WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: choice(y){y -> cons(x,y)} = choice(cons(x,y)) ->^+ choice(y) = C[choice(y) = choice(y){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 0 p(choice) = 4*x1 p(cons) = 1 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = 4 + 4*x1 p(if) = 4*x1 + x2 + x3 p(member) = 4*x1 p(negate) = 0 p(nil) = 0 p(sat) = 7 + 5*x1 p(satck) = 2 + x2 p(true) = 0 p(unsat) = 0 p(verify) = 0 Following rules are strictly oriented: choice(cons(x,xs)) = 4 + 4*x + 4*xs > x = x choice(cons(x,xs)) = 4 + 4*x + 4*xs > 4*xs = choice(xs) guess(cons(clause,cnf)) = 8 + 4*clause + 4*cnf > 5 + 4*clause + 4*cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 4 > 0 = nil() sat(cnf) = 7 + 5*cnf > 6 + 4*cnf = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + assign > assign = if(verify(assign),assign,unsat()) Following rules are (at-least) weakly oriented: eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() if(false(),t,e) = e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,cons(y,ys)) = 4*x >= 4*x = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 4*x >= 0 = false() negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) verify(cons(l,ls)) = 0 >= 0 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 0 p(choice) = x1 p(cons) = x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = x1 p(if) = x1 + 4*x2 + x3 p(member) = x1 p(negate) = 0 p(nil) = 0 p(sat) = 6 + 6*x1 p(satck) = 6 + 2*x1 + 4*x2 p(true) = 0 p(unsat) = 1 p(verify) = 4 Following rules are strictly oriented: verify(nil()) = 4 > 0 = true() Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = x + xs >= x = x choice(cons(x,xs)) = x + xs >= xs = choice(xs) eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = clause + cnf >= clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = e + 4*t >= e = e if(true(),t,e) = e + 4*t >= t = t member(x,cons(y,ys)) = x >= x = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = x >= 0 = false() negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) sat(cnf) = 6 + 6*cnf >= 6 + 6*cnf = satck(cnf,guess(cnf)) satck(cnf,assign) = 6 + 4*assign + 2*cnf >= 5 + 4*assign = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 4 >= 4 = if(member(negate(l),ls),false(),verify(ls)) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 1 p(choice) = x1 p(cons) = 7 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = x1 p(if) = x1 + 3*x2 + x3 p(member) = 2*x1 p(negate) = 0 p(nil) = 4 p(sat) = 4 + 6*x1 p(satck) = 4 + 2*x1 + 4*x2 p(true) = 0 p(unsat) = 1 p(verify) = 2 + x1 Following rules are strictly oriented: verify(cons(l,ls)) = 9 + l + ls > 2 + ls = if(member(negate(l),ls),false(),verify(ls)) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 7 + x + xs >= x = x choice(cons(x,xs)) = 7 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 7 + clause + cnf >= 7 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 4 >= 4 = nil() if(false(),t,e) = e + 3*t >= e = e if(true(),t,e) = e + 3*t >= t = t member(x,cons(y,ys)) = 2*x >= 2*x = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 2*x >= 0 = false() negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) sat(cnf) = 4 + 6*cnf >= 4 + 6*cnf = satck(cnf,guess(cnf)) satck(cnf,assign) = 4 + 4*assign + 2*cnf >= 3 + 4*assign = if(verify(assign),assign,unsat()) verify(nil()) = 6 >= 0 = true() ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 4 p(choice) = x1 p(cons) = 4 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = x1 p(if) = 4*x1 + 2*x2 + x3 p(member) = x1 p(negate) = 1 p(nil) = 0 p(sat) = 6 + 7*x1 p(satck) = 6 + 6*x2 p(true) = 0 p(unsat) = 1 p(verify) = 1 + x1 Following rules are strictly oriented: negate(0(x)) = 1 > 0 = 1(x) negate(1(x)) = 1 > 0 = 0(x) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 4 + x + xs >= x = x choice(cons(x,xs)) = 4 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 4 + clause + cnf >= 4 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = e + 2*t >= e = e if(true(),t,e) = e + 2*t >= t = t member(x,cons(y,ys)) = x >= x = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = x >= 0 = false() sat(cnf) = 6 + 7*cnf >= 6 + 6*cnf = satck(cnf,guess(cnf)) satck(cnf,assign) = 6 + 6*assign >= 5 + 6*assign = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 5 + l + ls >= 5 + ls = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 