WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),#1(y)) -> eq(x,y) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(O(x),O(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,dd(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(dd(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,dd/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,dd,false,nil,true,unsat} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(dd) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#1) = [0] p(O) = [1] x1 + [0] p(choice) = [2] x1 + [2] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [4] p(false) = [0] p(guess) = [2] x1 + [7] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(member) = [1] x1 + [5] p(negate) = [0] p(nil) = [0] p(sat) = [3] x1 + [0] p(satck) = [1] x1 + [1] x2 + [2] p(true) = [0] p(unsat) = [0] p(verify) = [6] Following rules are strictly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [2] > [1] x + [0] = x eq(#1(x),O(y)) = [4] > [0] = false() eq(O(x),#1(y)) = [4] > [0] = false() eq(nil(),nil()) = [4] > [0] = true() guess(nil()) = [7] > [0] = nil() if(false(),t,e) = [1] e + [1] t + [2] > [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [2] > [1] t + [0] = t member(x,nil()) = [1] x + [5] > [0] = false() verify(nil()) = [6] > [0] = true() Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [2] >= [2] xs + [2] = choice(xs) eq(#1(x),#1(y)) = [4] >= [4] = eq(x,y) eq(O(x),O(y)) = [4] >= [4] = eq(x,y) guess(dd(clause,cnf)) = [2] clause + [2] cnf + [7] >= [2] clause + [2] cnf + [9] = dd(choice(clause),guess(cnf)) member(x,dd(y,ys)) = [1] x + [5] >= [1] x + [11] = if(eq(x,y),true(),member(x,ys)) negate(#0(x)) = [0] >= [0] = #1(x) negate(#1(x)) = [0] >= [0] = #0(x) sat(cnf) = [3] cnf + [0] >= [3] cnf + [9] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [1] cnf + [2] >= [1] assign + [8] = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = [6] >= [13] = if(member(negate(l),ls),false(),verify(ls)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: choice(dd(x,xs)) -> choice(xs) eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) member(x,dd(y,ys)) -> if(eq(x,y),true(),member(x,ys)) 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(dd(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) - Weak TRS: choice(dd(x,xs)) -> x eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,nil()) -> false() 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,dd/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,dd,false,nil,true,unsat} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(dd) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#1) = [0] p(O) = [1] x1 + [0] p(choice) = [2] x1 + [4] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [2] p(false) = [1] p(guess) = [2] x1 + [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [4] p(member) = [1] x1 + [1] p(negate) = [5] p(nil) = [0] p(sat) = [2] x1 + [0] p(satck) = [1] x2 + [5] p(true) = [0] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: negate(#0(x)) = [5] > [0] = #1(x) negate(#1(x)) = [5] > [0] = #0(x) satck(cnf,assign) = [1] assign + [5] > [1] assign + [4] = if(verify(assign),assign,unsat()) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [4] >= [1] x + [0] = x choice(dd(x,xs)) = [2] x + [2] xs + [4] >= [2] xs + [4] = choice(xs) eq(#1(x),#1(y)) = [2] >= [2] = eq(x,y) eq(#1(x),O(y)) = [2] >= [1] = false() eq(O(x),#1(y)) = [2] >= [1] = false() eq(O(x),O(y)) = [2] >= [2] = eq(x,y) eq(nil(),nil()) = [2] >= [0] = true() guess(dd(clause,cnf)) = [2] clause + [2] cnf + [4] >= [2] clause + [2] cnf + [8] = dd(choice(clause),guess(cnf)) guess(nil()) = [4] >= [0] = nil() if(false(),t,e) = [1] e + [1] t + [5] >= [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [4] >= [1] t + [0] = t member(x,dd(y,ys)) = [1] x + [1] >= [1] x + [7] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [1] >= [1] = false() sat(cnf) = [2] cnf + [0] >= [2] cnf + [9] = satck(cnf,guess(cnf)) verify(dd(l,ls)) = [0] >= [11] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: