YES(O(1),O(n^1)) 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(n^1)). 0.00/0.65 0.00/0.65 Strict Trs: 0.00/0.65 { map(Cons(x, xs)) -> Cons(f(x), map(xs)) 0.00/0.65 , map(Nil()) -> Nil() 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , goal(xs) -> map(xs) } 0.00/0.65 Weak Trs: 0.00/0.65 { *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 The weightgap principle applies (using the following nonconstant 0.00/0.65 growth matrix-interpretation) 0.00/0.65 0.00/0.65 The following argument positions are usable: 0.00/0.65 Uargs(+) = {2}, Uargs(Cons) = {1, 2} 0.00/0.65 0.00/0.65 TcT has computed the following matrix interpretation satisfying 0.00/0.65 not(EDA) and not(IDA(1)). 0.00/0.65 0.00/0.65 [map](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [+](x1, x2) = [1] x2 + [0] 0.00/0.65 0.00/0.65 [S](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [f](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [Cons](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.65 0.00/0.65 [Nil] = [7] 0.00/0.65 0.00/0.65 [+Full](x1, x2) = [1] x1 + [1] x2 + [7] 0.00/0.65 0.00/0.65 [0] = [7] 0.00/0.65 0.00/0.65 [goal](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [*](x1, x2) = [1] x1 + [7] 0.00/0.65 0.00/0.65 The order satisfies the following ordering constraints: 0.00/0.65 0.00/0.65 [map(Cons(x, xs))] = [1] x + [1] xs + [8] 0.00/0.65 ? [1] x + [1] xs + [15] 0.00/0.65 = [Cons(f(x), map(xs))] 0.00/0.65 0.00/0.65 [map(Nil())] = [14] 0.00/0.65 > [7] 0.00/0.65 = [Nil()] 0.00/0.65 0.00/0.65 [f(x)] = [1] x + [7] 0.00/0.65 >= [1] x + [7] 0.00/0.65 = [*(x, x)] 0.00/0.65 0.00/0.65 [+Full(S(x), y)] = [1] x + [1] y + [14] 0.00/0.65 >= [1] x + [1] y + [14] 0.00/0.65 = [+Full(x, S(y))] 0.00/0.65 0.00/0.65 [+Full(0(), y)] = [1] y + [14] 0.00/0.65 > [1] y + [0] 0.00/0.65 = [y] 0.00/0.65 0.00/0.65 [goal(xs)] = [1] xs + [7] 0.00/0.65 >= [1] xs + [7] 0.00/0.65 = [map(xs)] 0.00/0.65 0.00/0.65 [*(x, S(S(y)))] = [1] x + [7] 0.00/0.65 >= [1] x + [7] 0.00/0.65 = [+(x, *(x, S(y)))] 0.00/0.65 0.00/0.65 [*(x, S(0()))] = [1] x + [7] 0.00/0.65 > [1] x + [0] 0.00/0.65 = [x] 0.00/0.65 0.00/0.65 [*(x, 0())] = [1] x + [7] 0.00/0.65 >= [7] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 [*(0(), y)] = [14] 0.00/0.65 > [7] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 0.00/0.65 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(n^1)). 0.00/0.65 0.00/0.65 Strict Trs: 0.00/0.65 { map(Cons(x, xs)) -> Cons(f(x), map(xs)) 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) 0.00/0.65 , goal(xs) -> map(xs) } 0.00/0.65 Weak Trs: 0.00/0.65 { map(Nil()) -> Nil() 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 The weightgap principle applies (using the following nonconstant 0.00/0.65 growth matrix-interpretation) 0.00/0.65 0.00/0.65 The following argument positions are usable: 0.00/0.65 Uargs(+) = {2}, Uargs(Cons) = {1, 2} 0.00/0.65 0.00/0.65 TcT has computed the following matrix interpretation satisfying 0.00/0.65 not(EDA) and not(IDA(1)). 