geq(x,y) = geq(x,y){}] ** Step 1.b:1: Ara. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 1, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(4) F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "div") :: ["A"(4) x "A"(0)] -(1)-> "A"(0) F (TrsFun "false") :: [] -(0)-> "A"(0) F (TrsFun "geq") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "if") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(4) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(4)] -(4)-> "A"(4) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) F (TrsFun "true") :: [] -(0)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) 2. Weak: geq(0(),s(Y)) -> false() ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: geq(0(),s(Y)) -> false() - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = [0] p(div) = [12] x1 + [0] p(false) = [0] p(geq) = [2] p(if) = [4] x1 + [8] x2 + [8] x3 + [0] p(minus) = [0] p(s) = [1] x1 + [2] p(true) = [2] Following rules are strictly oriented: geq(0(),s(Y)) = [2] > [0] = false() Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [0] >= [0] = 0() div(s(X),s(Y)) = [12] X + [24] >= [24] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [2] >= [2] = true() geq(s(X),s(Y)) = [2] >= [2] = geq(X,Y) if(false(),X,Y) = [8] X + [8] Y + [0] >= [1] Y + [0] = Y if(true(),X,Y) = [8] X + [8] Y + [8] >= [1] X + [0] = X minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [0] >= [0] = minus(X,Y) ** Step 1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))