/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),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff,if,leq,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: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff,if,leq,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: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: leq(x,y){x -> s(x),y -> s(y)} = leq(s(x),s(y)) ->^+ leq(x,y) = C[leq(x,y) = leq(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff,if,leq,p} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) if#(false(),X,Y) -> c_2() if#(true(),X,Y) -> c_3() leq#(0(),Y) -> c_4() leq#(s(X),0()) -> c_5() leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) p#(0()) -> c_7() p#(s(X)) -> c_8() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) if#(false(),X,Y) -> c_2() if#(true(),X,Y) -> c_3() leq#(0(),Y) -> c_4() leq#(s(X),0()) -> c_5() leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) p#(0()) -> c_7() p#(s(X)) -> c_8() - Weak TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/4,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,p#} and constructors {0,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,7,8} by application of Pre({2,3,4,5,7,8}) = {1,6}. Here rules are labelled as follows: 1: diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) 2: if#(false(),X,Y) -> c_2() 3: if#(true(),X,Y) -> c_3() 4: leq#(0(),Y) -> c_4() 5: leq#(s(X),0()) -> c_5() 6: leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) 7: p#(0()) -> c_7() 8: p#(s(X)) -> c_8() ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak DPs: if#(false(),X,Y) -> c_2() if#(true(),X,Y) -> c_3() leq#(0(),Y) -> c_4() leq#(s(X),0()) -> c_5() p#(0()) -> c_7() p#(s(X)) -> c_8() - Weak TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/4,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,p#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) -->_2 leq#(s(X),s(Y)) -> c_6(leq#(X,Y)):2 -->_4 p#(s(X)) -> c_8():8 -->_4 p#(0()) -> c_7():7 -->_2 leq#(s(X),0()) -> c_5():6 -->_2 leq#(0(),Y) -> c_4():5 -->_1 if#(true(),X,Y) -> c_3():4 -->_1 if#(false(),X,Y) -> c_2():3 -->_3 diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)):1 2:S:leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) -->_1 leq#(s(X),0()) -> c_5():6 -->_1 leq#(0(),Y) -> c_4():5 -->_1 leq#(s(X),s(Y)) -> c_6(leq#(X,Y)):2 3:W:if#(false(),X,Y) -> c_2() 4:W:if#(true(),X,Y) -> c_3() 5:W:leq#(0(),Y) -> c_4() 6:W:leq#(s(X),0()) -> c_5() 7:W:p#(0()) -> c_7() 8:W:p#(s(X)) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: if#(false(),X,Y) -> c_2() 4: if#(true(),X,Y) -> c_3() 7: p#(0()) -> c_7() 8: p#(s(X)) -> c_8() 5: leq#(0(),Y) -> c_4() 6: leq#(s(X),0()) -> c_5() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/4,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,p#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)) -->_2 leq#(s(X),s(Y)) -> c_6(leq#(X,Y)):2 -->_3 diff#(X,Y) -> c_1(if#(leq(X,Y),0(),s(diff(p(X),Y))),leq#(X,Y),diff#(p(X),Y),p#(X)):1 2:S:leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) -->_1 leq#(s(X),s(Y)) -> c_6(leq#(X,Y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak TRS: diff(X,Y) -> if(leq(X,Y),0(),s(diff(p(X),Y))) if(false(),X,Y) -> Y if(true(),X,Y) -> X leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,p#} and constructors {0,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(0()) -> 0() p(s(X)) -> X diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak TRS: p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,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 diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) and a lower component leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) Further, following extension rules are added to the lower component. diff#(X,Y) -> diff#(p(X),Y) diff#(X,Y) -> leq#(X,Y) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) - Weak TRS: p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,p#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)) -->_2 diff#(X,Y) -> c_1(leq#(X,Y),diff#(p(X),Y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: diff#(X,Y) -> c_1(diff#(p(X),Y)) *** Step 1.b:6.a:2: Failure MAYBE + Considered Problem: - Strict DPs: diff#(X,Y) -> c_1(diff#(p(X),Y)) - Weak TRS: p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,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: leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak DPs: diff#(X,Y) -> diff#(p(X),Y) diff#(X,Y) -> leq#(X,Y) - Weak TRS: p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,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_6) = {1} Following symbols are considered usable: {p,diff#,if#,leq#,p#} TcT has computed the following interpretation: p(0) = [7] p(diff) = [1] p(false) = [0] p(if) = [4] x1 + [2] x2 + [1] x3 + [1] p(leq) = [1] x1 + [1] x2 + [1] p(p) = [1] x1 + [0] p(s) = [1] x1 + [8] p(true) = [0] p(diff#) = [1] x1 + [8] x2 + [4] p(if#) = [1] x1 + [1] x3 + [2] p(leq#) = [1] x1 + [2] p(p#) = [1] x1 + [0] p(c_1) = [8] x2 + [2] p(c_2) = [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [8] p(c_6) = [1] x1 + [4] p(c_7) = [2] p(c_8) = [0] Following rules are strictly oriented: leq#(s(X),s(Y)) = [1] X + [10] > [1] X + [6] = c_6(leq#(X,Y)) Following rules are (at-least) weakly oriented: diff#(X,Y) = [1] X + [8] Y + [4] >= [1] X + [8] Y + [4] = diff#(p(X),Y) diff#(X,Y) = [1] X + [8] Y + [4] >= [1] X + [2] = leq#(X,Y) p(0()) = [7] >= [7] = 0() p(s(X)) = [1] X + [8] >= [1] X + [0] = X *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: diff#(X,Y) -> diff#(p(X),Y) diff#(X,Y) -> leq#(X,Y) leq#(s(X),s(Y)) -> c_6(leq#(X,Y)) - Weak TRS: p(0()) -> 0() p(s(X)) -> X - Signature: {diff/2,if/3,leq/2,p/1,diff#/2,if#/3,leq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/1,c_7/0,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {diff#,if#,leq#,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),?)