YES(O(1),O(n^3)) 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict Trs: 152.26/82.98 { terms(N) -> cons(recip(sqr(N))) 152.26/82.98 , sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) 152.26/82.98 , first(0(), X) -> nil() 152.26/82.98 , first(s(X), cons(Y)) -> cons(Y) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We add the following dependency tuples: 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 , sqr^#(0()) -> c_2() 152.26/82.98 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(0(), X) -> c_4() 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(0()) -> c_6() 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) 152.26/82.98 , first^#(0(), X) -> c_8() 152.26/82.98 , first^#(s(X), cons(Y)) -> c_9() } 152.26/82.98 152.26/82.98 and mark the set of starting terms. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 , sqr^#(0()) -> c_2() 152.26/82.98 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(0(), X) -> c_4() 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(0()) -> c_6() 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) 152.26/82.98 , first^#(0(), X) -> c_8() 152.26/82.98 , first^#(s(X), cons(Y)) -> c_9() } 152.26/82.98 Weak Trs: 152.26/82.98 { terms(N) -> cons(recip(sqr(N))) 152.26/82.98 , sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) 152.26/82.98 , first(0(), X) -> nil() 152.26/82.98 , first(s(X), cons(Y)) -> cons(Y) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We estimate the number of application of {2,4,6,8,9} by 152.26/82.98 applications of Pre({2,4,6,8,9}) = {1,3,5,7}. Here rules are 152.26/82.98 labeled as follows: 152.26/82.98 152.26/82.98 DPs: 152.26/82.98 { 1: terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 , 2: sqr^#(0()) -> c_2() 152.26/82.98 , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , 4: add^#(0(), X) -> c_4() 152.26/82.98 , 5: add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , 6: dbl^#(0()) -> c_6() 152.26/82.98 , 7: dbl^#(s(X)) -> c_7(dbl^#(X)) 152.26/82.98 , 8: first^#(0(), X) -> c_8() 152.26/82.98 , 9: first^#(s(X), cons(Y)) -> c_9() } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 Weak DPs: 152.26/82.98 { sqr^#(0()) -> c_2() 152.26/82.98 , add^#(0(), X) -> c_4() 152.26/82.98 , dbl^#(0()) -> c_6() 152.26/82.98 , first^#(0(), X) -> c_8() 152.26/82.98 , first^#(s(X), cons(Y)) -> c_9() } 152.26/82.98 Weak Trs: 152.26/82.98 { terms(N) -> cons(recip(sqr(N))) 152.26/82.98 , sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) 152.26/82.98 , first(0(), X) -> nil() 152.26/82.98 , first(s(X), cons(Y)) -> cons(Y) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 The following weak DPs constitute a sub-graph of the DG that is 152.26/82.98 closed under successors. The DPs are removed. 152.26/82.98 152.26/82.98 { sqr^#(0()) -> c_2() 152.26/82.98 , add^#(0(), X) -> c_4() 152.26/82.98 , dbl^#(0()) -> c_6() 152.26/82.98 , first^#(0(), X) -> c_8() 152.26/82.98 , first^#(s(X), cons(Y)) -> c_9() } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { terms(N) -> cons(recip(sqr(N))) 152.26/82.98 , sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) 152.26/82.98 , first(0(), X) -> nil() 152.26/82.98 , first(s(X), cons(Y)) -> cons(Y) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 Consider the dependency graph 152.26/82.98 152.26/82.98 1: terms^#(N) -> c_1(sqr^#(N)) 152.26/82.98 -->_1 sqr^#(s(X)) -> 152.26/82.98 c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2 152.26/82.98 152.26/82.98 2: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 -->_3 dbl^#(s(X)) -> c_7(dbl^#(X)) :4 152.26/82.98 -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3 152.26/82.98 -->_2 sqr^#(s(X)) -> 152.26/82.98 c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) :2 152.26/82.98 152.26/82.98 3: add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 -->_1 add^#(s(X), Y) -> c_5(add^#(X, Y)) :3 152.26/82.98 152.26/82.98 4: dbl^#(s(X)) -> c_7(dbl^#(X)) 152.26/82.98 -->_1 dbl^#(s(X)) -> c_7(dbl^#(X)) :4 152.26/82.98 152.26/82.98 152.26/82.98 Following roots of the dependency graph are removed, as the 152.26/82.98 considered set of starting terms is closed under reduction with 152.26/82.98 respect to these rules (modulo compound contexts). 