WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u) f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u)) perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs f#(0(),y,0(),u) -> c_1() f#(0(),y,s(z),u) -> c_2() f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(0()) -> c_5() perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(0(),y,0(),u) -> c_1() f#(0(),y,s(z),u) -> c_2() f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(0()) -> c_5() perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) - Strict TRS: f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u) f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u)) perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(0(),y,0(),u) -> c_1() f#(0(),y,s(z),u) -> c_2() f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(0()) -> c_5() perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(0(),y,0(),u) -> c_1() f#(0(),y,s(z),u) -> c_2() f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(0()) -> c_5() perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,5} by application of Pre({1,2,5}) = {4,6}. Here rules are labelled as follows: 1: f#(0(),y,0(),u) -> c_1() 2: f#(0(),y,s(z),u) -> c_2() 3: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) 4: f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) 5: perfectp#(0()) -> c_5() 6: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) - Weak DPs: f#(0(),y,0(),u) -> c_1() f#(0(),y,s(z),u) -> c_2() perfectp#(0()) -> c_5() - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) -->_2 f#(0(),y,s(z),u) -> c_2():5 -->_2 f#(0(),y,0(),u) -> c_1():4 -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) -->_1 f#(0(),y,s(z),u) -> c_2():5 -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 4:W:f#(0(),y,0(),u) -> c_1() 5:W:f#(0(),y,s(z),u) -> c_2() 6:W:perfectp#(0()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: perfectp#(0()) -> c_5() 4: f#(0(),y,0(),u) -> c_1() 5: f#(0(),y,s(z),u) -> c_2() * Step 5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)) -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) * Step 6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 2:S:f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2 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). [(3,perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))))] * Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + 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: 2: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) The strictly oriented rules are moved into the weak component. ** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + 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_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {f#,perfectp#} TcT has computed the following interpretation: p(0) = [4] p(f) = [1] x1 + [1] x3 + [0] p(false) = [1] p(if) = [1] x1 + [1] x3 + [4] p(le) = [1] x2 + [1] p(minus) = [1] p(perfectp) = [4] p(s) = [1] x1 + [4] p(true) = [4] p(f#) = [2] x1 + [1] x3 + [2] x4 + [0] p(perfectp#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [7] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] Following rules are strictly oriented: f#(s(x),s(y),z,u) = [2] u + [2] x + [1] z + [8] > [2] u + [2] x + [1] z + [0] = c_4(f#(x,u,z,u)) Following rules are (at-least) weakly oriented: f#(s(x),0(),z,u) = [2] u + [2] x + [1] z + [8] >= [2] u + [2] x + [8] = c_3(f#(x,u,minus(z,s(x)),u)) ** Step 7.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) - Weak DPs: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 7.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) - Weak DPs: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + 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: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) The strictly oriented rules are moved into the weak component. *** Step 7.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) - Weak DPs: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + 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_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {f#,perfectp#} TcT has computed the following interpretation: p(0) = [4] p(f) = [1] x2 + [1] x3 + [1] x4 + [2] p(false) = [2] p(if) = [1] x1 + [1] p(le) = [0] p(minus) = [1] x1 + [3] p(perfectp) = [1] x1 + [0] p(s) = [1] x1 + [14] p(true) = [0] p(f#) = [1] x1 + [2] x3 + [4] x4 + [2] p(perfectp#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [8] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: f#(s(x),0(),z,u) = [4] u + [1] x + [2] z + [16] > [4] u + [1] x + [2] z + [8] = c_3(f#(x,u,minus(z,s(x)),u)) Following rules are (at-least) weakly oriented: f#(s(x),s(y),z,u) = [4] u + [1] x + [2] z + [16] >= [4] u + [1] x + [2] z + [10] = c_4(f#(x,u,z,u)) *** Step 7.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 2:W:f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2 -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)) 2: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)) *** Step 7.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0 ,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))