YES(O(1),O(n^2)) 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(n^2)). 35.37/13.33 35.37/13.33 Strict Trs: 35.37/13.33 { *(x, *(y, z)) -> *(otimes(x, y), z) 35.37/13.33 , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) 35.37/13.33 , *(1(), y) -> y 35.37/13.33 , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 We add the following weak dependency pairs: 35.37/13.33 35.37/13.33 Strict DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 35.37/13.33 and mark the set of starting terms. 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(n^2)). 35.37/13.33 35.37/13.33 Strict DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 Strict Trs: 35.37/13.33 { *(x, *(y, z)) -> *(otimes(x, y), z) 35.37/13.33 , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) 35.37/13.33 , *(1(), y) -> y 35.37/13.33 , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 No rule is usable, rules are removed from the input problem. 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(n^2)). 35.37/13.33 35.37/13.33 Strict DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 The weightgap principle applies (using the following constant 35.37/13.33 growth matrix-interpretation) 35.37/13.33 35.37/13.33 The following argument positions are usable: 35.37/13.33 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.37/13.33 35.37/13.33 TcT has computed the following constructor-restricted matrix 35.37/13.33 interpretation. 35.37/13.33 35.37/13.33 [*](x1, x2) = [2 2] x1 + [2 2] x2 + [2] 35.37/13.33 [2 2] [2 2] [1] 35.37/13.33 35.37/13.33 [otimes](x1, x2) = [1 1] x1 + [1 1] x2 + [1] 35.37/13.33 [0 1] [0 1] [1] 35.37/13.33 35.37/13.33 [1] = [1] 35.37/13.33 [1] 35.37/13.33 35.37/13.33 [+](x1, x2) = [1 1] x1 + [1 1] x2 + [1] 35.37/13.33 [0 1] [0 1] [1] 35.37/13.33 35.37/13.33 [oplus](x1, x2) = [1 2] x1 + [1 2] x2 + [1] 35.37/13.33 [0 1] [0 1] [1] 35.37/13.33 35.37/13.33 [*^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] 35.37/13.33 [2 1] [1 1] [1] 35.37/13.33 35.37/13.33 [c_1](x1) = [1 0] x1 + [1] 35.37/13.33 [0 1] [2] 35.37/13.33 35.37/13.33 [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 35.37/13.33 [0 1] [0 1] [1] 35.37/13.33 35.37/13.33 [c_3](x1) = [0 0] x1 + [0] 35.37/13.33 [1 1] [0] 35.37/13.33 35.37/13.33 [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 35.37/13.33 [0 1] [0 1] [1] 35.37/13.33 35.37/13.33 The order satisfies the following ordering constraints: 35.37/13.33 35.37/13.33 [*^#(x, *(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] 35.37/13.33 [2 1] [4 4] [4 4] [4] 35.37/13.33 ? [0 0] x + [0 0] y + [0 0] z + [2] 35.37/13.33 [2 3] [2 3] [1 1] [6] 35.37/13.33 = [c_1(*^#(otimes(x, y), z))] 35.37/13.33 35.37/13.33 [*^#(x, oplus(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] 35.37/13.33 [2 1] [1 3] [1 3] [3] 35.37/13.33 ? [0 0] x + [0 0] y + [0 0] z + [3] 35.37/13.33 [4 2] [1 1] [1 1] [3] 35.37/13.33 = [c_2(*^#(x, y), *^#(x, z))] 35.37/13.33 35.37/13.33 [*^#(1(), y)] = [0 0] y + [1] 35.37/13.33 [1 1] [4] 35.37/13.33 > [0 0] y + [0] 35.37/13.33 [1 1] [0] 35.37/13.33 = [c_3(y)] 35.37/13.33 35.37/13.33 [*^#(+(x, y), z)] = [0 0] x + [0 0] y + [0 0] z + [1] 35.37/13.33 [2 3] [2 3] [1 1] [4] 35.37/13.33 ? [0 0] x + [0 0] y + [0 0] z + [3] 35.37/13.33 [2 1] [2 1] [2 2] [3] 35.37/13.33 = [c_4(*^#(x, z), *^#(y, z))] 35.37/13.33 35.37/13.33 35.37/13.33 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(n^2)). 