WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ='(.(z0,z1),.(u(),v())) -> c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) -> c7(='(z1,v())) ='(.(z0,z1),nil()) -> c4() ='(nil(),.(z0,z1)) -> c5() ='(nil(),nil()) -> c3() DEL(.(z0,.(z1,z2))) -> c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) -> c2(DEL(.(z1,z2))) F(true(),z0,z1,z2) -> c1(DEL(.(z1,z2))) - Weak TRS: =(.(z0,z1),.(u(),v())) -> and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) -> false() =(nil(),.(z0,z1)) -> false() =(nil(),nil()) -> true() del(.(z0,.(z1,z2))) -> f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) -> .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) -> del(.(z1,z2)) - Signature: {=/2,='/2,DEL/1,F/4,del/1,f/4} / {./2,and/2,c/2,c1/1,c2/1,c3/0,c4/0,c5/0,c6/1,c7/1,false/0,nil/0,true/0,u/0 ,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,=',DEL,F,del,f} and constructors {.,and,c,c1,c2,c3,c4 ,c5,c6,c7,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: ='(.(z0,z1),.(u(),v())) -> c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) -> c7(='(z1,v())) ='(.(z0,z1),nil()) -> c4() ='(nil(),.(z0,z1)) -> c5() ='(nil(),nil()) -> c3() DEL(.(z0,.(z1,z2))) -> c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) -> c2(DEL(.(z1,z2))) F(true(),z0,z1,z2) -> c1(DEL(.(z1,z2))) - Weak TRS: =(.(z0,z1),.(u(),v())) -> and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) -> false() =(nil(),.(z0,z1)) -> false() =(nil(),nil()) -> true() del(.(z0,.(z1,z2))) -> f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) -> .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) -> del(.(z1,z2)) - Signature: {=/2,='/2,DEL/1,F/4,del/1,f/4} / {./2,and/2,c/2,c1/1,c2/1,c3/0,c4/0,c5/0,c6/1,c7/1,false/0,nil/0,true/0,u/0 ,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,=',DEL,F,del,f} and constructors {.,and,c,c1,c2,c3,c4 ,c5,c6,c7,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}, uargs(c) = {1,2}, uargs(c1) = {1}, uargs(c2) = {1}, uargs(f) = {1} Following symbols are considered usable: {=,=',DEL,F,del,f} TcT has computed the following interpretation: p(.) = x1 + x2 p(=) = 2*x1 p(=') = 0 p(DEL) = 4*x1 p(F) = 2*x1 + 4*x3 + 4*x4 p(and) = x2 p(c) = x1 + x2 p(c1) = x1 p(c2) = x1 p(c3) = 0 p(c4) = 0 p(c5) = 0 p(c6) = x1 p(c7) = x1 p(del) = 1 + 5*x1 p(f) = 1 + 2*x1 + x2 + 5*x3 + 5*x4 p(false) = 0 p(nil) = 4 p(true) = 2 p(u) = 0 p(v) = 0 Following rules are strictly oriented: F(true(),z0,z1,z2) = 4 + 4*z1 + 4*z2 > 4*z1 + 4*z2 = c1(DEL(.(z1,z2))) Following rules are (at-least) weakly oriented: =(.(z0,z1),.(u(),v())) = 2*z0 + 2*z1 >= 2*z1 = and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) = 2*z0 + 2*z1 >= 0 = false() =(nil(),.(z0,z1)) = 8 >= 0 = false() =(nil(),nil()) = 8 >= 2 = true() ='(.(z0,z1),.(u(),v())) = 0 >= 0 = c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) = 0 >= 0 = c7(='(z1,v())) ='(.(z0,z1),nil()) = 0 >= 0 = c4() ='(nil(),.(z0,z1)) = 0 >= 0 = c5() ='(nil(),nil()) = 0 >= 0 = c3() DEL(.(z0,.(z1,z2))) = 4*z0 + 4*z1 + 4*z2 >= 4*z0 + 4*z1 + 4*z2 = c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) = 4*z1 + 4*z2 >= 4*z1 + 4*z2 = c2(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) = 1 + 5*z0 + 5*z1 + 5*z2 >= 1 + 5*z0 + 5*z1 + 5*z2 = f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) = 1 + z0 + 5*z1 + 5*z2 >= 1 + z0 + 5*z1 + 5*z2 = .