WORST_CASE(?,O(n^3)) * Step 1: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = 0 p(a__div) = 4 + x1 + 4*x2 p(a__geq) = 0 p(a__if) = x1 + 4*x2 + 4*x3 p(a__minus) = 0 p(div) = 1 + x1 + x2 p(false) = 0 p(geq) = 0 p(if) = x1 + x2 + x3 p(mark) = 4*x1 p(minus) = 0 p(s) = x1 p(true) = 0 Following rules are strictly oriented: a__div(X1,X2) = 4 + X1 + 4*X2 > 1 + X1 + X2 = div(X1,X2) a__div(0(),s(Y)) = 4 + 4*Y > 0 = 0() Following rules are (at-least) weakly oriented: a__div(s(X),s(Y)) = 4 + X + 4*Y >= 4 + 4*Y = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = 0 >= 0 = true() a__geq(X1,X2) = 0 >= 0 = geq(X1,X2) a__geq(0(),s(Y)) = 0 >= 0 = false() a__geq(s(X),s(Y)) = 0 >= 0 = a__geq(X,Y) a__if(X1,X2,X3) = X1 + 4*X2 + 4*X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 4*X + 4*Y >= 4*Y = mark(Y) a__if(true(),X,Y) = 4*X + 4*Y >= 4*X = mark(X) a__minus(X1,X2) = 0 >= 0 = minus(X1,X2) a__minus(0(),Y) = 0 >= 0 = 0() a__minus(s(X),s(Y)) = 0 >= 0 = a__minus(X,Y) mark(0()) = 0 >= 0 = 0() mark(div(X1,X2)) = 4 + 4*X1 + 4*X2 >= 4 + 4*X1 + 4*X2 = a__div(mark(X1),X2) mark(false()) = 0 >= 0 = false() mark(geq(X1,X2)) = 0 >= 0 = a__geq(X1,X2) mark(if(X1,X2,X3)) = 4*X1 + 4*X2 + 4*X3 >= 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = 0 >= 0 = a__minus(X1,X2) mark(s(X)) = 4*X >= 4*X = s(mark(X)) mark(true()) = 0 >= 0 = true() * Step 2: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = 0 p(a__div) = x1 p(a__geq) = x1 p(a__if) = x1 + x2 + x3 p(a__minus) = 0 p(div) = x1 p(false) = 0 p(geq) = x1 p(if) = x1 + x2 + x3 p(mark) = x1 p(minus) = 0 p(s) = 1 + x1 p(true) = 0 Following rules are strictly oriented: a__geq(s(X),s(Y)) = 1 + X > X = a__geq(X,Y) Following rules are (at-least) weakly oriented: a__div(X1,X2) = X1 >= X1 = div(X1,X2) a__div(0(),s(Y)) = 0 >= 0 = 0() a__div(s(X),s(Y)) = 1 + X >= 1 + X = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = X >= 0 = true() a__geq(X1,X2) = X1 >= X1 = geq(X1,X2) a__geq(0(),s(Y)) = 0 >= 0 = false() a__if(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = X + Y >= Y = mark(Y) a__if(true(),X,Y) = X + Y >= X = mark(X) a__minus(X1,X2) = 0 >= 0 = minus(X1,X2) a__minus(0(),Y) = 0 >= 0 = 0() a__minus(s(X),s(Y)) = 0 >= 0 = a__minus(X,Y) mark(0()) = 0 >= 0 = 0() mark(div(X1,X2)) = X1 >= X1 = a__div(mark(X1),X2) mark(false()) = 0 >= 0 = false() mark(geq(X1,X2)) = X1 >= X1 = a__geq(X1,X2) mark(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = 0 >= 0 = a__minus(X1,X2) mark(s(X)) = 1 + X >= 1 + X = s(mark(X)) mark(true()) = 0 >= 0 = true() * Step 3: NaturalPI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(s(X),s(Y)) -> a__geq(X,Y) - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = 0 p(a__div) = 4 + x1 + x2 p(a__geq) = 0 p(a__if) = x1 + x2 + x3 p(a__minus) = x1 p(div) = 4 + x1 + x2 p(false) = 0 p(geq) = 0 p(if) = x1 + x2 + x3 p(mark) = x1 p(minus) = x1 p(s) = 4 + x1 p(true) = 0 Following rules are strictly oriented: a__minus(s(X),s(Y)) = 4 + X > X = a__minus(X,Y) Following rules are (at-least) weakly oriented: a__div(X1,X2) = 4 + X1 + X2 >= 4 + X1 + X2 = div(X1,X2) a__div(0(),s(Y)) = 8 + Y >= 0 = 0() a__div(s(X),s(Y)) = 12 + X + Y >= 12 + X + Y = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = 0 >= 0 = true() a__geq(X1,X2) = 0 >= 0 = geq(X1,X2) a__geq(0(),s(Y)) = 0 >= 0 = false() a__geq(s(X),s(Y)) = 0 >= 0 = a__geq(X,Y) a__if(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = X + Y >= Y = mark(Y) a__if(true(),X,Y) = X + Y >= X = mark(X) a__minus(X1,X2) = X1 >= X1 = minus(X1,X2) a__minus(0(),Y) = 0 >= 0 = 0() mark(0()) = 0 >= 0 = 0() mark(div(X1,X2)) = 4 + X1 + X2 >= 4 + X1 + X2 = a__div(mark(X1),X2) mark(false()) = 0 >= 0 = false() mark(geq(X1,X2)) = 0 >= 0 = a__geq(X1,X2) mark(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = X1 >= X1 = a__minus(X1,X2) mark(s(X)) = 4 + X >= 4 + X = s(mark(X)) mark(true()) = 0 >= 0 = true() * Step 4: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0] [1] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [1] [0] p(geq) = [0] [0] [0] p(if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [1] [0] Following rules are strictly oriented: mark(false()) = [1] [2] [0] > [0] [1] [0] = false() mark(true()) = [1] [2] [0] > [0] [1] [0] = true() Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [0] [1] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [1] [0] >= [0] [1] [0] = true() a__geq(X1,X2) = [0] [1] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [1] [0] >= [0] [1] [0] = false() a__geq(s(X),s(Y)) = [0] [1] [0] >= [0] [1] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [1] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0] [0 