MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(0()) -> s(s(0())) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,minus,p,plus} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,0()) -> c_3() minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(0(),y) -> c_5() minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(0()) -> c_7() p#(s(s(x))) -> c_8(p#(s(x))) plus#(0(),y) -> c_9() plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,0()) -> c_3() minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(0(),y) -> c_5() minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(0()) -> c_7() p#(s(s(x))) -> c_8(p#(s(x))) plus#(0(),y) -> c_9() plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(0()) -> s(s(0())) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,0()) -> c_3() minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(0(),y) -> c_5() minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(0()) -> c_7() p#(s(s(x))) -> c_8(p#(s(x))) plus#(0(),y) -> c_9() plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,0()) -> c_3() minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(0(),y) -> c_5() minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(0()) -> c_7() p#(s(s(x))) -> c_8(p#(s(x))) plus#(0(),y) -> c_9() plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,5,7,9} by application of Pre({3,5,7,9}) = {1,2,4,6,10}. Here rules are labelled as follows: 1: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) 2: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) 3: minus#(x,0()) -> c_3() 4: minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) 5: minus#(0(),y) -> c_5() 6: minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) 7: p#(0()) -> c_7() 8: p#(s(s(x))) -> c_8(p#(s(x))) 9: plus#(0(),y) -> c_9() 10: plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) -> c_8(p#(s(x))) plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak DPs: minus#(x,0()) -> c_3() minus#(0(),y) -> c_5() p#(0()) -> c_7() plus#(0(),y) -> c_9() - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) -->_1 plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))):6 -->_3 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_2 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_1 plus#(0(),y) -> c_9():10 -->_3 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 -->_2 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 2:S:div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) -->_2 minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))):4 -->_2 minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)):3 -->_2 minus#(0(),y) -> c_5():8 -->_2 minus#(x,0()) -> c_3():7 -->_1 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_1 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 3:S:minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) -->_2 minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))):4 -->_1 minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))):4 -->_2 minus#(0(),y) -> c_5():8 -->_1 minus#(0(),y) -> c_5():8 -->_2 minus#(x,0()) -> c_3():7 -->_1 minus#(x,0()) -> c_3():7 -->_2 minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)):3 -->_1 minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)):3 4:S:minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) -->_3 p#(s(s(x))) -> c_8(p#(s(x))):5 -->_2 p#(s(s(x))) -> c_8(p#(s(x))):5 -->_1 minus#(0(),y) -> c_5():8 -->_1 minus#(x,0()) -> c_3():7 -->_1 minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))):4 -->_1 minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)):3 5:S:p#(s(s(x))) -> c_8(p#(s(x))) -->_1 p#(s(s(x))) -> c_8(p#(s(x))):5 6:S:plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) -->_1 