/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: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {help,if,lt,plus,quot} 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: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {help,if,lt,plus,quot} 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: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {help,if,lt,plus,quot} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: lt(x,y){x -> s(x),y -> s(y)} = lt(s(x),s(y)) ->^+ lt(x,y) = C[lt(x,y) = lt(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {help,if,lt,plus,quot} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(false(),x,s(y),c) -> c_2() if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(x,0()) -> c_4() lt#(0(),s(y)) -> c_5() lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,0()) -> c_7() plus#(x,s(y)) -> c_8(plus#(x,y)) quot#(x,s(y)) -> c_9(help#(x,s(y),0())) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(false(),x,s(y),c) -> c_2() if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(x,0()) -> c_4() lt#(0(),s(y)) -> c_5() lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,0()) -> c_7() plus#(x,s(y)) -> c_8(plus#(x,y)) quot#(x,s(y)) -> c_9(help#(x,s(y),0())) - Weak TRS: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,5,7} by application of Pre({2,4,5,7}) = {1,6,8}. Here rules are labelled as follows: 1: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) 2: if#(false(),x,s(y),c) -> c_2() 3: if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) 4: lt#(x,0()) -> c_4() 5: lt#(0(),s(y)) -> c_5() 6: lt#(s(x),s(y)) -> c_6(lt#(x,y)) 7: plus#(x,0()) -> c_7() 8: plus#(x,s(y)) -> c_8(plus#(x,y)) 9: quot#(x,s(y)) -> c_9(help#(x,s(y),0())) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) quot#(x,s(y)) -> c_9(help#(x,s(y),0())) - Weak DPs: if#(false(),x,s(y),c) -> c_2() lt#(x,0()) -> c_4() lt#(0(),s(y)) -> c_5() plus#(x,0()) -> c_7() - Weak TRS: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) -->_2 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3 -->_1 if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))):2 -->_2 lt#(0(),s(y)) -> c_5():8 -->_2 lt#(x,0()) -> c_4():7 -->_1 if#(false(),x,s(y),c) -> c_2():6 2:S:if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) -->_2 plus#(x,s(y)) -> c_8(plus#(x,y)):4 -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1 3:S:lt#(s(x),s(y)) -> c_6(lt#(x,y)) -->_1 lt#(0(),s(y)) -> c_5():8 -->_1 lt#(x,0()) -> c_4():7 -->_1 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3 4:S:plus#(x,s(y)) -> c_8(plus#(x,y)) -->_1 plus#(x,0()) -> c_7():9 -->_1 plus#(x,s(y)) -> c_8(plus#(x,y)):4 5:S:quot#(x,s(y)) -> c_9(help#(x,s(y),0())) -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1 6:W:if#(false(),x,s(y),c) -> c_2() 7:W:lt#(x,0()) -> c_4() 8:W:lt#(0(),s(y)) -> c_5() 9:W:plus#(x,0()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: if#(false(),x,s(y),c) -> c_2() 9: plus#(x,0()) -> c_7() 7: lt#(x,0()) -> c_4() 8: lt#(0(),s(y)) -> c_5() ** Step 1.b:4: RemoveHeads. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) quot#(x,s(y)) -> c_9(help#(x,s(y),0())) - Weak TRS: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) -->_2 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3 -->_1 if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))):2 2:S:if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) -->_2 plus#(x,s(y)) -> c_8(plus#(x,y)):4 -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1 3:S:lt#(s(x),s(y)) -> c_6(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_6(lt#(x,y)):3 4:S:plus#(x,s(y)) -> c_8(plus#(x,y)) -->_1 plus#(x,s(y)) -> c_8(plus#(x,y)):4 5:S:quot#(x,s(y)) -> c_9(help#(x,s(y),0())) -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,quot#(x,s(y)) -> c_9(help#(x,s(y),0())))] ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) - Weak TRS: help(x,s(y),c) -> if(lt(c,x),x,s(y),c) if(false(),x,s(y),c) -> 0() if(true(),x,s(y),c) -> s(help(x,s(y),plus(c,s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) quot(x,s(y)) -> help(x,s(y),0()) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} 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 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) and a lower component lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) Further, following extension rules are added to the lower component. help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c) help#(x,s(y),c) -> lt#(c,x) if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) -> plus#(c,s(y)) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)) -->_1 if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))):2 2:S:if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y))),plus#(c,s(y))) -->_1 help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c),lt#(c,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y)))) *** Step 