choice(dd(x,xs)) -> choice(xs) eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) member(x,dd(y,ys)) -> if(eq(x,y),true(),member(x,ys)) sat(cnf) -> satck(cnf,guess(cnf)) verify(dd(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) - Weak TRS: choice(dd(x,xs)) -> x eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,nil()) -> false() negate(#0(x)) -> #1(x) negate(#1(x)) -> #0(x) 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,dd/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,dd,false,nil,true,unsat} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(dd) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [5] p(#1) = [5] p(O) = [0] p(choice) = [2] x1 + [1] p(dd) = [1] x1 + [1] x2 + [4] p(eq) = [0] p(false) = [0] p(guess) = [2] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(member) = [1] x1 + [0] p(negate) = [5] p(nil) = [0] p(sat) = [2] x1 + [1] p(satck) = [1] x2 + [7] p(true) = [0] p(unsat) = [2] p(verify) = [5] Following rules are strictly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [9] > [2] xs + [1] = choice(xs) guess(dd(clause,cnf)) = [2] clause + [2] cnf + [8] > [2] clause + [2] cnf + [5] = dd(choice(clause),guess(cnf)) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [9] >= [1] x + [0] = x eq(#1(x),#1(y)) = [0] >= [0] = eq(x,y) eq(#1(x),O(y)) = [0] >= [0] = false() eq(O(x),#1(y)) = [0] >= [0] = false() eq(O(x),O(y)) = [0] >= [0] = eq(x,y) eq(nil(),nil()) = [0] >= [0] = true() guess(nil()) = [0] >= [0] = nil() if(false(),t,e) = [1] e + [1] t + [0] >= [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [0] >= [1] t + [0] = t member(x,dd(y,ys)) = [1] x + [0] >= [1] x + [0] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [0] >= [0] = false() negate(#0(x)) = [5] >= [5] = #1(x) negate(#1(x)) = [5] >= [5] = #0(x) sat(cnf) = [2] cnf + [1] >= [2] cnf + [7] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [7] >= [1] assign + [7] = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = [5] >= [10] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [5] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) member(x,dd(y,ys)) -> if(eq(x,y),true(),member(x,ys)) sat(cnf) -> satck(cnf,guess(cnf)) verify(dd(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,nil()) -> false() negate(#0(x)) -> #1(x) negate(#1(x)) -> #0(x) 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,dd/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,dd,false,nil,true,unsat} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(dd) = {1,2}, uargs(if) = {1,3}, uargs(member) = {1}, uargs(satck) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [6] p(#1) = [6] p(O) = [1] x1 + [0] p(choice) = [2] x1 + [0] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [1] p(false) = [1] p(guess) = [3] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(member) = [1] x1 + [1] p(negate) = [6] p(nil) = [1] p(sat) = [3] x1 + [7] p(satck) = [1] x2 + [6] p(true) = [0] p(unsat) = [0] p(verify) = [4] Following rules are strictly oriented: sat(cnf) = [3] cnf + [7] > [3] cnf + [6] = satck(cnf,guess(cnf)) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [0] >= [1] x + [0] = x choice(dd(x,xs)) = [2] x + [2] xs + [0] >= [2] xs + [0] = choice(xs) eq(#1(x),#1(y)) = [1] >= [1] = eq(x,y) eq(#1(x),O(y)) = [1] >= [1] = false() eq(O(x),#1(y)) = [1] >= [1] = false() eq(O(x),O(y)) = [1] >= [1] = eq(x,y) eq(nil(),nil()) = [1] >= [0] = true() guess(dd(clause,cnf)) = [3] clause + [3] cnf + [0] >= [2] clause + [3] cnf + [0] = dd(choice(clause),guess(cnf)) guess(nil()) = [3] >= [1] = nil() if(false(),t,e) = [1] e + [1] t + [2] >= [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [1] >= [1] t + [0] = t member(x,dd(y,ys)) = [1] x + [1] >= [1] x + [3] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [1] >= [1] = false() negate(#0(x)) = [6] >= [6] = #1(x) negate(#1(x)) = [6] >= [6] = #0(x) satck(cnf,assign) = [1] assign + [6] >= [1] assign + [5] = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = [4] >= [13] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [4] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) member(x,dd(y,ys)) -> if(eq(x,y),true(),member(x,ys)) verify(dd(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t 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(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,dd/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,dd,false,nil,true,unsat} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(dd) = {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) = [8] p(choice) = [2] x1 + [1] p(dd) = [1] x1 + [1] x2 + [2] p(eq) = [0] p(false) = [0] p(guess) = [2] x1 + [2] p(if) = [4] x1 + [1] x2 + [1] x3 + [0] p(member) = [2] x1 + [0] p(negate) = [0] p(nil) = [12] p(sat) = [15] x1 + [15] p(satck) = [4] x1 + [5] x2 + [5] p(true) = [0] p(unsat) = [0] p(verify) = [1] x1 + [0] Following rules are strictly oriented: verify(dd(l,ls)) = [1] l + [1] ls + [2] > [1] ls + [0] = if(member(negate(l),ls),false(),verify(ls)) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = [2] x + [2] xs + [5] >= [1] x + [0] = x choice(dd(x,xs)) = [2] x + [2] xs + [5] >= [2] xs + [1] = choice(xs) eq(#1(x),#1(y)) = [0] >= [0] = eq(x,y) eq(#1(x),O(y)) = [0] >= [0] = false() eq(O(x),#1(y)) = [0] >= [0] = false() eq(O(x),O(y)) = [0] >= [0] = eq(x,y) eq(nil(),nil()) = [0] >= [0] = true() guess(dd(clause,cnf)) = [2] clause + [2] cnf + [6] >= [2] clause + [2] cnf + [5] = dd(choice(clause),guess(cnf)) guess(nil()) = [26] >= [12] = nil() if(false(),t,e) = [1] e + [1] t + [0] >= [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [0] >= [1] t + [0] = t member(x,dd(y,ys)) = [2] x + [0] >= [2] x + [0] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [2] x + [0] >= [0] = false() negate(#0(x)) = [0] >= [0] = #1(x) negate(#1(x)) = [0] >= [0] = #0(x) sat(cnf) = [15] cnf + [15] >= [14] cnf + [15] = satck(cnf,guess(cnf)) satck(cnf,assign) = [5] assign + [4] cnf + [5] >= [5] assign + [0] = if(verify(assign),assign,unsat()) verify(nil()) = [12] >= [0] = true() * Step 6: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) member(x,dd(y,ys)) -> if(eq(x,y),true(),member(x,ys)) - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t 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(dd(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,dd/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,dd,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(dd) = {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(dd) = 1 + x1 + x2 p(eq) = 0 p(false) = 0 p(guess) = x1 p(if) = x1 + x2 + x3 p(member) = x1 + x1*x2 + x2 p(negate) = 1 p(nil) = 0 p(sat) = 2 + 2*x1 + 2*x1^2 p(satck) = 2 + 2*x2 + 2*x2^2 p(true) = 0 p(unsat) = 1 p(verify) = 2*x1^2 Following rules are strictly oriented: member(x,dd(y,ys)) = 1 + 2*x + x*y + x*ys + y + ys > x + x*ys + ys = if(eq(x,y),true(),member(x,ys)) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = 1 + x + xs >= x = x choice(dd(x,xs)) = 1 + x + xs >= xs = choice(xs) eq(#1(x),#1(y)) = 0 >= 0 = eq(x,y) eq(#1(x),O(y)) = 0 >= 0 = false() eq(O(x),#1(y)) = 0 >= 0 = false() eq(O(x),O(y)) = 0 >= 0 = eq(x,y) eq(nil(),nil()) = 0 >= 0 = true() guess(dd(clause,cnf)) = 1 + clause + cnf >= 1 + clause + cnf = dd(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,nil()) = x >= 0 = false() negate(#0(x)) = 1 >= 0 = #1(x) negate(#1(x)) = 1 >= 0 = #0(x) sat(cnf) = 2 + 2*cnf + 2*cnf^2 >= 2 + 2*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 2*assign + 2*assign^2 >= 1 + assign + 2*assign^2 = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = 2 + 4*l + 4*l*ls + 2*l^2 + 4*ls + 2*ls^2 >= 1 + 2*ls + 2*ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() * Step 7: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(#1(x),#1(y)) -> eq(x,y) eq(O(x),O(y)) -> eq(x,y) - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,dd(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(dd(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,dd/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,dd,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(dd) = {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(dd) = 