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) = [1] p(div) = [1] x1 + [8] x2 + [1] p(false) = [0] p(geq) = [13] p(if) = [1] x1 + [1] x2 + [4] x3 + [1] p(minus) = [0] p(s) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: div(0(),s(Y)) = [8] Y + [10] > [1] = 0() geq(X,0()) = [13] > [0] = true() geq(0(),s(Y)) = [13] > [0] = false() if(false(),X,Y) = [1] X + [4] Y + [1] > [1] Y + [0] = Y if(true(),X,Y) = [1] X + [4] Y + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: div(s(X),s(Y)) = [1] X + [8] Y + [10] >= [8] Y + [28] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(s(X),s(Y)) = [13] >= [13] = geq(X,Y) minus(0(),Y) = [0] >= [1] = 0() minus(s(X),s(Y)) = [0] >= [0] = 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(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(0(),Y) -> 0() 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 - 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 + [8] x2 + [0] p(false) = [3] p(geq) = [8] p(if) = [1] x1 + [1] x2 + [8] x3 + [9] p(minus) = [1] p(s) = [1] x1 + [0] p(true) = [1] Following rules are strictly oriented: minus(0(),Y) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [8] Y + [0] >= [0] = 0() div(s(X),s(Y)) = [1] X + [8] Y + [0] >= [8] Y + [18] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [8] >= [1] = true() geq(0(),s(Y)) = [8] >= [3] = false() geq(s(X),s(Y)) = [8] >= [8] = geq(X,Y) if(false(),X,Y) = [1] X + [8] Y + [12] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [8] Y + [10] >= [1] X + [0] = X minus(s(X),s(Y)) = [1] >= [1] = minus(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^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: 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) = [8] p(div) = [1] x1 + [1] x2 + [0] p(false) = [10] p(geq) = [1] x1 + [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [6] p(minus) = [9] p(s) = [1] x1 + [1] p(true) = [2] Following rules are strictly oriented: geq(s(X),s(Y)) = [1] X + [5] > [1] X + [4] = geq(X,Y) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [1] Y + [9] >= [8] = 0() div(s(X),s(Y)) = [1] X + [1] Y + [2] >= [1] X + [1] Y + [29] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = [1] X + [4] >= [2] = true() geq(0(),s(Y)) = [12] >= [10] = false() if(false(),X,Y) = [1] X + [1] Y + [16] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [1] Y + [8] >= [1] X + [0] = X minus(0(),Y) = [9] >= [8] = 0() minus(s(X),s(Y)) = [9] >= [9] = minus(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^1)) + Considered Problem: - Strict TRS: div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: div(0(),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) = [5] p(div) = [1] x1 + [8] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [3] 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)) = [13] >= [5] = 0() div(s(X),s(Y)) = [1] X + [16] >= [1] X + [31] = 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 + [3] Y + [0] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [3] Y + [0] >= [1] X + [0] = X minus(0(),Y) = [5] >= [5] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: 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()) - Weak TRS: div(0(),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: 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) = [2] x1 + [1] x2 + [1] p(false) = [0] p(geq) = [0] p(if) = [4] x1 + [1] x2 + [1] x3 + [0] p(minus) = [0] p(s) = [1] x1 + [2] p(true) = [0] Following rules are strictly oriented: div(s(X),s(Y)) = [2] X + [1] Y + [7] > [1] Y + [5] = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [1] Y + [3] >= [0] = 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 + [1] Y + [0] >= [1] Y + [0] = Y if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = X minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [0] >= [0] = minus(X,Y) * 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))