0.00/0.65 0.00/0.65 [map](x1) = [1] x1 + [0] 0.00/0.65 0.00/0.65 [+](x1, x2) = [1] x2 + [0] 0.00/0.65 0.00/0.65 [S](x1) = [1] x1 + [0] 0.00/0.65 0.00/0.65 [f](x1) = [1] x1 + [0] 0.00/0.65 0.00/0.65 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [Nil] = [7] 0.00/0.65 0.00/0.65 [+Full](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [0] = [7] 0.00/0.65 0.00/0.65 [goal](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [*](x1, x2) = [1] x1 + [7] 0.00/0.65 0.00/0.65 The order satisfies the following ordering constraints: 0.00/0.65 0.00/0.65 [map(Cons(x, xs))] = [1] x + [1] xs + [0] 0.00/0.65 >= [1] x + [1] xs + [0] 0.00/0.65 = [Cons(f(x), map(xs))] 0.00/0.65 0.00/0.65 [map(Nil())] = [7] 0.00/0.65 >= [7] 0.00/0.65 = [Nil()] 0.00/0.65 0.00/0.65 [f(x)] = [1] x + [0] 0.00/0.65 ? [1] x + [7] 0.00/0.65 = [*(x, x)] 0.00/0.65 0.00/0.65 [+Full(S(x), y)] = [1] x + [1] y + [0] 0.00/0.65 >= [1] x + [1] y + [0] 0.00/0.65 = [+Full(x, S(y))] 0.00/0.65 0.00/0.65 [+Full(0(), y)] = [1] y + [7] 0.00/0.65 > [1] y + [0] 0.00/0.65 = [y] 0.00/0.65 0.00/0.65 [goal(xs)] = [1] xs + [7] 0.00/0.65 > [1] xs + [0] 0.00/0.65 = [map(xs)] 0.00/0.65 0.00/0.65 [*(x, S(S(y)))] = [1] x + [7] 0.00/0.65 >= [1] x + [7] 0.00/0.65 = [+(x, *(x, S(y)))] 0.00/0.65 0.00/0.65 [*(x, S(0()))] = [1] x + [7] 0.00/0.65 > [1] x + [0] 0.00/0.65 = [x] 0.00/0.65 0.00/0.65 [*(x, 0())] = [1] x + [7] 0.00/0.65 >= [7] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 [*(0(), y)] = [14] 0.00/0.65 > [7] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 0.00/0.65 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(n^1)). 0.00/0.65 0.00/0.65 Strict Trs: 0.00/0.65 { map(Cons(x, xs)) -> Cons(f(x), map(xs)) 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) } 0.00/0.65 Weak Trs: 0.00/0.65 { map(Nil()) -> Nil() 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , goal(xs) -> map(xs) 0.00/0.65 , *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 The weightgap principle applies (using the following nonconstant 0.00/0.65 growth matrix-interpretation) 0.00/0.65 0.00/0.65 The following argument positions are usable: 0.00/0.65 Uargs(+) = {2}, Uargs(Cons) = {1, 2} 0.00/0.65 0.00/0.65 TcT has computed the following matrix interpretation satisfying 0.00/0.65 not(EDA) and not(IDA(1)). 0.00/0.65 0.00/0.65 [map](x1) = [1] x1 + [0] 0.00/0.65 0.00/0.65 [+](x1, x2) = [1] x2 + [0] 0.00/0.65 0.00/0.65 [S](x1) = [1] x1 + [0] 0.00/0.65 0.00/0.65 [f](x1) = [1] x1 + [1] 0.00/0.65 0.00/0.65 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [Nil] = [7] 0.00/0.65 0.00/0.65 [+Full](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [0] = [0] 0.00/0.65 0.00/0.65 [goal](x1) = [1] x1 + [7] 0.00/0.65 0.00/0.65 [*](x1, x2) = [1] x1 + [0] 0.00/0.65 0.00/0.65 The order satisfies the following ordering constraints: 0.00/0.65 0.00/0.65 [map(Cons(x, xs))] = [1] x + [1] xs + [0] 0.00/0.65 ? [1] x + [1] xs + [1] 0.00/0.65 = [Cons(f(x), map(xs))] 0.00/0.65 0.00/0.65 [map(Nil())] = [7] 0.00/0.65 >= [7] 0.00/0.65 = [Nil()] 0.00/0.65 0.00/0.65 [f(x)] = [1] x + [1] 0.00/0.65 > [1] x + [0] 0.00/0.65 = [*(x, x)] 0.00/0.65 0.00/0.65 [+Full(S(x), y)] = [1] x + [1] y + [0] 0.00/0.65 >= [1] x + [1] y + [0] 0.00/0.65 = [+Full(x, S(y))] 0.00/0.65 0.00/0.65 [+Full(0(), y)] = [1] y + [0] 0.00/0.65 >= [1] y + [0] 0.00/0.65 = [y] 0.00/0.65 0.00/0.65 [goal(xs)] = [1] xs + [7] 