WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(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) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(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) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(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) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: {if_mod,le,minus,mod} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if_mod) = 10 + x1 + x2 p(le) = 0 p(minus) = 5 + x1 p(mod) = 13 + x1 p(s) = 15 + x1 p(true) = 0 Following rules are strictly oriented: if_mod(false(),s(x),s(y)) = 25 + x > 15 + x = s(x) if_mod(true(),s(x),s(y)) = 25 + x > 18 + x = mod(minus(x,y),s(y)) minus(x,0()) = 5 + x > x = x minus(s(x),s(y)) = 20 + x > 5 + x = minus(x,y) mod(0(),y) = 13 > 0 = 0() mod(s(x),0()) = 28 + x > 0 = 0() mod(s(x),s(y)) = 28 + x > 25 + x = if_mod(le(y,x),s(x),s(y)) Following rules are (at-least) weakly oriented: le(0(),y) = 0 >= 0 = true() le(s(x),0()) = 0 >= 0 = false() le(s(x),s(y)) = 0 >= 0 = le(x,y) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: {if_mod,le,minus,mod} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if_mod) = x1 + x2 p(le) = 1 p(minus) = 6 + x1 p(mod) = 1 + x1 p(s) = 8 + x1 p(true) = 0 Following rules are strictly oriented: le(0(),y) = 1 > 0 = true() le(s(x),0()) = 1 > 0 = false() Following rules are (at-least) weakly oriented: if_mod(false(),s(x),s(y)) = 8 + x >= 8 + x = s(x) if_mod(true(),s(x),s(y)) = 8 + x >= 7 + x = mod(minus(x,y),s(y)) le(s(x),s(y)) = 1 >= 1 = le(x,y) minus(x,0()) = 6 + x >= x = x minus(s(x),s(y)) = 14 + x >= 6 + x = minus(x,y) mod(0(),y) = 1 >= 0 = 0() mod(s(x),0()) = 9 + x >= 0 = 0() mod(s(x),s(y)) = 9 + x >= 9 + x = if_mod(le(y,x),s(x),s(y)) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: le(s(x),s(y)) -> le(x,y) - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(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) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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(if_mod) = {1}, uargs(mod) = {1} Following symbols are considered usable: {if_mod,le,minus,mod} TcT has computed the following interpretation: p(0) = 0 p(false) = 0 p(if_mod) = 1 + 4*x1 + x2^2 p(le) = x2 p(minus) = x1 p(mod) = 4*x1 + x1^2 p(s) = 2 + x1 p(true) = 0 Following rules are strictly oriented: le(s(x),s(y)) = 2 + y > y = le(x,y) Following rules are (at-least) weakly oriented: if_mod(false(),s(x),s(y)) = 5 + 4*x + x^2 >= 2 + x = s(x) if_mod(true(),s(x),s(y)) = 5 + 4*x + x^2 >= 4*x + x^2 = mod(minus(x,y),s(y)) le(0(),y) = y >= 0 = true() le(s(x),0()) = 0 >= 0 = false() minus(x,0()) = x >= x = x minus(s(x),s(y)) = 2 + x >= x = minus(x,y) mod(0(),y) = 0 >= 0 = 0() mod(s(x),0()) = 12 + 8*x + x^2 >= 0 = 0() mod(s(x),s(y)) = 12 + 8*x + x^2 >= 5 + 8*x + x^2 = if_mod(le(y,x),s(x),s(y)) ** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(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) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod} 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))