WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum WORST_CASE(Omega(n^1),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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: NaturalPI 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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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 p(false) = 0 p(ifMinus) = x1 p(le) = 0 p(minus) = 0 p(quot) = 6 + 8*x1 p(s) = x1 p(true) = 0 Following rules are strictly oriented: quot(0(),s(Y)) = 6 > 0 = 0() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = 0 >= 0 = s(minus(X,Y)) ifMinus(true(),s(X),Y) = 0 >= 0 = 0() le(0(),Y) = 0 >= 0 = true() le(s(X),0()) = 0 >= 0 = false() le(s(X),s(Y)) = 0 >= 0 = le(X,Y) minus(0(),Y) = 0 >= 0 = 0() minus(s(X),Y) = 0 >= 0 = ifMinus(le(s(X),Y),s(X),Y) quot(s(X),s(Y)) = 6 + 8*X >= 6 = s(quot(minus(X,Y),s(Y))) ** Step 1.b:2: NaturalMI 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(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Weak TRS: quot(0(),s(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) = [0] p(false) = [0] p(ifMinus) = [4] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [4] x2 + [6] p(s) = [1] x1 + [4] p(true) = [0] Following rules are strictly oriented: ifMinus(true(),s(X),Y) = [1] X + [4] > [0] = 0() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [4] >= [1] X + [4] = s(minus(X,Y)) le(0(),Y) = [0] >= [0] = true() le(s(X),0()) = [0] >= [0] = false() le(s(X),s(Y)) = [0] >= [0] = le(X,Y) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),Y) = [1] X + [4] >= [1] X + [4] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [4] Y + [22] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [4] Y + [26] >= [1] X + [4] Y + [26] = s(quot(minus(X,Y),s(Y))) ** Step 1.b:3: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) 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(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y))) - Weak TRS: ifMinus(true(),s(X),Y) -> 0() quot(0(),s(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) = [0] p(false) = [0] p(ifMinus) = [8] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [4] x1 + [0] p(s) = [1] x1 + [4] p(true) = [0] Following rules are strictly oriented: quot(s(X),s(Y)) = [4] X + [16] > [4] X + [4] = s(quot(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [4] >= [1] X + [4] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [4] >= [0] = 0() le(0(),Y) = [0] >= [0] = true() le(s(X),0()) = [0] >= [0] = false() le(s(X),s(Y)) = [0] >= [0] = le(X,Y) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),Y) = [1] X + [4] >= [1] X + [4] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [0] >= [0] = 0() ** Step 1.b:4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) 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) - Weak TRS: ifMinus(true(),s(X),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 = 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) = [0] p(false) = [0] p(ifMinus) = [8] x1 + [1] x2 + [1] p(le) = [0] p(minus) = [1] x1 + [1] p(quot) = [4] x1 + [8] p(s) = [1] x1 + [5] p(true) = [0] Following rules are strictly oriented: minus(0(),Y) = [1] > [0] = 0() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [6] >= [1] X + [6] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [6] >= [0] = 0() le(0(),Y) = [0] >= [0] = true() le(s(X),0()) = [0] >= [0] = false() le(s(X),s(Y)) = [0] >= [0] = le(X,Y) minus(s(X),Y) = [1] X + [6] >= [1] X + [6] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [8] >= [0] = 0() quot(s(X),s(Y)) = [4] X + [28] >= [4] X + [17] = s(quot(minus(X,Y),s(Y))) ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) 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(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: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): 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) = 1 p(false) = 1 p(ifMinus) = 4*x1 + x2 p(le) = 1 p(minus) = 4 + x1 p(quot) = 4 + 2*x1 + x2 p(s) = 8 + x1 p(true) = 0 Following rules are strictly oriented: le(0(),Y) = 1 > 0 = true() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = 12 + X >= 12 + X = s(minus(X,Y)) ifMinus(true(),s(X),Y) = 8 + X >= 1 = 0() le(s(X),0()) = 1 >= 1 = false() le(s(X),s(Y)) = 1 >= 1 = le(X,Y) minus(0(),Y) = 5 >= 1 = 0() minus(s(X),Y) = 12 + X >= 12 + X = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = 14 + Y >= 1 = 0() quot(s(X),s(Y)) = 28 + 2*X + Y >= 28 + 2*X + Y = s(quot(minus(X,Y),s(Y))) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) 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(true(),s(X),Y) -> 0() le(0(),Y) -> true() 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 = 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) = [2] [2] [0] p(false) = [0] [1] [0] p(ifMinus) = [1 2 0] [1 1 0] [0 0 0] [0] [0 0 2] x1 + [0 2 0] x2 + [0 3 0] x3 + [1] [0 0 0] [0 1 0] [0 0 0] [2] p(le) = [0 0 0] [0] [0 0 0] x2 + [1] [1 0 0] [2] p(minus) = [1 0 1] [0 0 0] [0] [0 0 2] x1 + [2 3 0] x2 + [2] [0 0 1] [0 0 0] [0] p(quot) = [2 0 0] [0] [0 1 2] x1 + [2] [0 0 1] [0] p(s) = [1 0 2] [0] [0 0 1] x1 + [0] [0 0 1] [2] p(true) = [0] [1] [1] Following rules are strictly oriented: ifMinus(false(),s(X),Y) = [1 0 3] [0 0 0] [2] [0 0 2] X + [0 3 0] Y + [1] [0 0 1] [0 0 0] [2] > [1 0 3] [0] [0 0 1] X + [0] [0 0 1] [2] = s(minus(X,Y)) Following rules are (at-least) weakly oriented: ifMinus(true(),s(X),Y) = [1 0 3] [0 0 0] [2] [0 0 2] X + [0 3 0] Y + [3] [0 0 1] [0 0 0] [2] >= [2] [2] [0] = 0() le(0(),Y) = [0 0 0] [0] [0 0 0] Y + [1] [1 0 0] [2] >= [0] [1] [1] = true() le(s(X),0()) = [0] [1] [4] >= [0] [1] [0] = false() le(s(X),s(Y)) = [0 0 0] [0] [0 0 0] Y + [1] [1 0 2] [2] >= [0 0 0] [0] [0 0 0] Y + [1] [1 0 0] [2] = le(X,Y) minus(0(),Y) = [0 0 0] [2] [2 3 0] Y + [2] [0 0 0] [0] >= [2] [2] [0] = 0() minus(s(X),Y) = [1 0 3] [0 0 0] [2] [0 0 2] X + [2 3 0] Y + [6] [0 0 1] [0 0 0] [2] >= [1 0 3] [0 0 0] [2] [0 0 2] X + [2 3 0] Y + [5] [0 0 1] [0 0 0] [2] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [4] [4] [0] >= [2] [2] [0] = 0() quot(s(X),s(Y)) = [2 0 4] [0] [0 0 3] X + [6] [0 0 1] [2] >= [2 0 4] [0] [0 0 1] X + [0] [0 0 1] [2] = s(quot(minus(X,Y),s(Y))) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: 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() le(0(),Y) -> true() 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 = 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) = [0] [0] [0] p(ifMinus) = [2 0 0] [1 