152.26/82.98 152.26/82.98 { terms^#(N) -> c_1(sqr^#(N)) } 152.26/82.98 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { terms(N) -> cons(recip(sqr(N))) 152.26/82.98 , sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) 152.26/82.98 , first(0(), X) -> nil() 152.26/82.98 , first(s(X), cons(Y)) -> cons(Y) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We replace rewrite rules by usable rules: 152.26/82.98 152.26/82.98 Weak Usable Rules: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 152.26/82.98 , dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We use the processor 'matrix interpretation of dimension 2' to 152.26/82.98 orient following rules strictly. 152.26/82.98 152.26/82.98 DPs: 152.26/82.98 { 3: dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 152.26/82.98 Sub-proof: 152.26/82.98 ---------- 152.26/82.98 The following argument positions are usable: 152.26/82.98 Uargs(c_3) = {1, 2, 3}, Uargs(c_5) = {1}, Uargs(c_7) = {1} 152.26/82.98 152.26/82.98 TcT has computed the following constructor-based matrix 152.26/82.98 interpretation satisfying not(EDA). 152.26/82.98 152.26/82.98 [sqr](x1) = [0] 152.26/82.98 [0] 152.26/82.98 152.26/82.98 [0] = [0] 152.26/82.98 [0] 152.26/82.98 152.26/82.98 [s](x1) = [1 4] x1 + [0] 152.26/82.98 [0 1] [1] 152.26/82.98 152.26/82.98 [add](x1, x2) = [0] 152.26/82.98 [0] 152.26/82.98 152.26/82.98 [dbl](x1) = [0] 152.26/82.98 [1] 152.26/82.98 152.26/82.98 [sqr^#](x1) = [1 0] x1 + [0] 152.26/82.98 [0 0] [0] 152.26/82.98 152.26/82.98 [c_3](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [2 0] x3 + [0] 152.26/82.98 [0 0] [0 0] [0 0] [0] 152.26/82.98 152.26/82.98 [add^#](x1, x2) = [0] 152.26/82.98 [4] 152.26/82.98 152.26/82.98 [dbl^#](x1) = [0 1] x1 + [0] 152.26/82.98 [0 0] [4] 152.26/82.98 152.26/82.98 [c_5](x1) = [4 0] x1 + [0] 152.26/82.98 [0 0] [3] 152.26/82.98 152.26/82.98 [c_7](x1) = [1 0] x1 + [0] 152.26/82.98 [0 0] [3] 152.26/82.98 152.26/82.98 The order satisfies the following ordering constraints: 152.26/82.98 152.26/82.98 [sqr(0())] = [0] 152.26/82.98 [0] 152.26/82.98 >= [0] 152.26/82.98 [0] 152.26/82.98 = [0()] 152.26/82.98 152.26/82.98 [sqr(s(X))] = [0] 152.26/82.98 [0] 152.26/82.98 ? [0] 152.26/82.98 [1] 152.26/82.98 = [s(add(sqr(X), dbl(X)))] 152.26/82.98 152.26/82.98 [add(0(), X)] = [0] 152.26/82.98 [0] 152.26/82.98 ? [1 0] X + [0] 152.26/82.98 [0 1] [0] 152.26/82.98 = [X] 152.26/82.98 152.26/82.98 [add(s(X), Y)] = [0] 152.26/82.98 [0] 152.26/82.98 ? [0] 152.26/82.98 [1] 152.26/82.98 = [s(add(X, Y))] 152.26/82.98 152.26/82.98 [dbl(0())] = [0] 152.26/82.98 [1] 152.26/82.98 >= [0] 152.26/82.98 [0] 152.26/82.98 = [0()] 152.26/82.98 152.26/82.98 [dbl(s(X))] = [0] 152.26/82.98 [1] 152.26/82.98 ? [12] 152.26/82.98 [3] 152.26/82.98 = [s(s(dbl(X)))] 152.26/82.98 152.26/82.98 [sqr^#(s(X))] = [1 4] X + [0] 152.26/82.98 [0 0] [0] 152.26/82.98 >= [1 2] X + [0] 152.26/82.98 [0 0] [0] 152.26/82.98 = [c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X))] 152.26/82.98 152.26/82.98 [add^#(s(X), Y)] = [0] 152.26/82.98 [4] 152.26/82.98 >= [0] 152.26/82.98 [3] 152.26/82.98 = [c_5(add^#(X, Y))] 152.26/82.98 152.26/82.98 [dbl^#(s(X))] = [0 1] X + [1] 152.26/82.98 [0 0] [4] 152.26/82.98 > [0 1] X + [0] 152.26/82.98 [0 0] [3] 152.26/82.98 = [c_7(dbl^#(X))] 152.26/82.98 152.26/82.98 152.26/82.98 The strictly oriented rules are moved into the weak component. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) } 152.26/82.98 Weak DPs: { dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 The following weak DPs constitute a sub-graph of the DG that is 152.26/82.98 closed under successors. The DPs are removed. 