1 1] X1 + [1] [0 0 1] [0] >= [0 1 0] [0] [0 1 1] X1 + [1] [0 0 1] [0] = a__div(mark(X1),X2) mark(geq(X1,X2)) = [0] [1] [0] >= [0] [1] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [1] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 0] [1] = s(mark(X)) * Step 5: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 1 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__geq) = [0 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(a__if) = [1 0 0] [1 0 1] [1 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(a__minus) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(div) = [1 1 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(if) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(s) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(true) = [0] [0] [0] Following rules are strictly oriented: a__geq(X,0()) = [0 0 0] [1] [0 0 0] X + [0] [0 0 1] [0] > [0] [0] [0] = true() a__geq(0(),s(Y)) = [1] [0] [0] > [0] [0] [0] = false() Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 1 0] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 0] [0 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 1 0] [0 0 1] [1] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [0] >= [1 1 0] [0 0 1] [1] [0 0 0] X + [0 0 0] Y + [1] [0 0 1] [0 0 1] [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X1,X2) = [0 0 0] [1] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [1] [0 0 0] X1 + [0] [0 0 1] [0] = geq(X1,X2) a__geq(s(X),s(Y)) = [0 0 0] [1] [0 0 0] X + [0] [0 0 1] [0] >= [0 0 0] [1] [0 0 0] X + [0] [0 0 1] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [1 0 1] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1 0 1] [1 0 1] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [1 0 1] [1 0 1] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [1 1 0] [0] [0 0 0] X1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X1 + [0] [0 0 0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [1 1 0] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [1 1 1] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 1 1] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [0] >= [0 0 0] [1] [0 0 0] X1 + [0] [0 0 1] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1 0 1] [1 0 1] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 1] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1 1 0] [0] [0 0 0] X1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 0 0] X1 + [0] [0 0 0] [0] = a__minus(X1,X2) mark(s(X)) = [1 1 1] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 1 1] [0] [0 0 0] X + [1] [0 0 1] [0] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0 1 1] [1] [0 1 1] x1 + [0 1 1] x2 + [1] [0 0 1] [0 0 0] [0] p(a__geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [1] [1] [0] p(div) = [1 0 0] [0 0 0] [1] [0 1 1] x1 + [0 1 1] x2 + [1] [0 0 1] [0 0 0] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(if) = [0 0 0] [0 1 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [1] [1] [0] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__minus(0(),Y) = [1] [1] [0] > [0] [0] [0] = 0() Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0 1 1] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 0] [0] >= [1 0 0] [0 0 0] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 0] [0] = div(X1,X2) a__div(0(),s(Y)) = [0 1 2] [2] [0 1 2] Y + [2] [0 0 0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 1] [0 1 2] [3] [0 1 2] X + [0 1 2] Y + [3] [0 0 1] [0 0 0] [1] >= [0 0 0] [0 1 2] [3] [0 1 0] X + [0 1 2] Y + [3] [0 0 0] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 1 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [1] [1] [0] >= [1] [1] [0] = minus(X1,X2) a__minus(s(X),s(Y)) = [1] [1] [0] >= [1] [1] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0 1 1] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 0] [0] >= [0 1 1] [0 1 1] [1] [0 1 1] X1 + [0 1 1] X2 + [1] [0 0 1] [0 0 0] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] [1] [0] >= [1] [1] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 1] [0] [0 1 1] X + [0] [0 0 1] [1] >= [0 1 0] [0] [0 1 1] X + [0] [0 0 1] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0] [0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [1 0 0] [0 0 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__div(s(X),s(Y)) = [1 0 0] [1] [0 1 0] X + [1] [0 0 0] [1] > [0] [0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 0] [1] [0 1 1] x1 + [1] [0 0 1] [0] p(a__geq) = [0] [0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [1] [0 1 1] x1 + [1] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [1 0 0] [0 0 