YES(O(1),O(n^2)) 124.17/60.80 YES(O(1),O(n^2)) 124.17/60.80 124.17/60.80 We are left with following problem, upon which TcT provides the 124.17/60.80 certificate YES(O(1),O(n^2)). 124.17/60.80 124.17/60.80 Strict Trs: 124.17/60.80 { sqr(0()) -> 0() 124.17/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.17/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.17/60.80 , +(x, 0()) -> x 124.17/60.80 , +(x, s(y)) -> s(+(x, y)) 124.17/60.80 , double(0()) -> 0() 124.17/60.80 , double(s(x)) -> s(s(double(x))) } 124.17/60.80 Obligation: 124.17/60.80 innermost runtime complexity 124.17/60.80 Answer: 124.17/60.80 YES(O(1),O(n^2)) 124.17/60.80 124.17/60.80 We add the following dependency tuples: 124.17/60.80 124.17/60.80 Strict DPs: 124.17/60.80 { sqr^#(0()) -> c_1() 124.17/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.17/60.80 , sqr^#(s(x)) -> 124.17/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.17/60.80 , +^#(x, 0()) -> c_4() 124.17/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) 124.17/60.80 , double^#(0()) -> c_6() 124.17/60.80 , double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 124.17/60.80 and mark the set of starting terms. 124.17/60.80 124.17/60.80 We are left with following problem, upon which TcT provides the 124.17/60.80 certificate YES(O(1),O(n^2)). 124.17/60.80 124.17/60.80 Strict DPs: 124.17/60.80 { sqr^#(0()) -> c_1() 124.17/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.17/60.80 , sqr^#(s(x)) -> 124.17/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.17/60.80 , +^#(x, 0()) -> c_4() 124.17/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) 124.17/60.80 , double^#(0()) -> c_6() 124.17/60.80 , double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 Weak Trs: 124.17/60.80 { sqr(0()) -> 0() 124.17/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.17/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.17/60.80 , +(x, 0()) -> x 124.17/60.80 , +(x, s(y)) -> s(+(x, y)) 124.17/60.80 , double(0()) -> 0() 124.17/60.80 , double(s(x)) -> s(s(double(x))) } 124.17/60.80 Obligation: 124.17/60.80 innermost runtime complexity 124.17/60.80 Answer: 124.17/60.80 YES(O(1),O(n^2)) 124.17/60.80 124.17/60.80 We estimate the number of application of {1,4,6} by applications of 124.17/60.80 Pre({1,4,6}) = {2,3,5,7}. Here rules are labeled as follows: 124.17/60.80 124.17/60.80 DPs: 124.17/60.80 { 1: sqr^#(0()) -> c_1() 124.17/60.80 , 2: sqr^#(s(x)) -> 124.17/60.80 c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.17/60.80 , 3: sqr^#(s(x)) -> 124.17/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.17/60.80 , 4: +^#(x, 0()) -> c_4() 124.17/60.80 , 5: +^#(x, s(y)) -> c_5(+^#(x, y)) 124.17/60.80 , 6: double^#(0()) -> c_6() 124.17/60.80 , 7: double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 124.17/60.80 We are left with following problem, upon which TcT provides the 124.17/60.80 certificate YES(O(1),O(n^2)). 124.17/60.80 124.17/60.80 Strict DPs: 124.17/60.80 { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.17/60.80 , sqr^#(s(x)) -> 124.17/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.17/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) 124.17/60.80 , double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 Weak DPs: 124.17/60.80 { sqr^#(0()) -> c_1() 124.17/60.80 , +^#(x, 0()) -> c_4() 124.17/60.80 , double^#(0()) -> c_6() } 124.17/60.80 Weak Trs: 124.17/60.80 { sqr(0()) -> 0() 124.17/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.17/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.17/60.80 , +(x, 0()) -> x 124.17/60.80 , +(x, s(y)) -> s(+(x, y)) 124.17/60.80 , double(0()) -> 0() 124.17/60.80 , double(s(x)) -> s(s(double(x))) } 124.17/60.80 Obligation: 124.17/60.80 innermost runtime complexity 124.17/60.80 Answer: 124.17/60.80 YES(O(1),O(n^2)) 124.17/60.80 124.17/60.80 The following weak DPs constitute a sub-graph of the DG that is 124.17/60.80 closed under successors. The DPs are removed. 