YES(O(1),O(n^1)) 9.77/2.83 YES(O(1),O(n^1)) 9.77/2.83 9.77/2.83 We are left with following problem, upon which TcT provides the 9.77/2.83 certificate YES(O(1),O(n^1)). 9.77/2.83 9.77/2.83 Strict Trs: 9.77/2.83 { a__and(X1, X2) -> and(X1, X2) 9.77/2.83 , a__and(true(), X) -> mark(X) 9.77/2.83 , a__and(false(), Y) -> false() 9.77/2.83 , mark(true()) -> true() 9.77/2.83 , mark(false()) -> false() 9.77/2.83 , mark(0()) -> 0() 9.77/2.83 , mark(s(X)) -> s(X) 9.77/2.83 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.83 , mark(nil()) -> nil() 9.77/2.83 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.83 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.83 , mark(from(X)) -> a__from(X) 9.77/2.83 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.83 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.83 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.83 , a__if(true(), X, Y) -> mark(X) 9.77/2.83 , a__if(false(), X, Y) -> mark(Y) 9.77/2.83 , a__add(X1, X2) -> add(X1, X2) 9.77/2.83 , a__add(0(), X) -> mark(X) 9.77/2.83 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.83 , a__first(X1, X2) -> first(X1, X2) 9.77/2.83 , a__first(0(), X) -> nil() 9.77/2.83 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.83 , a__from(X) -> cons(X, from(s(X))) 9.77/2.83 , a__from(X) -> from(X) } 9.77/2.83 Obligation: 9.77/2.83 innermost runtime complexity 9.77/2.83 Answer: 9.77/2.83 YES(O(1),O(n^1)) 9.77/2.83 9.77/2.83 The weightgap principle applies (using the following nonconstant 9.77/2.83 growth matrix-interpretation) 9.77/2.83 9.77/2.83 The following argument positions are usable: 9.77/2.83 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.83 Uargs(a__first) = {1, 2} 9.77/2.83 9.77/2.83 TcT has computed the following matrix interpretation satisfying 9.77/2.83 not(EDA) and not(IDA(1)). 9.77/2.83 9.77/2.83 [a__and](x1, x2) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [true] = [0] 9.77/2.83 9.77/2.83 [mark](x1) = [0] 9.77/2.83 9.77/2.83 [false] = [0] 9.77/2.83 9.77/2.83 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [a__add](x1, x2) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [0] = [0] 9.77/2.83 9.77/2.83 [s](x1) = [0] 9.77/2.83 9.77/2.83 [add](x1, x2) = [1] x1 + [5] 9.77/2.83 9.77/2.83 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.83 9.77/2.83 [nil] = [7] 9.77/2.83 9.77/2.83 [cons](x1, x2) = [0] 9.77/2.83 9.77/2.83 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.83 9.77/2.83 [a__from](x1) = [5] 9.77/2.83 9.77/2.83 [from](x1) = [5] 9.77/2.83 9.77/2.83 [and](x1, x2) = [1] x1 + [7] 9.77/2.83 9.77/2.83 [if](x1, x2, x3) = [1] x1 + [7] 9.77/2.83 9.77/2.83 The order satisfies the following ordering constraints: 9.77/2.83 9.77/2.83 [a__and(X1, X2)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [7] 9.77/2.83 = [and(X1, X2)] 9.77/2.83 9.77/2.83 [a__and(true(), X)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__and(false(), Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(true())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [true()] 9.77/2.83 9.77/2.83 [mark(false())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(0())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [0()] 9.77/2.83 9.77/2.83 [mark(s(X))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(X)] 9.77/2.83 9.77/2.83 [mark(add(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__add(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(nil())] = [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [mark(cons(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(X1, X2)] 9.77/2.83 9.77/2.83 [mark(first(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__first(mark(X1), mark(X2))] 9.77/2.83 9.77/2.83 [mark(from(X))] = [0] 9.77/2.83 ? [5] 9.77/2.83 = [a__from(X)] 9.77/2.83 9.77/2.83 [mark(and(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__and(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(if(X1, X2, X3))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__if(mark(X1), X2, X3)] 9.77/2.83 9.77/2.83 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [7] 9.77/2.83 = [if(X1, X2, X3)] 9.77/2.83 9.77/2.83 [a__if(true(), X, Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__if(false(), X, Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(Y)] 9.77/2.83 9.77/2.83 [a__add(X1, X2)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [5] 9.77/2.83 = [add(X1, X2)] 9.77/2.83 9.77/2.83 [a__add(0(), X)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__add(s(X), Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(add(X, Y))] 9.77/2.83 9.77/2.83 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.83 ? [1] X1 + [1] X2 + [5] 9.77/2.83 = [first(X1, X2)] 9.77/2.83 9.77/2.83 [a__first(0(), X)] = [1] X + [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(Y, first(X, Z))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 > [0] 9.77/2.83 = [cons(X, from(s(X)))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 >= [5] 9.77/2.83 = [from(X)] 9.77/2.83 9.77/2.83 9.77/2.83 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.83 9.77/2.83 We are left with following problem, upon which TcT provides the 9.77/2.83 certificate YES(O(1),O(n^1)). 9.77/2.83 9.77/2.83 Strict Trs: 9.77/2.83 { a__and(X1, X2) -> and(X1, X2) 9.77/2.83 , a__and(true(), X) -> mark(X) 9.77/2.83 , a__and(false(), Y) -> false() 9.77/2.83 , mark(true()) -> true() 9.77/2.83 , mark(false()) -> false() 9.77/2.83 , mark(0()) -> 0() 9.77/2.83 , mark(s(X)) -> s(X) 9.77/2.83 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.83 , mark(nil()) -> nil() 9.77/2.83 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.83 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.83 , mark(from(X)) -> a__from(X) 9.77/2.83 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.83 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.83 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.83 , a__if(true(), X, Y) -> mark(X) 9.77/2.83 , a__if(false(), X, Y) -> mark(Y) 9.77/2.83 , a__add(X1, X2) -> add(X1, X2) 9.77/2.83 , a__add(0(), X) -> mark(X) 9.77/2.83 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.83 , a__first(X1, X2) -> first(X1, X2) 9.77/2.83 , a__first(0(), X) -> nil() 9.77/2.83 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.83 , a__from(X) -> from(X) } 9.77/2.83 Weak Trs: { a__from(X) -> cons(X, from(s(X))) } 9.77/2.83 Obligation: 9.77/2.83 innermost runtime complexity 9.77/2.83 Answer: 9.77/2.83 YES(O(1),O(n^1)) 9.77/2.83 9.77/2.83 The weightgap principle applies (using the following nonconstant 9.77/2.83 growth matrix-interpretation) 9.77/2.83 9.77/2.83 The following argument positions are usable: 9.77/2.83 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.83 Uargs(a__first) = {1, 2} 9.77/2.83 9.77/2.83 TcT has computed the following matrix interpretation satisfying 9.77/2.83 not(EDA) and not(IDA(1)). 9.77/2.83 9.77/2.83 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.83 9.77/2.83 [true] = [0] 9.77/2.83 9.77/2.83 [mark](x1) = [0] 9.77/2.83 9.77/2.83 [false] = [0] 9.77/2.83 9.77/2.83 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [a__add](x1, x2) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [0] = [0] 9.77/2.83 9.77/2.83 [s](x1) = [0] 9.77/2.83 9.77/2.83 [add](x1, x2) = [1] x1 + [5] 9.77/2.83 9.77/2.83 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.83 9.77/2.83 [nil] = [7] 9.77/2.83 9.77/2.83 [cons](x1, x2) = [0] 9.77/2.83 9.77/2.83 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.83 9.77/2.83 [a__from](x1) = [5] 9.77/2.83 9.77/2.83 [from](x1) = [5] 9.77/2.83 9.77/2.83 [and](x1, x2) = [1] x1 + [3] 9.77/2.83 9.77/2.83 [if](x1, x2, x3) = [1] x1 + [7] 9.77/2.83 9.77/2.83 The order satisfies the following ordering constraints: 9.77/2.83 9.77/2.83 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.83 > [1] X1 + [3] 9.77/2.83 = [and(X1, X2)] 9.77/2.83 9.77/2.83 [a__and(true(), X)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__and(false(), Y)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(true())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [true()] 9.77/2.83 9.77/2.83 [mark(false())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(0())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [0()] 9.77/2.83 9.77/2.83 [mark(s(X))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(X)] 