YES(O(1),O(n^3)) 120.05/50.05 YES(O(1),O(n^3)) 120.05/50.05 120.05/50.05 We are left with following problem, upon which TcT provides the 120.05/50.05 certificate YES(O(1),O(n^3)). 120.05/50.05 120.05/50.05 Strict Trs: 120.05/50.05 { +(0(), y) -> y 120.05/50.05 , +(s(x), y) -> s(+(x, y)) 120.05/50.05 , +(p(x), y) -> p(+(x, y)) 120.05/50.05 , minus(0()) -> 0() 120.05/50.05 , minus(s(x)) -> p(minus(x)) 120.05/50.05 , minus(p(x)) -> s(minus(x)) 120.05/50.05 , *(0(), y) -> 0() 120.05/50.05 , *(s(x), y) -> +(*(x, y), y) 120.05/50.05 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.05 Obligation: 120.05/50.05 innermost runtime complexity 120.05/50.05 Answer: 120.05/50.05 YES(O(1),O(n^3)) 120.05/50.05 120.05/50.05 We add the following dependency tuples: 120.05/50.05 120.05/50.05 Strict DPs: 120.05/50.05 { +^#(0(), y) -> c_1() 120.05/50.05 , +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.05 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.05 , minus^#(0()) -> c_4() 120.05/50.05 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.05 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.05 , *^#(0(), y) -> c_7() 120.05/50.05 , *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.05 , *^#(p(x), y) -> 120.05/50.05 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.05 120.05/50.05 and mark the set of starting terms. 120.05/50.05 120.05/50.05 We are left with following problem, upon which TcT provides the 120.05/50.05 certificate YES(O(1),O(n^3)). 120.05/50.05 120.05/50.05 Strict DPs: 120.05/50.05 { +^#(0(), y) -> c_1() 120.05/50.05 , +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(0()) -> c_4() 120.05/50.06 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , *^#(0(), y) -> c_7() 120.05/50.06 , *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^3)) 120.05/50.06 120.05/50.06 We estimate the number of application of {1,4,7} by applications of 120.05/50.06 Pre({1,4,7}) = {2,3,5,6,8,9}. Here rules are labeled as follows: 120.05/50.06 120.05/50.06 DPs: 120.05/50.06 { 1: +^#(0(), y) -> c_1() 120.05/50.06 , 2: +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , 3: +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , 4: minus^#(0()) -> c_4() 120.05/50.06 , 5: minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , 6: minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , 7: *^#(0(), y) -> c_7() 120.05/50.06 , 8: *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , 9: *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^3)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Weak DPs: 120.05/50.06 { +^#(0(), y) -> c_1() 120.05/50.06 , minus^#(0()) -> c_4() 120.05/50.06 , *^#(0(), y) -> c_7() } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^3)) 120.05/50.06 120.05/50.06 The following weak DPs constitute a sub-graph of the DG that is 120.05/50.06 closed under successors. The DPs are removed. 120.05/50.06 120.05/50.06 { +^#(0(), y) -> c_1() 120.05/50.06 , minus^#(0()) -> c_4() 120.05/50.06 , *^#(0(), y) -> c_7() } 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^3)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^3)) 120.05/50.06 120.05/50.06 We decompose the input problem according to the dependency graph 120.05/50.06 into the upper component 120.05/50.06 120.05/50.06 { *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 120.05/50.06 and lower component 120.05/50.06 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) } 120.05/50.06 120.05/50.06 Further, following extension rules are added to the lower 120.05/50.06 component. 120.05/50.06 120.05/50.06 { *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> minus^#(y) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 120.05/50.06 TcT solves the upper component with certificate YES(O(1),O(n^1)). 