124.17/60.80 124.17/60.80 { sqr^#(0()) -> c_1() 124.17/60.80 , +^#(x, 0()) -> c_4() 124.17/60.80 , double^#(0()) -> c_6() } 124.17/60.80 124.17/60.80 We are left with following problem, upon which TcT provides the 124.17/60.80 certificate YES(O(1),O(n^2)). 124.17/60.80 124.17/60.80 Strict DPs: 124.17/60.80 { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.17/60.80 , sqr^#(s(x)) -> 124.17/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.17/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) 124.17/60.80 , double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 Weak Trs: 124.17/60.80 { sqr(0()) -> 0() 124.17/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.17/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.17/60.80 , +(x, 0()) -> x 124.17/60.80 , +(x, s(y)) -> s(+(x, y)) 124.17/60.80 , double(0()) -> 0() 124.17/60.80 , double(s(x)) -> s(s(double(x))) } 124.17/60.80 Obligation: 124.17/60.80 innermost runtime complexity 124.17/60.80 Answer: 124.17/60.80 YES(O(1),O(n^2)) 124.17/60.80 124.17/60.80 We use the processor 'matrix interpretation of dimension 2' to 124.17/60.80 orient following rules strictly. 124.17/60.80 124.17/60.80 DPs: 124.17/60.80 { 4: double^#(s(x)) -> c_7(double^#(x)) } 124.17/60.80 124.17/60.80 Sub-proof: 124.17/60.80 ---------- 124.17/60.80 The following argument positions are usable: 124.17/60.80 Uargs(c_2) = {1, 2, 3}, Uargs(c_3) = {1, 2, 3}, Uargs(c_5) = {1}, 124.17/60.80 Uargs(c_7) = {1} 124.17/60.80 124.17/60.80 TcT has computed the following constructor-based matrix 124.17/60.80 interpretation satisfying not(EDA). 124.17/60.80 124.17/60.80 [sqr](x1) = [0] 124.17/60.80 [0] 124.17/60.80 124.17/60.80 [0] = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [s](x1) = [1 7] x1 + [0] 124.48/60.80 [0 1] [2] 124.48/60.80 124.48/60.80 [+](x1, x2) = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [double](x1) = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [sqr^#](x1) = [1 1] x1 + [0] 124.48/60.80 [2 0] [0] 124.48/60.80 124.48/60.80 [c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [2 0] x3 + [2] 124.48/60.80 [0 0] [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [+^#](x1, x2) = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [double^#](x1) = [0 2] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c_3](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [2] 124.48/60.80 [0 0] [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [c_5](x1) = [2 0] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c_7](x1) = [1 0] x1 + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 The order satisfies the following ordering constraints: 124.48/60.80 124.48/60.80 [sqr(0())] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [0()] 124.48/60.80 124.48/60.80 [sqr(s(x))] = [0] 124.48/60.80 [0] 124.48/60.80 ? [0] 124.48/60.80 [2] 124.48/60.80 = [s(+(sqr(x), double(x)))] 124.48/60.80 124.48/60.80 [sqr(s(x))] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [+(sqr(x), s(double(x)))] 124.48/60.80 124.48/60.80 [+(x, 0())] = [0] 124.48/60.80 [0] 124.48/60.80 ? [1 0] x + [0] 124.48/60.80 [0 1] [0] 124.48/60.80 = [x] 124.48/60.80 124.48/60.80 [+(x, s(y))] = [0] 124.48/60.80 [0] 124.48/60.80 ? [0] 124.48/60.80 [2] 124.48/60.80 = [s(+(x, y))] 