/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^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + 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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,if_minus,le,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: 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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,if_minus,le,minus} 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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,if_minus,le,minus} 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^3)) + 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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,if_minus,le,minus} 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) if_minus#(true(),s(x),y) -> c_7() le#(0(),y) -> c_8() le#(s(x),0()) -> c_9() le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(0(),y) -> c_11() minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) if_minus#(true(),s(x),y) -> c_7() le#(0(),y) -> c_8() le#(s(x),0()) -> c_9() le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(0(),y) -> c_11() minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,7,8,9,11} by application of Pre({1,2,7,8,9,11}) = {3,4,5,6,10,12}. 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: if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) 7: if_minus#(true(),s(x),y) -> c_7() 8: le#(0(),y) -> c_8() 9: le#(s(x),0()) -> c_9() 10: le#(s(x),s(y)) -> c_10(le#(x,y)) 11: minus#(0(),y) -> c_11() 12: minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) - Weak DPs: gcd#(0(),y) -> c_1() gcd#(s(x),0()) -> c_2() if_minus#(true(),s(x),y) -> c_7() le#(0(),y) -> c_8() le#(s(x),0()) -> c_9() minus#(0(),y) -> 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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_10(le#(x,y)):5 -->_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_9():11 -->_2 le#(0(),y) -> c_8():10 2:S:if_gcd#(false(),s(x),s(y)) -> c_4(gcd#(minus(y,x),s(x)),minus#(y,x)) -->_2 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6 -->_2 minus#(0(),y) -> c_11():12 -->_1 gcd#(0(),y) -> c_1():7 -->_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#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6 -->_2 minus#(0(),y) -> c_11():12 -->_1 gcd#(0(),y) -> c_1():7 -->_1 gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y)),le#(y,x)):1 4:S:if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) -->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):6 -->_1 minus#(0(),y) -> c_11():12 5:S:le#(s(x),s(y)) -> c_10(le#(x,y)) -->_1 le#(s(x),0()) -> c_9():11 -->_1 le#(0(),y) -> c_8():10 -->_1 le#(s(x),s(y)) -> c_10(le#(x,y)):5 6:S:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_2 le#(s(x),0()) -> c_9():11 -->_1 if_minus#(true(),s(x),y) -> c_7():9 -->_2 le#(s(x),s(y)) -> c_10(le#(x,y)):5 -->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):4 7:W:gcd#(0(),y) -> c_1() 8:W:gcd#(s(x),0()) -> c_2() 9:W:if_minus#(true(),s(x),y) -> c_7() 10:W:le#(0(),y) -> c_8() 11:W:le#(s(x),0()) -> c_9() 12:W:minus#(0(),y) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: gcd#(s(x),0()) -> c_2() 7: gcd#(0(),y) -> c_1() 12: minus#(0(),y) -> c_11() 9: if_minus#(true(),s(x),y) -> c_7() 10: le#(0(),y) -> c_8() 11: le#(s(x),0()) -> c_9() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^3)) + 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(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)) if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) ** Step 1.b:5: DecomposeDG. WORST_CASE(?,O(n^3)) + 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)) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(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:5.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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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:5.a:2: WeightGap. 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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {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) = [2] p(false) = [1] p(gcd) = [1] x2 + [1] p(if_gcd) = [1] x1 + [1] x2 + [4] p(if_minus) = [1] x1 + [1] p(le) = [1] p(minus) = [2] p(s) = [1] x1 + [0] p(true) = [1] p(gcd#) = [1] x1 + [7] p(if_gcd#) = [1] x1 + [1] x2 + [3] p(if_minus#) = [1] x1 + [4] x2 + [4] x3 + [0] p(le#) = [0] p(minus#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [4] p(c_5) = [1] x1 + [4] p(c_6) = [4] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: gcd#(s(x),s(y)) = [1] x + [7] > [1] x + [4] = 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)) = [1] x + [4] >= [13] = c_4(gcd#(minus(y,x),s(x))) if_gcd#(true(),s(x),s(y)) = [1] x + [4] >= [13] = c_5(gcd#(minus(x,y),s(y))) if_minus(false(),s(x),y) = [2] >= [2] = s(minus(x,y)) if_minus(true(),s(x),y) = [2] >= [2] = 0() le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(0(),y) = [2] >= [2] = 0() minus(s(x),y) = [2] >= [2] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {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) = [0] p(false) = [0] p(gcd) = [1] x2 + [1] p(if_gcd) = [1] x1 + [2] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [2] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [2] p(if_minus#) = [1] x1 + [1] x2 + [1] x3 + [1] p(le#) = [4] x2 + [1] p(minus#) = [4] x1 + [4] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [4] p(c_5) = [1] x1 + [0] p(c_6) = [2] x1 + [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x2 + [4] Following rules are strictly oriented: if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [4] > [1] x + [1] y + [3] = c_5(gcd#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [7] = c_4(gcd#(minus(y,x),s(x))) if_minus(false(),s(x),y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [1] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.a:4: 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))) - Weak DPs: gcd#(s(x),s(y)) -> c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(true(),s(x),s(y)) -> c_5(gcd#(minus(x,y),s(y))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {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) = [0] p(false) = [0] p(gcd) = [2] p(if_gcd) = [1] x1 + [1] x2 + [1] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [6] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [5] p(if_minus#) = [0] p(le#) = [2] x1 + [1] x2 + [0] p(minus#) = [2] x1 + [4] x2 + [0] p(c_1) = [1] p(c_2) = [4] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [4] p(c_9) = [1] p(c_10) = [1] p(c_11) = [4] p(c_12) = [1] Following rules are strictly oriented: if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [13] > [1] x + [1] y + [12] = c_4(gcd#(minus(y,x),s(x))) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [14] >= [1] x + [1] y + [14] = c_3(if_gcd#(le(y,x),s(x),s(y))) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [13] >= [1] x + [1] y + [10] = c_5(gcd#(minus(x,y),s(y))) if_minus(false(),s(x),y) = [1] x + [4] >= [1] x + [4] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [4] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [4] >= [1] x + [4] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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)) -> 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) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) and a lower component le#(s(x),s(y)) -> c_10(le#(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) if_minus#(false(),s(x),y) -> minus#(x,y) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) **** Step 1.b:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> le#(y,x) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) - Weak DPs: 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#(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(s(x),s(y)) -> le#(y,x) 2:S:if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) -->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):3 3:S:minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_1 if_minus#(false(),s(x),y) -> c_6(minus#(x,y)):2 4:W:gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) -->_1 if_gcd#(true(),s(x),s(y)) -> minus#(x,y):8 -->_1 if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)):7 -->_1 if_gcd#(false(),s(x),s(y)) -> minus#(y,x):6 -->_1 if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)):5 5:W:if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) -->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):4 -->_1 gcd#(s(x),s(y)) -> le#(y,x):1 6:W:if_gcd#(false(),s(x),s(y)) -> minus#(y,x) -->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):3 7:W:if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) -->_1 gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)):4 -->_1 gcd#(s(x),s(y)) -> le#(y,x):1 8:W:if_gcd#(true(),s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):3 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_minus#(le(s(x),y),s(x),y)) **** Step 1.b:5.b:1.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> le#(y,x) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)) - Weak DPs: 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#(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(if_minus#) = {1}, uargs(c_6) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [2] x1 + [0] p(if_gcd) = [4] x1 + [1] x2 + [4] x3 + [4] p(if_minus) = [1] x1 + [0] p(le) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(gcd#) = [1] x1 + [0] p(if_gcd#) = [1] x1 + [1] x2 + [0] p(if_minus#) = [1] x1 + [1] p(le#) = [1] x2 + [2] p(minus#) = [0] p(c_1) = [0] p(c_2) = [4] p(c_3) = [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x2 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: if_minus#(false(),s(x),y) = [1] > [0] = c_6(minus#(x,y)) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) = [1] x + [0] >= [1] x + [2] = le#(y,x) if_gcd#(false(),s(x),s(y)) = [1] x + [0] >= [0] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [1] x + [0] >= [0] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [1] x + [0] >= [0] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [1] x + [0] >= [0] = minus#(x,y) minus#(s(x),y) = [0] >= [1] = c_12(if_minus#(le(s(x),y),s(x),y)) if_minus(false(),s(x),y) = [0] >= [0] = s(minus(x,y)) if_minus(true(),s(x),y) = [0] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [0] >= [0] = if_minus(le(s(x),y),s(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:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> le#(y,x) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)) - Weak DPs: 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#(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) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(if_minus#) = {1}, uargs(c_6) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [1] x2 + [1] p(if_gcd) = [1] x2 + [1] p(if_minus) = [1] x1 + [0] p(le) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(gcd#) = [1] x1 + [4] p(if_gcd#) = [1] x1 + [4] p(if_minus#) = [1] x1 + [4] p(le#) = [1] x2 + [0] p(minus#) = [0] p(c_1) = [1] p(c_2) = [2] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [2] x1 + [1] x2 + [0] p(c_6) = [1] x1 + [4] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [7] Following rules are strictly oriented: gcd#(s(x),s(y)) = [1] x + [4] > [1] x + [0] = le#(y,x) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [4] >= [4] = if_gcd#(le(y,x),s(x),s(y)) if_gcd#(false(),s(x),s(y)) = [4] >= [4] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [4] >= [0] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [4] >= [4] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [4] >= [0] = minus#(x,y) if_minus#(false(),s(x),y) = [4] >= [4] = c_6(minus#(x,y)) minus#(s(x),y) = [0] >= [11] = c_12(if_minus#(le(s(x),y),s(x),y)) if_minus(false(),s(x),y) = [0] >= [0] = s(minus(x,y)) if_minus(true(),s(x),y) = [0] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [0] >= [0] = if_minus(le(s(x),y),s(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_minus#(le(s(x),y),s(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) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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_minus) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if_gcd#) = {1}, uargs(if_minus#) = {1}, uargs(c_6) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [2] p(if_gcd) = [4] x3 + [1] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [4] p(if_gcd#) = [1] x1 + [1] x2 + [1] x3 + [4] p(if_minus#) = [1] x1 + [1] x2 + [0] p(le#) = [1] x1 + [1] p(minus#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [1] p(c_5) = [1] x2 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] x1 + [0] p(c_11) = [4] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: minus#(s(x),y) = [1] x + [5] > [1] x + [4] = c_12(if_minus#(le(s(x),y),s(x),y)) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [1] x + [1] y + [12] >= [1] x + [1] y + [12] = if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) = [1] x + [1] y + [12] >= [1] y + [1] = le#(y,x) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [12] >= [1] x + [1] y + [8] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [1] x + [1] y + [12] >= [1] y + [1] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [12] >= [1] x + [1] y + [8] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [1] x + [1] y + [12] >= [1] x + [1] = minus#(x,y) if_minus#(false(),s(x),y) = [1] x + [4] >= [1] x + [1] = c_6(minus#(x,y)) if_minus(false(),s(x),y) = [1] x + [4] >= [1] x + [4] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [4] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [4] >= [1] x + [4] = if_minus(le(s(x),y),s(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: 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) if_minus#(false(),s(x),y) -> c_6(minus#(x,y)) minus#(s(x),y) -> c_12(if_minus#(le(s(x),y),s(x),y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,minus#} 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: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_10(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) if_minus#(false(),s(x),y) -> minus#(x,y) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,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(c_10) = {1} Following symbols are considered usable: {if_minus,minus,gcd#,if_gcd#,if_minus#,le#,minus#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [1] x1 + [2] x2 + [0] p(if_gcd) = [2] x2 + [0] p(if_minus) = [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(gcd#) = [2] x1 + [2] x2 + [0] p(if_gcd#) = [2] x2 + [2] x3 + [0] p(if_minus#) = [1] x3 + [1] p(le#) = [1] x2 + [0] p(minus#) = [1] x2 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [4] x1 + [1] x2 + [2] p(c_4) = [1] p(c_5) = [1] x1 + [1] x2 + [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [1] p(c_11) = [0] p(c_12) = [0] Following rules are strictly oriented: le#(s(x),s(y)) = [1] y + [2] > [1] y + [1] = c_10(le#(x,y)) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [2] x + [2] y + [8] >= [2] x + [2] y + [8] = if_gcd#(le(y,x),s(x),s(y)) gcd#(s(x),s(y)) = [2] x + [2] y + [8] >= [1] x + [0] = le#(y,x) if_gcd#(false(),s(x),s(y)) = [2] x + [2] y + [8] >= [2] x + [2] y + [4] = gcd#(minus(y,x),s(x)) if_gcd#(false(),s(x),s(y)) = [2] x + [2] y + [8] >= [1] x + [1] = minus#(y,x) if_gcd#(true(),s(x),s(y)) = [2] x + [2] y + [8] >= [2] x + [2] y + [4] = gcd#(minus(x,y),s(y)) if_gcd#(true(),s(x),s(y)) = [2] x + [2] y + [8] >= [1] y + [1] = minus#(x,y) if_minus#(false(),s(x),y) = [1] y + [1] >= [1] y + [1] = minus#(x,y) minus#(s(x),y) = [1] y + [1] >= [1] y + [1] = if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) = [1] y + [1] >= [1] y + [0] = le#(s(x),y) if_minus(false(),s(x),y) = [1] x + [2] >= [1] x + [2] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [2] >= [0] = 0() minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [2] >= [1] x + [2] = if_minus(le(s(x),y),s(x),y) **** Step 1.b:5.b:1.b:2: 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) if_minus#(false(),s(x),y) -> minus#(x,y) le#(s(x),s(y)) -> c_10(le#(x,y)) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {gcd/2,if_gcd/3,if_minus/3,le/2,minus/2,gcd#/2,if_gcd#/3,if_minus#/3,le#/2,minus#/2} / {0/0,false/0,s/1 ,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if_gcd#,if_minus#,le#,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^3))