120.05/50.06 120.05/50.06 Sub-proof: 120.05/50.06 ---------- 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^1)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^1)) 120.05/50.06 120.05/50.06 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 120.05/50.06 to orient following rules strictly. 120.05/50.06 120.05/50.06 DPs: 120.05/50.06 { 1: *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , 2: *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , *(0(), y) -> 0() } 120.05/50.06 120.05/50.06 Sub-proof: 120.05/50.06 ---------- 120.05/50.06 The input was oriented with the instance of 'Small Polynomial Path 120.05/50.06 Order (PS,1-bounded)' as induced by the safe mapping 120.05/50.06 120.05/50.06 safe(+) = {}, safe(0) = {}, safe(s) = {1}, safe(p) = {1}, 120.05/50.06 safe(minus) = {}, safe(*) = {}, safe(+^#) = {}, safe(minus^#) = {}, 120.05/50.06 safe(*^#) = {}, safe(c_8) = {}, safe(c_9) = {} 120.05/50.06 120.05/50.06 and precedence 120.05/50.06 120.05/50.06 minus > +, * > +, *^# > +, minus ~ * . 120.05/50.06 120.05/50.06 Following symbols are considered recursive: 120.05/50.06 120.05/50.06 {*^#} 120.05/50.06 120.05/50.06 The recursion depth is 1. 120.05/50.06 120.05/50.06 Further, following argument filtering is employed: 120.05/50.06 120.05/50.06 pi(+) = [1, 2], pi(0) = [], pi(s) = [1], pi(p) = [1], 120.05/50.06 pi(minus) = [], pi(*) = [], pi(+^#) = [], pi(minus^#) = [1], 120.05/50.06 pi(*^#) = [1, 2], pi(c_8) = [1, 2], pi(c_9) = [1, 2, 3] 120.05/50.06 120.05/50.06 Usable defined function symbols are a subset of: 120.05/50.06 120.05/50.06 {+^#, minus^#, *^#} 120.05/50.06 120.05/50.06 For your convenience, here are the satisfied ordering constraints: 120.05/50.06 120.05/50.06 pi(*^#(s(x), y)) = *^#(s(; x), y;) 120.05/50.06 > c_8(+^#(), *^#(x, y;);) 120.05/50.06 = pi(c_8(+^#(*(x, y), y), *^#(x, y))) 120.05/50.06 120.05/50.06 pi(*^#(p(x), y)) = *^#(p(; x), y;) 120.05/50.06 > c_9(+^#(), *^#(x, y;), minus^#(y;);) 120.05/50.06 = pi(c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y))) 120.05/50.06 120.05/50.06 120.05/50.06 The strictly oriented rules are moved into the weak component. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Weak DPs: 120.05/50.06 { *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 The following weak DPs constitute a sub-graph of the DG that is 120.05/50.06 closed under successors. The DPs are removed. 120.05/50.06 120.05/50.06 { *^#(s(x), y) -> c_8(+^#(*(x, y), y), *^#(x, y)) 120.05/50.06 , *^#(p(x), y) -> 120.05/50.06 c_9(+^#(*(x, y), minus(y)), *^#(x, y), minus^#(y)) } 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 No rule is usable, rules are removed from the input problem. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Rules: Empty 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 Empty rules are trivially bounded 120.05/50.06 120.05/50.06 We return to the main proof. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^2)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) } 120.05/50.06 Weak DPs: 120.05/50.06 { *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> minus^#(y) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^2)) 120.05/50.06 120.05/50.06 We use the processor 'matrix interpretation of dimension 1' to 120.05/50.06 orient following rules strictly. 120.05/50.06 120.05/50.06 DPs: 120.05/50.06 { 3: minus^#(s(x)) -> c_5(minus^#(x)) } 120.05/50.06 120.05/50.06 Sub-proof: 120.05/50.06 ---------- 120.05/50.06 The following argument positions are usable: 120.05/50.06 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1}, 120.05/50.06 Uargs(c_6) = {1} 120.05/50.06 120.05/50.06 TcT has computed the following constructor-based matrix 120.05/50.06 interpretation satisfying not(EDA). 