1 0] [0 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 0] [2 3 0] [0] p(le) = [0] [0] [0] p(minus) = [1 1 0] [0 0 0] [1] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [2 3 0] [0] p(quot) = [2 0 0] [0] [0 1 0] x1 + [0] [1 2 0] [0] p(s) = [1 2 0] [2] [0 1 0] x1 + [1] [0 0 0] [0] p(true) = [0] [0] [0] Following rules are strictly oriented: minus(s(X),Y) = [1 3 0] [0 0 0] [4] [0 1 0] X + [0 0 0] Y + [1] [0 0 0] [2 3 0] [0] > [1 3 0] [0 0 0] [3] [0 1 0] X + [0 0 0] Y + [1] [0 0 0] [2 3 0] [0] = ifMinus(le(s(X),Y),s(X),Y) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1 3 0] [0 0 0] [3] [0 1 0] X + [0 0 0] Y + [1] [0 0 0] [2 3 0] [0] >= [1 3 0] [3] [0 1 0] X + [1] [0 0 0] [0] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1 3 0] [0 0 0] [3] [0 1 0] X + [0 0 0] Y + [1] [0 0 0] [2 3 0] [0] >= [0] [0] [0] = 0() le(0(),Y) = [0] [0] [0] >= [0] [0] [0] = true() le(s(X),0()) = [0] [0] [0] >= [0] [0] [0] = false() le(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = le(X,Y) minus(0(),Y) = [0 0 0] [1] [0 0 0] Y + [0] [2 3 0] [0] >= [0] [0] [0] = 0() quot(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() quot(s(X),s(Y)) = [2 4 0] [4] [0 1 0] X + [1] [1 4 0] [4] >= [2 4 0] [4] [0 1 0] X + [1] [0 0 0] [0] = s(quot(minus(X,Y),s(Y))) ** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: le(s(X),0()) -> false() 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() 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) = [0] [0] [0] p(ifMinus) = [1 0 0] [1 0 1] [0 0 0] [0] [0 2 1] x1 + [2 0 0] x2 + [0 2 0] x3 + [0] [0 0 0] [0 0 1] [0 0 0] [0] p(le) = [0 0 0] [0 0 0] [1] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 3] [2 0 1] [3] p(minus) = [1 0 1] [0 0 0] [1] [2 0 3] x1 + [2 2 3] x2 + [3] [0 0 1] [0 0 0] [0] p(quot) = [2 0 2] [2 0 0] [0] [0 1 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 0] [0] p(s) = [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: le(s(X),0()) = [0 0 0] [1] [0 0 0] X + [0] [0 0 3] [6] > [0] [0] [0] = false() Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1 0 3] [0 0 0] [1] [2 0 4] X + [0 2 0] Y + [0] [0 0 1] [0 0 0] [1] >= [1 0 3] [1] [0 0 0] X + [0] [0 0 1] [1] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1 0 3] [0 0 0] [1] [2 0 4] X + [0 2 0] Y + [0] [0 0 1] [0 0 0] [1] >= [0] [0] [0] = 0() le(0(),Y) = [0 0 0] [1] [0 0 1] Y + [0] [2 0 1] [3] >= [0] [0] [0] = true() le(s(X),s(Y)) = [0 0 0] [0 0 0] [1] [0 0 0] X + [0 0 1] Y + [1] [0 0 3] [2 0 5] [7] >= [0 0 0] [0 0 0] [1] [0 0 0] X + [0 0 1] Y + [0] [0 0 3] [2 0 1] [3] = le(X,Y) minus(0(),Y) = [0 0 0] [1] [2 2 3] Y + [3] [0 0 0] [0] >= [0] [0] [0] = 0() minus(s(X),Y) = [1 0 3] [0 0 0] [2] [2 0 7] X + [2 2 3] Y + [6] [0 0 1] [0 0 0] [1] >= [1 0 3] [0 0 0] [2] [2 0 7] X + [2 2 3] Y + [6] [0 0 1] [0 0 0] [1] = ifMinus(le(s(X),Y),s(X),Y) quot(0(),s(Y)) = [2 0 4] [0] [0 0 1] Y + [1] [0 0 0] [0] >= [0] [0] [0] = 0() quot(s(X),s(Y)) = [2 0 6] [2 0 4] [2] [0 0 0] X + [0 0 1] Y + [1] [0 0 1] [0 0 0] [1] >= [2 0 6] [2 0 4] [2] [0 0 0] X + [0 0 0] Y + [0] [0 0 1] [0 0 0] [1] = s(quot(minus(X,Y),s(Y))) ** Step 1.b:9: 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 1.b:10: 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(Omega(n^1),O(n^3))