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: 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)) - 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: NaturalMI {miDimension = 2, miDegree = 2, 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) = [0] [0] p(1) = [3] [0] p(O) = [0 1] x1 + [4] [0 0] [1] p(choice) = [1 0] x1 + [0] [2 0] [3] p(cons) = [1 1] x1 + [1 2] x2 + [5] [0 0] [0 1] [4] p(eq) = [2] [4] p(false) = [1] [1] p(guess) = [3 0] x1 + [0] [0 1] [0] p(if) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [2 1] [0 1] [5] p(member) = [1 4] x1 + [0 1] x2 + [1] [3 0] [0 2] [3] p(negate) = [0 1] x1 + [3] [0 0] [1] p(nil) = [0] [0] p(sat) = [6 7] x1 + [0] [6 5] [7] p(satck) = [0 6] x1 + [2 1] x2 + [0] [0 4] [2 1] [7] p(true) = [0] [2] p(unsat) = [0] [2] p(verify) = [1 1] x1 + [0] [0 2] [2] Following rules are strictly oriented: member(x,cons(y,ys)) = [1 4] x + [0 1] ys + [5] [3 0] [0 2] [11] > [1 4] x + [0 1] ys + [3] [3 0] [0 2] [10] = if(eq(x,y),true(),member(x,ys)) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1 1] x + [1 2] xs + [5] [2 2] [2 4] [13] >= [1 0] x + [0] [0 1] [0] = x choice(cons(x,xs)) = [1 1] x + [1 2] xs + [5] [2 2] [2 4] [13] >= [1 0] xs + [0] [2 0] [3] = choice(xs) eq(0(x),1(y)) = [2] [4] >= [1] [1] = false() eq(1(x),0(y)) = [2] [4] >= [1] [1] = false() eq(1(x),1(y)) = [2] [4] >= [2] [4] = eq(x,y) eq(O(x),0(y)) = [2] [4] >= [2] [4] = eq(x,y) eq(nil(),nil()) = [2] [4] >= [0] [2] = true() guess(cons(clause,cnf)) = [3 3] clause + [3 6] cnf + [15] [0 0] [0 1] [4] >= [3 0] clause + [3 2] cnf + [8] [0 0] [0 1] [4] = cons(choice(clause),guess(cnf)) guess(nil()) = [0] [0] >= [0] [0] = nil() if(false(),t,e) = [1 0] e + [1 0] t + [1] [0 1] [2 1] [5] >= [1 0] e + [0] [0 1] [0] = e if(true(),t,e) = [1 0] e + [1 0] t + [0] [0 1] [2 1] [5] >= [1 0] t + [0] [0 1] [0] = t member(x,nil()) = [1 4] x + [1] [3 0] [3] >= [1] [1] = false() negate(0(x)) = [3] [1] >= [3] [0] = 1(x) negate(1(x)) = [3] [1] >= [0] [0] = 0(x) sat(cnf) = [6 7] cnf + [0] [6 5] [7] >= [6 7] cnf + [0] [6 5] [7] = satck(cnf,guess(cnf)) satck(cnf,assign) = [2 1] assign + [0 6] cnf + [0] [2 1] [0 4] [7] >= [2 1] assign + [0] [2 1] [7] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [1 1] l + [1 3] ls + [9] [0 0] [0 2] [10] >= [0 1] l + [1 2] ls + [9] [0 0] [0 2] [10] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [0] [2] >= [0] [2] = true() * Step 9: 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) - 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: NaturalMI {miDimension = 2, miDegree = 2, 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) = [0 2] x1 + [2] [0 1] [0] p(1) = [0 2] x1 + [1] [0 1] [4] p(O) = [1 0] x1 + [1] [0 0] [2] p(choice) = [1 0] x1 + [0] [0 1] [0] p(cons) = [1 4] x1 + [1 6] x2 + [4] [0 1] [0 1] [0] p(eq) = [0 1] x2 + [0] [0 1] [0] p(false) = [0] [0] p(guess) = [1 0] x1 + [0] [0 1] [0] p(if) = [2 2] x1 + [2 4] x2 + [1 0] x3 + [0] [0 0] [0 4] [0 1] [0] p(member) = [1 0] x1 + [0 4] x2 + [0] [0 0] [0 2] [4] p(negate) = [1 2] x1 + [0] [4 0] [0] p(nil) = [0] [0] p(sat) = [7 6] x1 + [3] [1 6] [3] p(satck) = [0 0] x1 + [6 4] x2 + [3] [1 2] [0 4] [2] p(true) = [0] [0] p(unsat) = [2] [2] p(verify) = [2 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: eq(1(x),1(y)) = [0 1] y + [4] [0 1] [4] > [0 1] y + [0] [0 1] [0] = eq(x,y) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1 4] x + [1 6] xs + [4] [0 1] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x choice(cons(x,xs)) = [1 4] x + [1 6] xs + [4] [0 1] [0 1] [0] >= [1 0] xs + [0] [0 1] [0] = choice(xs) eq(0(x),1(y)) = [0 1] y + [4] [0 1] [4] >= [0] [0] = false() eq(1(x),0(y)) = [0 1] y + [0] [0 1] [0] >= [0] [0] = false() eq(O(x),0(y)) = [0 1] y + [0] [0 1] [0] >= [0 1] y + [0] [0 