WORST_CASE(?,O(n^3)) * Step 1: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(0(),Y) -> 0() minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [9] p(ifMinus) = [1] x1 + [1] x2 + [6] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [10] p(true) = [0] Following rules are strictly oriented: ifMinus(false(),s(X),Y) = [1] X + [25] > [1] X + [10] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [16] > [0] = 0() Following rules are (at-least) weakly oriented: le(0(),Y) = [0] >= [0] = true() le(s(X),0()) = [0] >= [9] = false() le(s(X),s(Y)) = [0] >= [0] = le(X,Y) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),Y) = [1] X + [10] >= [1] X + [16] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [0] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [10] >= [1] X + [10] = s(quot(minus(X,Y),s(Y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(0(),Y) -> 0() minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {ifMinus,le,minus,quot} TcT has computed the following interpretation: p(0) = [13] p(false) = [1] p(ifMinus) = [1] x1 + [1] x2 + [2] p(le) = [1] p(minus) = [1] x1 + [3] p(quot) = [2] x1 + [1] p(s) = [1] x1 + [10] p(true) = [1] Following rules are strictly oriented: minus(0(),Y) = [16] > [13] = 0() quot(0(),s(Y)) = [27] > [13] = 0() quot(s(X),s(Y)) = [2] X + [21] > [2] X + [17] = s(quot(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [13] >= [1] X + [13] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [13] >= [13] = 0() le(0(),Y) = [1] >= [1] = true() le(s(X),0()) = [1] >= [1] = false() le(s(X),s(Y)) = [1] >= [1] = le(X,Y) minus(s(X),Y) = [1] X + [13] >= [1] X + [13] = ifMinus(le(s(X),Y),s(X),Y) * Step 3: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() minus(0(),Y) -> 0() quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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(ifMinus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [7] p(ifMinus) = [1] x1 + [1] p(le) = [9] p(minus) = [0] p(quot) = [1] x1 + [2] x2 + [7] p(s) = [1] x1 + [8] p(true) = [0] Following rules are strictly oriented: le(0(),Y) = [9] > [0] = true() le(s(X),0()) = [9] > [7] = false() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [8] >= [8] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] >= [0] = 0() le(s(X),s(Y)) = [9] >= [9] = le(X,Y) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),Y) = [0] >= [10] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [2] Y + [23] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [2] Y + [31] >= [2] Y + [31] = s(quot(minus(X,Y),s(Y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(s(X),s(Y)) -> le(X,Y) minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() le(0(),Y) -> true() le(s(X),0()) -> false() minus(0(),Y) -> 0() quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} and constructors {0,false,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(ifMinus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {ifMinus,le,minus,quot} TcT has computed the following interpretation: p(0) = [0] [2] p(false) = [0] [0] p(ifMinus) = [4 0] x1 + [1 2] x2 + [3] [0 0] [0 1] [0] p(le) = [0] [0] p(minus) = [1 2] x1 + [5] [0 1] [0] p(quot) = [2 5] x1 + [0 1] x2 + [0] [0 1] [0 0] [0] p(s) = [1 4] x1 + [0] [0 1] [2] p(true) = [0] [0] Following rules are strictly oriented: minus(s(X),Y) = [1 6] X + [9] [0 1] [2] > [1 6] X + [7] [0 1] [2] = ifMinus(le(s(X),Y),s(X),Y) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1 6] X + [7] [0 1] [2] >= [1 6] X + [5] [0 1] [2] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1 6] X + [7] [0 1] [2] >= [0] [2] = 0() le(0(),Y) = [0] [0] >= [0] [0] = true() le(s(X),0()) = [0] [0] >= [0] [0] = false() le(s(X),s(Y)) = [0] [0] >= [0] [0] = le(X,Y) minus(0(),Y) = [9] [2] >= [0] [2] = 0() quot(0(),s(Y)) = [0 1] Y + [12] [0 0] [2] >= [0] [2] = 0() quot(s(X),s(Y)) = [2 13] X + [0 1] Y + [12] [0 1] [0 0] [2] >= [2 13] X + [0 1] Y + [12] [0 1] [0 0] [2] = s(quot(minus(X,Y),s(Y))) * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(s(X),s(Y)) -> le(X,Y) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() le(0(),Y) -> true() le(s(X),0()) -> false() minus(0(),Y) -> 0() minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} and constructors {0,false,s,true} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(ifMinus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {ifMinus,le,minus,quot} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(false) = [2] [0] [0] p(ifMinus) = [1 0 0] [1 1 0] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [0] p(le) = [0 0 1] [0] [2 0 0] x1 + [0] [0 0 0] [0] p(minus) = [1 1 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(quot) = [2 2 0] [0 1 0] [2] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(s) = [1 2 0] [0] [0 1 2] x1 + [0] [0 0 1] [2] p(true) = [0] [0] [0] Following rules are strictly oriented: le(s(X),s(Y)) = [0 0 1] [2] [2 4 0] X + [0] [0 0 0] [0] > [0 0 1] [0] [2 0 0] X + [0] [0 0 0] [0] = le(X,Y) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1 3 2] [2] [0 1 2] X + [0] [0 0 1] [2] >= [1 3 1] [0] [0 1 2] X + [0] [0 0 1] [2] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1 3 2] [0] [0 1 2] X + [0] [0 0 1] [2] >= [0] [0] [0] = 0() le(0(),Y) = [0] [0] [0] >= [0] [0] [0] = true() le(s(X),0()) = [0 0 1] [2] [2 4 0] X + [0] [0 0 0] [0] >= [2] [0] [0] = false() minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() minus(s(X),Y) = [1 3 3] [2] [0 1 2] X + [0] [0 0 1] [2] >= [1 3 3] [2] [0 1 2] X + [0] [0 0 1] [2] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [0 1 2] [2] [0 0 0] Y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() quot(s(X),s(Y)) = [2 6 4] [0 1 2] [2] [0 1 2] X + [0 0 0] Y + [0] [0 0 1] [0 0 0] [2] >= [2 6 2] [0 1 2] [2] [0 1 2] X + [0 0 0] Y + [0] [0 0 1] [0 0 0] [2] = s(quot(minus(X,Y),s(Y))) * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(true(),s(X),Y) -> 0() le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(0(),Y) -> 0() minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Signature: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} 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))