152.26/82.98 152.26/82.98 { dbl^#(s(X)) -> c_7(dbl^#(X)) } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_5(add^#(X, Y)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 Due to missing edges in the dependency-graph, the right-hand sides 152.26/82.98 of following rules could be simplified: 152.26/82.98 152.26/82.98 { sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X)), sqr^#(X), dbl^#(X)) } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^3)). 152.26/82.98 152.26/82.98 Strict DPs: 152.26/82.98 { sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) 152.26/82.98 , add^#(s(X), Y) -> c_2(add^#(X, Y)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^3)) 152.26/82.98 152.26/82.98 We decompose the input problem according to the dependency graph 152.26/82.98 into the upper component 152.26/82.98 152.26/82.98 { sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) } 152.26/82.98 152.26/82.98 and lower component 152.26/82.98 152.26/82.98 { add^#(s(X), Y) -> c_2(add^#(X, Y)) } 152.26/82.98 152.26/82.98 Further, following extension rules are added to the lower 152.26/82.98 component. 152.26/82.98 152.26/82.98 { sqr^#(s(X)) -> sqr^#(X) 152.26/82.98 , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) } 152.26/82.98 152.26/82.98 TcT solves the upper component with certificate YES(O(1),O(n^1)). 152.26/82.98 152.26/82.98 Sub-proof: 152.26/82.98 ---------- 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^1)). 152.26/82.98 152.26/82.98 Strict DPs: { sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^1)) 152.26/82.98 152.26/82.98 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 152.26/82.98 to orient following rules strictly. 152.26/82.98 152.26/82.98 DPs: 152.26/82.98 { 1: sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) } 152.26/82.98 Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , add(0(), X) -> X } 152.26/82.98 152.26/82.98 Sub-proof: 152.26/82.98 ---------- 152.26/82.98 The input was oriented with the instance of 'Small Polynomial Path 152.26/82.98 Order (PS,1-bounded)' as induced by the safe mapping 152.26/82.98 152.26/82.98 safe(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(add) = {1, 2}, 152.26/82.98 safe(dbl) = {}, safe(sqr^#) = {}, safe(add^#) = {}, safe(c_1) = {} 152.26/82.98 152.26/82.98 and precedence 152.26/82.98 152.26/82.98 sqr^# > sqr, sqr^# > add, sqr^# > dbl, sqr ~ add, sqr ~ dbl, 152.26/82.98 add ~ dbl . 152.26/82.98 152.26/82.98 Following symbols are considered recursive: 152.26/82.98 152.26/82.98 {sqr^#} 152.26/82.98 152.26/82.98 The recursion depth is 1. 152.26/82.98 152.26/82.98 Further, following argument filtering is employed: 152.26/82.98 152.26/82.98 pi(sqr) = [], pi(0) = [], pi(s) = [1], pi(add) = [1, 2], 152.26/82.98 pi(dbl) = 1, pi(sqr^#) = [1], pi(add^#) = [], pi(c_1) = [1, 2] 152.26/82.98 152.26/82.98 Usable defined function symbols are a subset of: 152.26/82.98 152.26/82.98 {sqr^#, add^#} 152.26/82.98 152.26/82.98 For your convenience, here are the satisfied ordering constraints: 152.26/82.98 152.26/82.98 pi(sqr^#(s(X))) = sqr^#(s(; X);) 152.26/82.98 > c_1(add^#(), sqr^#(X;);) 152.26/82.98 = pi(c_1(add^#(sqr(X), dbl(X)), sqr^#(X))) 152.26/82.98 152.26/82.98 152.26/82.98 The strictly oriented rules are moved into the weak component. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Weak DPs: { sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 The following weak DPs constitute a sub-graph of the DG that is 152.26/82.98 closed under successors. The DPs are removed. 152.26/82.98 152.26/82.98 { sqr^#(s(X)) -> c_1(add^#(sqr(X), dbl(X)), sqr^#(X)) } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 No rule is usable, rules are removed from the input problem. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Rules: Empty 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 Empty rules are trivially bounded 152.26/82.98 152.26/82.98 We return to the main proof. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(n^2)). 