WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} 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^2)) + Considered Problem: - Strict TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(gcd) = 2 + x1 + x2 p(if_gcd) = 2 + 4*x1 + x2 + x3 p(le) = 0 p(minus) = x1 p(s) = x1 p(true) = 0 Following rules are strictly oriented: gcd(0(),y) = 2 + y > y = y gcd(s(x),0()) = 2 + x > x = s(x) Following rules are (at-least) weakly oriented: gcd(s(x),s(y)) = 2 + x + y >= 2 + x + y = if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) = 2 + x + y >= 2 + x + y = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 2 + x + y >= 2 + x + y = gcd(minus(x,y),s(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(x,0()) = x >= x = x minus(s(x),s(y)) = x >= x = minus(x,y) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(gcd) = 4 + 4*x1 + 4*x2 p(if_gcd) = 4 + 4*x1 + 4*x2 + 4*x3 p(le) = 0 p(minus) = x1 p(s) = 1 + x1 p(true) = 0 Following rules are strictly oriented: if_gcd(false(),s(x),s(y)) = 12 + 4*x + 4*y > 8 + 4*x + 4*y = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 12 + 4*x + 4*y > 8 + 4*x + 4*y = gcd(minus(x,y),s(y)) minus(s(x),s(y)) = 1 + x > x = minus(x,y) Following rules are (at-least) weakly oriented: gcd(0(),y) = 4 + 4*y >= y = y gcd(s(x),0()) = 8 + 4*x >= 1 + x = s(x) gcd(s(x),s(y)) = 12 + 4*x + 4*y >= 12 + 4*x + 4*y = if_gcd(le(y,x),s(x),s(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(x,0()) = x >= x = x ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 3 p(false) = 2 p(gcd) = 5 + x1 + x2 p(if_gcd) = 2*x1 + x2 + x3 p(le) = 2 p(minus) = x1 p(s) = 4 + x1 p(true) = 1 Following rules are strictly oriented: gcd(s(x),s(y)) = 13 + x + y > 12 + x + y = if_gcd(le(y,x),s(x),s(y)) le(0(),y) = 2 > 1 = true() Following rules are (at-least) weakly oriented: gcd(0(),y) = 8 + y >= y = y gcd(s(x),0()) = 12 + x >= 4 + x = s(x) if_gcd(false(),s(x),s(y)) = 12 + x + y >= 9 + x + y = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 10 + x + y >= 9 + x + y = gcd(minus(x,y),s(y)) le(s(x),0()) = 2 >= 2 = false() le(s(x),s(y)) = 2 >= 2 = le(x,y) minus(x,0()) = x >= x = x minus(s(x),s(y)) = 4 + x >= x = minus(x,y) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 0 p(false) = 2 p(gcd) = 5 + 4*x1 + 4*x2 p(if_gcd) = 1 + 2*x1 + 4*x2 + 4*x3 p(le) = 2 p(minus) = 1 + x1 p(s) = 1 + x1 p(true) = 2 Following rules are strictly oriented: minus(x,0()) = 1 + x > x = x Following rules are (at-least) weakly oriented: gcd(0(),y) = 5 + 4*y >= y = y gcd(s(x),0()) = 9 + 4*x >= 1 + x = s(x) gcd(s(x),s(y)) = 13 + 4*x + 4*y >= 13 + 4*x + 4*y = if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) = 13 + 4*x + 4*y >= 13 + 4*x + 4*y = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 13 + 4*x + 4*y >= 13 + 4*x + 4*y = gcd(minus(x,y),s(y)) le(0(),y) = 2 >= 2 = true() le(s(x),0()) = 2 >= 2 = false() le(s(x),s(y)) = 2 >= 2 = le(x,y) minus(s(x),s(y)) = 2 + x >= 1 + x = minus(x,y) ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(gcd) = 7 + x1 + x2 p(if_gcd) = 4 + x1 + x2 + x3 p(le) = 2 p(minus) = x1 p(s) = 4 + x1 p(true) = 0 Following rules are strictly oriented: le(s(x),0()) = 2 > 0 = false() Following rules are (at-least) weakly oriented: gcd(0(),y) = 7 + y >= y = y gcd(s(x),0()) = 11 + x >= 4 + x = s(x) gcd(s(x),s(y)) = 15 + x + y >= 14 + x + y = if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) = 12 + x + y >= 11 + x + y = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 12 + x + y >= 11 + x + y = gcd(minus(x,y),s(y)) le(0(),y) = 2 >= 0 = true() le(s(x),s(y)) = 2 >= 2 = le(x,y) minus(x,0()) = x >= x = x minus(s(x),s(y)) = 4 + x >= x = minus(x,y) ** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(s(x),s(y)) -> le(x,y) - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(gcd) = {1}, uargs(if_gcd) = {1} Following symbols are considered usable: {gcd,if_gcd,le,minus} TcT has computed the following interpretation: p(0) = 1 p(false) = 0 p(gcd) = 4*x1 + 5*x1*x2 + 2*x1^2 + 2*x2^2 p(if_gcd) = 1 + 4*x1 + 5*x2*x3 + 2*x2^2 + 2*x3^2 p(le) = x2 p(minus) = x1 p(s) = 1 + x1 p(true) = 0 Following rules are strictly oriented: le(s(x),s(y)) = 1 + y > y = le(x,y) Following rules are (at-least) weakly oriented: gcd(0(),y) = 6 + 5*y + 2*y^2 >= y = y gcd(s(x),0()) = 13 + 13*x + 2*x^2 >= 1 + x = s(x) gcd(s(x),s(y)) = 13 + 13*x + 5*x*y + 2*x^2 + 9*y + 2*y^2 >= 10 + 13*x + 5*x*y + 2*x^2 + 9*y + 2*y^2 = if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) = 10 + 9*x + 5*x*y + 2*x^2 + 9*y + 2*y^2 >= 2 + 4*x + 5*x*y + 2*x^2 + 9*y + 2*y^2 = gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) = 10 + 9*x + 5*x*y + 2*x^2 + 9*y + 2*y^2 >= 2 + 9*x + 5*x*y + 2*x^2 + 4*y + 2*y^2 = gcd(minus(x,y),s(y)) le(0(),y) = y >= 0 = true() le(s(x),0()) = 1 >= 0 = false() minus(x,0()) = x >= x = x minus(s(x),s(y)) = 1 + x >= x = minus(x,y) ** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: gcd(0(),y) -> y gcd(s(x),0()) -> s(x) gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y)) if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x)) if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if_gcd,le,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^2))