YES(O(1),O(n^2)) 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict Trs: 35.16/13.40 { *(x, *(y, z)) -> *(otimes(x, y), z) 35.16/13.40 , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) 35.16/13.40 , *(1(), y) -> y 35.16/13.40 , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 We add the following weak dependency pairs: 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(1(), y) -> c_3() 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 35.16/13.40 and mark the set of starting terms. 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(1(), y) -> c_3() 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Strict Trs: 35.16/13.40 { *(x, *(y, z)) -> *(otimes(x, y), z) 35.16/13.40 , *(x, oplus(y, z)) -> oplus(*(x, y), *(x, z)) 35.16/13.40 , *(1(), y) -> y 35.16/13.40 , *(+(x, y), z) -> oplus(*(x, z), *(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 No rule is usable, rules are removed from the input problem. 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(1(), y) -> c_3() 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 The weightgap principle applies (using the following constant 35.16/13.40 growth matrix-interpretation) 35.16/13.40 35.16/13.40 The following argument positions are usable: 35.16/13.40 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.16/13.40 35.16/13.40 TcT has computed the following constructor-restricted matrix 35.16/13.40 interpretation. 35.16/13.40 35.16/13.40 [*](x1, x2) = [1 1] x1 + [1 1] x2 + [1] 35.16/13.40 [2 2] [2 2] [1] 35.16/13.40 35.16/13.40 [otimes](x1, x2) = [1 2] x1 + [1 2] x2 + [2] 35.16/13.40 [0 1] [0 1] [1] 35.16/13.40 35.16/13.40 [1] = [2] 35.16/13.40 [1] 35.16/13.40 35.16/13.40 [+](x1, x2) = [1 2] x1 + [1 2] x2 + [2] 35.16/13.40 [0 1] [0 1] [1] 35.16/13.40 35.16/13.40 [oplus](x1, x2) = [1 1] x1 + [1 1] x2 + [1] 35.16/13.40 [0 1] [0 1] [1] 35.16/13.40 35.16/13.40 [*^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] 35.16/13.40 [1 2] [2 1] [1] 35.16/13.40 35.16/13.40 [c_1](x1) = [1 0] x1 + [1] 35.16/13.40 [0 1] [1] 35.16/13.40 35.16/13.40 [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 35.16/13.40 [0 1] [0 1] [1] 35.16/13.40 35.16/13.40 [c_3] = [0] 35.16/13.40 [1] 35.16/13.40 35.16/13.40 [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 35.16/13.40 [0 1] [0 1] [1] 35.16/13.40 35.16/13.40 The order satisfies the following ordering constraints: 35.16/13.40 35.16/13.40 [*^#(x, *(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] 35.16/13.40 [1 2] [4 4] [4 4] [4] 35.16/13.40 ? [0 0] x + [0 0] y + [0 0] z + [2] 35.16/13.40 [1 4] [1 4] [2 1] [6] 35.16/13.40 = [c_1(*^#(otimes(x, y), z))] 35.16/13.40 35.16/13.40 [*^#(x, oplus(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1] 35.16/13.40 [1 2] [2 3] [2 3] [4] 35.16/13.40 ? [0 0] x + [0 0] y + [0 0] z + [3] 35.16/13.40 [2 4] [2 1] [2 1] [3] 35.16/13.40 = [c_2(*^#(x, y), *^#(x, z))] 35.16/13.40 35.16/13.40 [*^#(1(), y)] = [0 0] y + [1] 35.16/13.40 [2 1] [5] 35.16/13.40 > [0] 35.16/13.40 [1] 35.16/13.40 = [c_3()] 35.16/13.40 35.16/13.40 [*^#(+(x, y), z)] = [0 0] x + [0 0] y + [0 0] z + [1] 35.16/13.40 [1 4] [1 4] [2 1] [5] 35.16/13.40 ? [0 0] x + [0 0] y + [0 0] z + [3] 35.16/13.40 [1 2] [1 2] [4 2] [3] 35.16/13.40 = [c_4(*^#(x, z), *^#(y, z))] 35.16/13.40 35.16/13.40 35.16/13.40 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Weak DPs: { *^#(1(), y) -> c_3() } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 The following weak DPs constitute a sub-graph of the DG that is 35.16/13.40 closed under successors. The DPs are removed. 35.16/13.40 35.16/13.40 { *^#(1(), y) -> c_3() } 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 We use the processor 'polynomial interpretation' to orient 35.16/13.40 following rules strictly. 35.16/13.40 35.16/13.40 DPs: 35.16/13.40 { 3: *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 35.16/13.40 Sub-proof: 35.16/13.40 ---------- 35.16/13.40 We consider the following typing: 35.16/13.40 35.16/13.40 * :: (b,c) -> c 35.16/13.40 otimes :: (a,b) -> a 35.16/13.40 + :: (a,a) -> a 35.16/13.40 oplus :: (c,c) -> c 35.16/13.40 *^# :: (a,c) -> d 35.16/13.40 c_1 :: d -> d 35.16/13.40 c_2 :: (d,d) -> d 35.16/13.40 c_4 :: (d,d) -> d 35.16/13.40 35.16/13.40 The following argument positions are considered usable: 35.16/13.40 35.16/13.40 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.16/13.40 35.16/13.40 TcT has computed the following constructor-restricted 35.16/13.40 typedpolynomial interpretation. 