120.05/50.06 120.05/50.06 [+](x1, x2) = [0] 120.05/50.06 120.05/50.06 [0] = [0] 120.05/50.06 120.05/50.06 [s](x1) = [1] x1 + [1] 120.05/50.06 120.05/50.06 [p](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [minus](x1) = [0] 120.05/50.06 120.05/50.06 [*](x1, x2) = [0] 120.05/50.06 120.05/50.06 [+^#](x1, x2) = [0] 120.05/50.06 120.05/50.06 [c_2](x1) = [2] x1 + [0] 120.05/50.06 120.05/50.06 [c_3](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [minus^#](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [c_5](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [c_6](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [*^#](x1, x2) = [7] x2 + [0] 120.05/50.06 120.05/50.06 The order satisfies the following ordering constraints: 120.05/50.06 120.05/50.06 [+(0(), y)] = [0] 120.05/50.06 ? [1] y + [0] 120.05/50.06 = [y] 120.05/50.06 120.05/50.06 [+(s(x), y)] = [0] 120.05/50.06 ? [1] 120.05/50.06 = [s(+(x, y))] 120.05/50.06 120.05/50.06 [+(p(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [p(+(x, y))] 120.05/50.06 120.05/50.06 [minus(0())] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [minus(s(x))] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [p(minus(x))] 120.05/50.06 120.05/50.06 [minus(p(x))] = [0] 120.05/50.06 ? [1] 120.05/50.06 = [s(minus(x))] 120.05/50.06 120.05/50.06 [*(0(), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [*(s(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [+(*(x, y), y)] 120.05/50.06 120.05/50.06 [*(p(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [+(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [+^#(s(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [c_2(+^#(x, y))] 120.05/50.06 120.05/50.06 [+^#(p(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [c_3(+^#(x, y))] 120.05/50.06 120.05/50.06 [minus^#(s(x))] = [1] x + [1] 120.05/50.06 > [1] x + [0] 120.05/50.06 = [c_5(minus^#(x))] 120.05/50.06 120.05/50.06 [minus^#(p(x))] = [1] x + [0] 120.05/50.06 >= [1] x + [0] 120.05/50.06 = [c_6(minus^#(x))] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = [7] y + [0] 120.05/50.06 >= [0] 120.05/50.06 = [+^#(*(x, y), y)] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = [7] y + [0] 120.05/50.06 >= [7] y + [0] 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [0] 120.05/50.06 = [+^#(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [1] y + [0] 120.05/50.06 = [minus^#(y)] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [7] y + [0] 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 120.05/50.06 The strictly oriented rules are moved into the weak component. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^2)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) } 120.05/50.06 Weak DPs: 120.05/50.06 { minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> minus^#(y) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^2)) 120.05/50.06 120.05/50.06 We use the processor 'matrix interpretation of dimension 1' to 120.05/50.06 orient following rules strictly. 120.05/50.06 120.05/50.06 DPs: 120.05/50.06 { 3: minus^#(p(x)) -> c_6(minus^#(x)) } 120.05/50.06 120.05/50.06 Sub-proof: 120.05/50.06 ---------- 120.05/50.06 The following argument positions are usable: 120.05/50.06 Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1}, 120.05/50.06 Uargs(c_6) = {1} 120.05/50.06 120.05/50.06 TcT has computed the following constructor-based matrix 120.05/50.06 interpretation satisfying not(EDA). 120.05/50.06 120.05/50.06 [+](x1, x2) = [0] 120.05/50.06 120.05/50.06 [0] = [0] 120.05/50.06 120.05/50.06 [s](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [p](x1) = [1] x1 + [7] 120.05/50.06 120.05/50.06 [minus](x1) = [0] 120.05/50.06 120.05/50.06 [*](x1, x2) = [0] 120.05/50.06 120.05/50.06 [+^#](x1, x2) = [0] 120.05/50.06 120.05/50.06 [c_2](x1) = [4] x1 + [0] 120.05/50.06 120.05/50.06 [c_3](x1) = [4] x1 + [0] 120.05/50.06 120.05/50.06 [minus^#](x1) = [2] x1 + [0] 120.05/50.06 120.05/50.06 [c_5](x1) = [1] x1 + [0] 120.05/50.06 120.05/50.06 [c_6](x1) = [1] x1 + [5] 120.05/50.06 120.05/50.06 [*^#](x1, x2) = [7] x2 + [0] 120.05/50.06 120.05/50.06 The order satisfies the following ordering constraints: 120.05/50.06 120.05/50.06 [+(0(), y)] = [0] 120.05/50.06 ? [1] y + [0] 120.05/50.06 = [y] 120.05/50.06 120.05/50.06 [+(s(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [s(+(x, y))] 120.05/50.06 120.05/50.06 [+(p(x), y)] = [0] 120.05/50.06 ? [7] 120.05/50.06 = [p(+(x, y))] 120.05/50.06 120.05/50.06 [minus(0())] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [minus(s(x))] = [0] 120.05/50.06 ? [7] 120.05/50.06 = [p(minus(x))] 120.05/50.06 120.05/50.06 [minus(p(x))] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [s(minus(x))] 120.05/50.06 120.05/50.06 [*(0(), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [*(s(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [+(*(x, y), y)] 120.05/50.06 120.05/50.06 [*(p(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [+(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [+^#(s(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [c_2(+^#(x, y))] 120.05/50.06 120.05/50.06 [+^#(p(x), y)] = [0] 120.05/50.06 >= [0] 120.05/50.06 = [c_3(+^#(x, y))] 120.05/50.06 120.05/50.06 [minus^#(s(x))] = [2] x + [0] 120.05/50.06 >= [2] x + [0] 120.05/50.06 = [c_5(minus^#(x))] 120.05/50.06 120.05/50.06 [minus^#(p(x))] = [2] x + [14] 120.05/50.06 > [2] x + [5] 120.05/50.06 = [c_6(minus^#(x))] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = [7] y + [0] 120.05/50.06 >= [0] 120.05/50.06 = [+^#(*(x, y), y)] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = [7] y + [0] 120.05/50.06 >= [7] y + [0] 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [0] 120.05/50.06 = [+^#(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [2] y + [0] 120.05/50.06 = [minus^#(y)] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = [7] y + [0] 120.05/50.06 >= [7] y + [0] 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 120.05/50.06 The strictly oriented rules are moved into the weak component. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^2)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) } 120.05/50.06 Weak DPs: 120.05/50.06 { minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> minus^#(y) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^2)) 120.05/50.06 120.05/50.06 The following weak DPs constitute a sub-graph of the DG that is 120.05/50.06 closed under successors. The DPs are removed. 120.05/50.06 120.05/50.06 { minus^#(s(x)) -> c_5(minus^#(x)) 120.05/50.06 , minus^#(p(x)) -> c_6(minus^#(x)) 120.05/50.06 , *^#(p(x), y) -> minus^#(y) } 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(n^2)). 120.05/50.06 120.05/50.06 Strict DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) } 120.05/50.06 Weak DPs: 120.05/50.06 { *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(n^2)) 120.05/50.06 120.05/50.06 We use the processor 'polynomial interpretation' to orient 120.05/50.06 following rules strictly. 120.05/50.06 120.05/50.06 DPs: 120.05/50.06 { 1: +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , 2: +^#(p(x), y) -> c_3(+^#(x, y)) } 120.05/50.06 120.05/50.06 Sub-proof: 120.05/50.06 ---------- 120.05/50.06 We consider the following typing: 120.05/50.06 120.05/50.06 + :: (a,a) -> a 120.05/50.06 0 :: a 120.05/50.06 s :: a -> a 120.05/50.06 p :: a -> a 120.05/50.06 minus :: a -> a 120.05/50.06 * :: (a,a) -> a 120.05/50.06 +^# :: (a,a) -> b 120.05/50.06 c_2 :: b -> b 120.05/50.06 c_3 :: b -> b 120.05/50.06 minus^# :: c -> d 120.05/50.06 c_5 :: e -> f 120.05/50.06 c_6 :: g -> h 120.05/50.06 *^# :: (a,a) -> b 120.05/50.06 120.05/50.06 The following argument positions are considered usable: 120.05/50.06 120.05/50.06 Uargs(c_2) = {1}, Uargs(c_3) = {1} 120.05/50.06 120.05/50.06 TcT has computed the following constructor-restricted 120.05/50.06 typedpolynomial interpretation. 