* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(0()) -> true() f(1()) -> false() f(s(x)) -> f(x) g(x,c(y)) -> g(x,g(s(c(y)),y)) g(s(x),s(y)) -> if(f(x),s(x),s(y)) if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,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(g) = {2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(1) = [2] p(c) = [1] x1 + [0] p(f) = [0] p(false) = [8] p(g) = [1] x2 + [15] p(if) = [1] x1 + [7] x2 + [4] x3 + [1] p(s) = [1] p(true) = [3] Following rules are strictly oriented: g(s(x),s(y)) = [16] > [12] = if(f(x),s(x),s(y)) if(false(),x,y) = [7] x + [4] y + [9] > [1] y + [0] = y if(true(),x,y) = [7] x + [4] y + [4] > [1] x + [0] = x Following rules are (at-least) weakly oriented: f(0()) = [0] >= [3] = true() f(1()) = [0] >= [8] = false() f(s(x)) = [0] >= [0] = f(x) g(x,c(y)) = [1] y + [15] >= [1] y + [30] = g(x,g(s(c(y)),y)) 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: f(0()) -> true() f(1()) -> false() f(s(x)) -> f(x) g(x,c(y)) -> g(x,g(s(c(y)),y)) - Weak TRS: g(s(x),s(y)) -> if(f(x),s(x),s(y)) if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,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(g) = {2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(1) = [1] p(c) = [1] x1 + [10] p(f) = [4] p(false) = [4] p(g) = [1] x2 + [9] p(if) = [1] x1 + [1] x2 + [2] x3 + [5] p(s) = [0] p(true) = [0] Following rules are strictly oriented: f(0()) = [4] > [0] = true() g(x,c(y)) = [1] y + [19] > [1] y + [18] = g(x,g(s(c(y)),y)) Following rules are (at-least) weakly oriented: f(1()) = [4] >= [4] = false() f(s(x)) = [4] >= [4] = f(x) g(s(x),s(y)) = [9] >= [9] = if(f(x),s(x),s(y)) if(false(),x,y) = [1] x + [2] y + [9] >= [1] y + [0] = y if(true(),x,y) = [1] x + [2] y + [5] >= [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^2)) + Considered Problem: - Strict TRS: f(1()) -> false() f(s(x)) -> f(x) - Weak TRS: f(0()) -> true() g(x,c(y)) -> g(x,g(s(c(y)),y)) g(s(x),s(y)) -> if(f(x),s(x),s(y)) if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,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(g) = {2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(1) = [0] p(c) = [1] x1 + [8] p(f) = [4] p(false) = [0] p(g) = [1] x2 + [8] p(if) = [1] x1 + [2] x2 + [2] x3 + [0] p(s) = [0] p(true) = [4] Following rules are strictly oriented: f(1()) = [4] > [0] = false() Following rules are (at-least) weakly oriented: f(0()) = [4] >= [4] = true() f(s(x)) = [4] >= [4] = f(x) g(x,c(y)) = [1] y + [16] >= [1] y + [16] = g(x,g(s(c(y)),y)) g(s(x),s(y)) = [8] >= [4] = if(f(x),s(x),s(y)) if(false(),x,y) = [2] x + [2] y + [0] >= [1] y + [0] = y if(true(),x,y) = [2] x + [2] y + [4] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(s(x)) -> f(x) - Weak TRS: f(0()) -> true() f(1()) -> false() g(x,c(y)) -> g(x,g(s(c(y)),y)) g(s(x),s(y)) -> if(f(x),s(x),s(y)) if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(g) = {2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] [2] p(1) = [0] [1] p(c) = [1 0] x_1 + [2] [0 0] [0] p(f) = [0 2] x_1 + [0] [0 1] [2] p(false) = [2] [3] p(g) = [3 0] x_1 + [1 0] x_2 + [0] [4 3] [4 0] [2] p(if) = [2 0] x_1 + [1 0] x_2 + [1 0] x_3 + [0] [4 1] [0 1] [0 2] [0] p(s) = [0 2] x_1 + [0] [0 1] [4] p(true) = [2] [0] Following rules are strictly oriented: f(s(x)) = [0 2] x + [8] [0 1] [6] > [0 2] x + [0] [0 1] [2] = f(x) Following rules are (at-least) weakly oriented: f(0()) = [4] [4] >= [2] [0] = true() f(1()) = [2] [3] >= [2] [3] = false() g(x,c(y)) = [3 0] x + [1 0] y + [2] [4 3] [4 0] [10] >= [3 0] x + [1 0] y + [0] [4 3] [4 0] [2] = g(x,g(s(c(y)),y)) g(s(x),s(y)) = [0 6] x + [0 2] y + [0] [0 11] [0 8] [14] >= [0 6] x + [0 2] y + [0] [0 10] [0 2] [14] = if(f(x),s(x),s(y)) if(false(),x,y) = [1 0] x + [1 0] y + [4] [0 1] [0 2] [11] >= [1 0] y + [0] [0 1] [0] = y if(true(),x,y) = [1 0] x + [1 0] y + [4] [0 1] [0 2] [8] >= [1 0] x + [0] [0 1] [0] = x * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(x)) -> f(x) g(x,c(y)) -> g(x,g(s(c(y)),y)) g(s(x),s(y)) -> if(f(x),s(x),s(y)) if(false(),x,y) -> y if(true(),x,y) -> x - Signature: {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))