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [1] [0 1 0] x1 + [1] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [2] [0 0 0] [0 0 1] [0 0 1] [0] > [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [2] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 0] [1] [0 1 1] X1 + [1] [0 0 1] [0] >= [0 0 0] [1] [0 1 1] X1 + [1] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [1] [1] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [2] [0 1 0] X + [3] [0 0 0] [1] >= [2] [3] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0 0 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [1] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [1] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [1] [0 1 1] X1 + [2] [0 0 1] [0] >= [0 1 0] [1] [0 1 1] X1 + [2] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [1] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0] [1] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(minus(X1,X2)) = [0] [1] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [1] [0 1 0] X + [2] [0 0 0] [1] >= [0 1 0] [1] [0 1 0] X + [2] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [1] [0] >= [0] [0] [0] = true() * Step 9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(geq(X1,X2)) -> a__geq(X1,X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0] [1] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0] [1] [0] p(if) = [1 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: mark(geq(X1,X2)) = [1] [1] [0] > [0] [1] [0] = a__geq(X1,X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [1] >= [0] [1] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [1] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [1] [0] >= [0] [1] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [1] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [1] [0] >= [0] [1] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 1 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 1 0] [0 0 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 0] [1] p(a__geq) = [1] [0] [0] p(a__if) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 1] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] p(a__minus) = [0] [0] [1] p(div) = [1 1 0] [0 0 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 1] [0 0 0] [1] p(false) = [0] [0] [0] p(geq) = [1] [0] [0] p(if) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 1 1] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] p(mark) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [1] p(s) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [0] p(true) = [1] [0] [0] Following rules are strictly oriented: a__if(true(),X,Y) = [1 0 0] [1 0 0] [1] [0 1 1] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [1] > [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 1 0] [0 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [1] >= [1 1 0] [0 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [1] = div(X1,X2) a__div(0(),s(Y)) = [0 0 0] [1] [0 0 1] Y + [2] [0 0 0] [1] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 1] [0 0 0] [2] [0 0 1] X + [0 0 1] Y + [3] [0 0 0] [0 0 0] [1] >= [2] [3] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1] [0] [0] >= [1] [0] [0] = true() a__geq(X1,X2) = [1] [0] [0] >= [1] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [1] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [1] [0] [0] >= [1] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] = if(X1,X2,X3) a__if(false(),X,Y) = [1 0 0] [1 0 0] [0] [0 1 1] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__minus(X1,X2) = [0] [0] [1] >= [0] [0] [1] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [1] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [1] >= [0] [0] [1] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [1 1 0] [0 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [1] >= [1 1 0] [0 0 0] [1] [0 1 0] X1 + [0 1 0] X2 + [1] [0 0 1] [0 0 0] [1] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [1] [0] [0] >= [1] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] X1 + [0 1 1] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [1] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [1] >= [0] [0] [1] = a__minus(X1,X2) mark(s(X)) = [1 0 0] [0] [0 0 1] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 1] X + [1] [0 0 0] [0] = s(mark(X)) mark(true()) = [1] [0] [0] >= [1] [0] [0] = true() * Step 11: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [1] [0] p(a__div) = [1 0 1] [1] [0 1 1] x1 + [0] [0 0 0] [1] p(a__geq) = [0] [0] [0] p(a__if) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(div) = [1 0 1] [1] [0 1 1] x1 + [0] [0 0 