124.48/60.80 124.48/60.80 [double(0())] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [0()] 124.48/60.80 124.48/60.80 [double(s(x))] = [0] 124.48/60.80 [0] 124.48/60.80 ? [14] 124.48/60.80 [4] 124.48/60.80 = [s(s(double(x)))] 124.48/60.80 124.48/60.80 [sqr^#(s(x))] = [1 8] x + [2] 124.48/60.80 [2 14] [0] 124.48/60.80 >= [1 5] x + [2] 124.48/60.80 [0 0] [0] 124.48/60.80 = [c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))] 124.48/60.80 124.48/60.80 [sqr^#(s(x))] = [1 8] x + [2] 124.48/60.80 [2 14] [0] 124.48/60.80 >= [1 3] x + [2] 124.48/60.80 [0 0] [0] 124.48/60.80 = [c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))] 124.48/60.80 124.48/60.80 [+^#(x, s(y))] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [c_5(+^#(x, y))] 124.48/60.80 124.48/60.80 [double^#(s(x))] = [0 2] x + [4] 124.48/60.80 [0 0] [0] 124.48/60.80 > [0 2] x + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 = [c_7(double^#(x))] 124.48/60.80 124.48/60.80 124.48/60.80 The strictly oriented rules are moved into the weak component. 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^2)). 124.48/60.80 124.48/60.80 Strict DPs: 124.48/60.80 { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.48/60.80 , sqr^#(s(x)) -> 124.48/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.48/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) } 124.48/60.80 Weak DPs: { double^#(s(x)) -> c_7(double^#(x)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^2)) 124.48/60.80 124.48/60.80 The following weak DPs constitute a sub-graph of the DG that is 124.48/60.80 closed under successors. The DPs are removed. 124.48/60.80 124.48/60.80 { double^#(s(x)) -> c_7(double^#(x)) } 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^2)). 124.48/60.80 124.48/60.80 Strict DPs: 124.48/60.80 { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.48/60.80 , sqr^#(s(x)) -> 124.48/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) 124.48/60.80 , +^#(x, s(y)) -> c_5(+^#(x, y)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^2)) 124.48/60.80 124.48/60.80 Due to missing edges in the dependency-graph, the right-hand sides 124.48/60.80 of following rules could be simplified: 124.48/60.80 124.48/60.80 { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)) 124.48/60.80 , sqr^#(s(x)) -> 124.48/60.80 c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) } 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^2)). 124.48/60.80 124.48/60.80 Strict DPs: 124.48/60.80 { sqr^#(s(x)) -> c_1(+^#(sqr(x), s(double(x))), sqr^#(x)) 124.48/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x)) 124.48/60.80 , +^#(x, s(y)) -> c_3(+^#(x, y)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^2)) 124.48/60.80 124.48/60.80 We use the processor 'matrix interpretation of dimension 2' to 124.48/60.80 orient following rules strictly. 124.48/60.80 124.48/60.80 DPs: 124.48/60.80 { 3: +^#(x, s(y)) -> c_3(+^#(x, y)) } 124.48/60.80 Trs: { double(0()) -> 0() } 124.48/60.80 124.48/60.80 Sub-proof: 124.48/60.80 ---------- 124.48/60.80 The following argument positions are usable: 124.48/60.80 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1, 2}, Uargs(c_3) = {1} 124.48/60.80 124.48/60.80 TcT has computed the following constructor-based matrix 124.48/60.80 interpretation satisfying not(EDA). 