WORST_CASE(?,O(n^2)) * Step 1: WeightGap 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: 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(cons) = {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) = [2] p(O) = [0] p(choice) = [1] x1 + [4] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(false) = [0] p(guess) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(member) = [1] x1 + [0] p(negate) = [0] p(nil) = [0] p(sat) = [1] x1 + [4] p(satck) = [1] x2 + [0] p(true) = [0] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [4] > [1] x + [0] = x sat(cnf) = [1] cnf + [4] > [1] cnf + [0] = satck(cnf,guess(cnf)) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [4] >= [1] xs + [4] = 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)) = [1] clause + [1] cnf + [0] >= [1] clause + [1] cnf + [4] = cons(choice(clause),guess(cnf)) 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,cons(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)) = [0] >= [2] = 1(x) negate(1(x)) = [0] >= [0] = 0(x) satck(cnf,assign) = [1] assign + [0] >= [1] assign + [0] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [0] >= [0] = 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 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Weak TRS: choice(cons(x,xs)) -> x sat(cnf) -> satck(cnf,guess(cnf)) - 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: 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(cons) = {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) = [0] p(choice) = [2] x1 + [3] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [4] p(false) = [0] p(guess) = [2] x1 + [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(member) = [1] x1 + [0] p(negate) = [0] p(nil) = [4] p(sat) = [2] x1 + [3] p(satck) = [1] x2 + [1] p(true) = [0] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: eq(0(x),1(y)) = [4] > [0] = false() eq(1(x),0(y)) = [4] > [0] = false() eq(nil(),nil()) = [4] > [0] = true() guess(nil()) = [9] > [4] = 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 Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [2] x + [2] xs + [3] >= [1] x + [0] = x choice(cons(x,xs)) = [2] x + [2] xs + [3] >= [2] xs + [3] = choice(xs) eq(1(x),1(y)) = [4] >= [4] = eq(x,y) eq(O(x),0(y)) = [4] >= [4] = eq(x,y) guess(cons(clause,cnf)) = [2] clause + [2] cnf + [1] >= [2] clause + [2] cnf + [4] = cons(choice(clause),guess(cnf)) member(x,cons(y,ys)) = [1] x + [0] >= [1] x + [6] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [0] >= [0] = false() negate(0(x)) = [0] >= [0] = 1(x) negate(1(x)) = [0] >= [0] = 0(x) sat(cnf) = [2] cnf + [3] >= [2] cnf + [2] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [1] >= [1] assign + [2] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [0] >= [2] = 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(cons(x,xs)) -> choice(xs) eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) 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) satck(cnf,assign) -> if(verify(assign),assign,unsat()) verify(cons(l,ls)) -> if(member(negate(l),ls),false(),verify(ls)) verify(nil()) -> true() - Weak TRS: choice(cons(x,xs)) -> x eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(nil(),nil()) -> true() guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t sat(cnf) -> satck(cnf,guess(cnf)) - 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: 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(cons) = {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) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [4] p(eq) = [1] p(false) = [0] p(guess) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(member) = [1] x1 + [0] p(negate) = [0] p(nil) = [0] p(sat) = [1] x1 + [0] p(satck) = [1] x2 + [0] p(true) = [1] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [5] > [1] xs + [1] = choice(xs) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [5] >= [1] x + [0] = x eq(0(x),1(y)) = [1] >= [0] = false() eq(1(x),0(y)) = [1] >= [0] = false() eq(1(x),1(y)) = [1] >= [1] = eq(x,y) eq(O(x),0(y)) = [1] >= [1] = eq(x,y) eq(nil(),nil()) = [1] >= [1] = true() guess(cons(clause,cnf)) = [1] clause + [1] cnf + [4] >= [1] clause + [1] cnf + [5] = cons(choice(clause),guess(cnf)) 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 + [1] >= [1] t + [0] = t member(x,cons(y,ys)) = [1] x + [0] >= [1] x + [2] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [0] >= [0] = false() negate(0(x)) = [0] >= [0] = 1(x) negate(1(x)) = [0] >= [0] = 0(x) sat(cnf) = [1] cnf + [0] >= [1] cnf + [0] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [0] >= [1] assign + [0] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [0] >= [0] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [0] >= [1] = 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),0(y)) -> eq(x,y) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) 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) satck(cnf,assign) -> if(verify(assign),assign,unsat()) 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) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() eq(nil(),nil()) -> true() guess(nil()) -> nil() if(false(),t,e) -> e if(true(),t,e) -> t sat(cnf) -> satck(cnf,guess(cnf)) - 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: 