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) = 5 + z0 + 5*z1 + 5*z2 >= 1 + 5*z1 + 5*z2 = del(.(z1,z2)) * Step 3: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ='(.(z0,z1),.(u(),v())) -> c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) -> c7(='(z1,v())) ='(.(z0,z1),nil()) -> c4() ='(nil(),.(z0,z1)) -> c5() ='(nil(),nil()) -> c3() DEL(.(z0,.(z1,z2))) -> c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) -> c2(DEL(.(z1,z2))) - Weak TRS: =(.(z0,z1),.(u(),v())) -> and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) -> false() =(nil(),.(z0,z1)) -> false() =(nil(),nil()) -> true() F(true(),z0,z1,z2) -> c1(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) -> f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) -> .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) -> del(.(z1,z2)) - Signature: {=/2,='/2,DEL/1,F/4,del/1,f/4} / {./2,and/2,c/2,c1/1,c2/1,c3/0,c4/0,c5/0,c6/1,c7/1,false/0,nil/0,true/0,u/0 ,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,=',DEL,F,del,f} and constructors {.,and,c,c1,c2,c3,c4 ,c5,c6,c7,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}, uargs(c) = {1,2}, uargs(c1) = {1}, uargs(c2) = {1}, uargs(f) = {1} Following symbols are considered usable: {=,=',DEL,F,del,f} TcT has computed the following interpretation: p(.) = x1 + x2 p(=) = 0 p(=') = 4*x1 p(DEL) = 5*x1 p(F) = 4*x1 + x2 + 5*x3 + 5*x4 p(and) = x1 p(c) = x1 + x2 p(c1) = x1 p(c2) = x1 p(c3) = 0 p(c4) = 0 p(c5) = 2 p(c6) = 0 p(c7) = x1 p(del) = 2*x1 p(f) = 4*x1 + 2*x2 + 2*x3 + 2*x4 p(false) = 0 p(nil) = 2 p(true) = 0 p(u) = 0 p(v) = 4 Following rules are strictly oriented: ='(nil(),.(z0,z1)) = 8 > 2 = c5() ='(nil(),nil()) = 8 > 0 = c3() Following rules are (at-least) weakly oriented: =(.(z0,z1),.(u(),v())) = 0 >= 0 = and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) = 0 >= 0 = false() =(nil(),.(z0,z1)) = 0 >= 0 = false() =(nil(),nil()) = 0 >= 0 = true() ='(.(z0,z1),.(u(),v())) = 4*z0 + 4*z1 >= 0 = c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) = 4*z0 + 4*z1 >= 4*z1 = c7(='(z1,v())) ='(.(z0,z1),nil()) = 4*z0 + 4*z1 >= 0 = c4() DEL(.(z0,.(z1,z2))) = 5*z0 + 5*z1 + 5*z2 >= 5*z0 + 5*z1 + 5*z2 = c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) = z0 + 5*z1 + 5*z2 >= 5*z1 + 5*z2 = c2(DEL(.(z1,z2))) F(true(),z0,z1,z2) = z0 + 5*z1 + 5*z2 >= 5*z1 + 5*z2 = c1(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) = 2*z0 + 2*z1 + 2*z2 >= 2*z0 + 2*z1 + 2*z2 = f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) = 2*z0 + 2*z1 + 2*z2 >= z0 + 2*z1 + 2*z2 = .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) = 2*z0 + 2*z1 + 2*z2 >= 2*z1 + 2*z2 = del(.(z1,z2)) * Step 4: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ='(.(z0,z1),.(u(),v())) -> c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) -> c7(='(z1,v())) ='(.(z0,z1),nil()) -> c4() DEL(.(z0,.(z1,z2))) -> c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) -> c2(DEL(.(z1,z2))) - Weak TRS: =(.(z0,z1),.(u(),v())) -> and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) -> false() =(nil(),.