WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(div) = [1] x1 + [9] x2 + [0] p(false) = [0] p(geq) = [15] p(if) = [1] x1 + [1] x2 + [2] x3 + [0] p(minus) = [1] x1 + [1] p(s) = [1] x1 + [0] p(true) = [8] Following rules are strictly oriented: geq(X,0()) = [15] > [8] = true() geq(0(),s(Y)) = [15] > [0] = false() if(true(),X,Y) = [1] X + [2] Y + [8] > [1] X + [0] = X minus(0(),Y) = [6] > [5] = 0() Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [9] Y + [5] >= [5] = 0() div(s(X),s(Y)) = [1] X + [9] Y + [0] >= [1] X + [9] Y + [26] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(s(X),s(Y)) = [15] >= [15] = geq(X,Y) if(false(),X,Y) = [1] X + [2] Y + [0] >= [1] Y + [0] = Y minus(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = minus(X,Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: geq(X,0()) -> true() geq(0(),s(Y)) -> false() if(true(),X,Y) -> X minus(0(),Y) -> 0() - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(div) = [1] x1 + [2] x2 + [10] p(false) = [1] p(geq) = [3] p(if) = [1] x1 + [1] x2 + [6] x3 + [0] p(minus) = [5] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: div(0(),s(Y)) = [2] Y + [12] > [2] = 0() if(false(),X,Y) = [1] X + [6] Y + [1] > [1] Y + [0] = Y Following rules are (at-least) weakly oriented: div(s(X),s(Y)) = [1] X + [2] Y + [10] >= [2] Y + [30] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [3] >= [0] = true() geq(0(),s(Y)) = [3] >= [1] = false() geq(s(X),s(Y)) = [3] >= [3] = geq(X,Y) if(true(),X,Y) = [1] X + [6] Y + [0] >= [1] X + [0] = X minus(0(),Y) = [5] >= [2] = 0() minus(s(X),s(Y)) = [5] >= [5] = minus(X,Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(s(X),s(Y)) -> geq(X,Y) minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: div(0(),s(Y)) -> 0() geq(X,0()) -> true() geq(0(),s(Y)) -> false() if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = [0] p(div) = [8] x1 + [4] p(false) = [0] p(geq) = [1] p(if) = [1] x1 + [1] x2 + [4] x3 + [1] p(minus) = [2] p(s) = [1] x1 + [3] p(true) = [1] Following rules are strictly oriented: div(s(X),s(Y)) = [8] X + [28] > [25] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [4] >= [0] = 0() geq(X,0()) = [1] >= [1] = true() geq(0(),s(Y)) = [1] >= [0] = false() geq(s(X),s(Y)) = [1] >= [1] = geq(X,Y) if(false(),X,Y) = [1] X + [4] Y + [1] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [4] Y + [2] >= [1] X + [0] = X minus(0(),Y) = [2] >= [0] = 0() minus(s(X),s(Y)) = [2] >= [2] = minus(X,Y) * Step 4: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: geq(s(X),s(Y)) -> geq(X,Y) minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = [0] p(div) = [12] x1 + [1] p(false) = [0] p(geq) = [4] x1 + [0] p(if) = [2] x1 + [8] x2 + [2] x3 + [1] p(minus) = [0] p(s) = [1] x1 + [2] p(true) = [0] Following rules are strictly oriented: geq(s(X),s(Y)) = [4] X + [8] > [4] X + [0] = geq(X,Y) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [1] >= [0] = 0() div(s(X),s(Y)) = [12] X + [25] >= [8] X + [25] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [4] X + [0] >= [0] = true() geq(0(),s(Y)) = [0] >= [0] = false() if(false(),X,Y) = [8] X + [2] Y + [1] >= [1] Y + [0] = Y if(true(),X,Y) = [8] X + [2] Y + [1] >= [1] X + [0] = X minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [0] >= [0] = minus(X,Y) * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [8] x3 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [8] p(true) = [0] Following rules are strictly oriented: minus(s(X),s(Y)) = [1] X + [8] > [1] X + [0] = minus(X,Y) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [0] >= [0] = 0() div(s(X),s(Y)) = [1] X + [8] >= [1] X + [8] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [0] >= [0] = true() geq(0(),s(Y)) = [0] >= [0] = false() geq(s(X),s(Y)) = [0] >= [0] = geq(X,Y) if(false(),X,Y) = [1] X + [8] Y + [0] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [8] Y + [0] >= [1] X + [0] = X minus(0(),Y) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))