1 >= 0 = true() ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 1 p(choice) = x1 p(cons) = 7 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = 1 + x1 p(if) = 4*x1 + x2 + x3 p(member) = 1 + x1 p(negate) = 0 p(nil) = 0 p(sat) = 7 + 6*x1 p(satck) = 2 + x1 + 5*x2 p(true) = 0 p(unsat) = 1 p(verify) = x1 Following rules are strictly oriented: member(x,nil()) = 1 + x > 0 = false() Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 7 + x + xs >= x = x choice(cons(x,xs)) = 7 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 8 + clause + cnf >= 8 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 1 >= 0 = nil() if(false(),t,e) = e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,cons(y,ys)) = 1 + x >= 1 + x = if(eq(x,y),true(),member(x,ys)) negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) sat(cnf) = 7 + 6*cnf >= 7 + 6*cnf = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 5*assign + cnf >= 1 + 5*assign = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 7 + l + ls >= 4 + ls = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() ** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = x1 p(1) = x1 p(O) = x1 p(choice) = x1 p(cons) = x1 + x2 p(eq) = x1*x2 p(false) = 0 p(guess) = x1 p(if) = x1 + x2 + x3 p(member) = 2*x1 + x1*x2 p(negate) = x1 p(nil) = 1 p(sat) = 3 + 3*x1 + 2*x1^2 p(satck) = 2 + x1^2 + 3*x2 + x2^2 p(true) = 0 p(unsat) = 0 p(verify) = 1 + 2*x1 + x1^2 Following rules are strictly oriented: eq(nil(),nil()) = 1 > 0 = true() Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = x + xs >= x = x choice(cons(x,xs)) = x + xs >= xs = choice(xs) eq(0(x),1(y)) = x*y >= 0 = false() eq(1(x),0(y)) = x*y >= 0 = false() eq(1(x),1(y)) = x*y >= x*y = eq(x,y) eq(O(x),0(y)) = x*y >= x*y = eq(x,y) guess(cons(clause,cnf)) = clause + cnf >= clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 1 >= 1 = nil() if(false(),t,e) = e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,cons(y,ys)) = 2*x + x*y + x*ys >= 2*x + x*y + x*ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 3*x >= 0 = false() negate(0(x)) = x >= x = 1(x) negate(1(x)) = x >= x = 0(x) sat(cnf) = 3 + 3*cnf + 2*cnf^2 >= 2 + 3*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 3*assign + assign^2 + cnf^2 >= 1 + 3*assign + assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 1 + 2*l + 2*l*ls + l^2 + 2*ls + ls^2 >= 1 + 2*l + l*ls + 2*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 4 >= 0 = true() ** Step 1.b:7: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 0 p(1) = 0 p(O) = 0 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = x1 p(if) = 1 + x1 + 2*x2 + x3 p(member) = x1 + 2*x2 p(negate) = 0 p(nil) = 0 p(sat) = 3 + 2*x1 + 3*x1^2 p(satck) = 2 + 2*x1^2 + 2*x2 + x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1^2 Following rules are strictly oriented: if(false(),t,e) = 1 + e + 2*t > e = e if(true(),t,e) = 1 + e + 2*t > t = t member(x,cons(y,ys)) = 4 + x + 2*y + 2*ys > 1 + x + 2*ys = if(eq(x,y),true(),member(x,ys)) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 2 + x + xs >= x = x choice(cons(x,xs)) = 2 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = 0 >= 0 = false() eq(1(x),0(y)) = 0 >= 0 = false() eq(1(x),1(y)) = 0 >= 0 = eq(x,y) eq(O(x),0(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() member(x,nil()) = x >= 0 = false() negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) sat(cnf) = 3 + 2*cnf + 3*cnf^2 >= 2 + 2*cnf + 3*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 2*assign + assign^2 + 2*cnf^2 >= 1 + 2*assign + assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 4 + 4*l + 2*l*ls + l^2 + 4*ls + ls^2 >= 1 + 2*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() ** Step 1.b:8: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = x1 p(1) = x1 p(O) = 1 + x1 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = 2*x1 p(false) = 0 p(guess) = x1 p(if) = x1 + x2 + x3 p(member) = x1 + x1*x2 + x2 p(negate) = x1 p(nil) = 1 p(sat) = 2 + 2*x1 + 2*x1^2 p(satck) = 1 + 2*x2 + 2*x2^2 p(true) = 1 p(unsat) = 1 p(verify) = x1^2 Following rules are strictly oriented: eq(O(x),0(y)) = 2 + 2*x > 2*x = eq(x,y) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 2 + x + xs >= x = x choice(cons(x,xs)) = 2 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = 2*x >= 0 = false() eq(1(x),0(y)) = 2*x >= 0 = false() eq(1(x),1(y)) = 2*x >= 2*x = eq(x,y) eq(nil(),nil()) = 2 >= 1 = true() guess(cons(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 1 >= 1 = nil() if(false(),t,e) = e + t >= e = e if(true(),t,e) = 1 + e + t >= t = t member(x,cons(y,ys)) = 2 + 3*x + x*y + x*ys + y + ys >= 1 + 3*x + x*ys + ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 1 + 2*x >= 0 = false() negate(0(x)) = x >= x = 1(x) negate(1(x)) = x >= x = 0(x) sat(cnf) = 2 + 2*cnf + 2*cnf^2 >= 1 + 2*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 1 + 2*assign + 2*assign^2 >= 1 + assign + assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 4 + 4*l + 2*l*ls + l^2 + 4*ls + ls^2 >= l + l*ls + ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 1 >= 1 = true() ** Step 1.b:9: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = 1 + x1 p(1) = x1 p(O) = 0 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = x2 p(false) = 0 p(guess) = x1 p(if) = 2 + x1 + 2*x2 + x3 p(member) = 1 + x1 + x2 p(negate) = 1 + x1 p(nil) = 0 p(sat) = 3 + 3*x1 + 2*x1^2 p(satck) = 2 + 3*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1 + x1^2 Following rules are strictly oriented: eq(1(x),0(y)) = 1 + y > 0 = false() Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 2 + x + xs >= x = x choice(cons(x,xs)) = 2 + x + xs >= xs = choice(xs) eq(0(x),1(y)) = y >= 0 = false() eq(1(x),1(y)) = y >= y = eq(x,y) eq(O(x),0(y)) = 1 + y >= y = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = 2 + e + 2*t >= e = e if(true(),t,e) = 2 + e + 2*t >= t = t member(x,cons(y,ys)) = 3 + x + y + ys >= 3 + x + y + ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 1 + x >= 0 = false() negate(0(x)) = 2 + x >= x = 1(x) negate(1(x)) = 1 + x >= 1 + x = 0(x) sat(cnf) = 3 + 3*cnf + 2*cnf^2 >= 2 + 3*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 3*assign + 2*assign^2 >= 2 + 3*assign + assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 6 + 5*l + 2*l*ls + l^2 + 5*ls + ls^2 >= 4 + l + 2*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() ** Step 1.b:10: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(x),1(y)) -> false() eq(1(x),1(y)) -> eq(x,y) - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(1(x),0(y)) -> false() eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + 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(cons) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: {choice,eq,guess,if,member,negate,sat,satck,verify} TcT has computed the following interpretation: p(0) = x1 p(1) = 1 + x1 p(O) = 0 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = x2 p(false) = 0 p(guess) = x1 p(if) = 2*x1 + 2*x2 + x3 p(member) = x1 + 2*x2 p(negate) = 1 + x1 p(nil) = 0 p(sat) = 2 + 2*x1 + 2*x1^2 p(satck) = 1 + 2*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1^2 Following rules are strictly oriented: eq(0(x),1(y)) = 1 + y > 0 = false() eq(1(x),1(y)) = 1 + y > y = eq(x,y) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = 2 + x + xs >= x = x choice(cons(x,xs)) = 2 + x + xs >= xs = choice(xs) eq(1(x),0(y)) = y >= 0 = false() eq(O(x),0(y)) = y >= y = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(cons(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = e + 2*t >= e = e if(true(),t,e) = e + 2*t >= t = t member(x,cons(y,ys)) = 4 + x + 2*y + 2*ys >= x + 2*y + 2*ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = x >= 0 = false() negate(0(x)) = 1 + x >= 1 + x = 1(x) negate(1(x)) = 2 + x >= x = 0(x) sat(cnf) = 2 + 2*cnf + 2*cnf^2 >= 1 + 2*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 1 + 2*assign + 2*assign^2 >= 2*assign + 2*assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 4 + 4*l + 2*l*ls + l^2 + 4*ls + ls^2 >= 2 + 2*l + 4*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() ** Step 1.b:11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: choice(cons(x,xs)) -> x choice(cons(x,xs)) -> choice(xs) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) member(x,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Signature: {choice/1,eq/2,guess/1,if/3,member/2,negate/1,sat/1,satck/2,verify/1} / {0/1,1/1,O/1,cons/2,false/0,nil/0 ,true/0,unsat/0} - Obligation: innermost runtime complexity wrt. defined symbols {choice,eq,guess,if,member,negate,sat,satck ,verify} and constructors {0,1,O,cons,false,nil,true,unsat} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))