(z0,z1)) -> false() =(nil(),nil()) -> true() ='(nil(),.(z0,z1)) -> c5() ='(nil(),nil()) -> c3() F(true(),z0,z1,z2) -> c1(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) -> f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) -> .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) -> del(.(z1,z2)) - Signature: {=/2,='/2,DEL/1,F/4,del/1,f/4} / {./2,and/2,c/2,c1/1,c2/1,c3/0,c4/0,c5/0,c6/1,c7/1,false/0,nil/0,true/0,u/0 ,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,=',DEL,F,del,f} and constructors {.,and,c,c1,c2,c3,c4 ,c5,c6,c7,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}, uargs(c) = {1,2}, uargs(c1) = {1}, uargs(c2) = {1}, uargs(f) = {1} Following symbols are considered usable: {=,=',DEL,F,del,f} TcT has computed the following interpretation: p(.) = 1 + x1 + x2 p(=) = 3 + 2*x1 p(=') = 1 + x1 p(DEL) = 3 + 5*x1 p(F) = 2*x1 + 5*x3 + 5*x4 p(and) = 2 p(c) = 4 + x1 + x2 p(c1) = 1 + x1 p(c2) = 1 + x1 p(c3) = 4 p(c4) = 0 p(c5) = 0 p(c6) = 0 p(c7) = 0 p(del) = 1 + 7*x1 p(f) = 3*x1 + x2 + 7*x3 + 7*x4 p(false) = 5 p(nil) = 4 p(true) = 5 p(u) = 2 p(v) = 0 Following rules are strictly oriented: ='(.(z0,z1),.(u(),v())) = 2 + z0 + z1 > 0 = c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) = 2 + z0 + z1 > 0 = c7(='(z1,v())) ='(.(z0,z1),nil()) = 2 + z0 + z1 > 0 = c4() DEL(.(z0,.(z1,z2))) = 13 + 5*z0 + 5*z1 + 5*z2 > 11 + 5*z0 + 5*z1 + 5*z2 = c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) = 10 + 5*z1 + 5*z2 > 9 + 5*z1 + 5*z2 = c2(DEL(.(z1,z2))) Following rules are (at-least) weakly oriented: =(.(z0,z1),.(u(),v())) = 5 + 2*z0 + 2*z1 >= 2 = and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) = 5 + 2*z0 + 2*z1 >= 5 = false() =(nil(),.(z0,z1)) = 11 >= 5 = false() =(nil(),nil()) = 11 >= 5 = true() ='(nil(),.(z0,z1)) = 5 >= 0 = c5() ='(nil(),nil()) = 5 >= 4 = c3() F(true(),z0,z1,z2) = 10 + 5*z1 + 5*z2 >= 9 + 5*z1 + 5*z2 = c1(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) = 15 + 7*z0 + 7*z1 + 7*z2 >= 9 + 7*z0 + 7*z1 + 7*z2 = f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) = 15 + z0 + 7*z1 + 7*z2 >= 9 + z0 + 7*z1 + 7*z2 = .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) = 15 + z0 + 7*z1 + 7*z2 >= 8 + 7*z1 + 7*z2 = del(.(z1,z2)) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: =(.(z0,z1),.(u(),v())) -> and(=(z0,u()),=(z1,v())) =(.(z0,z1),nil()) -> false() =(nil(),.(z0,z1)) -> false() =(nil(),nil()) -> true() ='(.(z0,z1),.(u(),v())) -> c6(='(z0,u())) ='(.(z0,z1),.(u(),v())) -> c7(='(z1,v())) ='(.(z0,z1),nil()) -> c4() ='(nil(),.(z0,z1)) -> c5() ='(nil(),nil()) -> c3() DEL(.(z0,.(z1,z2))) -> c(F(=(z0,z1),z0,z1,z2),='(z0,z1)) F(false(),z0,z1,z2) -> c2(DEL(.(z1,z2))) F(true(),z0,z1,z2) -> c1(DEL(.(z1,z2))) del(.(z0,.(z1,z2))) -> f(=(z0,z1),z0,z1,z2) f(false(),z0,z1,z2) -> .(z0,del(.(z1,z2))) f(true(),z0,z1,z2) -> del(.(z1,z2)) - Signature: {=/2,='/2,DEL/1,F/4,del/1,f/4} / {./2,and/2,c/2,c1/1,c2/1,c3/0,c4/0,c5/0,c6/1,c7/1,false/0,nil/0,true/0,u/0 ,v/0} - Obligation: innermost runtime complexity wrt. defined symbols {=,=',DEL,F,del,f} and constructors {.,and,c,c1,c2,c3,c4 ,c5,c6,c7,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))