WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [3] p(minus_active) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: mark(minus(x,y)) = [1] x + [3] > [1] x + [0] = minus_active(x,y) minus_active(s(x),s(y)) = [1] x + [1] > [1] x + [0] = minus_active(x,y) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [4] >= [4] = 0() div_active(s(x),s(y)) = [1] x + [1] >= [1] x + [8] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [4] >= [4] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(s(x)) = [1] x + [1] >= [1] x + [1] = s(mark(x)) minus_active(x,y) = [1] x + [0] >= [1] x + [3] = minus(x,y) minus_active(0(),y) = [4] >= [4] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() - Weak TRS: mark(minus(x,y)) -> minus_active(x,y) minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [1] x1 + [0] p(ge_active) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [7] p(minus) = [0] p(minus_active) = [7] p(s) = [1] x1 + [5] p(true) = [0] Following rules are strictly oriented: ge_active(0(),s(y)) = [4] > [0] = false() ge_active(s(x),s(y)) = [1] x + [5] > [1] x + [0] = ge_active(x,y) mark(0()) = [11] > [4] = 0() mark(ge(x,y)) = [1] x + [7] > [1] x + [0] = ge_active(x,y) minus_active(x,y) = [7] > [0] = minus(x,y) minus_active(0(),y) = [7] > [4] = 0() Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [4] >= [4] = 0() div_active(s(x),s(y)) = [1] x + [5] >= [1] x + [9] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] x + [0] >= [1] x + [0] = ge(x,y) ge_active(x,0()) = [1] x + [0] >= [0] = true() if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [7] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [7] = mark(x) mark(div(x,y)) = [1] x + [7] >= [1] x + [7] = div_active(mark(x),y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [7] >= [1] x + [1] y + [1] z + [7] = if_active(mark(x),y,z) mark(minus(x,y)) = [7] >= [7] = minus_active(x,y) mark(s(x)) = [1] x + [12] >= [1] x + [12] = s(mark(x)) minus_active(s(x),s(y)) = [7] >= [7] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(div(x,y)) -> div_active(mark(x),y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(s(x)) -> s(mark(x)) - Weak TRS: ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(div_active) = [1] x1 + [5] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [0] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: div_active(x,y) = [1] x + [5] > [0] = div(x,y) div_active(0(),s(y)) = [5] > [0] = 0() div_active(s(x),s(y)) = [1] x + [5] > [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) Following rules are (at-least) weakly oriented: ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [1] x + [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [0] = mark(y) if_active(true(),x,y) = [0] >= [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [0] >= [5] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [0] >= [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [0] >= [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(div(x,y)) -> div_active(mark(x),y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(s(x)) -> s(mark(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] x2 + [1] p(div_active) = [1] x1 + [1] x2 + [2] p(false) = [0] p(ge) = [1] p(ge_active) = [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [2] p(true) = [0] Following rules are strictly oriented: ge_active(x,0()) = [1] > [0] = true() Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [1] y + [2] >= [1] x + [1] y + [1] = div(x,y) div_active(0(),s(y)) = [1] y + [4] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [1] y + [6] >= [1] y + [6] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] >= [1] = ge(x,y) ge_active(0(),s(y)) = [1] >= [0] = false() ge_active(s(x),s(y)) = [1] >= [1] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [1] y + [1] >= [1] x + [1] y + [2] = div_active(mark(x),y) mark(ge(x,y)) = [1] >= [1] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [1] x + [2] >= [1] x + [2] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ge_active(x,y) -> ge(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(div(x,y)) -> div_active(mark(x),y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(s(x)) -> s(mark(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(div) = [1] x1 + [1] p(div_active) = [1] x1 + [4] p(false) = [0] p(ge) = [1] p(ge_active) = [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(minus_active) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: mark(if(x,y,z)) = [1] x + [1] y + [1] z + [1] > [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [4] >= [1] x + [1] = div(x,y) div_active(0(),s(y)) = [5] >= [1] = 0() div_active(s(x),s(y)) = [1] x + [4] >= [1] x + [3] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] >= [1] = ge(x,y) ge_active(x,0()) = [1] >= [0] = true() ge_active(0(),s(y)) = [1] >= [0] = false() ge_active(s(x),s(y)) = [1] >= [1] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [1] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [1] >= [1] = 0() mark(div(x,y)) = [1] x + [1] >= [1] x + [4] = div_active(mark(x),y) mark(ge(x,y)) = [1] >= [1] = ge_active(x,y) mark(minus(x,y)) = [1] x + [0] >= [1] x + [0] = minus_active(x,y) mark(s(x)) = [1] x + [0] >= [1] x + [0] = s(mark(x)) minus_active(x,y) = [1] x + [0] >= [1] x + [0] = minus(x,y) minus_active(0(),y) = [1] >= [1] = 0() minus_active(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ge_active(x,y) -> ge(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(div(x,y)) -> div_active(mark(x),y) mark(s(x)) -> s(mark(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(div_active) = [1] x1 + [2] p(false) = [0] p(ge) = [0] p(ge_active) = [2] p(if) = [0] p(if_active) = [1] x1 + [0] p(mark) = [3] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: ge_active(x,y) = [2] > [0] = ge(x,y) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [2] >= [0] = div(x,y) div_active(0(),s(y)) = [2] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [2] >= [2] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,0()) = [2] >= [0] = true() ge_active(0(),s(y)) = [2] >= [0] = false() ge_active(s(x),s(y)) = [2] >= [2] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [3] = mark(y) if_active(true(),x,y) = [0] >= [3] = mark(x) mark(0()) = [3] >= [0] = 0() mark(div(x,y)) = [3] >= [5] = div_active(mark(x),y) mark(ge(x,y)) = [3] >= [2] = ge_active(x,y) mark(if(x,y,z)) = [3] >= [3] = if_active(mark(x),y,z) mark(minus(x,y)) = [3] >= [0] = minus_active(x,y) mark(s(x)) = [3] >= [3] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(div(x,y)) -> div_active(mark(x),y) mark(s(x)) -> s(mark(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [2] p(false) = [0] p(ge) = [1] x1 + [2] p(ge_active) = [1] x1 + [2] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [2] Following rules are strictly oriented: if_active(true(),x,y) = [1] x + [1] y + [2] > [1] x + [0] = mark(x) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [2] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [2] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] x + [2] >= [1] x + [2] = ge(x,y) ge_active(x,0()) = [1] x + [2] >= [2] = true() ge_active(0(),s(y)) = [2] >= [0] = false() ge_active(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [2] = div_active(mark(x),y) mark(ge(x,y)) = [1] x + [2] >= [1] x + [2] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [1] x + [0] >= [1] x + [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) mark(div(x,y)) -> div_active(mark(x),y) mark(s(x)) -> s(mark(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [1] x2 + [0] p(div_active) = [1] x1 + [4] x2 + [6] p(false) = [3] p(ge) = [2] p(ge_active) = [3] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [4] x2 + [4] x3 + [0] p(mark) = [4] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [1] p(true) = [3] Following rules are strictly oriented: if_active(false(),x,y) = [4] x + [4] y + [3] > [4] y + [0] = mark(y) mark(s(x)) = [4] x + [4] > [4] x + [1] = s(mark(x)) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [4] y + [6] >= [1] x + [1] y + [0] = div(x,y) div_active(0(),s(y)) = [4] y + [10] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [4] y + [11] >= [4] y + [11] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [3] >= [2] = ge(x,y) ge_active(x,0()) = [3] >= [3] = true() ge_active(0(),s(y)) = [3] >= [3] = false() ge_active(s(x),s(y)) = [3] >= [3] = ge_active(x,y) if_active(x,y,z) = [1] x + [4] y + [4] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(true(),x,y) = [4] x + [4] y + [3] >= [4] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [4] x + [4] y + [0] >= [4] x + [4] y + [6] = div_active(mark(x),y) mark(ge(x,y)) = [8] >= [3] = ge_active(x,y) mark(if(x,y,z)) = [4] x + [4] y + [4] z + [0] >= [4] x + [4] y + [4] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: if_active(x,y,z) -> if(x,y,z) mark(div(x,y)) -> div_active(mark(x),y) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [2] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(if_active) = [1] x1 + [4] x2 + [4] x3 + [2] p(mark) = [4] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: if_active(x,y,z) = [1] x + [4] y + [4] z + [2] > [1] x + [1] y + [1] z + [1] = if(x,y,z) Following rules are (at-least) weakly oriented: div_active(x,y) = [1] x + [2] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [2] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [2] >= [2] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(false(),x,y) = [4] x + [4] y + [2] >= [4] y + [0] = mark(y) if_active(true(),x,y) = [4] x + [4] y + [2] >= [4] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [4] x + [0] >= [4] x + [2] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [4] x + [4] y + [4] z + [4] >= [4] x + [4] y + [4] z + [2] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [4] x + [0] >= [4] x + [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(div(x,y)) -> div_active(mark(x),y) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1} Following symbols are considered usable: {div_active,ge_active,if_active,mark,minus_active} TcT has computed the following interpretation: p(0) = [0] [0] p(div) = [1 2] x1 + [0 0] x2 + [2] [0 1] [0 1] [2] p(div_active) = [1 2] x1 + [0 0] x2 + [2] [0 1] [0 1] [2] p(false) = [1] [0] p(ge) = [1 0] x1 + [0] [0 0] [0] p(ge_active) = [1 0] x1 + [1] [0 0] [0] p(if) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1] [0 0] [0 1] [0 1] [2] p(if_active) = [1 0] x1 + [2 0] x2 + [2 0] x3 + [1] [0 0] [0 2] [0 2] [2] p(mark) = [2 0] x1 + [1] [0 2] [0] p(minus) = [0] [1] p(minus_active) = [0] [1] p(s) = [1 0] x1 + [0] [0 0] [4] p(true) = [0] [0] Following rules are strictly oriented: mark(div(x,y)) = [2 4] x + [0 0] y + [5] [0 2] [0 2] [4] > [2 4] x + [0 0] y + [3] [0 2] [0 1] [2] = div_active(mark(x),y) Following rules are (at-least) weakly oriented: div_active(x,y) = [1 2] x + [0 0] y + [2] [0 1] [0 1] [2] >= [1 2] x + [0 0] y + [2] [0 1] [0 1] [2] = div(x,y) div_active(0(),s(y)) = [2] [6] >= [0] [0] = 0() div_active(s(x),s(y)) = [1 0] x + [10] [0 0] [10] >= [1 0] x + [10] [0 0] [10] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1 0] x + [1] [0 0] [0] >= [1 0] x + [0] [0 0] [0] = ge(x,y) ge_active(x,0()) = [1 0] x + [1] [0 0] [0] >= [0] [0] = true() ge_active(0(),s(y)) = [1] [0] >= [1] [0] = false() ge_active(s(x),s(y)) = [1 0] x + [1] [0 0] [0] >= [1 0] x + [1] [0 0] [0] = ge_active(x,y) if_active(x,y,z) = [1 0] x + [2 0] y + [2 0] z + [1] [0 0] [0 2] [0 2] [2] >= [1 0] x + [1 0] y + [1 0] z + [1] [0 0] [0 1] [0 1] [2] = if(x,y,z) if_active(false(),x,y) = [2 0] x + [2 0] y + [2] [0 2] [0 2] [2] >= [2 0] y + [1] [0 2] [0] = mark(y) if_active(true(),x,y) = [2 0] x + [2 0] y + [1] [0 2] [0 2] [2] >= [2 0] x + [1] [0 2] [0] = mark(x) mark(0()) = [1] [0] >= [0] [0] = 0() mark(ge(x,y)) = [2 0] x + [1] [0 0] [0] >= [1 0] x + [1] [0 0] [0] = ge_active(x,y) mark(if(x,y,z)) = [2 0] x + [2 0] y + [2 0] z + [3] [0 0] [0 2] [0 2] [4] >= [2 0] x + [2 0] y + [2 0] z + [2] [0 0] [0 2] [0 2] [2] = if_active(mark(x),y,z) mark(minus(x,y)) = [1] [2] >= [0] [1] = minus_active(x,y) mark(s(x)) = [2 0] x + [1] [0 0] [8] >= [2 0] x + [1] [0 0] [4] = s(mark(x)) minus_active(x,y) = [0] [1] >= [0] [1] = minus(x,y) minus_active(0(),y) = [0] [1] >= [0] [0] = 0() minus_active(s(x),s(y)) = [0] [1] >= [0] [1] = minus_active(x,y) * Step 11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))