0] [1] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__if(false(),X,Y) = [1 0 0] [1 0 0] [1] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] > [1 0 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [1] [0 1 1] X1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 1 1] X1 + [0] [0 0 0] [1] = div(X1,X2) a__div(0(),s(Y)) = [1] [1] [1] >= [0] [1] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [2] [0 0 0] X + [1] [0 0 0] [1] >= [1 0 0] [2] [0 0 0] X + [1] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(true(),X,Y) = [1 0 0] [1 0 0] [1] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [1] [0] >= [0] [1] [0] = 0() a__minus(s(X),s(Y)) = [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [1] [0 0 0] [0] = a__minus(X,Y) mark(0()) = [0] [1] [0] >= [0] [1] [0] = 0() mark(div(X1,X2)) = [1 0 1] [1] [0 1 1] X1 + [0] [0 0 0] [1] >= [1 0 1] [1] [0 1 1] X1 + [0] [0 0 0] [1] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X1 + [1] [0 0 0] [0] = a__minus(X1,X2) mark(s(X)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 12: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 1] [0] p(a__geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(if) = [0 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: mark(div(X1,X2)) = [0 1 1] [0 1 0] [1] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 1] [0] > [0 1 1] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 1] [0] = a__div(mark(X1),X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 1] [0] >= [0 0 0] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [1] [0 0 0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0 1 0] [0] [0 1 0] Y + [1] [0 0 0] [1] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [2] [0 0 0] [0 0 0] [1] >= [0 0 0] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 0] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 13: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__minus(X1,X2) -> minus(X1,X2) mark(0()) -> 0() mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [1 1 0] [1] [1 0 1] x1 + [1 1 0] x2 + [0] [0 0 1] [0 0 0] [0] p(a__geq) = [0] [0] [0] p(a__if) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(a__minus) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(div) = [1 0 1] [1 1 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] p(mark) = [1 0 0] [1] [1 0 0] x1 + [0] [0 0 1] [0] p(minus) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(s) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: mark(0()) = [1] [0] [0] > [0] [0] [0] = 0() mark(minus(X1,X2)) = [1 0 1] [1] [1 0 1] X1 + [0] [0 0 0] [0] > [1 0 1] [0] [0 0 0] X1 + [0] [0 0 0] [0] = a__minus(X1,X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [1 1 0] [1] [1 0 1] X1 + [1 1 0] X2 + [0] [0 0 1] [0 0 0] [0] >= [1 0 1] [1 1 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [0] = div(X1,X2) a__div(0(),s(Y)) = [1 0 2] [1] [1 0 2] Y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 1] [1 0 2] [2] [1 0 1] X + [1 0 2] Y + [1] [0 0 0] [0 0 0] [1] >= [1 0 1] [1 0 2] [2] [1 0 1] X + [1 0 2] Y + [1] [0 0 0] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1 0 0] [1 0 0] [1] [1 0 0] X + [1 0 0] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1] [1 0 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [1 0 0] [1 0 0] [1] [1 0 0] X + [1 0 0] Y + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1] [1 0 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [1 0 1] [1] [0 0 0] X + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = a__minus(X,Y) mark(div(X1,X2)) = [1 0 1] [1 1 0] [2] [1 0 1] X1 + [1 1 0] X2 + [1] [0 0 1] [0 0 0] [0] >= [1 0 1] [1 1 0] [2] [1 0 1] X1 + [1 1 0] X2 + [1] [0 0 1] [0 0 0] [0] = a__div(mark(X1),X2) mark(false()) = [1] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [1] [0] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1 0 0] [1 0 0] [1 0 0] [2] [1 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [1 0 0] [2] [0 0 0] X1 + [1 0 0] X2 + [1 0 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(s(X)) = [1 0 1] [1] [1 0 1] X + [0] [0 0 0] [1] >= [1 0 1] [1] [0 0 1] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [1] [0] [0] >= [0] [0] [0] = true() * Step 14: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__minus(X1,X2) -> minus(X1,X2) mark(s(X)) -> s(mark(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0 0 1] [1] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(a__geq) = [0] [0] [0] p(a__if) = [1 0 0] [0 1 1] [0 1 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0 0 1] [0] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] p(mark) = [0 1 