/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^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} 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: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} 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: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} and constructors {0,false,s ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus,pred} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs gcd#(0(),y) -> c_1() gcd#(s(x),0()) -> c_2() gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,0()) -> c_9() minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_11() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(0(),y) -> c_1() gcd#(s(x),0()) -> c_2() gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,0()) -> c_9() minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_11() - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,6,7,9,11} by application of Pre({1,2,6,7,9,11}) = {3,4,5,8,10}. Here rules are labelled as follows: 1: gcd#(0(),y) -> c_1() 2: gcd#(s(x),0()) -> c_2() 3: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) 4: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) 5: if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) 6: le#(0(),y) -> c_6() 7: le#(s(x),0()) -> c_7() 8: le#(s(x),s(y)) -> c_8(le#(x,y)) 9: minus#(x,0()) -> c_9() 10: minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) 11: pred#(s(x)) -> c_11() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) - Weak DPs: gcd#(0(),y) -> c_1() gcd#(s(x),0()) -> c_2() le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() minus#(x,0()) -> c_9() pred#(s(x)) -> c_11() - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) -->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):4 -->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3 -->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2 -->_2 le#(s(x),0()) -> c_7():9 -->_2 le#(0(),y) -> c_6():8 2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 -->_2 minus#(x,0()) -> c_9():10 -->_1 gcd#(0(),y) -> c_1():6 -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 -->_2 minus#(x,0()) -> c_9():10 -->_1 gcd#(0(),y) -> c_1():6 -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 4:S:le#(s(x),s(y)) -> c_8(le#(x,y)) -->_1 le#(s(x),0()) -> c_7():9 -->_1 le#(0(),y) -> c_6():8 -->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):4 5:S:minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) -->_1 pred#(s(x)) -> c_11():11 -->_2 minus#(x,0()) -> c_9():10 -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 6:W:gcd#(0(),y) -> c_1() 7:W:gcd#(s(x),0()) -> c_2() 8:W:le#(0(),y) -> c_6() 9:W:le#(s(x),0()) -> c_7() 10:W:minus#(x,0()) -> c_9() 11:W:pred#(s(x)) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: gcd#(s(x),0()) -> c_2() 6: gcd#(0(),y) -> c_1() 10: minus#(x,0()) -> c_9() 11: pred#(s(x)) -> c_11() 8: le#(0(),y) -> c_6() 9: le#(s(x),0()) -> c_7() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) -->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):4 -->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3 -->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2 2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 4:S:le#(s(x),s(y)) -> c_8(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):4 5:S:minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_10(pred#(minus(x,y)),minus#(x,y)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(x,s(y)) -> c_10(minus#(x,y)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) and a lower component le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) Further, following extension rules are added to the lower component. gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) -> minus#(y,x) if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) -> minus#(x,y) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)) -->_1 if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)):3 -->_1 if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)):2 2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 3:S:if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y)),minus#(x,y)) -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) *** Step 1.b:6.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {le,minus,pred,gcd#,if_gcd#,le#,minus#,pred#} TcT has computed the following interpretation: p(0) = [0] p(false) = [1] p(gcd) = [0] p(if_gcd) = [1] x1 + [2] x3 + [2] p(le) = [1] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [1] p(gcd#) = [2] x1 + [2] x2 + [12] p(if_gcd#) = [8] x1 + [2] x2 + [2] x3 + [2] p(le#) = [1] x1 + [1] x2 + [0] p(minus#) = [1] x1 + [1] p(pred#) = [2] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [4] p(c_8) = [1] x1 + [1] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] Following rules are strictly oriented: gcd#(s(x),s(y)) = [2] x + [2] y + [16] > [2] x + [2] y + [15] = c_3(if_gcd#(le(y,x),s(x),s(y))) Following rules are (at-least) weakly oriented: if_gcd#(false(),s(x),s(y)) = [2] x + [2] y + [14] >= [2] x + [2] y + [14] = c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) = [2] x + [2] y + [14] >= [2] x + [2] y + [14] = c_5(gcd#(minus(x,y),s(y))) le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x *** Step 1.b:6.