/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: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} 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: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} 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: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: gt(x,y){x -> s(x),y -> s(y)} = gt(s(x),s(y)) ->^+ gt(x,y) = C[gt(x,y) = gt(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gt,if,if1,lt,minus,mod} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs gt#(0(),y) -> c_1() gt#(s(x),0()) -> c_2() gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(false(),x,y) -> c_4() if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) if1#(true(),x,y) -> c_7() lt#(x,0()) -> c_8() lt#(0(),s(x)) -> c_9() lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(0(),y) -> c_11() minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,0()) -> c_13() mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: gt#(0(),y) -> c_1() gt#(s(x),0()) -> c_2() gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(false(),x,y) -> c_4() if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) if1#(true(),x,y) -> c_7() lt#(x,0()) -> c_8() lt#(0(),s(x)) -> c_9() lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(0(),y) -> c_11() minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,0()) -> c_13() mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,7,8,9,11,13} by application of Pre({1,2,4,7,8,9,11,13}) = {3,5,6,10,12,14}. Here rules are labelled as follows: 1: gt#(0(),y) -> c_1() 2: gt#(s(x),0()) -> c_2() 3: gt#(s(x),s(y)) -> c_3(gt#(x,y)) 4: if#(false(),x,y) -> c_4() 5: if#(true(),x,y) -> c_5(minus#(x,y)) 6: if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) 7: if1#(true(),x,y) -> c_7() 8: lt#(x,0()) -> c_8() 9: lt#(0(),s(x)) -> c_9() 10: lt#(s(x),s(y)) -> c_10(lt#(x,y)) 11: minus#(0(),y) -> c_11() 12: minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) 13: mod#(x,0()) -> c_13() 14: mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak DPs: gt#(0(),y) -> c_1() gt#(s(x),0()) -> c_2() if#(false(),x,y) -> c_4() if1#(true(),x,y) -> c_7() lt#(x,0()) -> c_8() lt#(0(),s(x)) -> c_9() minus#(0(),y) -> c_11() mod#(x,0()) -> c_13() - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gt#(s(x),s(y)) -> c_3(gt#(x,y)) -->_1 gt#(s(x),0()) -> c_2():8 -->_1 gt#(0(),y) -> c_1():7 -->_1 gt#(s(x),s(y)) -> c_3(gt#(x,y)):1 2:S:if#(true(),x,y) -> c_5(minus#(x,y)) -->_1 minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)):5 -->_1 minus#(0(),y) -> c_11():13 3:S:if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) -->_1 mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):6 -->_2 minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)):5 -->_1 mod#(x,0()) -> c_13():14 -->_2 minus#(0(),y) -> c_11():13 4:S:lt#(s(x),s(y)) -> c_10(lt#(x,y)) -->_1 lt#(0(),s(x)) -> c_9():12 -->_1 lt#(x,0()) -> c_8():11 -->_1 lt#(s(x),s(y)) -> c_10(lt#(x,y)):4 5:S:minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) -->_1 if#(false(),x,y) -> c_4():9 -->_2 gt#(s(x),0()) -> c_2():8 -->_1 if#(true(),x,y) -> c_5(minus#(x,y)):2 -->_2 gt#(s(x),s(y)) -> c_3(gt#(x,y)):1 6:S:mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) -->_2 lt#(0(),s(x)) -> c_9():12 -->_1 if1#(true(),x,y) -> c_7():10 -->_2 lt#(s(x),s(y)) -> c_10(lt#(x,y)):4 -->_1 if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)):3 7:W:gt#(0(),y) -> c_1() 8:W:gt#(s(x),0()) -> c_2() 9:W:if#(false(),x,y) -> c_4() 10:W:if1#(true(),x,y) -> c_7() 11:W:lt#(x,0()) -> c_8() 12:W:lt#(0(),s(x)) -> c_9() 13:W:minus#(0(),y) -> c_11() 14:W:mod#(x,0()) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: mod#(x,0()) -> c_13() 11: lt#(x,0()) -> c_8() 10: if1#(true(),x,y) -> c_7() 12: lt#(0(),s(x)) -> c_9() 13: minus#(0(),y) -> c_11() 9: if#(false(),x,y) -> c_4() 7: gt#(0(),y) -> c_1() 8: gt#(s(x),0()) -> c_2() ** Step 1.b:4: UsableRules. MAYBE + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> mod(minus(x,y),y) if1(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if1(lt(x,s(y)),x,s(y)) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) ** Step 1.b:5: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} 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 if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) and a lower component gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) Further, following extension rules are added to the lower component. if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) *** Step 1.b:5.