* Step 1: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if) = [1] x1 + [2] x2 + [1] x3 + [0] p(le) = [0] p(minus) = [6] x1 + [1] x2 + [0] p(p) = [1] x1 + [5] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: p(0()) = [5] > [0] = 0() p(s(x)) = [1] x + [5] > [1] x + [0] = x Following rules are (at-least) weakly oriented: if(false(),x,y) = [2] x + [1] y + [0] >= [1] y + [0] = y if(true(),x,y) = [2] x + [1] y + [0] >= [1] x + [0] = x le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(x,0()) = [6] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [6] x + [1] y + [0] >= [6] x + [1] y + [10] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) - Weak TRS: p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if) = [1] x1 + [2] x2 + [1] x3 + [0] p(le) = [4] p(minus) = [1] x1 + [1] x2 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: le(0(),y) = [4] > [0] = true() le(s(x),0()) = [4] > [0] = false() Following rules are (at-least) weakly oriented: if(false(),x,y) = [2] x + [1] y + [0] >= [1] y + [0] = y if(true(),x,y) = [2] x + [1] y + [0] >= [1] x + [0] = x le(s(x),s(y)) = [4] >= [4] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [4] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = [0] >= [0] = 0() p(s(x)) = [1] x + [0] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if) = [1] x1 + [8] x2 + [1] x3 + [0] p(le) = [0] p(minus) = [8] x1 + [1] x2 + [1] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: minus(x,0()) = [8] x + [1] > [1] x + [0] = x Following rules are (at-least) weakly oriented: if(false(),x,y) = [8] x + [1] y + [0] >= [1] y + [0] = y if(true(),x,y) = [8] x + [1] y + [0] >= [1] x + [0] = x le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(x,s(y)) = [8] x + [1] y + [1] >= [8] x + [1] y + [1] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = [0] >= [0] = 0() p(s(x)) = [1] x + [0] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(s(x),s(y)) -> le(x,y) minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() minus(x,0()) -> x p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} 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(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(le) = [1] p(minus) = [8] x1 + [1] x2 + [0] p(p) = [1] x1 + [8] p(s) = [1] x1 + [8] p(true) = [1] Following rules are strictly oriented: if(false(),x,y) = [1] x + [1] y + [1] > [1] y + [0] = y if(true(),x,y) = [1] x + [1] y + [1] > [1] x + [0] = x Following rules are (at-least) weakly oriented: le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [8] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [8] x + [1] y + [8] >= [8] x + [1] y + [25] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = [8] >= [0] = 0() p(s(x)) = [1] x + [16] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: MI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(s(x),s(y)) -> le(x,y) minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) - Weak TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() minus(x,0()) -> x p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))): The following argument positions are considered usable: uargs(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [0] p(false) = [0] [0] [0] p(if) = [1 0 0] [2 0 2] [1 0 0] [1] [0 0 0] x_1 + [0 2 1] x_2 + [0 1 0] x_3 + [2] [1 0 0] [1 1 2] [0 0 1] [0] p(le) = [0 0 0] [0 0 0] [0] [2 1 0] x_1 + [0 0 2] x_2 + [3] [1 0 0] [0 0 2] [0] p(minus) = [3 2 0] [2 0 1] [3] [3 2 0] x_1 + [2 1 3] x_2 + [1] [3 2 1] [3 0 3] [0] p(p) = [1 0 0] [0] [1 0 0] x_1 + [2] [0 1 0] [0] p(s) = [1 1 0] [0] [0 0 1] x_1 + [0] [0 0 1] [2] p(true) = [0] [3] [0] Following rules are strictly oriented: minus(x,s(y)) = [3 2 0] [2 2 1] [5] [3 2 0] x + [2 2 4] y + [7] [3 2 1] [3 3 3] [6] > [3 2 0] [2 2 1] [4] [3 2 0] x + [2 2 1] y + [7] [3 2 0] [3 3 3] [3] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) Following rules are (at-least) weakly oriented: if(false(),x,y) = [2 0 2] [1 0 0] [1] [0 2 1] x + [0 1 0] y + [2] [1 1 2] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] y + [0] [0 0 1] [0] = y if(true(),x,y) = [2 0 2] [1 0 0] [1] [0 2 1] x + [0 1 0] y + [2] [1 1 2] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x le(0(),y) = [0 0 0] [0] [0 0 2] y + [3] [0 0 2] [0] >= [0] [3] [0] = true() le(s(x),0()) = [0 0 0] [0] [2 2 1] x + [3] [1 1 0] [0] >= [0] [0] [0] = false() le(s(x),s(y)) = [0 0 0] [0 0 0] [0] [2 2 1] x + [0 0 2] y + [7] [1 1 0] [0 0 2] [4] >= [0 0 0] [0 0 0] [0] [2 1 0] x + [0 0 2] y + [3] [1 0 0] [0 0 2] [0] = le(x,y) minus(x,0()) = [3 2 0] [3] [3 2 0] x + [1] [3 2 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x p(0()) = [0] [2] [0] >= [0] [0] [0] = 0() p(s(x)) = [1 1 0] [0] [1 1 0] x + [2] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x * Step 6: MI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(s(x),s(y)) -> le(x,y) - Weak TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} and constructors {0,false,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))): The following argument positions are considered usable: uargs(if) = {1,3}, uargs(minus) = {2}, uargs(p) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(false) = [0] [0] [1] [0] p(if) = [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] x_1 + [0 1 0 0] x_2 + [0 1 0 0] x_3 + [0] [0 0 0 0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 1] [0 0 0 1] [0] p(le) = [0 0 0 0] [0 0 0 1] [0] [1 0 0 1] x_1 + [0 0 0 0] x_2 + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 1 0] [0 0 0 1] [1] p(minus) = [1 1 0 0] [1 0 1 0] [0] [1 1 1 0] x_1 + [1 0 1 0] x_2 + [1] [1 1 1 0] [1 0 1 0] [0] [1 1 1 1] [1 1 0 0] [0] p(p) = [1 0 0 0] [0] [1 0 0 0] x_1 + [1] [0 1 0 0] [0] [0 0 1 0] [0] p(s) = [1 1 0 1] [1] [0 0 1 0] x_1 + [0] [0 0 1 1] [1] [0 0 0 1] [1] p(true) = [0] [0] [0] [1] Following rules are strictly oriented: le(s(x),s(y)) = [0 0 0 0] [0 0 0 1] [1] [1 1 0 2] x + [0 0 0 0] y + [3] [0 0 0 0] [0 0 1 2] [3] [0 0 1 1] [0 0 0 1] [3] > [0 0 0 0] [0 0 0 1] [0] [1 0 0 1] x + [0 0 0 0] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 1 0] [0 0 0 1] [1] = le(x,y) Following rules are (at-least) weakly oriented: if(false(),x,y) = [1 0 0 0] [1 0 0 0] [0] [0 1 0 0] x + [0 1 0 0] y + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] y + [0] [0 0 1 0] [0] [0 0 0 1] [0] = y if(true(),x,y) = [1 0 0 0] [1 0 0 0] [0] [0 1 0 0] x + [0 1 0 0] y + [0] [0 0 1 0] [0 0 1 0] [0] [0 0 0 1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x le(0(),y) = [0 0 0 1] [0] [0 0 0 0] y + [1] [0 0 1 1] [1] [0 0 0 1] [1] >= [0] [0] [0] [1] = true() le(s(x),0()) = [0 0 0 0] [0] [1 1 0 2] x + [3] [0 0 0 0] [1] [0 0 1 1] [2] >= [0] [0] [1] [0] = false() minus(x,0()) = [1 1 0 0] [0] [1 1 1 0] x + [1] [1 1 1 0] [0] [1 1 1 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x minus(x,s(y)) = [1 1 0 0] [1 1 1 2] [2] [1 1 1 0] x + [1 1 1 2] y + [3] [1 1 1 0] [1 1 1 2] [2] [1 1 1 1] [1 1 1 1] [1] >= [1 1 0 0] [1 1 1 2] [2] [1 1 0 0] x + [1 1 1 1] y + [2] [1 1 1 0] [1 1 1 1] [2] [1 1 1 0] [1 1 1 1] [1] = if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) = [0] [1] [0] [0] >= [0] [0] [0] [0] = 0() p(s(x)) = [1 1 0 1] [1] [1 1 0 1] x + [2] [0 0 1 0] [0] [0 0 1 1] [1] >= [1 0 0 0] [0] [0 1 0 0] x + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(x,s(y)) -> if(le(x,s(y)),0(),p(minus(x,p(s(y))))) p(0()) -> 0() p(s(x)) -> x - Signature: {if/3,le/2,minus/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {if,le,minus,p} 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^3))