124.48/60.80 124.48/60.80 [sqr](x1) = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [0] = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [s](x1) = [1 2] x1 + [1] 124.48/60.80 [0 1] [1] 124.48/60.80 124.48/60.80 [+](x1, x2) = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [double](x1) = [4 0] x1 + [7] 124.48/60.80 [0 2] [0] 124.48/60.80 124.48/60.80 [sqr^#](x1) = [1 0] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c_2](x1, x2, x3) = [7 7] x1 + [7 7] x2 + [7 7] x3 + [0] 124.48/60.80 [0 0] [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [+^#](x1, x2) = [0 1] x2 + [0] 124.48/60.80 [0 0] [4] 124.48/60.80 124.48/60.80 [double^#](x1) = [7 7] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c_3](x1, x2, x3) = [7 7] x1 + [7 7] x2 + [7 7] x3 + [0] 124.48/60.80 [0 0] [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [c_5](x1) = [7 7] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c_7](x1) = [7 7] x1 + [0] 124.48/60.80 [0 0] [0] 124.48/60.80 124.48/60.80 [c] = [0] 124.48/60.80 [0] 124.48/60.80 124.48/60.80 [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 124.48/60.80 [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 124.48/60.80 [0 0] [0 0] [0] 124.48/60.80 124.48/60.80 [c_3](x1) = [1 0] x1 + [0] 124.48/60.80 [0 0] [3] 124.48/60.80 124.48/60.80 The order satisfies the following ordering constraints: 124.48/60.80 124.48/60.80 [sqr(0())] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [0()] 124.48/60.80 124.48/60.80 [sqr(s(x))] = [0] 124.48/60.80 [0] 124.48/60.80 ? [1] 124.48/60.80 [1] 124.48/60.80 = [s(+(sqr(x), double(x)))] 124.48/60.80 124.48/60.80 [sqr(s(x))] = [0] 124.48/60.80 [0] 124.48/60.80 >= [0] 124.48/60.80 [0] 124.48/60.80 = [+(sqr(x), s(double(x)))] 124.48/60.80 124.48/60.80 [+(x, 0())] = [0] 124.48/60.80 [0] 124.48/60.80 ? [1 0] x + [0] 124.48/60.80 [0 1] [0] 124.48/60.80 = [x] 124.48/60.80 124.48/60.80 [+(x, s(y))] = [0] 124.48/60.80 [0] 124.48/60.80 ? [1] 124.48/60.80 [1] 124.48/60.80 = [s(+(x, y))] 124.48/60.80 124.48/60.80 [double(0())] = [7] 124.48/60.80 [0] 124.48/60.80 > [0] 124.48/60.80 [0] 124.48/60.80 = [0()] 124.48/60.80 124.48/60.80 [double(s(x))] = [4 8] x + [11] 124.48/60.80 [0 2] [2] 124.48/60.80 >= [4 8] x + [11] 124.48/60.80 [0 2] [2] 124.48/60.80 = [s(s(double(x)))] 124.48/60.80 124.48/60.80 [sqr^#(s(x))] = [1 2] x + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 >= [1 2] x + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 = [c_1(+^#(sqr(x), s(double(x))), sqr^#(x))] 124.48/60.80 124.48/60.80 [sqr^#(s(x))] = [1 2] x + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 >= [1 2] x + [1] 124.48/60.80 [0 0] [0] 124.48/60.80 = [c_2(+^#(sqr(x), double(x)), sqr^#(x))] 124.48/60.80 124.48/60.80 [+^#(x, s(y))] = [0 1] y + [1] 124.48/60.80 [0 0] [4] 124.48/60.80 > [0 1] y + [0] 124.48/60.80 [0 0] [3] 124.48/60.80 = [c_3(+^#(x, y))] 124.48/60.80 124.48/60.80 124.48/60.80 The strictly oriented rules are moved into the weak component. 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^1)). 124.48/60.80 124.48/60.80 Strict DPs: 124.48/60.80 { sqr^#(s(x)) -> c_1(+^#(sqr(x), s(double(x))), sqr^#(x)) 124.48/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x)) } 124.48/60.80 Weak DPs: { +^#(x, s(y)) -> c_3(+^#(x, y)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^1)) 124.48/60.80 124.48/60.80 The following weak DPs constitute a sub-graph of the DG that is 124.48/60.80 closed under successors. The DPs are removed. 124.48/60.80 124.48/60.80 { +^#(x, s(y)) -> c_3(+^#(x, y)) } 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^1)). 124.48/60.80 124.48/60.80 Strict DPs: 124.48/60.80 { sqr^#(s(x)) -> c_1(+^#(sqr(x), s(double(x))), sqr^#(x)) 124.48/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^1)) 124.48/60.80 124.48/60.80 Due to missing edges in the dependency-graph, the right-hand sides 124.48/60.80 of following rules could be simplified: 124.48/60.80 124.48/60.80 { sqr^#(s(x)) -> c_1(+^#(sqr(x), s(double(x))), sqr^#(x)) 124.48/60.80 , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x)) } 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^1)). 