0.00/0.65 > [1] xs + [0] 0.00/0.65 = [map(xs)] 0.00/0.65 0.00/0.65 [*(x, S(S(y)))] = [1] x + [0] 0.00/0.65 >= [1] x + [0] 0.00/0.65 = [+(x, *(x, S(y)))] 0.00/0.65 0.00/0.65 [*(x, S(0()))] = [1] x + [0] 0.00/0.65 >= [1] x + [0] 0.00/0.65 = [x] 0.00/0.65 0.00/0.65 [*(x, 0())] = [1] x + [0] 0.00/0.65 >= [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 [*(0(), y)] = [0] 0.00/0.65 >= [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 0.00/0.65 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(n^1)). 0.00/0.65 0.00/0.65 Strict Trs: 0.00/0.65 { map(Cons(x, xs)) -> Cons(f(x), map(xs)) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) } 0.00/0.65 Weak Trs: 0.00/0.65 { map(Nil()) -> Nil() 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , goal(xs) -> map(xs) 0.00/0.65 , *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.65 orient following rules strictly. 0.00/0.65 0.00/0.65 Trs: { +Full(S(x), y) -> +Full(x, S(y)) } 0.00/0.65 0.00/0.65 The induced complexity on above rules (modulo remaining rules) is 0.00/0.65 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.65 component(s). 0.00/0.65 0.00/0.65 Sub-proof: 0.00/0.65 ---------- 0.00/0.65 The following argument positions are usable: 0.00/0.65 Uargs(+) = {2}, Uargs(Cons) = {1, 2} 0.00/0.65 0.00/0.65 TcT has computed the following constructor-based matrix 0.00/0.65 interpretation satisfying not(EDA). 0.00/0.65 0.00/0.65 [map](x1) = [4] x1 + [0] 0.00/0.65 0.00/0.65 [+](x1, x2) = [1] x2 + [3] 0.00/0.65 0.00/0.65 [S](x1) = [1] x1 + [4] 0.00/0.65 0.00/0.65 [f](x1) = [4] x1 + [0] 0.00/0.65 0.00/0.65 [Cons](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [Nil] = [2] 0.00/0.65 0.00/0.65 [+Full](x1, x2) = [2] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [0] = [0] 0.00/0.65 0.00/0.65 [goal](x1) = [7] x1 + [7] 0.00/0.65 0.00/0.65 [*](x1, x2) = [3] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 The order satisfies the following ordering constraints: 0.00/0.65 0.00/0.65 [map(Cons(x, xs))] = [4] x + [4] xs + [0] 0.00/0.65 >= [4] x + [4] xs + [0] 0.00/0.65 = [Cons(f(x), map(xs))] 0.00/0.65 0.00/0.65 [map(Nil())] = [8] 0.00/0.65 > [2] 0.00/0.65 = [Nil()] 0.00/0.65 0.00/0.65 [f(x)] = [4] x + [0] 0.00/0.65 >= [4] x + [0] 0.00/0.65 = [*(x, x)] 0.00/0.65 0.00/0.65 [+Full(S(x), y)] = [2] x + [1] y + [8] 0.00/0.65 > [2] x + [1] y + [4] 0.00/0.65 = [+Full(x, S(y))] 0.00/0.65 0.00/0.65 [+Full(0(), y)] = [1] y + [0] 0.00/0.65 >= [1] y + [0] 0.00/0.65 = [y] 0.00/0.65 0.00/0.65 [goal(xs)] = [7] xs + [7] 0.00/0.65 > [4] xs + [0] 0.00/0.65 = [map(xs)] 0.00/0.65 0.00/0.65 [*(x, S(S(y)))] = [3] x + [1] y + [8] 0.00/0.65 > [3] x + [1] y + [7] 0.00/0.65 = [+(x, *(x, S(y)))] 0.00/0.65 0.00/0.65 [*(x, S(0()))] = [3] x + [4] 0.00/0.65 > [1] x + [0] 0.00/0.65 = [x] 0.00/0.65 0.00/0.65 [*(x, 0())] = [3] x + [0] 0.00/0.65 >= [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 [*(0(), y)] = [1] y + [0] 0.00/0.65 >= [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 0.00/0.65 We return to the main proof. 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(n^1)). 