* Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: circ(s,id()) -> s circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a subst(a,id()) -> a - Signature: {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ,msubst,subst} and constructors {cons,id,lift} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs circ#(s,id()) -> c_1() circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) circ#(id(),s) -> c_5() msubst#(a,id()) -> c_6() subst#(a,id()) -> c_7() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(s,id()) -> c_1() circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) circ#(id(),s) -> c_5() msubst#(a,id()) -> c_6() subst#(a,id()) -> c_7() - Strict TRS: circ(s,id()) -> s circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a subst(a,id()) -> a - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: circ#(s,id()) -> c_1() circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) circ#(id(),s) -> c_5() msubst#(a,id()) -> c_6() subst#(a,id()) -> c_7() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(s,id()) -> c_1() circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) circ#(id(),s) -> c_5() msubst#(a,id()) -> c_6() subst#(a,id()) -> c_7() - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6,7} by application of Pre({1,5,6,7}) = {2,3,4}. Here rules are labelled as follows: 1: circ#(s,id()) -> c_1() 2: circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) 3: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) 4: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) 5: circ#(id(),s) -> c_5() 6: msubst#(a,id()) -> c_6() 7: subst#(a,id()) -> c_7() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Weak DPs: circ#(s,id()) -> c_1() circ#(id(),s) -> c_5() msubst#(a,id()) -> c_6() subst#(a,id()) -> c_7() - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 msubst#(a,id()) -> c_6():6 -->_2 circ#(id(),s) -> c_5():5 -->_2 circ#(s,id()) -> c_1():4 -->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(id(),s) -> c_5():5 -->_1 circ#(s,id()) -> c_1():4 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) -->_1 circ#(id(),s) -> c_5():5 -->_1 circ#(s,id()) -> c_1():4 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 4:W:circ#(s,id()) -> c_1() 5:W:circ#(id(),s) -> c_5() 6:W:msubst#(a,id()) -> c_6() 7:W:subst#(a,id()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: subst#(a,id()) -> c_7() 6: msubst#(a,id()) -> c_6() 4: circ#(s,id()) -> c_1() 5: circ#(id(),s) -> c_5() * Step 5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)) -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_2 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 2:S:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 3:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(msubst#(a,t),circ#(s,t)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: circ#(cons(a,s),t) -> c_2(circ#(s,t)) * Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(cons(a,s),t) -> c_2(circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + 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: circ#(cons(a,s),t) -> c_2(circ#(s,t)) 2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) 3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) The strictly oriented rules are moved into the weak component. ** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(cons(a,s),t) -> c_2(circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + 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_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {circ#,msubst#,subst#} TcT has computed the following interpretation: p(circ) = [1] x2 + [1] p(cons) = [1] x1 + [1] x2 + [1] p(id) = [1] p(lift) = [3] p(msubst) = [8] x1 + [1] x2 + [1] p(subst) = [2] x1 + [4] p(circ#) = [1] x1 + [1] x2 + [1] p(msubst#) = [1] x1 + [0] p(subst#) = [1] x2 + [0] p(c_1) = [8] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [5] p(c_5) = [1] p(c_6) = [1] p(c_7) = [0] Following rules are strictly oriented: circ#(cons(a,s),t) = [1] a + [1] s + [1] t + [2] > [1] s + [1] t + [1] = c_2(circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) = [1] a + [1] s + [1] t + [6] > [1] s + [1] t + [1] = c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) = [1] s + [1] t + [9] > [1] s + [1] t + [6] = c_4(circ#(s,t)) Following rules are (at-least) weakly oriented: ** Step 6.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: circ#(cons(a,s),t) -> c_2(circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: circ#(cons(a,s),t) -> c_2(circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:circ#(cons(a,s),t) -> c_2(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1 2:W:circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1 3:W:circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_2(circ#(s,t)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: circ#(cons(a,s),t) -> c_2(circ#(s,t)) 3: circ#(cons(lift(),s),cons(lift(),t)) -> c_4(circ#(s,t)) 2: circ#(cons(lift(),s),cons(a,t)) -> c_3(circ#(s,t)) ** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))