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)) -->_1 mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):2 2:S:mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) -->_1 if1#(false(),x,y) -> c_6(mod#(minus(x,y),y),minus#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: if1#(false(),x,y) -> c_6(mod#(minus(x,y),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y))) *** Step 1.b:5.a:2: WeightGap. MAYBE + Considered Problem: - Strict DPs: if1#(false(),x,y) -> c_6(mod#(minus(x,y),y)) mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y))) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(if) = {1}, uargs(s) = {1}, uargs(if1#) = {1}, uargs(mod#) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [0] p(lt) = [3] p(minus) = [0] p(mod) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(gt#) = [1] x2 + [1] p(if#) = [1] x1 + [2] x3 + [2] p(if1#) = [1] x1 + [1] p(lt#) = [2] x2 + [2] p(minus#) = [1] x1 + [0] p(mod#) = [1] x1 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [2] p(c_10) = [4] x1 + [4] p(c_11) = [2] p(c_12) = [2] x2 + [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] Following rules are strictly oriented: if1#(false(),x,y) = [1] > [0] = c_6(mod#(minus(x,y),y)) Following rules are (at-least) weakly oriented: mod#(x,s(y)) = [1] x + [0] >= [4] = c_14(if1#(lt(x,s(y)),x,s(y))) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [0] = 0() if(true(),x,y) = [0] >= [0] = s(minus(x,y)) lt(x,0()) = [3] >= [0] = false() lt(0(),s(x)) = [3] >= [0] = true() lt(s(x),s(y)) = [3] >= [3] = lt(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [0] >= [0] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.a:3: Failure MAYBE + Considered Problem: - Strict DPs: mod#(x,s(y)) -> c_14(if1#(lt(x,s(y)),x,s(y))) - Weak DPs: if1#(false(),x,y) -> c_6(mod#(minus(x,y),y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:5.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> c_5(minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak DPs: if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} 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 if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) and a lower component gt#(s(x),s(y)) -> c_3(gt#(x,y)) Further, following extension rules are added to the lower component. if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> lt#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) **** Step 1.b:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if#(true(),x,y) -> c_5(minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak DPs: if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if#(true(),x,y) -> c_5(minus#(x,y)) -->_1 minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)):3 2:S:lt#(s(x),s(y)) -> c_10(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_10(lt#(x,y)):2 3:S:minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)) -->_1 if#(true(),x,y) -> c_5(minus#(x,y)):1 4:W:if1#(false(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y),gt#(s(x),y)):3 5:W:if1#(false(),x,y) -> mod#(minus(x,y),y) -->_1 mod#(x,s(y)) -> lt#(x,s(y)):7 -->_1 mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)):6 6:W:mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) -->_1 if1#(false(),x,y) -> mod#(minus(x,y),y):5 -->_1 if1#(false(),x,y) -> minus#(x,y):4 7:W:mod#(x,s(y)) -> lt#(x,s(y)) -->_1 lt#(s(x),s(y)) -> c_10(lt#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y)) **** Step 1.b:5.