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(cons) = {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 + [1] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [7] p(false) = [0] p(guess) = [2] x1 + [5] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(member) = [1] x1 + [4] p(negate) = [6] p(nil) = [5] p(sat) = [2] x1 + [6] p(satck) = [1] x2 + [0] p(true) = [0] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: member(x,nil()) = [1] x + [4] > [0] = false() negate(0(x)) = [6] > [0] = 1(x) negate(1(x)) = [6] > [0] = 0(x) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [2] x + [2] xs + [1] >= [1] x + [0] = x choice(cons(x,xs)) = [2] x + [2] xs + [1] >= [2] xs + [1] = choice(xs) eq(0(x),1(y)) = [7] >= [0] = false() eq(1(x),0(y)) = [7] >= [0] = false() eq(1(x),1(y)) = [7] >= [7] = eq(x,y) eq(O(x),0(y)) = [7] >= [7] = eq(x,y) eq(nil(),nil()) = [7] >= [0] = true() guess(cons(clause,cnf)) = [2] clause + [2] cnf + [5] >= [2] clause + [2] cnf + [6] = cons(choice(clause),guess(cnf)) guess(nil()) = [15] >= [5] = nil() if(false(),t,e) = [1] e + [1] t + [1] >= [1] e + [0] = e if(true(),t,e) = [1] e + [1] t + [1] >= [1] t + [0] = t member(x,cons(y,ys)) = [1] x + [4] >= [1] x + [12] = if(eq(x,y),true(),member(x,ys)) sat(cnf) = [2] cnf + [6] >= [2] cnf + [5] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [0] >= [1] assign + [1] = if(verify(assign),assign,unsat()) verify(cons(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 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) guess(cons(clause,cnf)) -> cons(choice(clause),guess(cnf)) member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) 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) eq(0(x),1(y)) -> false() eq(1(x),0(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) sat(cnf) -> satck(cnf,guess(cnf)) - 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: 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(cons) = {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) = [5] x1 + [0] p(cons) = [1] x1 + [1] x2 + [1] p(eq) = [5] p(false) = [0] p(guess) = [5] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(member) = [1] x1 + [0] p(negate) = [0] p(nil) = [0] p(sat) = [5] x1 + [0] p(satck) = [1] x2 + [0] p(true) = [5] p(unsat) = [0] p(verify) = [7] Following rules are strictly oriented: guess(cons(clause,cnf)) = [5] clause + [5] cnf + [5] > [5] clause + [5] cnf + [1] = cons(choice(clause),guess(cnf)) verify(nil()) = [7] > [5] = true() Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [5] x + [5] xs + [5] >= [1] x + [0] = x choice(cons(x,xs)) = [5] x + [5] xs + [5] >= [5] xs + [0] = choice(xs) eq(0(x),1(y)) = [5] >= [0] = false() eq(1(x),0(y)) = [5] >= [0] = false() eq(1(x),1(y)) = [5] >= [5] = eq(x,y) eq(O(x),0(y)) = [5] >= [5] = eq(x,y) eq(nil(),nil()) = [5] >= [5] = 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 + [5] >= [1] t + [0] = t member(x,cons(y,ys)) = [1] x + [0] >= [1] x + [10] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [0] >= [0] = false() negate(0(x)) = [0] >= [0] = 1(x) negate(1(x)) = [0] >= [0] = 0(x) sat(cnf) = [5] cnf + [0] >= [5] cnf + [0] = satck(cnf,guess(cnf)) satck(cnf,assign) = [1] assign + [0] >= [1] assign + [7] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [7] >= [7] = 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 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) satck(cnf,assign) -> if(verify(assign),assign,unsat()) 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) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() 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,nil()) -> false() negate(0(x)) -> 1(x) negate(1(x)) -> 0(x) sat(cnf) -> satck(cnf,guess(cnf)) 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: 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(cons) = {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) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(false) = [0] p(guess) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(member) = [1] x1 + [0] p(negate) = [0] p(nil) = [0] p(sat) = [1] x1 + [4] p(satck) = [1] x2 + [4] p(true) = [0] p(unsat) = [0] p(verify) = [0] Following rules are strictly oriented: satck(cnf,assign) = [1] assign + [4] > [1] assign + [0] = if(verify(assign),assign,unsat()) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [0] = x choice(cons(x,xs)) = [1] x + [1] xs + [0] >= [1] xs + [0] = 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)) = [1] clause + [1] cnf + [0] >= [1] clause + [1] cnf + [0] = cons(choice(clause),guess(cnf)) 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,cons(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)) = [0] >= [0] = 1(x) negate(1(x)) = [0] >= [0] = 0(x) sat(cnf) = [1] cnf + [4] >= [1] cnf + [4] = satck(cnf,guess(cnf)) verify(cons(l,ls)) = [0] >= [0] = 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 7: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(1(x),1(y)) -> eq(x,y) eq(O(x),0(y)) -> eq(x,y) member(x,cons(y,ys)) -> if(eq(x,y),true(),member(x,ys)) 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) eq(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() 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,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,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: 