1.b:6.a:2: WeightGap. MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c)) if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y)))) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} 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(s) = {1}, uargs(help#) = {3}, uargs(if#) = {1}, uargs(c_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(help) = [0] p(if) = [0] p(lt) = [9] p(plus) = [1] x1 + [0] p(quot) = [0] p(s) = [1] x1 + [0] p(true) = [1] p(help#) = [1] x3 + [0] p(if#) = [1] x1 + [1] x4 + [0] p(lt#) = [0] p(plus#) = [0] p(quot#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] Following rules are strictly oriented: if#(true(),x,s(y),c) = [1] c + [1] > [1] c + [0] = c_3(help#(x,s(y),plus(c,s(y)))) Following rules are (at-least) weakly oriented: help#(x,s(y),c) = [1] c + [0] >= [1] c + [9] = c_1(if#(lt(c,x),x,s(y),c)) lt(x,0()) = [9] >= [0] = false() lt(0(),s(y)) = [9] >= [1] = true() lt(s(x),s(y)) = [9] >= [9] = lt(x,y) plus(x,0()) = [1] x + [0] >= [1] x + [0] = x plus(x,s(y)) = [1] x + [0] >= [1] x + [0] = s(plus(x,y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: Failure MAYBE + Considered Problem: - Strict DPs: help#(x,s(y),c) -> c_1(if#(lt(c,x),x,s(y),c)) - Weak DPs: if#(true(),x,s(y),c) -> c_3(help#(x,s(y),plus(c,s(y)))) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0 ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) - Weak DPs: help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c) help#(x,s(y),c) -> lt#(c,x) if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) -> plus#(c,s(y)) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {help#,if#,lt#,plus#,quot#} TcT has computed the following interpretation: p(0) = [3] p(false) = [0] p(help) = [1] x2 + [1] x3 + [2] p(if) = [1] x2 + [0] p(lt) = [4] x1 + [8] x2 + [0] p(plus) = [5] x2 + [6] p(quot) = [2] x1 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(help#) = [1] x1 + [8] x2 + [12] p(if#) = [1] x2 + [8] x3 + [12] p(lt#) = [0] p(plus#) = [4] x2 + [0] p(quot#) = [1] x1 + [0] p(c_1) = [8] x1 + [1] p(c_2) = [1] p(c_3) = [1] x1 + [1] x2 + [1] p(c_4) = [1] p(c_5) = [2] p(c_6) = [8] x1 + [0] p(c_7) = [2] p(c_8) = [1] x1 + [0] p(c_9) = [1] Following rules are strictly oriented: plus#(x,s(y)) = [4] y + [8] > [4] y + [0] = c_8(plus#(x,y)) Following rules are (at-least) weakly oriented: help#(x,s(y),c) = [1] x + [8] y + [28] >= [1] x + [8] y + [28] = if#(lt(c,x),x,s(y),c) help#(x,s(y),c) = [1] x + [8] y + [28] >= [0] = lt#(c,x) if#(true(),x,s(y),c) = [1] x + [8] y + [28] >= [1] x + [8] y + [28] = help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) = [1] x + [8] y + [28] >= [4] y + [8] = plus#(c,s(y)) lt#(s(x),s(y)) = [0] >= [0] = c_6(lt#(x,y)) *** Step 1.b:6.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt#(s(x),s(y)) -> c_6(lt#(x,y)) - Weak DPs: help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c) help#(x,s(y),c) -> lt#(c,x) if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) -> plus#(c,s(y)) plus#(x,s(y)) -> c_8(plus#(x,y)) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} and constructors {0,false,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {help#,if#,lt#,plus#,quot#} TcT has computed the following interpretation: p(0) = [4] p(false) = [1] p(help) = [1] x1 + [2] x2 + [1] x3 + [1] p(if) = [1] x2 + [1] x4 + [4] p(lt) = [2] x2 + [4] p(plus) = [9] x1 + [3] p(quot) = [1] x1 + [2] x2 + [1] p(s) = [1] x1 + [4] p(true) = [0] p(help#) = [2] x1 + [1] x2 + [9] p(if#) = [2] x2 + [1] x3 + [9] p(lt#) = [2] x2 + [0] p(plus#) = [6] p(quot#) = [1] x1 + [1] p(c_1) = [2] x2 + [1] p(c_2) = [2] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] p(c_6) = [1] x1 + [6] p(c_7) = [2] p(c_8) = [1] x1 + [0] p(c_9) = [0] Following rules are strictly oriented: lt#(s(x),s(y)) = [2] y + [8] > [2] y + [6] = c_6(lt#(x,y)) Following rules are (at-least) weakly oriented: help#(x,s(y),c) = [2] x + [1] y + [13] >= [2] x + [1] y + [13] = if#(lt(c,x),x,s(y),c) help#(x,s(y),c) = [2] x + [1] y + [13] >= [2] x + [0] = lt#(c,x) if#(true(),x,s(y),c) = [2] x + [1] y + [13] >= [2] x + [1] y + [13] = help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) = [2] x + [1] y + [13] >= [6] = plus#(c,s(y)) plus#(x,s(y)) = [6] >= [6] = c_8(plus#(x,y)) *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: help#(x,s(y),c) -> if#(lt(c,x),x,s(y),c) help#(x,s(y),c) -> lt#(c,x) if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) if#(true(),x,s(y),c) -> plus#(c,s(y)) lt#(s(x),s(y)) -> c_6(lt#(x,y)) plus#(x,s(y)) -> c_8(plus#(x,y)) - Weak TRS: lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Signature: {help/3,if/4,lt/2,plus/2,quot/2,help#/3,if#/4,lt#/2,plus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/2,c_2/0 ,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {help#,if#,lt#,plus#,quot#} 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),?)