9.77/2.83 9.77/2.83 [mark(add(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__add(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(nil())] = [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [mark(cons(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(X1, X2)] 9.77/2.83 9.77/2.83 [mark(first(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__first(mark(X1), mark(X2))] 9.77/2.83 9.77/2.83 [mark(from(X))] = [0] 9.77/2.83 ? [5] 9.77/2.83 = [a__from(X)] 9.77/2.83 9.77/2.83 [mark(and(X1, X2))] = [0] 9.77/2.83 ? [4] 9.77/2.83 = [a__and(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(if(X1, X2, X3))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__if(mark(X1), X2, X3)] 9.77/2.83 9.77/2.83 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [7] 9.77/2.83 = [if(X1, X2, X3)] 9.77/2.83 9.77/2.83 [a__if(true(), X, Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__if(false(), X, Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(Y)] 9.77/2.83 9.77/2.83 [a__add(X1, X2)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [5] 9.77/2.83 = [add(X1, X2)] 9.77/2.83 9.77/2.83 [a__add(0(), X)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__add(s(X), Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(add(X, Y))] 9.77/2.83 9.77/2.83 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.83 ? [1] X1 + [1] X2 + [5] 9.77/2.83 = [first(X1, X2)] 9.77/2.83 9.77/2.83 [a__first(0(), X)] = [1] X + [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(Y, first(X, Z))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 > [0] 9.77/2.83 = [cons(X, from(s(X)))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 >= [5] 9.77/2.83 = [from(X)] 9.77/2.83 9.77/2.83 9.77/2.83 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.83 9.77/2.83 We are left with following problem, upon which TcT provides the 9.77/2.83 certificate YES(O(1),O(n^1)). 9.77/2.83 9.77/2.83 Strict Trs: 9.77/2.83 { mark(true()) -> true() 9.77/2.83 , mark(false()) -> false() 9.77/2.83 , mark(0()) -> 0() 9.77/2.83 , mark(s(X)) -> s(X) 9.77/2.83 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.83 , mark(nil()) -> nil() 9.77/2.83 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.83 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.83 , mark(from(X)) -> a__from(X) 9.77/2.83 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.83 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.83 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.83 , a__if(true(), X, Y) -> mark(X) 9.77/2.83 , a__if(false(), X, Y) -> mark(Y) 9.77/2.83 , a__add(X1, X2) -> add(X1, X2) 9.77/2.83 , a__add(0(), X) -> mark(X) 9.77/2.83 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.83 , a__first(X1, X2) -> first(X1, X2) 9.77/2.83 , a__first(0(), X) -> nil() 9.77/2.83 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.83 , a__from(X) -> from(X) } 9.77/2.83 Weak Trs: 9.77/2.83 { a__and(X1, X2) -> and(X1, X2) 9.77/2.83 , a__and(true(), X) -> mark(X) 9.77/2.83 , a__and(false(), Y) -> false() 9.77/2.83 , a__from(X) -> cons(X, from(s(X))) } 9.77/2.83 Obligation: 9.77/2.83 innermost runtime complexity 9.77/2.83 Answer: 9.77/2.83 YES(O(1),O(n^1)) 9.77/2.83 9.77/2.83 The weightgap principle applies (using the following nonconstant 9.77/2.83 growth matrix-interpretation) 9.77/2.83 9.77/2.83 The following argument positions are usable: 9.77/2.83 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.83 Uargs(a__first) = {1, 2} 9.77/2.83 9.77/2.83 TcT has computed the following matrix interpretation satisfying 9.77/2.83 not(EDA) and not(IDA(1)). 