35.37/13.33 35.37/13.33 Strict DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 Weak DPs: { *^#(1(), y) -> c_3(y) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 We use the processor 'polynomial interpretation' to orient 35.37/13.33 following rules strictly. 35.37/13.33 35.37/13.33 DPs: 35.37/13.33 { 3: *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 35.37/13.33 Sub-proof: 35.37/13.33 ---------- 35.37/13.33 We consider the following typing: 35.37/13.33 35.37/13.33 * :: (b,c) -> c 35.37/13.33 otimes :: (a,b) -> a 35.37/13.33 1 :: a 35.37/13.33 + :: (a,a) -> a 35.37/13.33 oplus :: (c,c) -> c 35.37/13.33 *^# :: (a,c) -> d 35.37/13.33 c_1 :: d -> d 35.37/13.33 c_2 :: (d,d) -> d 35.37/13.33 c_3 :: c -> d 35.37/13.33 c_4 :: (d,d) -> d 35.37/13.33 35.37/13.33 The following argument positions are considered usable: 35.37/13.33 35.37/13.33 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.37/13.33 35.37/13.33 TcT has computed the following constructor-restricted 35.37/13.33 typedpolynomial interpretation. 35.37/13.33 35.37/13.33 [*](x1, x2) = 0 35.37/13.33 35.37/13.33 [otimes](x1, x2) = 0 35.37/13.33 35.37/13.33 [1]() = 0 35.37/13.33 35.37/13.33 [+](x1, x2) = 2 + x1 + x2 35.37/13.33 35.37/13.33 [oplus](x1, x2) = 2 + x1 + x2 35.37/13.33 35.37/13.33 [*^#](x1, x2) = 2*x1 + x1*x2 35.37/13.33 35.37/13.33 [c_1](x1) = 2*x1 35.37/13.33 35.37/13.33 [c_2](x1, x2) = x1 + x2 35.37/13.33 35.37/13.33 [c_3](x1) = 0 35.37/13.33 35.37/13.33 [c_4](x1, x2) = x1 + x2 35.37/13.33 35.37/13.33 35.37/13.33 This order satisfies the following ordering constraints. 35.37/13.33 35.37/13.33 [*^#(x, *(y, z))] = 2*x 35.37/13.33 >= 35.37/13.33 = [c_1(*^#(otimes(x, y), z))] 35.37/13.33 35.37/13.33 [*^#(x, oplus(y, z))] = 4*x + x*y + x*z 35.37/13.33 >= 4*x + x*y + x*z 35.37/13.33 = [c_2(*^#(x, y), *^#(x, z))] 35.37/13.33 35.37/13.33 [*^#(1(), y)] = 35.37/13.33 >= 35.37/13.33 = [c_3(y)] 35.37/13.33 35.37/13.33 [*^#(+(x, y), z)] = 4 + 2*x + 2*y + 2*z + x*z + y*z 35.37/13.33 > 2*x + x*z + 2*y + y*z 35.37/13.33 = [c_4(*^#(x, z), *^#(y, z))] 35.37/13.33 35.37/13.33 35.37/13.33 The strictly oriented rules are moved into the weak component. 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(n^2)). 35.37/13.33 35.37/13.33 Strict DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) } 35.37/13.33 Weak DPs: 35.37/13.33 { *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(n^2)) 35.37/13.33 35.37/13.33 We use the processor 'polynomial interpretation' to orient 35.37/13.33 following rules strictly. 