WORST_CASE(?,O(n^1)) * Step 1: Sum WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v())) =(.(x,y),nil()) -> false() =(nil(),.(y,z)) -> false() =(nil(),nil()) -> true() del(.(x,.(y,z))) -> f(=(x,y),x,y,z) f(false(),x,y,z) -> .(x,del(.(y,z))) f(true(),x,y,z) -> del(.(y,z)) - Signature: {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v())) =(.(x,y),nil()) -> false() =(nil(),.(y,z)) -> false() =(nil(),nil()) -> true() del(.(x,.(y,z))) -> f(=(x,y),x,y,z) f(false(),x,y,z) -> .(x,del(.(y,z))) f(true(),x,y,z) -> del(.(y,z)) - Signature: {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v} + 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(.) = {2}, uargs(f) = {1} Following symbols are considered usable: {=,del,f} TcT has computed the following interpretation: p(.) = 2 + x2 p(=) = 1 p(and) = 0 p(del) = 6 + 4*x1 p(f) = 8 + 8*x1 + 4*x4 p(false) = 1 p(nil) = 0 p(true) = 1 p(u) = 1 p(v) = 0 Following rules are strictly oriented: =(.(x,y),.(u(),v())) = 1 > 0 = and(=(x,u()),=(y,v())) del(.(x,.(y,z))) = 22 + 4*z > 16 + 4*z = f(=(x,y),x,y,z) f(true(),x,y,z) = 16 + 4*z > 14 + 4*z = del(.(y,z)) Following rules are (at-least) weakly oriented: =(.(x,y),nil()) = 1 >= 1 = false() =(nil(),.(y,z)) = 1 >= 1 = false() =(nil(),nil()) = 1 >= 1 = true() f(false(),x,y,z) = 16 + 4*z >= 16 + 4*z = .(x,del(.(y,z))) * Step 3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: =(.(x,y),nil()) -> false() =(nil(),.(y,z)) -> false() =(nil(),nil()) -> true() f(false(),x,y,z) -> .(x,del(.(y,z))) - Weak TRS: =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v())) del(.(x,.(y,z))) -> f(=(x,y),x,y,z) f(true(),x,y,z) -> del(.(y,z)) - Signature: {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v} + 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(.) = {2}, uargs(f) = {1} Following symbols are considered usable: {=,del,f} TcT has computed the following interpretation: p(.) = 4 + x2 p(=) = 4 p(and) = 0 p(del) = 2 + 2*x1 p(f) = 4*x1 + 2*x4 p(false) = 4 p(nil) = 2 p(true) = 3 p(u) = 2 p(v) = 8 Following rules are strictly oriented: =(nil(),nil()) = 4 > 3 = true() f(false(),x,y,z) = 16 + 2*z > 14 + 2*z = .(x,del(.(y,z))) Following rules are (at-least) weakly oriented: =(.(x,y),.(u(),v())) = 4 >= 0 = and(=(x,u()),=(y,v())) =(.(x,y),nil()) = 4 >= 4 = false() =(nil(),.(y,z)) = 4 >= 4 = false() del(.(x,.(y,z))) = 18 + 2*z >= 16 + 2*z = f(=(x,y),x,y,z) f(true(),x,y,z) = 12 + 2*z >= 10 + 2*z = del(.(y,z)) * Step 4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: =(.(x,y),nil()) -> false() =(nil(),.(y,z)) -> false() - Weak TRS: =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v())) =(nil(),nil()) -> true() del(.(x,.(y,z))) -> f(=(x,y),x,y,z) f(false(),x,y,z) -> .(x,del(.(y,z))) f(true(),x,y,z) -> del(.(y,z)) - Signature: {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v} + 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(.) = {2}, uargs(f) = {1} Following symbols are considered usable: {=,del,f} TcT has computed the following interpretation: p(.) = 4 + x1 + x2 p(=) = 1 p(and) = 0 p(del) = 1 + 2*x1 p(f) = 13 + 2*x1 + x2 + 2*x3 + 2*x4 p(false) = 0 p(nil) = 1 p(true) = 1 p(u) = 10 p(v) = 3 Following rules are strictly oriented: =(.(x,y),nil()) = 1 > 0 = false() =(nil(),.(y,z)) = 1 > 0 = false() Following rules are (at-least) weakly oriented: =(.(x,y),.(u(),v())) = 1 >= 0 = and(=(x,u()),=(y,v())) =(nil(),nil()) = 1 >= 1 = true() del(.(x,.(y,z))) = 17 + 2*x + 2*y + 2*z >= 15 + x + 2*y + 2*z = f(=(x,y),x,y,z) f(false(),x,y,z) = 13 + x + 2*y + 2*z >= 13 + x + 2*y + 2*z = .(x,del(.(y,z))) f(true(),x,y,z) = 15 + x + 2*y + 2*z >= 9 + 2*y + 2*z = del(.(y,z)) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v())) =(.(x,y),nil()) -> false() =(nil(),.(y,z)) -> false() =(nil(),nil()) -> true() del(.(x,.(y,z))) -> f(=(x,y),x,y,z) f(false(),x,y,z) -> .(x,del(.(y,z))) f(true(),x,y,z) -> del(.(y,z)) - Signature: {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))