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(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) = [9] p(1) = [3] p(O) = [1] p(choice) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [4] p(eq) = [0] p(false) = [0] p(guess) = [1] x1 + [0] p(if) = [2] x1 + [4] x2 + [1] x3 + [0] p(member) = [1] x1 + [1] p(negate) = [2] x1 + [3] p(nil) = [0] p(sat) = [12] x1 + [0] p(satck) = [12] x2 + [0] p(true) = [0] p(unsat) = [0] p(verify) = [4] x1 + [0] Following rules are strictly oriented: verify(cons(l,ls)) = [4] l + [4] ls + [16] > [4] l + [4] ls + [8] = if(member(negate(l),ls),false(),verify(ls)) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1] x + [1] xs + [4] >= [1] x + [0] = x choice(cons(x,xs)) = [1] x + [1] xs + [4] >= [1] xs + [0] = 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)) = [1] clause + [1] cnf + [4] >= [1] clause + [1] cnf + [4] = cons(choice(clause),guess(cnf)) guess(nil()) = [0] >= [0] = nil() if(false(),t,e) = [1] e + [4] t + [0] >= [1] e + [0] = e if(true(),t,e) = [1] e + [4] t + [0] >= [1] t + [0] = t member(x,cons(y,ys)) = [1] x + [1] >= [1] x + [1] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1] x + [1] >= [0] = false() negate(0(x)) = [21] >= [3] = 1(x) negate(1(x)) = [9] >= [9] = 0(x) sat(cnf) = [12] cnf + [0] >= [12] cnf + [0] = satck(cnf,guess(cnf)) satck(cnf,assign) = [12] assign + [0] >= [12] assign + [0] = if(verify(assign),assign,unsat()) 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) eq(O(x),0(y)) -> eq(x,y) 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(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() 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,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) = 2*x1 + x2 + x3 p(member) = 2*x1 + x2 p(negate) = 0 p(nil) = 0 p(sat) = 2 + 3*x1 + 3*x1^2 p(satck) = 1 + x1^2 + 3*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1 + x1^2 Following rules are strictly oriented: member(x,cons(y,ys)) = 2 + 2*x + y + ys > 2*x + 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() if(false(),t,e) = e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,nil()) = 2*x >= 0 = false() negate(0(x)) = 0 >= 0 = 1(x) negate(1(x)) = 0 >= 0 = 0(x) sat(cnf) = 2 + 3*cnf + 3*cnf^2 >= 1 + 3*cnf + 3*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 1 + 3*assign + 2*assign^2 + cnf^2 >= 3*assign + 2*assign^2 = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = 6 + 5*l + 2*l*ls + l^2 + 5*ls + ls^2 >= 3*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() * Step 9: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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(0(x),1(y)) -> false() eq(1(x),0(y)) -> false() 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) = 1 + x1 p(O) = x1 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = 1 + x1 p(false) = 2 p(guess) = x1 p(if) = 2*x1 + x2 + x3 p(member) = 2*x1 + x1*x2 + x2 p(negate) = x1 p(nil) = 2 p(sat) = 3*x1 + 3*x1^2 p(satck) = x1^2 + 2*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1^2 Following rules are strictly oriented: eq(1(x),1(y)) = 2 + x > 1 + 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 >= 2 = false() eq(1(x),0(y)) = 2 + x >= 2 = false() eq(O(x),0(y)) = 1 + x >= 1 + x = eq(x,y) eq(nil(),nil()) = 3 >= 0 = true() guess(cons(clause,cnf)) = 2 + clause + cnf >= 2 + clause + cnf = cons(choice(clause),guess(cnf)) guess(nil()) = 2 >= 2 = nil() if(false(),t,e) = 4 + e + t >= e = e if(true(),t,e) = e + t >= t = t member(x,cons(y,ys)) = 2 + 4*x + x*y + x*ys + y + ys >= 2 + 4*x + x*ys + ys = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = 2 + 4*x >= 2 = false() negate(0(x)) = 1 + x >= 1 + x = 1(x) negate(1(x)) = 1 + x >= 1 + x = 0(x) sat(cnf) = 3*cnf + 3*cnf^2 >= 2*cnf + 3*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2*assign + 2*assign^2 + cnf^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 + 4*l + 2*l*ls + 2*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 4 >= 0 = true() * Step 10: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(O(x),0(y)) -> eq(x,y) - 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(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) = 2 + x1 p(choice) = x1 p(cons) = 2 + x1 + x2 p(eq) = x1 p(false) = 0 p(guess) = x1 p(if) = 2*x1 + 2*x2 + 2*x2*x3 + x3 p(member) = 1 + 2*x1 + x1*x2 p(negate) = x1 p(nil) = 0 p(sat) = 3*x1 + 3*x1^2 p(satck) = x1*x2 + 2*x2 + 2*x2^2 p(true) = 0 p(unsat) = 0 p(verify) = x1^2 Following rules are strictly oriented: eq(O(x),0(y)) = 2 + x > 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)) = x >= 0 = false() eq(1(x),0(y)) = x >= 0 = false() eq(1(x),1(y)) = x >= x = 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*e*t + 2*t >= e = e if(true(),t,e) = e + 2*e*t + 2*t >= t = t member(x,cons(y,ys)) = 1 + 4*x + x*y + x*ys >= 1 + 4*x + x*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) = 3*cnf + 3*cnf^2 >= 2*cnf + 3*cnf^2 = satck(cnf,guess(cnf)) satck(cnf,assign) = 2*assign + assign*cnf + 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 + 4*l + 2*l*ls + ls^2 = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = 0 >= 0 = true() * Step 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(?,O(n^2))