WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: geq(x,y){x -> s(x),y -> s(y)} = geq(s(x),s(y)) ->^+ geq(x,y) = C[geq(x,y) = geq(x,y){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = 0 p(div) = 8 + 4*x1 p(false) = 0 p(geq) = 0 p(if) = 8*x1 + x2 + x3 p(minus) = x1 p(s) = x1 p(true) = 0 Following rules are strictly oriented: div(0(),s(Y)) = 8 > 0 = 0() Following rules are (at-least) weakly oriented: div(s(X),s(Y)) = 8 + 4*X >= 8 + 4*X = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = 0 >= 0 = true() geq(0(),s(Y)) = 0 >= 0 = false() geq(s(X),s(Y)) = 0 >= 0 = geq(X,Y) if(false(),X,Y) = X + Y >= Y = Y if(true(),X,Y) = X + Y >= X = X minus(0(),Y) = 0 >= 0 = 0() minus(s(X),s(Y)) = X >= X = minus(X,Y) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Weak TRS: div(0(),s(Y)) -> 0() - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = 0 p(div) = 4*x1 p(false) = 0 p(geq) = 0 p(if) = 6 + 8*x1 + x2 + x3 p(minus) = x1 p(s) = 2 + x1 p(true) = 0 Following rules are strictly oriented: if(false(),X,Y) = 6 + X + Y > Y = Y if(true(),X,Y) = 6 + X + Y > X = X minus(s(X),s(Y)) = 2 + X > X = minus(X,Y) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = 0 >= 0 = 0() div(s(X),s(Y)) = 8 + 4*X >= 8 + 4*X = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = 0 >= 0 = true() geq(0(),s(Y)) = 0 >= 0 = false() geq(s(X),s(Y)) = 0 >= 0 = geq(X,Y) minus(0(),Y) = 0 >= 0 = 0() ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) minus(0(),Y) -> 0() - Weak TRS: div(0(),s(Y)) -> 0() if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = 0 p(div) = 12*x1 p(false) = 0 p(geq) = 1 p(if) = x1 + 8*x2 + 8*x3 p(minus) = 0 p(s) = 1 + x1 p(true) = 0 Following rules are strictly oriented: div(s(X),s(Y)) = 12 + 12*X > 9 = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = 1 > 0 = true() geq(0(),s(Y)) = 1 > 0 = false() Following rules are (at-least) weakly oriented: div(0(),s(Y)) = 0 >= 0 = 0() geq(s(X),s(Y)) = 1 >= 1 = geq(X,Y) if(false(),X,Y) = 8*X + 8*Y >= Y = Y if(true(),X,Y) = 8*X + 8*Y >= X = X minus(0(),Y) = 0 >= 0 = 0() minus(s(X),s(Y)) = 0 >= 0 = minus(X,Y) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: geq(s(X),s(Y)) -> geq(X,Y) minus(0(),Y) -> 0() - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = 0 p(div) = 4*x1 p(false) = 0 p(geq) = 4*x1 p(if) = x1 + 4*x2 + 12*x3 p(minus) = 0 p(s) = 4 + x1 p(true) = 0 Following rules are strictly oriented: geq(s(X),s(Y)) = 16 + 4*X > 4*X = geq(X,Y) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = 0 >= 0 = 0() div(s(X),s(Y)) = 16 + 4*X >= 16 + 4*X = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = 4*X >= 0 = true() geq(0(),s(Y)) = 0 >= 0 = false() if(false(),X,Y) = 4*X + 12*Y >= Y = Y if(true(),X,Y) = 4*X + 12*Y >= X = X minus(0(),Y) = 0 >= 0 = 0() minus(s(X),s(Y)) = 0 >= 0 = minus(X,Y) ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(0(),Y) -> 0() - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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(div) = {1}, uargs(if) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {div,geq,if,minus} TcT has computed the following interpretation: p(0) = 0 p(div) = 4*x1 p(false) = 0 p(geq) = 0 p(if) = 1 + 2*x1 + x2 + 8*x3 p(minus) = 5 + x1 p(s) = 7 + x1 p(true) = 0 Following rules are strictly oriented: minus(0(),Y) = 5 > 0 = 0() Following rules are (at-least) weakly oriented: div(0(),s(Y)) = 0 >= 0 = 0() div(s(X),s(Y)) = 28 + 4*X >= 28 + 4*X = if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) = 0 >= 0 = true() geq(0(),s(Y)) = 0 >= 0 = false() geq(s(X),s(Y)) = 0 >= 0 = geq(X,Y) if(false(),X,Y) = 1 + X + 8*Y >= Y = Y if(true(),X,Y) = 1 + X + 8*Y >= X = X minus(s(X),s(Y)) = 12 + X >= 5 + X = minus(X,Y) ** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} 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^1))