/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,gr,p} 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: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,gr,p} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,gr,p} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: gr(x,y){x -> s(x),y -> s(y)} = gr(s(x),s(y)) ->^+ gr(x,y) = C[gr(x,y) = gr(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,gr,p} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) gr#(0(),x) -> c_2() gr#(s(x),0()) -> c_3() gr#(s(x),s(y)) -> c_4(gr#(x,y)) p#(0()) -> c_5() p#(s(x)) -> c_6() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) gr#(0(),x) -> c_2() gr#(s(x),0()) -> c_3() gr#(s(x),s(y)) -> c_4(gr#(x,y)) p#(0()) -> c_5() p#(s(x)) -> c_6() - Weak TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,5,6} by application of Pre({2,3,5,6}) = {1,4}. Here rules are labelled as follows: 1: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) 2: gr#(0(),x) -> c_2() 3: gr#(s(x),0()) -> c_3() 4: gr#(s(x),s(y)) -> c_4(gr#(x,y)) 5: p#(0()) -> c_5() 6: p#(s(x)) -> c_6() ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak DPs: gr#(0(),x) -> c_2() gr#(s(x),0()) -> c_3() p#(0()) -> c_5() p#(s(x)) -> c_6() - Weak TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) -->_2 gr#(s(x),s(y)) -> c_4(gr#(x,y)):2 -->_3 p#(s(x)) -> c_6():6 -->_3 p#(0()) -> c_5():5 -->_2 gr#(s(x),0()) -> c_3():4 -->_2 gr#(0(),x) -> c_2():3 -->_1 cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)):1 2:S:gr#(s(x),s(y)) -> c_4(gr#(x,y)) -->_1 gr#(s(x),0()) -> c_3():4 -->_1 gr#(0(),x) -> c_2():3 -->_1 gr#(s(x),s(y)) -> c_4(gr#(x,y)):2 3:W:gr#(0(),x) -> c_2() 4:W:gr#(s(x),0()) -> c_3() 5:W:p#(0()) -> c_5() 6:W:p#(s(x)) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: p#(0()) -> c_5() 6: p#(s(x)) -> c_6() 3: gr#(0(),x) -> c_2() 4: gr#(s(x),0()) -> c_3() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/3,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)) -->_2 gr#(s(x),s(y)) -> c_4(gr#(x,y)):2 -->_1 cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y),p#(x)):1 2:S:gr#(s(x),s(y)) -> c_4(gr#(x,y)) -->_1 gr#(s(x),s(y)) -> c_4(gr#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak TRS: cond(true(),x,y) -> cond(gr(x,y),p(x),s(y)) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) gr#(s(x),s(y)) -> c_4(gr#(x,y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) and a lower component gr#(s(x),s(y)) -> c_4(gr#(x,y)) Further, following extension rules are added to the lower component. cond#(true(),x,y) -> cond#(gr(x,y),p(x),s(y)) cond#(true(),x,y) -> gr#(x,y) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)) -->_1 cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y)),gr#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y))) *** Step 1.b:6.a:2: Failure MAYBE + Considered Problem: - Strict DPs: cond#(true(),x,y) -> c_1(cond#(gr(x,y),p(x),s(y))) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak DPs: cond#(true(),x,y) -> cond#(gr(x,y),p(x),s(y)) cond#(true(),x,y) -> gr#(x,y) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} 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(c_4) = {1} Following symbols are considered usable: {gr,p,cond#,gr#,p#} TcT has computed the following interpretation: p(0) = [0] p(cond) = [2] x1 + [1] x3 + [0] p(false) = [2] p(gr) = [2] p(p) = [1] x1 + [0] p(s) = [1] x1 + [3] p(true) = [2] p(cond#) = [8] x1 + [4] x2 + [2] p(gr#) = [4] x1 + [0] p(p#) = [1] x1 + [2] p(c_1) = [1] x2 + [4] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] Following rules are strictly oriented: gr#(s(x),s(y)) = [4] x + [12] > [4] x + [0] = c_4(gr#(x,y)) Following rules are (at-least) weakly oriented: cond#(true(),x,y) = [4] x + [18] >= [4] x + [18] = cond#(gr(x,y),p(x),s(y)) cond#(true(),x,y) = [4] x + [18] >= [4] x + [0] = gr#(x,y) gr(0(),x) = [2] >= [2] = false() gr(s(x),0()) = [2] >= [2] = true() gr(s(x),s(y)) = [2] >= [2] = gr(x,y) p(0()) = [0] >= [0] = 0() p(s(x)) = [1] x + [3] >= [1] x + [0] = x *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond#(true(),x,y) -> cond#(gr(x,y),p(x),s(y)) cond#(true(),x,y) -> gr#(x,y) gr#(s(x),s(y)) -> c_4(gr#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond/3,gr/2,p/1,cond#/3,gr#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,gr#,p#} 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),?)