plus#(0(),y) -> c_9():10 -->_1 plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))):6 -->_2 minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))):4 7:W:minus#(x,0()) -> c_3() 8:W:minus#(0(),y) -> c_5() 9:W:p#(0()) -> c_7() 10:W:plus#(0(),y) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: p#(0()) -> c_7() 7: minus#(x,0()) -> c_3() 8: minus#(0(),y) -> c_5() 10: plus#(0(),y) -> c_9() * Step 5: DecomposeDG MAYBE + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) -> c_8(p#(s(x))) plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) and a lower component minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) -> c_8(p#(s(x))) plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) Further, following extension rules are added to the lower component. div#(plus(x,y),z) -> div#(x,z) div#(plus(x,y),z) -> div#(y,z) div#(plus(x,y),z) -> plus#(div(x,z),div(y,z)) div#(s(x),s(y)) -> div#(minus(x,y),s(y)) div#(s(x),s(y)) -> minus#(x,y) ** Step 5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) The strictly oriented rules are moved into the weak component. *** Step 5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {1} Following symbols are considered usable: {minus,p,div#,minus#,p#,plus#} TcT has computed the following interpretation: p(0) = [10] p(div) = [2] x1 + [2] x2 + [0] p(minus) = [1] x1 + [0] p(p) = [0] p(plus) = [2] x1 + [1] x2 + [8] p(s) = [1] x1 + [0] p(div#) = [2] x1 + [0] p(minus#) = [1] x2 + [1] p(p#) = [1] p(plus#) = [2] p(c_1) = [4] x1 + [2] x2 + [1] x3 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [1] x2 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [1] x3 + [1] p(c_7) = [4] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x2 + [2] Following rules are strictly oriented: div#(plus(x,y),z) = [4] x + [2] y + [16] > [4] x + [2] y + [8] = c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) Following rules are (at-least) weakly oriented: div#(s(x),s(y)) = [2] x + [0] >= [2] x + [0] = c_2(div#(minus(x,y),s(y)),minus#(x,y)) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,plus(y,z)) = [1] x + [0] >= [1] x + [0] = minus(minus(x,y),z) minus(0(),y) = [10] >= [10] = 0() minus(s(x),s(y)) = [1] x + [0] >= [0] = minus(p(s(x)),p(s(y))) p(s(s(x))) = [0] >= [0] = s(p(s(x))) *** Step 5.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_2) = {1} Following symbols are considered usable: {minus,p,div#,minus#,p#,plus#} TcT has computed the following interpretation: p(0) = [2] p(div) = [3] x1 + [2] p(minus) = [1] x1 + [0] p(p) = [1] x1 + [0] p(plus) = [6] x1 + [1] x2 + [3] p(s) = [1] x1 + [2] p(div#) = [4] x1 + [4] p(minus#) = [8] p(p#) = [8] x1 + [1] p(plus#) = [0] p(c_1) = [8] x1 + [2] x2 + [1] x3 + [4] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] x1 + [1] x2 + [2] x3 + [0] p(c_7) = [0] p(c_8) = [8] x1 + [1] p(c_9) = [8] p(c_10) = [1] x1 + [1] Following rules are strictly oriented: div#(s(x),s(y)) = [4] x + [12] > [4] x + [5] = c_2(div#(minus(x,y),s(y)),minus#(x,y)) Following rules are (at-least) weakly oriented: div#(plus(x,y),z) = [24] x + [4] y + [16] >= [8] x + [4] y + [16] = c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,plus(y,z)) = [1] x + [0] >= [1] x + [0] = minus(minus(x,y),z) minus(0(),y) = [2] >= [2] = 0() minus(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = minus(p(s(x)),p(s(y))) p(s(s(x))) = [1] x + [4] >= [1] x + [4] = s(p(s(x))) **** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) -->_3 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_2 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_3 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 -->_2 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 2:W:div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) -->_1 div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)):2 -->_1 div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: div#(plus(x,y),z) -> c_1(plus#(div(x,z),div(y,z)),div#(x,z),div#(y,z)) 2: div#(s(x),s(y)) -> c_2(div#(minus(x,y),s(y)),minus#(x,y)) **** Step 5.