124.48/60.80 124.48/60.80 Strict DPs: { sqr^#(s(x)) -> c_1(sqr^#(x)) } 124.48/60.80 Weak Trs: 124.48/60.80 { sqr(0()) -> 0() 124.48/60.80 , sqr(s(x)) -> s(+(sqr(x), double(x))) 124.48/60.80 , sqr(s(x)) -> +(sqr(x), s(double(x))) 124.48/60.80 , +(x, 0()) -> x 124.48/60.80 , +(x, s(y)) -> s(+(x, y)) 124.48/60.80 , double(0()) -> 0() 124.48/60.80 , double(s(x)) -> s(s(double(x))) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^1)) 124.48/60.80 124.48/60.80 No rule is usable, rules are removed from the input problem. 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(n^1)). 124.48/60.80 124.48/60.80 Strict DPs: { sqr^#(s(x)) -> c_1(sqr^#(x)) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(n^1)) 124.48/60.80 124.48/60.80 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 124.48/60.80 to orient following rules strictly. 124.48/60.80 124.48/60.80 DPs: 124.48/60.80 { 1: sqr^#(s(x)) -> c_1(sqr^#(x)) } 124.48/60.80 124.48/60.80 Sub-proof: 124.48/60.80 ---------- 124.48/60.80 The input was oriented with the instance of 'Small Polynomial Path 124.48/60.80 Order (PS,1-bounded)' as induced by the safe mapping 124.48/60.80 124.48/60.80 safe(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {}, 124.48/60.80 safe(double) = {}, safe(sqr^#) = {}, safe(c_2) = {}, 124.48/60.80 safe(+^#) = {}, safe(double^#) = {}, safe(c_3) = {}, 124.48/60.80 safe(c_5) = {}, safe(c_7) = {}, safe(c) = {}, safe(c_1) = {}, 124.48/60.80 safe(c_2) = {}, safe(c_3) = {}, safe(c) = {}, safe(c_1) = {} 124.48/60.80 124.48/60.80 and precedence 124.48/60.80 124.48/60.80 empty . 124.48/60.80 124.48/60.80 Following symbols are considered recursive: 124.48/60.80 124.48/60.80 {sqr^#} 124.48/60.80 124.48/60.80 The recursion depth is 1. 124.48/60.80 124.48/60.80 Further, following argument filtering is employed: 124.48/60.80 124.48/60.80 pi(sqr) = [], pi(0) = [], pi(s) = [1], pi(+) = [], pi(double) = [], 124.48/60.80 pi(sqr^#) = [1], pi(c_2) = [], pi(+^#) = [], pi(double^#) = [], 124.48/60.80 pi(c_3) = [], pi(c_5) = [], pi(c_7) = [], pi(c) = [], pi(c_1) = [], 124.48/60.80 pi(c_2) = [], pi(c_3) = [], pi(c) = [], pi(c_1) = [1] 124.48/60.80 124.48/60.80 Usable defined function symbols are a subset of: 124.48/60.80 124.48/60.80 {sqr^#, +^#, double^#} 124.48/60.80 124.48/60.80 For your convenience, here are the satisfied ordering constraints: 124.48/60.80 124.48/60.80 pi(sqr^#(s(x))) = sqr^#(s(; x);) 124.48/60.80 > c_1(sqr^#(x;);) 124.48/60.80 = pi(c_1(sqr^#(x))) 124.48/60.80 124.48/60.80 124.48/60.80 The strictly oriented rules are moved into the weak component. 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(1)). 124.48/60.80 124.48/60.80 Weak DPs: { sqr^#(s(x)) -> c_1(sqr^#(x)) } 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(1)) 124.48/60.80 124.48/60.80 The following weak DPs constitute a sub-graph of the DG that is 124.48/60.80 closed under successors. The DPs are removed. 124.48/60.80 124.48/60.80 { sqr^#(s(x)) -> c_1(sqr^#(x)) } 124.48/60.80 124.48/60.80 We are left with following problem, upon which TcT provides the 124.48/60.80 certificate YES(O(1),O(1)). 124.48/60.80 124.48/60.80 Rules: Empty 124.48/60.80 Obligation: 124.48/60.80 innermost runtime complexity 124.48/60.80 Answer: 124.48/60.80 YES(O(1),O(1)) 124.48/60.80 124.48/60.80 Empty rules are trivially bounded 124.48/60.80 124.48/60.80 Hurray, we answered YES(O(1),O(n^2)) 124.48/60.83 EOF