* Step 1: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: gcd(x,0()) -> x gcd(0(),y) -> y gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y))) - Signature: {gcd/2} / {-/2,0/0,2,if/3,s/1} - Obligation: runtime complexity wrt. defined symbols {gcd} and constructors {-,0,<,if,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following weak dependency pairs: Strict DPs gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))) - Strict TRS: gcd(x,0()) -> x gcd(0(),y) -> y gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y))) - Signature: {gcd/2,gcd#/2} / {-/2,0/0,2,if/3,s/1,c_1/1,c_2/1,c_3/4} - Obligation: runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))) * Step 3: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))) - Signature: {gcd/2,gcd#/2} / {-/2,0/0,2,if/3,s/1,c_1/1,c_2/1,c_3/4} - Obligation: runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(x,0()) -> c_1(x) -->_1 gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))):3 -->_1 gcd#(0(),y) -> c_2(y):2 -->_1 gcd#(x,0()) -> c_1(x):1 2:S:gcd#(0(),y) -> c_2(y) -->_1 gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))):3 -->_1 gcd#(0(),y) -> c_2(y):2 -->_1 gcd#(x,0()) -> c_1(x):1 3:S:gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))) -->_2 gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))):3 -->_1 gcd#(s(x),s(y)) -> c_3(x,y,gcd#(s(x),-(y,x)),gcd#(-(x,y),s(y))):3 -->_2 gcd#(0(),y) -> c_2(y):2 -->_1 gcd#(0(),y) -> c_2(y):2 -->_2 gcd#(x,0()) -> c_1(x):1 -->_1 gcd#(x,0()) -> c_1(x):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: gcd#(s(x),s(y)) -> c_3(x,y) * Step 4: MI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y) - Signature: {gcd/2,gcd#/2} / {-/2,0/0,2,if/3,s/1,c_1/1,c_2/1,c_3/2} - Obligation: runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 0))), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 0))): The following argument positions are considered usable: none Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [0] p(0) = [8] p(<) = [0] p(gcd) = [2] p(if) = [0] p(s) = [12] p(gcd#) = [1] x_1 + [11] p(c_1) = [1] x_1 + [0] p(c_2) = [7] p(c_3) = [0] Following rules are strictly oriented: gcd#(x,0()) = [1] x + [11] > [1] x + [0] = c_1(x) gcd#(0(),y) = [19] > [7] = c_2(y) gcd#(s(x),s(y)) = [23] > [0] = c_3(x,y) Following rules are (at-least) weakly oriented: * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(x,0()) -> c_1(x) gcd#(0(),y) -> c_2(y) gcd#(s(x),s(y)) -> c_3(x,y) - Signature: {gcd/2,gcd#/2} / {-/2,0/0,2,if/3,s/1,c_1/1,c_2/1,c_3/2} - Obligation: runtime complexity wrt. defined symbols {gcd#} and constructors {-,0,<,if,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))