WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(x,0()) -> x -(0(),y) -> 0() -(s(x),s(y)) -> -(x,y) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2} / {+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *#(0(),y) -> c_1() *#(s(x),y) -> c_2(*#(x,y)) -#(x,0()) -> c_3() -#(0(),y) -> c_4() -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,0()) -> c_6() exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(0(),y) -> c_1() *#(s(x),y) -> c_2(*#(x,y)) -#(x,0()) -> c_3() -#(0(),y) -> c_4() -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,0()) -> c_6() exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(x,0()) -> x -(0(),y) -> 0() -(s(x),s(y)) -> -(x,y) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6} by application of Pre({1,3,4,6}) = {2,5,7}. Here rules are labelled as follows: 1: *#(0(),y) -> c_1() 2: *#(s(x),y) -> c_2(*#(x,y)) 3: -#(x,0()) -> c_3() 4: -#(0(),y) -> c_4() 5: -#(s(x),s(y)) -> c_5(-#(x,y)) 6: exp#(x,0()) -> c_6() 7: exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(*#(x,y)) -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) - Weak DPs: *#(0(),y) -> c_1() -#(x,0()) -> c_3() -#(0(),y) -> c_4() exp#(x,0()) -> c_6() - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(x,0()) -> x -(0(),y) -> 0() -(s(x),s(y)) -> -(x,y) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*#(s(x),y) -> c_2(*#(x,y)) -->_1 *#(0(),y) -> c_1():4 -->_1 *#(s(x),y) -> c_2(*#(x,y)):1 2:S:-#(s(x),s(y)) -> c_5(-#(x,y)) -->_1 -#(0(),y) -> c_4():6 -->_1 -#(x,0()) -> c_3():5 -->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):2 3:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) -->_2 exp#(x,0()) -> c_6():7 -->_1 *#(0(),y) -> c_1():4 -->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):3 -->_1 *#(s(x),y) -> c_2(*#(x,y)):1 4:W:*#(0(),y) -> c_1() 5:W:-#(x,0()) -> c_3() 6:W:-#(0(),y) -> c_4() 7:W:exp#(x,0()) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: exp#(x,0()) -> c_6() 5: -#(x,0()) -> c_3() 6: -#(0(),y) -> c_4() 4: *#(0(),y) -> c_1() * Step 4: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(*#(x,y)) -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(x,0()) -> x -(0(),y) -> 0() -(s(x),s(y)) -> -(x,y) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) *#(s(x),y) -> c_2(*#(x,y)) -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) * Step 5: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(*#(x,y)) -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + 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 -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) and a lower component *#(s(x),y) -> c_2(*#(x,y)) Further, following extension rules are added to the lower component. -#(s(x),s(y)) -> -#(x,y) exp#(x,s(y)) -> *#(x,exp(x,y)) exp#(x,s(y)) -> exp#(x,y) ** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:-#(s(x),s(y)) -> c_5(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1 2:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)) -->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: exp#(x,s(y)) -> c_7(exp#(x,y)) ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(exp#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(exp#(x,y)) ** Step 5.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(exp#(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + 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(c_5) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [0] p(+) = [1] x1 + [1] x2 + [0] p(-) = [0] p(0) = [0] p(exp) = [0] p(s) = [1] x1 + [2] p(*#) = [0] p(-#) = [2] x2 + [0] p(exp#) = [8] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] Following rules are strictly oriented: -#(s(x),s(y)) = [2] y + [4] > [2] y + [0] = c_5(-#(x,y)) exp#(x,s(y)) = [8] y + [16] > [8] y + [0] = c_7(exp#(x,y)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(s(x),s(y)) -> c_5(-#(x,y)) exp#(x,s(y)) -> c_7(exp#(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(*#(x,y)) - Weak DPs: -#(s(x),s(y)) -> -#(x,y) exp#(x,s(y)) -> *#(x,exp(x,y)) exp#(x,s(y)) -> exp#(x,y) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*#(s(x),y) -> c_2(*#(x,y)) -->_1 *#(s(x),y) -> c_2(*#(x,y)):1 2:W:-#(s(x),s(y)) -> -#(x,y) -->_1 -#(s(x),s(y)) -> -#(x,y):2 3:W:exp#(x,s(y)) -> *#(x,exp(x,y)) -->_1 *#(s(x),y) -> c_2(*#(x,y)):1 4:W:exp#(x,s(y)) -> exp#(x,y) -->_1 exp#(x,s(y)) -> exp#(x,y):4 -->_1 exp#(x,s(y)) -> *#(x,exp(x,y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: -#(s(x),s(y)) -> -#(x,y) ** Step 5.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(*#(x,y)) - Weak DPs: exp#(x,s(y)) -> *#(x,exp(x,y)) exp#(x,s(y)) -> exp#(x,y) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + 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(*) = {2}, uargs(+) = {2}, uargs(*#) = {2}, uargs(c_2) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x2 + [2] p(+) = [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [0] p(0) = [2] p(exp) = [1] x1 + [1] x2 + [8] p(s) = [1] x1 + [2] p(*#) = [7] x1 + [1] x2 + [7] p(-#) = [1] p(exp#) = [9] x1 + [2] x2 + [13] p(c_1) = [1] p(c_2) = [1] x1 + [9] p(c_3) = [2] p(c_4) = [1] p(c_5) = [1] x1 + [1] p(c_6) = [0] p(c_7) = [1] x1 + [1] Following rules are strictly oriented: *#(s(x),y) = [7] x + [1] y + [21] > [7] x + [1] y + [16] = c_2(*#(x,y)) Following rules are (at-least) weakly oriented: exp#(x,s(y)) = [9] x + [2] y + [17] >= [8] x + [1] y + [15] = *#(x,exp(x,y)) exp#(x,s(y)) = [9] x + [2] y + [17] >= [9] x + [2] y + [13] = exp#(x,y) *(0(),y) = [1] y + [2] >= [2] = 0() *(s(x),y) = [1] y + [2] >= [1] y + [2] = +(y,*(x,y)) exp(x,0()) = [1] x + [10] >= [4] = s(0()) exp(x,s(y)) = [1] x + [1] y + [10] >= [1] x + [1] y + [10] = *(x,exp(x,y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(s(x),y) -> c_2(*#(x,y)) exp#(x,s(y)) -> *#(x,exp(x,y)) exp#(x,s(y)) -> exp#(x,y) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))