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if#(true(),x,y) -> c_5(minus#(x,y)) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y)) - Weak DPs: if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} 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_5) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {if,lt,minus,gt#,if#,if1#,lt#,minus#,mod#} TcT has computed the following interpretation: p(0) = [4] p(false) = [2] p(gt) = [0] p(if) = [1] x2 + [4] p(if1) = [1] x1 + [2] x2 + [1] x3 + [0] p(lt) = [2] p(minus) = [1] x1 + [0] p(mod) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [4] p(true) = [0] p(gt#) = [1] x1 + [0] p(if#) = [0] p(if1#) = [2] x1 + [8] x2 + [5] x3 + [5] p(lt#) = [5] x1 + [2] x2 + [0] p(minus#) = [0] p(mod#) = [8] x1 + [5] x2 + [9] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [4] p(c_5) = [8] x1 + [0] p(c_6) = [1] x2 + [1] p(c_7) = [4] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [14] p(c_11) = [0] p(c_12) = [2] x1 + [0] p(c_13) = [0] p(c_14) = [1] x2 + [1] Following rules are strictly oriented: lt#(s(x),s(y)) = [5] x + [2] y + [28] > [5] x + [2] y + [14] = c_10(lt#(x,y)) Following rules are (at-least) weakly oriented: if#(true(),x,y) = [0] >= [0] = c_5(minus#(x,y)) if1#(false(),x,y) = [8] x + [5] y + [9] >= [0] = minus#(x,y) if1#(false(),x,y) = [8] x + [5] y + [9] >= [8] x + [5] y + [9] = mod#(minus(x,y),y) minus#(s(x),y) = [0] >= [0] = c_12(if#(gt(s(x),y),x,y)) mod#(x,s(y)) = [8] x + [5] y + [29] >= [8] x + [5] y + [29] = if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) = [8] x + [5] y + [29] >= [5] x + [2] y + [8] = lt#(x,s(y)) if(false(),x,y) = [1] x + [4] >= [4] = 0() if(true(),x,y) = [1] x + [4] >= [1] x + [4] = s(minus(x,y)) lt(x,0()) = [2] >= [2] = false() lt(0(),s(x)) = [2] >= [0] = true() lt(s(x),s(y)) = [2] >= [2] = lt(x,y) minus(0(),y) = [4] >= [4] = 0() minus(s(x),y) = [1] x + [4] >= [1] x + [4] = if(gt(s(x),y),x,y) **** Step 1.b:5.b:1.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if#(true(),x,y) -> c_5(minus#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y)) - Weak DPs: if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> c_10(lt#(x,y)) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(if) = {1}, uargs(s) = {1}, uargs(if#) = {1}, uargs(if1#) = {1}, uargs(mod#) = {1}, uargs(c_5) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [0] p(lt) = [1] x1 + [0] p(minus) = [0] p(mod) = [2] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] p(gt#) = [1] x1 + [0] p(if#) = [1] x1 + [4] p(if1#) = [1] x1 + [1] p(lt#) = [1] p(minus#) = [0] p(mod#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [4] p(c_13) = [0] p(c_14) = [4] Following rules are strictly oriented: if#(true(),x,y) = [4] > [0] = c_5(minus#(x,y)) Following rules are (at-least) weakly oriented: if1#(false(),x,y) = [1] >= [0] = minus#(x,y) if1#(false(),x,y) = [1] >= [1] = mod#(minus(x,y),y) lt#(s(x),s(y)) = [1] >= [1] = c_10(lt#(x,y)) minus#(s(x),y) = [0] >= [8] = c_12(if#(gt(s(x),y),x,y)) mod#(x,s(y)) = [1] x + [1] >= [1] x + [1] = if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) = [1] x + [1] >= [1] = lt#(x,s(y)) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [0] = 0() if(true(),x,y) = [0] >= [0] = s(minus(x,y)) lt(x,0()) = [1] x + [0] >= [0] = false() lt(0(),s(x)) = [0] >= [0] = true() lt(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = lt(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [0] >= [0] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:5.b:1.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y)) - Weak DPs: if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> c_10(lt#(x,y)) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(if) = {1}, uargs(s) = {1}, uargs(if#) = {1}, uargs(if1#) = {1}, uargs(mod#) = {1}, uargs(c_5) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(gt) = [0] p(if) = [1] x1 + [1] x2 + [1] p(if1) = [4] x1 + [1] x2 + [2] x3 + [0] p(lt) = [0] p(minus) = [1] x1 + [0] p(mod) = [0] p(s) = [1] x1 + [1] p(true) = [0] p(gt#) = [4] x2 + [0] p(if#) = [1] x1 + [1] x2 + [0] p(if1#) = [1] x1 + [1] x2 + [1] p(lt#) = [1] x1 + [1] p(minus#) = [1] x1 + [0] p(mod#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [2] p(c_4) = [4] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [4] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [2] p(c_14) = [0] Following rules are strictly oriented: minus#(s(x),y) = [1] x + [1] > [1] x + [0] = c_12(if#(gt(s(x),y),x,y)) Following rules are (at-least) weakly oriented: if#(true(),x,y) = [1] x + [0] >= [1] x + [0] = c_5(minus#(x,y)) if1#(false(),x,y) = [1] x + [1] >= [1] x + [0] = minus#(x,y) if1#(false(),x,y) = [1] x + [1] >= [1] x + [1] = mod#(minus(x,y),y) lt#(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = c_10(lt#(x,y)) mod#(x,s(y)) = [1] x + [1] >= [1] x + [1] = if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) = [1] x + [1] >= [1] x + [1] = lt#(x,s(y)) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [1] x + [1] >= [1] = 0() if(true(),x,y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) lt(x,0()) = [0] >= [0] = false() lt(0(),s(x)) = [0] >= [0] = true() lt(s(x),s(y)) = [0] >= [0] = lt(x,y) minus(0(),y) = [1] >= [1] = 0() minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:5.b:1.