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) - Weak DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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(pred) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(gcd) = [1] x1 + [1] p(if_gcd) = [1] x1 + [2] x3 + [1] p(le) = [0] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [5] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [5] p(le#) = [2] x1 + [4] x2 + [2] p(minus#) = [1] x1 + [0] p(pred#) = [0] p(c_1) = [4] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [4] p(c_10) = [4] x1 + [0] p(c_11) = [0] Following rules are strictly oriented: if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [7] > [1] x + [1] y + [6] = c_4(gcd#(minus(y,x),s(x))) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [7] >= [1] x + [1] y + [7] = c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [7] >= [1] x + [1] y + [10] = c_5(gcd#(minus(x,y),s(y))) le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) - Weak DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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(pred) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(gcd) = [1] x1 + [1] x2 + [0] p(if_gcd) = [2] x2 + [0] p(le) = [1] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [1] p(gcd#) = [1] x1 + [1] x2 + [7] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [6] p(le#) = [1] x1 + [2] x2 + [0] p(minus#) = [4] p(pred#) = [4] x1 + [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [1] p(c_9) = [0] p(c_10) = [0] p(c_11) = [4] Following rules are strictly oriented: if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [9] > [1] x + [1] y + [8] = c_5(gcd#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [9] = c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [8] = c_4(gcd#(minus(y,x),s(x))) le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [0] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} and constructors {0,false ,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) - Weak DPs: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) -> minus#(y,x) if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) -> minus#(x,y) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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(pred) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [1] p(gcd) = [0] p(if_gcd) = [2] x2 + [0] p(le) = [1] p(minus) = [1] x1 + [1] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [1] p(gcd#) = [1] x1 + [1] x2 + [4] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [0] p(le#) = [0] p(minus#) = [1] x2 + [0] p(pred#) = [1] x1 + [1] p(c_1) = [4] p(c_2) = [4] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] x1 + [2] p(c_9) = [4] p(c_10) = [1] x1 + [0] p(c_11) = [1] Following rules are strictly oriented: minus#(x,s(y)) = [1] y + [4] > [1] y + [0] = c_10(minus#(x,y)) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [12] >= [1] x + [1] y + [9] = if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) = [1] x + [1] y + [12] >= [0] = le#(y,x) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [9] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [0] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [9] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [9] >= [1] y + [0] = minus#(x,y) le#(s(x),s(y)) = [0] >= [2] = c_8(le#(x,y)) le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [1] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [1] >= [1] x + [1] = pred(minus(x,y)) pred(s(x)) = [1] x + [4] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_8(le#(x,y)) - Weak DPs: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) -> minus#(y,x) if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) -> minus#(x,y) minus#(x,s(y)) -> c_10(minus#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {minus,pred,gcd#,if_gcd#,le#,minus#,pred#} TcT has computed the following interpretation: p(0) = [3] p(false) = [0] p(gcd) = [1] x1 + [8] x2 + [0] p(if_gcd) = [1] x1 + [8] p(le) = [1] x1 + [6] x2 + [5] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [1] p(gcd#) = [2] x1 + [2] x2 + [9] p(if_gcd#) = [2] x2 + [2] x3 + [9] p(le#) = [1] x1 + [0] p(minus#) = [0] p(pred#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [4] x1 + [8] x2 + [4] p(c_4) = [2] x1 + [1] x2 + [0] p(c_5) = [1] x2 + [1] p(c_6) = [0] p(c_7) = [8] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [2] x1 + [0] p(c_11) = [0] Following rules are strictly oriented: le#(s(x),s(y)) = [1] x + [1] > [1] x + [0] = c_8(le#(x,y)) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [2] x + [2] y + [13] >= [2] x + [2] y + [13] = if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) = [2] x + [2] y + [13] >= [1] y + [0] = le#(y,x) if_gcd#(false(),s(x),s(y)) = [2] x + [2] y + [13] >= [2] x + [2] y + [11] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [2] x + [2] y + [13] >= [0] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [2] x + [2] y + [13] >= [2] x + [2] y + [11] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [2] x + [2] y + [13] >= [0] = minus#(x,y) minus#(x,s(y)) = [0] >= [0] = c_10(minus#(x,y)) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) -> le#(y,x) if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) -> minus#(y,x) if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) -> minus#(x,y) le#(s(x),s(y)) -> c_8(le#(x,y)) minus#(x,s(y)) -> c_10(minus#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x - Signature: {gcd/2,if_gcd/3,le/2,minus/2,pred/1,gcd#/2,if_gcd#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0 ,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,le#,minus#,pred#} 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^2))