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: mark(s(X)) = [0 1 1] [1] [0 1 1] X + [0] [0 0 0] [1] > [0 1 1] [0] [0 1 1] X + [0] [0 0 0] [1] = s(mark(X)) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0 0 1] [1] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 1] [0] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [2] [1] [1] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [3] [0 1 1] X + [2] [0 0 0] [2] >= [3] [2] [2] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 1] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 1] [0 1 1] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 1] [0 1 1] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [1] >= [0 1 1] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 2] [0 0 1] [1] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [0 1 2] [0 0 1] [1] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 1] [0 1 1] [0 1 1] [1] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 1 1] [0 1 1] [0 1 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 1] [0 0 1] [0 0 1] [1] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 15: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__minus(X1,X2) -> minus(X1,X2) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] > [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [1] [0 1 0] X + [1] [0 0 0] [1] >= [0 0 0] [1] [0 1 0] X + [1] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__geq(X,Y) a__if(false(),X,Y) = [0 1 0] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [1] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [1] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 16: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__geq(X1,X2) -> geq(X1,X2) a__minus(X1,X2) -> minus(X1,X2) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0 0 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [1] [0] p(geq) = [0 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(if) = [0 0 0] [0 0 0] [0 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__geq(X1,X2) = [0 0 0] [1] [0 1 0] X1 + [1] [0 0 0] [0] > [0 0 0] [0] [0 1 0] X1 + [1] [0 0 0] [0] = geq(X1,X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [1] [0 1 0] X + [1] [0 0 0] [1] >= [0 0 0] [1] [0 1 0] X + [1] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(0(),s(Y)) = [1] [1] [0] >= [0] [1] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [1] [0 1 0] X + [1] [0 0 0] [0] >= [0 0 0] [1] [0 1 0] X + [1] [0 0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 1] Y + [1] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 1 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [1] [1] [0] >= [0] [1] [0] = false() mark(geq(X1,X2)) = [0 1 0] [1] [0 1 0] X1 + [1] [0 0 0] [0] >= [0 0 0] [1] [0 1 0] X1 + [1] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 1] [0 1 0] [0 1 1] [0] [0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 17: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__minus(X1,X2) -> minus(X1,X2) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [1 0 1] [1] [0 1 1] x1 + [1] [0 0 1] [0] p(a__geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__if) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [1] [1] [0] p(div) = [0 0 0] [1] [0 1 1] x1 + [1] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(if) = [0 0 0] [0 1 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [1] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(true) = [0] [0] [0] Following rules are strictly oriented: a__minus(X1,X2) = [1] [1] [0] > [0] [1] [0] = minus(X1,X2) Following rules are (at-least) weakly oriented: a__div(X1,X2) = [1 0 1] [1] [0 1 1] X1 + [1] [0 0 1] [0] >= [0 0 0] [1] [0 1 1] X1 + [1] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [1] [1] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [1 0 0] [2] [0 1 0] X + [2] [0 0 0] [1] >= [0 0 0] [2] [0 1 0] X + [2] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 0 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 1 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 1 0] X + [0 1 0] Y + [0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(0(),Y) = [1] [1] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [1] [1] [0] >= [1] [1] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 1 1] [1] [0 1 1] X1 + [1] [0 0 1] [0] >= [0 1 1] [1] [0 1 1] X1 + [1] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] [1] [0] >= [1] [1] [0] = a__minus(X1,X2) mark(s(X)) = [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] >= [0 1 0] [0] [0 1 0] X + [0] [0 0 0] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() * Step 18: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))