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if#(true(),x,y) -> c_5(minus#(x,y)) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> c_10(lt#(x,y)) minus#(s(x),y) -> c_12(if#(gt(s(x),y),x,y)) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) - Weak DPs: if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) lt#(s(x),s(y)) -> lt#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) mod#(x,s(y)) -> lt#(x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gt#(s(x),s(y)) -> c_3(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_3(gt#(x,y)):1 2:W:if#(true(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> if#(gt(s(x),y),x,y):7 -->_1 minus#(s(x),y) -> gt#(s(x),y):6 3:W:if1#(false(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> if#(gt(s(x),y),x,y):7 -->_1 minus#(s(x),y) -> gt#(s(x),y):6 4:W:if1#(false(),x,y) -> mod#(minus(x,y),y) -->_1 mod#(x,s(y)) -> lt#(x,s(y)):9 -->_1 mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)):8 5:W:lt#(s(x),s(y)) -> lt#(x,y) -->_1 lt#(s(x),s(y)) -> lt#(x,y):5 6:W:minus#(s(x),y) -> gt#(s(x),y) -->_1 gt#(s(x),s(y)) -> c_3(gt#(x,y)):1 7:W:minus#(s(x),y) -> if#(gt(s(x),y),x,y) -->_1 if#(true(),x,y) -> minus#(x,y):2 8:W:mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) -->_1 if1#(false(),x,y) -> mod#(minus(x,y),y):4 -->_1 if1#(false(),x,y) -> minus#(x,y):3 9:W:mod#(x,s(y)) -> lt#(x,s(y)) -->_1 lt#(s(x),s(y)) -> lt#(x,y):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: mod#(x,s(y)) -> lt#(x,s(y)) 5: lt#(s(x),s(y)) -> lt#(x,y) **** Step 1.b:5.b:1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) - Weak DPs: if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} 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_3) = {1} Following symbols are considered usable: {lt,gt#,if#,if1#,lt#,minus#,mod#} TcT has computed the following interpretation: p(0) = [0] p(false) = [1] p(gt) = [1] x2 + [0] p(if) = [8] x1 + [6] x2 + [6] x3 + [0] p(if1) = [4] x1 + [1] x3 + [0] p(lt) = [1] p(minus) = [2] x2 + [0] p(mod) = [1] x1 + [1] p(s) = [1] x1 + [2] p(true) = [0] p(gt#) = [8] x2 + [4] p(if#) = [8] x3 + [4] p(if1#) = [8] x1 + [8] x3 + [0] p(lt#) = [2] x1 + [1] x2 + [1] p(minus#) = [8] x2 + [4] p(mod#) = [8] x2 + [8] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] x1 + [8] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [2] x2 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] p(c_10) = [4] p(c_11) = [1] p(c_12) = [8] x2 + [1] p(c_13) = [2] p(c_14) = [2] x2 + [1] Following rules are strictly oriented: gt#(s(x),s(y)) = [8] y + [20] > [8] y + [12] = c_3(gt#(x,y)) Following rules are (at-least) weakly oriented: if#(true(),x,y) = [8] y + [4] >= [8] y + [4] = minus#(x,y) if1#(false(),x,y) = [8] y + [8] >= [8] y + [4] = minus#(x,y) if1#(false(),x,y) = [8] y + [8] >= [8] y + [8] = mod#(minus(x,y),y) minus#(s(x),y) = [8] y + [4] >= [8] y + [4] = gt#(s(x),y) minus#(s(x),y) = [8] y + [4] >= [8] y + [4] = if#(gt(s(x),y),x,y) mod#(x,s(y)) = [8] y + [24] >= [8] y + [24] = if1#(lt(x,s(y)),x,s(y)) lt(x,0()) = [1] >= [1] = false() lt(0(),s(x)) = [1] >= [0] = true() lt(s(x),s(y)) = [1] >= [1] = lt(x,y) **** Step 1.b:5.b:1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gt#(s(x),s(y)) -> c_3(gt#(x,y)) if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> minus#(x,y) if1#(false(),x,y) -> mod#(minus(x,y),y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) mod#(x,s(y)) -> if1#(lt(x,s(y)),x,s(y)) - Weak TRS: gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gt/2,if/3,if1/3,lt/2,minus/2,mod/2,gt#/2,if#/3,if1#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0 ,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gt#,if#,if1#,lt#,minus#,mod#} 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),?)