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: NaturalMI MAYBE + Considered Problem: - Strict DPs: minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) -> c_8(p#(s(x))) plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak DPs: div#(plus(x,y),z) -> div#(x,z) div#(plus(x,y),z) -> div#(y,z) div#(plus(x,y),z) -> plus#(div(x,z),div(y,z)) div#(s(x),s(y)) -> div#(minus(x,y),s(y)) div#(s(x),s(y)) -> minus#(x,y) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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_4) = {1,2}, uargs(c_6) = {1,2,3}, uargs(c_8) = {1}, uargs(c_10) = {1,2} Following symbols are considered usable: {p,div#,minus#,p#,plus#} TcT has computed the following interpretation: p(0) = [0] [0] p(div) = [0] [0] p(minus) = [0 2] x1 + [0] [2 0] [2] p(p) = [0] [0] p(plus) = [2 2] x1 + [2 2] x2 + [3] [0 0] [0 0] [0] p(s) = [1 0] x1 + [0] [0 0] [0] p(div#) = [2 0] x2 + [3] [0 0] [2] p(minus#) = [2 0] x2 + [0] [0 0] [0] p(p#) = [0] [0] p(plus#) = [1] [2] p(c_1) = [0] [0] p(c_2) = [0] [0] p(c_3) = [0] [0] p(c_4) = [1 0] x1 + [2 0] x2 + [3] [0 0] [0 0] [0] p(c_5) = [0] [0] p(c_6) = [2 0] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 0] [0 0] [0] p(c_7) = [0] [0] p(c_8) = [2 0] x1 + [0] [0 0] [0] p(c_9) = [0] [0] p(c_10) = [1 0] x1 + [2 0] x2 + [0] [0 0] [0 0] [0] Following rules are strictly oriented: minus#(x,plus(y,z)) = [4 4] y + [4 4] z + [6] [0 0] [0 0] [0] > [4 0] y + [2 0] z + [3] [0 0] [0 0] [0] = c_4(minus#(minus(x,y),z),minus#(x,y)) Following rules are (at-least) weakly oriented: div#(plus(x,y),z) = [2 0] z + [3] [0 0] [2] >= [2 0] z + [3] [0 0] [2] = div#(x,z) div#(plus(x,y),z) = [2 0] z + [3] [0 0] [2] >= [2 0] z + [3] [0 0] [2] = div#(y,z) div#(plus(x,y),z) = [2 0] z + [3] [0 0] [2] >= [1] [2] = plus#(div(x,z),div(y,z)) div#(s(x),s(y)) = [2 0] y + [3] [0 0] [2] >= [2 0] y + [3] [0 0] [2] = div#(minus(x,y),s(y)) div#(s(x),s(y)) = [2 0] y + [3] [0 0] [2] >= [2 0] y + [0] [0 0] [0] = minus#(x,y) minus#(s(x),s(y)) = [2 0] y + [0] [0 0] [0] >= [0] [0] = c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) = [0] [0] >= [0] [0] = c_8(p#(s(x))) plus#(s(x),y) = [1] [2] >= [1] [0] = c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) p(s(s(x))) = [0] [0] >= [0] [0] = s(p(s(x))) ** Step 5.b:2: Failure MAYBE + Considered Problem: - Strict DPs: minus#(s(x),s(y)) -> c_6(minus#(p(s(x)),p(s(y))),p#(s(x)),p#(s(y))) p#(s(s(x))) -> c_8(p#(s(x))) plus#(s(x),y) -> c_10(plus#(y,minus(s(x),s(0()))),minus#(s(x),s(0()))) - Weak DPs: div#(plus(x,y),z) -> div#(x,z) div#(plus(x,y),z) -> div#(y,z) div#(plus(x,y),z) -> plus#(div(x,z),div(y,z)) div#(s(x),s(y)) -> div#(minus(x,y),s(y)) div#(s(x),s(y)) -> minus#(x,y) minus#(x,plus(y,z)) -> c_4(minus#(minus(x,y),z),minus#(x,y)) - Weak TRS: div(plus(x,y),z) -> plus(div(x,z),div(y,z)) div(s(x),s(y)) -> s(div(minus(x,y),s(y))) minus(x,0()) -> x minus(x,plus(y,z)) -> minus(minus(x,y),z) minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(p(s(x)),p(s(y))) p(s(s(x))) -> s(p(s(x))) plus(0(),y) -> y plus(s(x),y) -> s(plus(y,minus(s(x),s(0())))) - Signature: {div/2,minus/2,p/1,plus/2,div#/2,minus#/2,p#/1,plus#/2} / {0/0,s/1,c_1/3,c_2/2,c_3/0,c_4/2,c_5/0,c_6/3,c_7/0 ,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {div#,minus#,p#,plus#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE