WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [1] p(a__geq) = [0] p(a__if) = [1] x1 + [3] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: a__div(X1,X2) = [1] X1 + [1] > [0] = div(X1,X2) a__div(0(),s(Y)) = [1] > [0] = 0() a__if(X1,X2,X3) = [1] X1 + [3] > [1] X1 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [3] > [0] = mark(Y) a__if(true(),X,Y) = [3] > [0] = mark(X) Following rules are (at-least) weakly oriented: a__div(s(X),s(Y)) = [1] X + [1] >= [3] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [1] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [3] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() 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: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(a__div) = [1] x1 + [7] p(a__geq) = [6] p(a__if) = [1] x1 + [4] p(a__minus) = [0] p(div) = [1] x1 + [5] p(false) = [7] p(geq) = [0] p(if) = [1] x1 + [1] p(mark) = [2] p(minus) = [1] p(s) = [1] x1 + [0] p(true) = [5] Following rules are strictly oriented: a__geq(X,0()) = [6] > [5] = true() a__geq(X1,X2) = [6] > [0] = geq(X1,X2) mark(minus(X1,X2)) = [2] > [0] = a__minus(X1,X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [7] >= [1] X1 + [5] = div(X1,X2) a__div(0(),s(Y)) = [9] >= [2] = 0() a__div(s(X),s(Y)) = [1] X + [7] >= [10] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(0(),s(Y)) = [6] >= [7] = false() a__geq(s(X),s(Y)) = [6] >= [6] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [4] >= [1] X1 + [1] = if(X1,X2,X3) a__if(false(),X,Y) = [11] >= [2] = mark(Y) a__if(true(),X,Y) = [9] >= [2] = mark(X) a__minus(X1,X2) = [0] >= [1] = minus(X1,X2) a__minus(0(),Y) = [0] >= [2] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [2] >= [2] = 0() mark(div(X1,X2)) = [2] >= [9] = a__div(mark(X1),X2) mark(false()) = [2] >= [7] = false() mark(geq(X1,X2)) = [2] >= [6] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2] >= [6] = a__if(mark(X1),X2,X3) mark(s(X)) = [2] >= [2] = s(mark(X)) mark(true()) = [2] >= [5] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(minus(X1,X2)) -> a__minus(X1,X2) - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [1] p(a__minus) = [1] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [1] p(mark) = [1] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: a__minus(X1,X2) = [1] > [0] = minus(X1,X2) a__minus(0(),Y) = [1] > [0] = 0() mark(0()) = [1] > [0] = 0() mark(false()) = [1] > [0] = false() mark(geq(X1,X2)) = [1] > [0] = a__geq(X1,X2) mark(true()) = [1] > [0] = true() Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] >= [1] = if(X1,X2,X3) a__if(false(),X,Y) = [1] >= [1] = mark(Y) a__if(true(),X,Y) = [1] >= [1] = mark(X) a__minus(s(X),s(Y)) = [1] >= [1] = a__minus(X,Y) mark(div(X1,X2)) = [1] >= [1] = a__div(mark(X1),X2) mark(if(X1,X2,X3)) = [1] >= [2] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] >= [1] = a__minus(X1,X2) mark(s(X)) = [1] >= [1] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [1] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: a__div(s(X),s(Y)) = [1] X + [2] > [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [1] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [1] >= [0] = 0() a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [1] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [1] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [7] p(a__geq) = [1] p(a__if) = [1] x1 + [2] p(a__minus) = [1] p(div) = [1] x1 + [7] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [1] p(minus) = [1] p(s) = [1] x1 + [0] p(true) = [1] Following rules are strictly oriented: a__geq(0(),s(Y)) = [1] > [0] = false() Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [7] >= [1] X1 + [7] = div(X1,X2) a__div(0(),s(Y)) = [7] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [7] >= [3] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1] >= [1] = true() a__geq(X1,X2) = [1] >= [0] = geq(X1,X2) a__geq(s(X),s(Y)) = [1] >= [1] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [2] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [2] >= [1] = mark(Y) a__if(true(),X,Y) = [3] >= [1] = mark(X) a__minus(X1,X2) = [1] >= [1] = minus(X1,X2) a__minus(0(),Y) = [1] >= [0] = 0() a__minus(s(X),s(Y)) = [1] >= [1] = a__minus(X,Y) mark(0()) = [1] >= [0] = 0() mark(div(X1,X2)) = [1] >= [8] = a__div(mark(X1),X2) mark(false()) = [1] >= [0] = false() mark(geq(X1,X2)) = [1] >= [1] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] >= [3] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] >= [1] = a__minus(X1,X2) mark(s(X)) = [1] >= [1] = s(mark(X)) mark(true()) = [1] >= [1] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [3] p(a__geq) = [1] x1 + [0] p(a__if) = [1] x1 + [4] x2 + [4] x3 + [0] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [4] x1 + [0] p(minus) = [0] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: a__geq(s(X),s(Y)) = [1] X + [1] > [1] X + [0] = a__geq(X,Y) mark(s(X)) = [4] X + [4] > [4] X + [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [3] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [3] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1] X + [0] >= [0] = true() a__geq(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [4] X + [4] Y + [0] >= [4] Y + [0] = mark(Y) a__if(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [4] X1 + [0] >= [4] X1 + [3] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [4] X1 + [0] >= [1] X1 + [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0] >= [4] X1 + [4] X2 + [4] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [2] p(a__geq) = [0] p(a__if) = [1] x1 + [4] x2 + [4] x3 + [2] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(mark) = [4] x1 + [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [4] > [4] X1 + [4] X2 + [4] X3 + [2] = a__if(mark(X1),X2,X3) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [2] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [2] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [2] >= [2] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [2] >= [1] X1 + [1] X2 + [1] X3 + [1] = if(X1,X2,X3) a__if(false(),X,Y) = [4] X + [4] Y + [2] >= [4] Y + [0] = mark(Y) a__if(true(),X,Y) = [4] X + [4] Y + [2] >= [4] X + [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [4] X1 + [0] >= [4] X1 + [2] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [4] X + [0] >= [4] X + [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__div) = [1] x1 + [1] p(a__geq) = [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus) = [1] x1 + [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [5] p(true) = [0] Following rules are strictly oriented: a__minus(s(X),s(Y)) = [1] X + [5] > [1] X + [0] = a__minus(X,Y) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1] X1 + [1] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [2] >= [1] = 0() a__div(s(X),s(Y)) = [1] X + [6] >= [1] X + [6] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__minus(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = minus(X1,X2) a__minus(0(),Y) = [1] >= [1] = 0() mark(0()) = [1] >= [1] = 0() mark(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [1] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__minus(X1,X2) mark(s(X)) = [1] X + [5] >= [1] X + [5] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: mark(div(X1,X2)) -> a__div(mark(X1),X2) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 7] x1 + [1] [0 0] [2] p(a__geq) = [1 1] x1 + [0] [0 0] [0] p(a__if) = [1 4] x1 + [4 1] x2 + [4 1] x3 + [2] [0 0] [0 1] [0 1] [0] p(a__minus) = [0] [0] p(div) = [1 2] x1 + [0] [0 0] [2] p(false) = [0] [0] p(geq) = [1 1] x1 + [0] [0 0] [0] p(if) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [2] [0 0] [0 1] [0 1] [0] p(mark) = [4 1] x1 + [0] [0 1] [0] p(minus) = [0] [0] p(s) = [1 1] x1 + [0] [0 0] [2] p(true) = [0] [0] Following rules are strictly oriented: mark(div(X1,X2)) = [4 8] X1 + [2] [0 0] [2] > [4 8] X1 + [1] [0 0] [2] = a__div(mark(X1),X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 7] X1 + [1] [0 0] [2] >= [1 2] X1 + [0] [0 0] [2] = div(X1,X2) a__div(0(),s(Y)) = [1] [2] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 1] X + [15] [0 0] [2] >= [1 1] X + [12] [0 0] [2] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1 1] X + [0] [0 0] [0] >= [0] [0] = true() a__geq(X1,X2) = [1 1] X1 + [0] [0 0] [0] >= [1 1] X1 + [0] [0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] >= [0] [0] = false() a__geq(s(X),s(Y)) = [1 1] X + [2] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 4] X1 + [4 1] X2 + [4 1] X3 + [2] [0 0] [0 1] [0 1] [0] >= [1 2] X1 + [1 0] X2 + [1 0] X3 + [2] [0 0] [0 1] [0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [4 1] X + [4 1] Y + [2] [0 1] [0 1] [0] >= [4 1] Y + [0] [0 1] [0] = mark(Y) a__if(true(),X,Y) = [4 1] X + [4 1] Y + [2] [0 1] [0 1] [0] >= [4 1] X + [0] [0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] >= [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] >= [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] >= [0] [0] = 0() mark(false()) = [0] [0] >= [0] [0] = false() mark(geq(X1,X2)) = [4 4] X1 + [0] [0 0] [0] >= [1 1] X1 + [0] [0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [4 8] X1 + [4 1] X2 + [4 1] X3 + [8] [0 0] [0 1] [0 1] [0] >= [4 5] X1 + [4 1] X2 + [4 1] X3 + [2] [0 0] [0 1] [0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] >= [0] [0] = a__minus(X1,X2) mark(s(X)) = [4 4] X + [2] [0 0] [2] >= [4 2] X + [0] [0 0] [2] = s(mark(X)) mark(true()) = [0] [0] >= [0] [0] = true() * Step 10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))