0.00/0.65 0.00/0.65 Strict Trs: { map(Cons(x, xs)) -> Cons(f(x), map(xs)) } 0.00/0.65 Weak Trs: 0.00/0.65 { map(Nil()) -> Nil() 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , goal(xs) -> map(xs) 0.00/0.65 , *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(n^1)) 0.00/0.65 0.00/0.65 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.65 orient following rules strictly. 0.00/0.65 0.00/0.65 Trs: { map(Cons(x, xs)) -> Cons(f(x), map(xs)) } 0.00/0.65 0.00/0.65 The induced complexity on above rules (modulo remaining rules) is 0.00/0.65 YES(?,O(n^1)) . These rules are moved into the corresponding weak 0.00/0.65 component(s). 0.00/0.65 0.00/0.65 Sub-proof: 0.00/0.65 ---------- 0.00/0.65 The following argument positions are usable: 0.00/0.65 Uargs(+) = {2}, Uargs(Cons) = {1, 2} 0.00/0.65 0.00/0.65 TcT has computed the following constructor-based matrix 0.00/0.65 interpretation satisfying not(EDA). 0.00/0.65 0.00/0.65 [map](x1) = [3] x1 + [0] 0.00/0.65 0.00/0.65 [+](x1, x2) = [1] x2 + [0] 0.00/0.65 0.00/0.65 [S](x1) = [1] x1 + [3] 0.00/0.65 0.00/0.65 [f](x1) = [3] x1 + [4] 0.00/0.65 0.00/0.65 [Cons](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.65 0.00/0.65 [Nil] = [3] 0.00/0.65 0.00/0.65 [+Full](x1, x2) = [3] x1 + [1] x2 + [0] 0.00/0.65 0.00/0.65 [0] = [0] 0.00/0.65 0.00/0.65 [goal](x1) = [7] x1 + [7] 0.00/0.65 0.00/0.65 [*](x1, x2) = [2] x1 + [1] x2 + [4] 0.00/0.65 0.00/0.65 The order satisfies the following ordering constraints: 0.00/0.65 0.00/0.65 [map(Cons(x, xs))] = [3] x + [3] xs + [12] 0.00/0.65 > [3] x + [3] xs + [8] 0.00/0.65 = [Cons(f(x), map(xs))] 0.00/0.65 0.00/0.65 [map(Nil())] = [9] 0.00/0.65 > [3] 0.00/0.65 = [Nil()] 0.00/0.65 0.00/0.65 [f(x)] = [3] x + [4] 0.00/0.65 >= [3] x + [4] 0.00/0.65 = [*(x, x)] 0.00/0.65 0.00/0.65 [+Full(S(x), y)] = [3] x + [1] y + [9] 0.00/0.65 > [3] x + [1] y + [3] 0.00/0.65 = [+Full(x, S(y))] 0.00/0.65 0.00/0.65 [+Full(0(), y)] = [1] y + [0] 0.00/0.65 >= [1] y + [0] 0.00/0.65 = [y] 0.00/0.65 0.00/0.65 [goal(xs)] = [7] xs + [7] 0.00/0.65 > [3] xs + [0] 0.00/0.65 = [map(xs)] 0.00/0.65 0.00/0.65 [*(x, S(S(y)))] = [2] x + [1] y + [10] 0.00/0.65 > [2] x + [1] y + [7] 0.00/0.65 = [+(x, *(x, S(y)))] 0.00/0.65 0.00/0.65 [*(x, S(0()))] = [2] x + [7] 0.00/0.65 > [1] x + [0] 0.00/0.65 = [x] 0.00/0.65 0.00/0.65 [*(x, 0())] = [2] x + [4] 0.00/0.65 > [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 [*(0(), y)] = [1] y + [4] 0.00/0.65 > [0] 0.00/0.65 = [0()] 0.00/0.65 0.00/0.65 0.00/0.65 We return to the main proof. 0.00/0.65 0.00/0.65 We are left with following problem, upon which TcT provides the 0.00/0.65 certificate YES(O(1),O(1)). 0.00/0.65 0.00/0.65 Weak Trs: 0.00/0.65 { map(Cons(x, xs)) -> Cons(f(x), map(xs)) 0.00/0.65 , map(Nil()) -> Nil() 0.00/0.65 , f(x) -> *(x, x) 0.00/0.65 , +Full(S(x), y) -> +Full(x, S(y)) 0.00/0.65 , +Full(0(), y) -> y 0.00/0.65 , goal(xs) -> map(xs) 0.00/0.65 , *(x, S(S(y))) -> +(x, *(x, S(y))) 0.00/0.65 , *(x, S(0())) -> x 0.00/0.65 , *(x, 0()) -> 0() 0.00/0.65 , *(0(), y) -> 0() } 0.00/0.65 Obligation: 0.00/0.65 innermost runtime complexity 0.00/0.65 Answer: 0.00/0.65 YES(O(1),O(1)) 0.00/0.65 0.00/0.65 Empty rules are trivially bounded 0.00/0.65 0.00/0.65 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.65 EOF