1 + x1 + x2 p(eq) = x2 p(false) = 0 p(guess) = x1 p(if) = 1 + x1 + x2 + x3 p(member) = x1 + x2 p(negate) = 2*x1 p(nil) = 0 p(sat) = 2 + 3*x1 + 3*x1^2 p(satck) = 2 + x1^2 + 2*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = 2*x1^2 Following rules are strictly oriented: eq(O(x),O(y)) = 1 + y > y = eq(x,y) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = 1 + x + xs >= x = x choice(dd(x,xs)) = 1 + x + xs >= xs = choice(xs) eq(#1(x),#1(y)) = y >= y = eq(x,y) eq(#1(x),O(y)) = 1 + y >= 0 = false() eq(O(x),#1(y)) = y >= 0 = false() eq(nil(),nil()) = 0 >= 0 = true() guess(dd(clause,cnf)) = 1 + clause + cnf >= 1 + clause + cnf = dd(choice(clause),guess(cnf)) guess(nil()) = 0 >= 0 = nil() if(false(),t,e) = 1 + e + t >= e = e if(true(),t,e) = 1 + e + t >= t = t member(x,dd(y,ys)) = 1 + x + y + ys >= 1 + x + y + ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = x >= 0 = false() negate(#0(x)) = 2*x >= x = #1(x) negate(#1(x)) = 2*x >= x = #0(x) sat(cnf) = 2 + 3*cnf + 3*cnf^2 >= 2 + 2*cnf + 3*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2 + 2*assign + 2*assign^2 + cnf^2 >= 1 + assign + 2*assign^2 = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = 2 + 4*l + 4*l*ls + 2*l^2 + 4*ls + 2*ls^2 >= 1 + 2*l + ls + 2*ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() * Step 8: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(#1(x),#1(y)) -> eq(x,y) - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(O(x),O(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,dd(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(dd(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,dd/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,dd,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(dd) = {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) = 2 + x1 p(#1) = 2 + x1 p(O) = x1 p(choice) = x1 p(dd) = 2 + x1 + x2 p(eq) = x2 p(false) = 0 p(guess) = x1 p(if) = 1 + 2*x1 + 3*x2 + x3 p(member) = 1 + x1 + 2*x2 p(negate) = x1 p(nil) = 1 p(sat) = 1 + 3*x1 + 2*x1^2 p(satck) = 1 + 3*x2 + 2*x2^2 p(true) = 1 p(unsat) = 0 p(verify) = x1^2 Following rules are strictly oriented: eq(#1(x),#1(y)) = 2 + y > y = eq(x,y) Following rules are (at-least) weakly oriented: choice(dd(x,xs)) = 2 + x + xs >= x = x choice(dd(x,xs)) = 2 + x + xs >= xs = choice(xs) eq(#1(x),O(y)) = y >= 0 = false() eq(O(x),#1(y)) = 2 + y >= 0 = false() eq(O(x),O(y)) = y >= y = eq(x,y) eq(nil(),nil()) = 1 >= 1 = true() guess(dd(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = dd(choice(clause),guess(cnf)) guess(nil()) = 1 >= 1 = nil() if(false(),t,e) = 1 + e + 3*t >= e = e if(true(),t,e) = 3 + e + 3*t >= t = t member(x,dd(y,ys)) = 5 + x + 2*y + 2*ys >= 5 + x + 2*y + 2*ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 3 + x >= 0 = false() negate(#0(x)) = 2 + x >= 2 + x = #1(x) negate(#1(x)) = 2 + x >= 2 + x = #0(x) sat(cnf) = 1 + 3*cnf + 2*cnf^2 >= 1 + 3*cnf + 2*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 1 + 3*assign + 2*assign^2 >= 1 + 3*assign + 2*assign^2 = if(verify(assign),assign,unsat()) verify(dd(l,ls)) = 4 + 4*l + 2*l*ls + l^2 + 4*ls + ls^2 >= 3 + 2*l + 4*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 1 >= 1 = true() * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: choice(dd(x,xs)) -> x choice(dd(x,xs)) -> choice(xs) eq(#1(x),#1(y)) -> eq(x,y) eq(#1(x),O(y)) -> false() eq(O(x),#1(y)) -> false() eq(O(x),O(y)) -> eq(x,y) eq(nil(),nil()) -> true() guess(dd(clause,cnf)) -> dd(choice(clause),guess(cnf)) guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t member(x,dd(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(dd(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,dd/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,dd,false,nil,true,unsat} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))