1] [0] = eq(x,y) eq(nil(),nil()) = [0] [0] >= [0] [0] = true() guess(cons(clause,cnf)) = [1 4] clause + [1 6] cnf + [4] [0 1] [0 1] [0] >= [1 4] clause + [1 6] cnf + [4] [0 1] [0 1] [0] = cons(choice(clause),guess(cnf)) guess(nil()) = [0] [0] >= [0] [0] = nil() if(false(),t,e) = [1 0] e + [2 4] t + [0] [0 1] [0 4] [0] >= [1 0] e + [0] [0 1] [0] = e if(true(),t,e) = [1 0] e + [2 4] t + [0] [0 1] [0 4] [0] >= [1 0] t + [0] [0 1] [0] = t member(x,cons(y,ys)) = [1 0] x + [0 4] y + [0 4] ys + [0] [0 0] [0 2] [0 2] [4] >= [1 0] x + [0 4] y + [0 4] ys + [0] [0 0] [0 0] [0 2] [4] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [1 0] x + [0] [0 0] [4] >= [0] [0] = false() negate(0(x)) = [0 4] x + [2] [0 8] [8] >= [0 2] x + [1] [0 1] [4] = 1(x) negate(1(x)) = [0 4] x + [9] [0 8] [4] >= [0 2] x + [2] [0 1] [0] = 0(x) sat(cnf) = [7 6] cnf + [3] [1 6] [3] >= [6 4] cnf + [3] [1 6] [2] = satck(cnf,guess(cnf)) satck(cnf,assign) = [6 4] assign + [0 0] cnf + [3] [0 4] [1 2] [2] >= [6 4] assign + [2] [0 4] [2] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [2 8] l + [2 12] ls + [8] [0 0] [0 0] [0] >= [2 4] l + [2 12] ls + [8] [0 0] [0 0] [0] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [0] [0] >= [0] [0] = true() * Step 10: NaturalMI 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: NaturalMI {miDimension = 2, miDegree = 2, 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) = [0 0] x1 + [0] [0 1] [1] p(1) = [0 0] x1 + [0] [0 1] [4] p(O) = [0] [4] p(choice) = [1 0] x1 + [5] [0 1] [0] p(cons) = [1 0] x1 + [1 2] x2 + [7] [0 1] [0 1] [5] p(eq) = [0 1] x2 + [0] [0 0] [0] p(false) = [0] [0] p(guess) = [1 1] x1 + [0] [0 1] [0] p(if) = [2 0] x1 + [1 2] x2 + [1 2] x3 + [0] [0 0] [1 1] [0 1] [0] p(member) = [2 0] x1 + [0 2] x2 + [0] [0 0] [0 0] [5] p(negate) = [0 0] x1 + [0] [0 2] [4] p(nil) = [4] [2] p(sat) = [7 7] x1 + [7] [7 7] [1] p(satck) = [2 0] x1 + [5 2] x2 + [7] [4 0] [2 4] [1] p(true) = [0] [0] p(unsat) = [3] [1] p(verify) = [2 0] x1 + [1] [0 0] [0] Following rules are strictly oriented: eq(O(x),0(y)) = [0 1] y + [1] [0 0] [0] > [0 1] y + [0] [0 0] [0] = eq(x,y) Following rules are (at-least) weakly oriented: choice(cons(x,xs)) = [1 0] x + [1 2] xs + [12] [0 1] [0 1] [5] >= [1 0] x + [0] [0 1] [0] = x choice(cons(x,xs)) = [1 0] x + [1 2] xs + [12] [0 1] [0 1] [5] >= [1 0] xs + [5] [0 1] [0] = choice(xs) eq(0(x),1(y)) = [0 1] y + [4] [0 0] [0] >= [0] [0] = false() eq(1(x),0(y)) = [0 1] y + [1] [0 0] [0] >= [0] [0] = false() eq(1(x),1(y)) = [0 1] y + [4] [0 0] [0] >= [0 1] y + [0] [0 0] [0] = eq(x,y) eq(nil(),nil()) = [2] [0] >= [0] [0] = true() guess(cons(clause,cnf)) = [1 1] clause + [1 3] cnf + [12] [0 1] [0 1] [5] >= [1 0] clause + [1 3] cnf + [12] [0 1] [0 1] [5] = cons(choice(clause),guess(cnf)) guess(nil()) = [6] [2] >= [4] [2] = nil() if(false(),t,e) = [1 2] e + [1 2] t + [0] [0 1] [1 1] [0] >= [1 0] e + [0] [0 1] [0] = e if(true(),t,e) = [1 2] e + [1 2] t + [0] [0 1] [1 1] [0] >= [1 0] t + [0] [0 1] [0] = t member(x,cons(y,ys)) = [2 0] x + [0 2] y + [0 2] ys + [10] [0 0] [0 0] [0 0] [5] >= [2 0] x + [0 2] y + [0 2] ys + [10] [0 0] [0 0] [0 0] [5] = if(eq(x,y),true(),member(x,ys)) member(x,nil()) = [2 0] x + [4] [0 0] [5] >= [0] [0] = false() negate(0(x)) = [0 0] x + [0] [0 2] [6] >= [0 0] x + [0] [0 1] [4] = 1(x) negate(1(x)) = [0 0] x + [0] [0 2] [12] >= [0 0] x + [0] [0 1] [1] = 0(x) sat(cnf) = [7 7] cnf + [7] [7 7] [1] >= [7 7] cnf + [7] [6 6] [1] = satck(cnf,guess(cnf)) satck(cnf,assign) = [5 2] assign + [2 0] cnf + [7] [2 4] [4 0] [1] >= [5 2] assign + [7] [1 1] [1] = if(verify(assign),assign,unsat()) verify(cons(l,ls)) = [2 0] l + [2 4] ls + [15] [0 0] [0 0] [0] >= [2 4] ls + [1] [0 0] [0] = if(member(negate(l),ls),false(),verify(ls)) verify(nil()) = [9] [0] >= [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))