35.16/13.40 35.16/13.40 [*](x1, x2) = 0 35.16/13.40 35.16/13.40 [otimes](x1, x2) = 0 35.16/13.40 35.16/13.40 [+](x1, x2) = 1 + x1 + x2 35.16/13.40 35.16/13.40 [oplus](x1, x2) = 1 + x1 + x2 35.16/13.40 35.16/13.40 [*^#](x1, x2) = x1 + x1*x2 35.16/13.40 35.16/13.40 [c_1](x1) = 2*x1 35.16/13.40 35.16/13.40 [c_2](x1, x2) = x1 + x2 35.16/13.40 35.16/13.40 [c_4](x1, x2) = x1 + x2 35.16/13.40 35.16/13.40 35.16/13.40 This order satisfies the following ordering constraints. 35.16/13.40 35.16/13.40 [*^#(x, *(y, z))] = x 35.16/13.40 >= 35.16/13.40 = [c_1(*^#(otimes(x, y), z))] 35.16/13.40 35.16/13.40 [*^#(x, oplus(y, z))] = 2*x + x*y + x*z 35.16/13.40 >= 2*x + x*y + x*z 35.16/13.40 = [c_2(*^#(x, y), *^#(x, z))] 35.16/13.40 35.16/13.40 [*^#(+(x, y), z)] = 1 + x + y + z + x*z + y*z 35.16/13.40 > x + x*z + y + y*z 35.16/13.40 = [c_4(*^#(x, z), *^#(y, z))] 35.16/13.40 35.16/13.40 35.16/13.40 The strictly oriented rules are moved into the weak component. 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(n^2)). 35.16/13.40 35.16/13.40 Strict DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) } 35.16/13.40 Weak DPs: { *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(n^2)) 35.16/13.40 35.16/13.40 We use the processor 'polynomial interpretation' to orient 35.16/13.40 following rules strictly. 35.16/13.40 35.16/13.40 DPs: 35.16/13.40 { 1: *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , 2: *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) } 35.16/13.40 35.16/13.40 Sub-proof: 35.16/13.40 ---------- 35.16/13.40 We consider the following typing: 35.16/13.40 35.16/13.40 * :: (b,c) -> c 35.16/13.40 otimes :: (a,b) -> a 35.16/13.40 + :: (a,a) -> a 35.16/13.40 oplus :: (c,c) -> c 35.16/13.40 *^# :: (a,c) -> d 35.16/13.40 c_1 :: d -> d 35.16/13.40 c_2 :: (d,d) -> d 35.16/13.40 c_4 :: (d,d) -> d 35.16/13.40 35.16/13.40 The following argument positions are considered usable: 35.16/13.40 35.16/13.40 Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_4) = {1, 2} 35.16/13.40 35.16/13.40 TcT has computed the following constructor-restricted 35.16/13.40 typedpolynomial interpretation. 35.16/13.40 35.16/13.40 [*](x1, x2) = 2 + 2*x1 + 2*x1^2 + 2*x2 35.16/13.40 35.16/13.40 [otimes](x1, x2) = 0 35.16/13.40 35.16/13.40 [+](x1, x2) = 2 + x1 + x2 35.16/13.40 35.16/13.40 [oplus](x1, x2) = 2 + x1 + x2 35.16/13.40 35.16/13.40 [*^#](x1, x2) = 2*x1*x2 + 2*x2 35.16/13.40 35.16/13.40 [c_1](x1) = x1 35.16/13.40 35.16/13.40 [c_2](x1, x2) = x1 + x2 35.16/13.40 35.16/13.40 [c_4](x1, x2) = x1 + x2 35.16/13.40 35.16/13.40 35.16/13.40 This order satisfies the following ordering constraints. 35.16/13.40 35.16/13.40 [*^#(x, *(y, z))] = 4*x + 4*x*y + 4*x*y^2 + 4*x*z + 4 + 4*y + 4*y^2 + 4*z 35.16/13.40 > 2*z 35.16/13.40 = [c_1(*^#(otimes(x, y), z))] 35.16/13.40 35.16/13.40 [*^#(x, oplus(y, z))] = 4*x + 2*x*y + 2*x*z + 4 + 2*y + 2*z 35.16/13.40 > 2*x*y + 2*y + 2*x*z + 2*z 35.16/13.40 = [c_2(*^#(x, y), *^#(x, z))] 35.16/13.40 35.16/13.40 [*^#(+(x, y), z)] = 6*z + 2*x*z + 2*y*z 35.16/13.40 >= 2*x*z + 4*z + 2*y*z 35.16/13.40 = [c_4(*^#(x, z), *^#(y, z))] 35.16/13.40 35.16/13.40 35.16/13.40 The strictly oriented rules are moved into the weak component. 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(1)). 35.16/13.40 35.16/13.40 Weak DPs: 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(1)) 35.16/13.40 35.16/13.40 The following weak DPs constitute a sub-graph of the DG that is 35.16/13.40 closed under successors. The DPs are removed. 35.16/13.40 35.16/13.40 { *^#(x, *(y, z)) -> c_1(*^#(otimes(x, y), z)) 35.16/13.40 , *^#(x, oplus(y, z)) -> c_2(*^#(x, y), *^#(x, z)) 35.16/13.40 , *^#(+(x, y), z) -> c_4(*^#(x, z), *^#(y, z)) } 35.16/13.40 35.16/13.40 We are left with following problem, upon which TcT provides the 35.16/13.40 certificate YES(O(1),O(1)). 35.16/13.40 35.16/13.40 Rules: Empty 35.16/13.40 Obligation: 35.16/13.40 innermost runtime complexity 35.16/13.40 Answer: 35.16/13.40 YES(O(1),O(1)) 35.16/13.40 35.16/13.40 Empty rules are trivially bounded 35.16/13.40 35.16/13.40 Hurray, we answered YES(O(1),O(n^2)) 35.16/13.42 EOF