WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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) = {1,3}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [5] p(0) = [8] p(div) = [1] x1 + [1] x2 + [0] p(false) = [0] p(if) = [1] x1 + [2] x2 + [1] x3 + [4] p(lt) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: -(x,0()) = [1] x + [5] > [1] x + [0] = x -(0(),s(y)) = [13] > [8] = 0() if(false(),x,y) = [2] x + [1] y + [4] > [1] y + [0] = y if(true(),x,y) = [2] x + [1] y + [4] > [1] x + [0] = x Following rules are (at-least) weakly oriented: -(s(x),s(y)) = [1] x + [5] >= [1] x + [5] = -(x,y) div(x,0()) = [1] x + [8] >= [8] = 0() div(0(),y) = [1] y + [8] >= [8] = 0() div(s(x),s(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [25] = if(lt(x,y),0(),s(div(-(x,y),s(y)))) lt(x,0()) = [0] >= [0] = false() lt(0(),s(y)) = [0] >= [0] = true() lt(s(x),s(y)) = [0] >= [0] = lt(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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) = {1,3}, uargs(s) = {1} Following symbols are considered usable: {-,div,if,lt} TcT has computed the following interpretation: p(-) = [1] x1 + [0] p(0) = [0] p(div) = [2] x1 + [4] p(false) = [0] p(if) = [8] x1 + [1] x2 + [1] x3 + [0] p(lt) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: div(x,0()) = [2] x + [4] > [0] = 0() div(0(),y) = [4] > [0] = 0() Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [0] >= [1] x + [0] = x -(0(),s(y)) = [0] >= [0] = 0() -(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = -(x,y) div(s(x),s(y)) = [2] x + [4] >= [2] x + [4] = if(lt(x,y),0(),s(div(-(x,y),s(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 lt(x,0()) = [0] >= [0] = false() lt(0(),s(y)) = [0] >= [0] = true() lt(s(x),s(y)) = [0] >= [0] = lt(x,y) * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(s(x),s(y)) -> -(x,y) div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() div(x,0()) -> 0() div(0(),y) -> 0() if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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) = {1,3}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [2] p(0) = [0] p(div) = [1] x1 + [8] x2 + [9] p(false) = [1] p(if) = [1] x1 + [2] x2 + [1] x3 + [13] p(lt) = [3] p(s) = [1] x1 + [0] p(true) = [3] Following rules are strictly oriented: lt(x,0()) = [3] > [1] = false() Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [2] >= [1] x + [0] = x -(0(),s(y)) = [2] >= [0] = 0() -(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = -(x,y) div(x,0()) = [1] x + [9] >= [0] = 0() div(0(),y) = [8] y + [9] >= [0] = 0() div(s(x),s(y)) = [1] x + [8] y + [9] >= [1] x + [8] y + [27] = if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) = [2] x + [1] y + [14] >= [1] y + [0] = y if(true(),x,y) = [2] x + [1] y + [16] >= [1] x + [0] = x lt(0(),s(y)) = [3] >= [3] = true() lt(s(x),s(y)) = [3] >= [3] = lt(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: -(s(x),s(y)) -> -(x,y) div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() div(x,0()) -> 0() div(0(),y) -> 0() if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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) = {1,3}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [2] p(0) = [0] p(div) = [1] x1 + [2] x2 + [7] p(false) = [2] p(if) = [1] x1 + [4] x2 + [1] x3 + [0] p(lt) = [10] p(s) = [1] x1 + [4] p(true) = [8] Following rules are strictly oriented: -(s(x),s(y)) = [1] x + [6] > [1] x + [2] = -(x,y) lt(0(),s(y)) = [10] > [8] = true() Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [2] >= [1] x + [0] = x -(0(),s(y)) = [2] >= [0] = 0() div(x,0()) = [1] x + [7] >= [0] = 0() div(0(),y) = [2] y + [7] >= [0] = 0() div(s(x),s(y)) = [1] x + [2] y + [19] >= [1] x + [2] y + [31] = if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) = [4] x + [1] y + [2] >= [1] y + [0] = y if(true(),x,y) = [4] x + [1] y + [8] >= [1] x + [0] = x lt(x,0()) = [10] >= [2] = false() lt(s(x),s(y)) = [10] >= [10] = lt(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) lt(s(x),s(y)) -> lt(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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) = {1,3}, uargs(s) = {1} Following symbols are considered usable: {-,div,if,lt} TcT has computed the following interpretation: p(-) = [1] x1 + [0] p(0) = [0] p(div) = [2] x1 + [0] p(false) = [1] p(if) = [2] x1 + [2] x2 + [1] x3 + [2] p(lt) = [1] p(s) = [1] x1 + [8] p(true) = [1] Following rules are strictly oriented: div(s(x),s(y)) = [2] x + [16] > [2] x + [12] = if(lt(x,y),0(),s(div(-(x,y),s(y)))) Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [0] >= [1] x + [0] = x -(0(),s(y)) = [0] >= [0] = 0() -(s(x),s(y)) = [1] x + [8] >= [1] x + [0] = -(x,y) div(x,0()) = [2] x + [0] >= [0] = 0() div(0(),y) = [0] >= [0] = 0() if(false(),x,y) = [2] x + [1] y + [4] >= [1] y + [0] = y if(true(),x,y) = [2] x + [1] y + [4] >= [1] x + [0] = x lt(x,0()) = [1] >= [1] = false() lt(0(),s(y)) = [1] >= [1] = true() lt(s(x),s(y)) = [1] >= [1] = lt(x,y) * Step 6: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: lt(s(x),s(y)) -> lt(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(div) = {1}, uargs(if) = {1,3}, uargs(s) = {1} Following symbols are considered usable: {-,div,if,lt} TcT has computed the following interpretation: p(-) = x1 p(0) = 0 p(div) = x1 + x1^2 p(false) = 0 p(if) = x1 + x2 + x3 p(lt) = x1 p(s) = 1 + x1 p(true) = 0 Following rules are strictly oriented: lt(s(x),s(y)) = 1 + x > x = lt(x,y) Following rules are (at-least) weakly oriented: -(x,0()) = x >= x = x -(0(),s(y)) = 0 >= 0 = 0() -(s(x),s(y)) = 1 + x >= x = -(x,y) div(x,0()) = x + x^2 >= 0 = 0() div(0(),y) = 0 >= 0 = 0() div(s(x),s(y)) = 2 + 3*x + x^2 >= 1 + 2*x + x^2 = if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) = x + y >= y = y if(true(),x,y) = x + y >= x = x lt(x,0()) = x >= 0 = false() lt(0(),s(y)) = 0 >= 0 = true() * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} 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^2))