120.05/50.06 120.05/50.06 [+](x1, x2) = x1 + 2*x2 120.05/50.06 120.05/50.06 [0]() = 0 120.05/50.06 120.05/50.06 [s](x1) = 2 + x1 120.05/50.06 120.05/50.06 [p](x1) = 2 + x1 120.05/50.06 120.05/50.06 [minus](x1) = x1 120.05/50.06 120.05/50.06 [*](x1, x2) = 2*x1*x2 120.05/50.06 120.05/50.06 [+^#](x1, x2) = 2 + x1 120.05/50.06 120.05/50.06 [c_2](x1) = x1 120.05/50.06 120.05/50.06 [c_3](x1) = x1 120.05/50.06 120.05/50.06 [minus^#](x1) = 3*x1 + 3*x1^2 120.05/50.06 120.05/50.06 [c_5](x1) = 3*x1 + 3*x1^2 120.05/50.06 120.05/50.06 [c_6](x1) = 3*x1 + 3*x1^2 120.05/50.06 120.05/50.06 [*^#](x1, x2) = 2 + 2*x1*x2 + 3*x2^2 120.05/50.06 120.05/50.06 120.05/50.06 This order satisfies the following ordering constraints. 120.05/50.06 120.05/50.06 [+(0(), y)] = 2*y 120.05/50.06 >= y 120.05/50.06 = [y] 120.05/50.06 120.05/50.06 [+(s(x), y)] = 2 + x + 2*y 120.05/50.06 >= 2 + x + 2*y 120.05/50.06 = [s(+(x, y))] 120.05/50.06 120.05/50.06 [+(p(x), y)] = 2 + x + 2*y 120.05/50.06 >= 2 + x + 2*y 120.05/50.06 = [p(+(x, y))] 120.05/50.06 120.05/50.06 [minus(0())] = 120.05/50.06 >= 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [minus(s(x))] = 2 + x 120.05/50.06 >= 2 + x 120.05/50.06 = [p(minus(x))] 120.05/50.06 120.05/50.06 [minus(p(x))] = 2 + x 120.05/50.06 >= 2 + x 120.05/50.06 = [s(minus(x))] 120.05/50.06 120.05/50.06 [*(0(), y)] = 120.05/50.06 >= 120.05/50.06 = [0()] 120.05/50.06 120.05/50.06 [*(s(x), y)] = 4*y + 2*x*y 120.05/50.06 >= 2*x*y + 2*y 120.05/50.06 = [+(*(x, y), y)] 120.05/50.06 120.05/50.06 [*(p(x), y)] = 4*y + 2*x*y 120.05/50.06 >= 2*x*y + 2*y 120.05/50.06 = [+(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [+^#(s(x), y)] = 4 + x 120.05/50.06 > 2 + x 120.05/50.06 = [c_2(+^#(x, y))] 120.05/50.06 120.05/50.06 [+^#(p(x), y)] = 4 + x 120.05/50.06 > 2 + x 120.05/50.06 = [c_3(+^#(x, y))] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = 2 + 4*y + 2*x*y + 3*y^2 120.05/50.06 >= 2 + 2*x*y 120.05/50.06 = [+^#(*(x, y), y)] 120.05/50.06 120.05/50.06 [*^#(s(x), y)] = 2 + 4*y + 2*x*y + 3*y^2 120.05/50.06 >= 2 + 2*x*y + 3*y^2 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = 2 + 4*y + 2*x*y + 3*y^2 120.05/50.06 >= 2 + 2*x*y 120.05/50.06 = [+^#(*(x, y), minus(y))] 120.05/50.06 120.05/50.06 [*^#(p(x), y)] = 2 + 4*y + 2*x*y + 3*y^2 120.05/50.06 >= 2 + 2*x*y + 3*y^2 120.05/50.06 = [*^#(x, y)] 120.05/50.06 120.05/50.06 120.05/50.06 The strictly oriented rules are moved into the weak component. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Weak DPs: 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 The following weak DPs constitute a sub-graph of the DG that is 120.05/50.06 closed under successors. The DPs are removed. 120.05/50.06 120.05/50.06 { +^#(s(x), y) -> c_2(+^#(x, y)) 120.05/50.06 , +^#(p(x), y) -> c_3(+^#(x, y)) 120.05/50.06 , *^#(s(x), y) -> +^#(*(x, y), y) 120.05/50.06 , *^#(s(x), y) -> *^#(x, y) 120.05/50.06 , *^#(p(x), y) -> +^#(*(x, y), minus(y)) 120.05/50.06 , *^#(p(x), y) -> *^#(x, y) } 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Weak Trs: 120.05/50.06 { +(0(), y) -> y 120.05/50.06 , +(s(x), y) -> s(+(x, y)) 120.05/50.06 , +(p(x), y) -> p(+(x, y)) 120.05/50.06 , minus(0()) -> 0() 120.05/50.06 , minus(s(x)) -> p(minus(x)) 120.05/50.06 , minus(p(x)) -> s(minus(x)) 120.05/50.06 , *(0(), y) -> 0() 120.05/50.06 , *(s(x), y) -> +(*(x, y), y) 120.05/50.06 , *(p(x), y) -> +(*(x, y), minus(y)) } 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 No rule is usable, rules are removed from the input problem. 120.05/50.06 120.05/50.06 We are left with following problem, upon which TcT provides the 120.05/50.06 certificate YES(O(1),O(1)). 120.05/50.06 120.05/50.06 Rules: Empty 120.05/50.06 Obligation: 120.05/50.06 innermost runtime complexity 120.05/50.06 Answer: 120.05/50.06 YES(O(1),O(1)) 120.05/50.06 120.05/50.06 Empty rules are trivially bounded 120.05/50.06 120.05/50.06 Hurray, we answered YES(O(1),O(n^3)) 120.05/50.07 EOF