35.37/13.33 35.37/13.33 DPs: 35.37/13.33 { 1: *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , 2: *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , 4: *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 35.37/13.33 Sub-proof: 35.37/13.33 ---------- 35.37/13.33 We consider the following typing: 35.37/13.33 35.37/13.33 * :: (b,c) -> c 35.37/13.33 otimes :: (a,b) -> a 35.37/13.33 1 :: a 35.37/13.33 + :: (a,a) -> a 35.37/13.33 oplus :: (c,c) -> c 35.37/13.33 *^# :: (a,c) -> d 35.37/13.33 c_1 :: d -> d 35.37/13.33 c_2 :: (d,d) -> d 35.37/13.33 c_3 :: c -> d 35.37/13.33 c_4 :: (d,d) -> d 35.37/13.33 35.37/13.33 The following argument positions are considered usable: 35.37/13.33 35.37/13.33 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.37/13.33 35.37/13.33 TcT has computed the following constructor-restricted 35.37/13.33 typedpolynomial interpretation. 35.37/13.33 35.37/13.33 [*](x1, x2) = 2 + 2*x1 + 2*x1*x2 + 2*x1^2 + 2*x2 35.37/13.33 35.37/13.33 [otimes](x1, x2) = 0 35.37/13.33 35.37/13.33 [1]() = 0 35.37/13.33 35.37/13.33 [+](x1, x2) = 2 + x1 + x2 35.37/13.33 35.37/13.33 [oplus](x1, x2) = 2 + x1 + x2 35.37/13.33 35.37/13.33 [*^#](x1, x2) = 2*x1 + 2*x1*x2 + 2*x2 35.37/13.33 35.37/13.33 [c_1](x1) = 2*x1 35.37/13.33 35.37/13.33 [c_2](x1, x2) = x1 + x2 35.37/13.33 35.37/13.33 [c_3](x1) = x1 35.37/13.33 35.37/13.33 [c_4](x1, x2) = x1 + x2 35.37/13.33 35.37/13.33 35.37/13.33 This order satisfies the following ordering constraints. 35.37/13.33 35.37/13.33 [*^#(x, *(y, z))] = 6*x + 4*x*y + 4*x*y*z + 4*x*y^2 + 4*x*z + 4 + 4*y + 4*y*z + 4*y^2 + 4*z 35.37/13.33 > 4*z 35.37/13.33 = [c_1(*^#(otimes(x, y), z))] 35.37/13.33 35.37/13.33 [*^#(x, oplus(y, z))] = 6*x + 2*x*y + 2*x*z + 4 + 2*y + 2*z 35.37/13.33 > 4*x + 2*x*y + 2*y + 2*x*z + 2*z 35.37/13.33 = [c_2(*^#(x, y), *^#(x, z))] 35.37/13.33 35.37/13.33 [*^#(1(), y)] = 2*y 35.37/13.33 >= y 35.37/13.33 = [c_3(y)] 35.37/13.33 35.37/13.33 [*^#(+(x, y), z)] = 4 + 2*x + 2*y + 6*z + 2*x*z + 2*y*z 35.37/13.33 > 2*x + 2*x*z + 4*z + 2*y + 2*y*z 35.37/13.33 = [c_4(*^#(x, z), *^#(y, z))] 35.37/13.33 35.37/13.33 35.37/13.33 The strictly oriented rules are moved into the weak component. 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(1)). 35.37/13.33 35.37/13.33 Weak DPs: 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(1)) 35.37/13.33 35.37/13.33 The following weak DPs constitute a sub-graph of the DG that is 35.37/13.33 closed under successors. The DPs are removed. 35.37/13.33 35.37/13.33 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.37/13.33 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.37/13.33 , *^#(1(), y) -> c_3(y) 35.37/13.33 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.37/13.33 35.37/13.33 We are left with following problem, upon which TcT provides the 35.37/13.33 certificate YES(O(1),O(1)). 35.37/13.33 35.37/13.33 Rules: Empty 35.37/13.33 Obligation: 35.37/13.33 runtime complexity 35.37/13.33 Answer: 35.37/13.33 YES(O(1),O(1)) 35.37/13.33 35.37/13.33 Empty rules are trivially bounded 35.37/13.33 35.37/13.33 Hurray, we answered YES(O(1),O(n^2)) 35.37/13.36 EOF