152.26/82.98 152.26/82.98 Strict DPs: { add^#(s(X), Y) -> c_2(add^#(X, Y)) } 152.26/82.98 Weak DPs: 152.26/82.98 { sqr^#(s(X)) -> sqr^#(X) 152.26/82.98 , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(n^2)) 152.26/82.98 152.26/82.98 We use the processor 'polynomial interpretation' to orient 152.26/82.98 following rules strictly. 152.26/82.98 152.26/82.98 DPs: 152.26/82.98 { 1: add^#(s(X), Y) -> c_2(add^#(X, Y)) 152.26/82.98 , 2: sqr^#(s(X)) -> sqr^#(X) 152.26/82.98 , 3: sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) } 152.26/82.98 Trs: { dbl(0()) -> 0() } 152.26/82.98 152.26/82.98 Sub-proof: 152.26/82.98 ---------- 152.26/82.98 We consider the following typing: 152.26/82.98 152.26/82.98 sqr :: a -> a 152.26/82.98 0 :: a 152.26/82.98 s :: a -> a 152.26/82.98 add :: (a,a) -> a 152.26/82.98 dbl :: a -> a 152.26/82.98 sqr^# :: a -> b 152.26/82.98 add^# :: (a,a) -> b 152.26/82.98 c_2 :: b -> b 152.26/82.98 152.26/82.98 The following argument positions are considered usable: 152.26/82.98 152.26/82.98 Uargs(c_2) = {1} 152.26/82.98 152.26/82.98 TcT has computed the following constructor-restricted 152.26/82.98 typedpolynomial interpretation. 152.26/82.98 152.26/82.98 [sqr](x1) = x1 + 2*x1^2 152.26/82.98 152.26/82.98 [0]() = 0 152.26/82.98 152.26/82.98 [s](x1) = 1 + x1 152.26/82.98 152.26/82.98 [add](x1, x2) = x1 + x2 152.26/82.98 152.26/82.98 [dbl](x1) = 2 + 2*x1 152.26/82.98 152.26/82.98 [sqr^#](x1) = 1 + 2*x1^2 152.26/82.98 152.26/82.98 [add^#](x1, x2) = x1 152.26/82.98 152.26/82.98 [c_2](x1) = x1 152.26/82.98 152.26/82.98 152.26/82.98 This order satisfies the following ordering constraints. 152.26/82.98 152.26/82.98 [sqr(0())] = 152.26/82.98 >= 152.26/82.98 = [0()] 152.26/82.98 152.26/82.98 [sqr(s(X))] = 3 + 5*X + 2*X^2 152.26/82.98 >= 3 + 3*X + 2*X^2 152.26/82.98 = [s(add(sqr(X), dbl(X)))] 152.26/82.98 152.26/82.98 [add(0(), X)] = X 152.26/82.98 >= X 152.26/82.98 = [X] 152.26/82.98 152.26/82.98 [add(s(X), Y)] = 1 + X + Y 152.26/82.98 >= 1 + X + Y 152.26/82.98 = [s(add(X, Y))] 152.26/82.98 152.26/82.98 [dbl(0())] = 2 152.26/82.98 > 152.26/82.98 = [0()] 152.26/82.98 152.26/82.98 [dbl(s(X))] = 4 + 2*X 152.26/82.98 >= 4 + 2*X 152.26/82.98 = [s(s(dbl(X)))] 152.26/82.98 152.26/82.98 [sqr^#(s(X))] = 3 + 4*X + 2*X^2 152.26/82.98 > 1 + 2*X^2 152.26/82.98 = [sqr^#(X)] 152.26/82.98 152.26/82.98 [sqr^#(s(X))] = 3 + 4*X + 2*X^2 152.26/82.98 > X + 2*X^2 152.26/82.98 = [add^#(sqr(X), dbl(X))] 152.26/82.98 152.26/82.98 [add^#(s(X), Y)] = 1 + X 152.26/82.98 > X 152.26/82.98 = [c_2(add^#(X, Y))] 152.26/82.98 152.26/82.98 152.26/82.98 The strictly oriented rules are moved into the weak component. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Weak DPs: 152.26/82.98 { sqr^#(s(X)) -> sqr^#(X) 152.26/82.98 , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) 152.26/82.98 , add^#(s(X), Y) -> c_2(add^#(X, Y)) } 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 The following weak DPs constitute a sub-graph of the DG that is 152.26/82.98 closed under successors. The DPs are removed. 152.26/82.98 152.26/82.98 { sqr^#(s(X)) -> sqr^#(X) 152.26/82.98 , sqr^#(s(X)) -> add^#(sqr(X), dbl(X)) 152.26/82.98 , add^#(s(X), Y) -> c_2(add^#(X, Y)) } 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Weak Trs: 152.26/82.98 { sqr(0()) -> 0() 152.26/82.98 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 152.26/82.98 , add(0(), X) -> X 152.26/82.98 , add(s(X), Y) -> s(add(X, Y)) 152.26/82.98 , dbl(0()) -> 0() 152.26/82.98 , dbl(s(X)) -> s(s(dbl(X))) } 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 No rule is usable, rules are removed from the input problem. 152.26/82.98 152.26/82.98 We are left with following problem, upon which TcT provides the 152.26/82.98 certificate YES(O(1),O(1)). 152.26/82.98 152.26/82.98 Rules: Empty 152.26/82.98 Obligation: 152.26/82.98 innermost runtime complexity 152.26/82.98 Answer: 152.26/82.98 YES(O(1),O(1)) 152.26/82.98 152.26/82.98 Empty rules are trivially bounded 152.26/82.98 152.26/82.98 Hurray, we answered YES(O(1),O(n^3)) 152.54/83.00 EOF