9.77/2.83 9.77/2.83 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.83 9.77/2.83 [true] = [0] 9.77/2.83 9.77/2.83 [mark](x1) = [0] 9.77/2.83 9.77/2.83 [false] = [0] 9.77/2.83 9.77/2.83 [a__if](x1, x2, x3) = [1] x1 + [4] 9.77/2.83 9.77/2.83 [a__add](x1, x2) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [0] = [0] 9.77/2.83 9.77/2.83 [s](x1) = [0] 9.77/2.83 9.77/2.83 [add](x1, x2) = [1] x1 + [5] 9.77/2.83 9.77/2.83 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.83 9.77/2.83 [nil] = [7] 9.77/2.83 9.77/2.83 [cons](x1, x2) = [0] 9.77/2.83 9.77/2.83 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.83 9.77/2.83 [a__from](x1) = [5] 9.77/2.83 9.77/2.83 [from](x1) = [5] 9.77/2.83 9.77/2.83 [and](x1, x2) = [1] x1 + [3] 9.77/2.83 9.77/2.83 [if](x1, x2, x3) = [1] x1 + [7] 9.77/2.83 9.77/2.83 The order satisfies the following ordering constraints: 9.77/2.83 9.77/2.83 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.83 > [1] X1 + [3] 9.77/2.83 = [and(X1, X2)] 9.77/2.83 9.77/2.83 [a__and(true(), X)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__and(false(), Y)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(true())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [true()] 9.77/2.83 9.77/2.83 [mark(false())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [false()] 9.77/2.83 9.77/2.83 [mark(0())] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [0()] 9.77/2.83 9.77/2.83 [mark(s(X))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(X)] 9.77/2.83 9.77/2.83 [mark(add(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__add(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(nil())] = [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [mark(cons(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(X1, X2)] 9.77/2.83 9.77/2.83 [mark(first(X1, X2))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [a__first(mark(X1), mark(X2))] 9.77/2.83 9.77/2.83 [mark(from(X))] = [0] 9.77/2.83 ? [5] 9.77/2.83 = [a__from(X)] 9.77/2.83 9.77/2.83 [mark(and(X1, X2))] = [0] 9.77/2.83 ? [4] 9.77/2.83 = [a__and(mark(X1), X2)] 9.77/2.83 9.77/2.83 [mark(if(X1, X2, X3))] = [0] 9.77/2.83 ? [4] 9.77/2.83 = [a__if(mark(X1), X2, X3)] 9.77/2.83 9.77/2.83 [a__if(X1, X2, X3)] = [1] X1 + [4] 9.77/2.83 ? [1] X1 + [7] 9.77/2.83 = [if(X1, X2, X3)] 9.77/2.83 9.77/2.83 [a__if(true(), X, Y)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__if(false(), X, Y)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [mark(Y)] 9.77/2.83 9.77/2.83 [a__add(X1, X2)] = [1] X1 + [0] 9.77/2.83 ? [1] X1 + [5] 9.77/2.83 = [add(X1, X2)] 9.77/2.83 9.77/2.83 [a__add(0(), X)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__add(s(X), Y)] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [s(add(X, Y))] 9.77/2.83 9.77/2.83 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.83 ? [1] X1 + [1] X2 + [5] 9.77/2.83 = [first(X1, X2)] 9.77/2.83 9.77/2.83 [a__first(0(), X)] = [1] X + [0] 9.77/2.83 ? [7] 9.77/2.83 = [nil()] 9.77/2.83 9.77/2.83 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.83 >= [0] 9.77/2.83 = [cons(Y, first(X, Z))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 > [0] 9.77/2.83 = [cons(X, from(s(X)))] 9.77/2.83 9.77/2.83 [a__from(X)] = [5] 9.77/2.83 >= [5] 9.77/2.83 = [from(X)] 9.77/2.83 9.77/2.83 9.77/2.83 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.83 9.77/2.83 We are left with following problem, upon which TcT provides the 9.77/2.83 certificate YES(O(1),O(n^1)). 9.77/2.83 9.77/2.83 Strict Trs: 9.77/2.83 { mark(true()) -> true() 9.77/2.83 , mark(false()) -> false() 9.77/2.83 , mark(0()) -> 0() 9.77/2.83 , mark(s(X)) -> s(X) 9.77/2.83 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.83 , mark(nil()) -> nil() 9.77/2.83 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.83 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.83 , mark(from(X)) -> a__from(X) 9.77/2.83 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.83 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.83 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.83 , a__add(X1, X2) -> add(X1, X2) 9.77/2.83 , a__add(0(), X) -> mark(X) 9.77/2.83 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.83 , a__first(X1, X2) -> first(X1, X2) 9.77/2.83 , a__first(0(), X) -> nil() 9.77/2.83 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.83 , a__from(X) -> from(X) } 9.77/2.83 Weak Trs: 9.77/2.83 { a__and(X1, X2) -> and(X1, X2) 9.77/2.83 , a__and(true(), X) -> mark(X) 9.77/2.83 , a__and(false(), Y) -> false() 9.77/2.83 , a__if(true(), X, Y) -> mark(X) 9.77/2.83 , a__if(false(), X, Y) -> mark(Y) 9.77/2.83 , a__from(X) -> cons(X, from(s(X))) } 9.77/2.83 Obligation: 9.77/2.83 innermost runtime complexity 9.77/2.83 Answer: 9.77/2.83 YES(O(1),O(n^1)) 9.77/2.83 9.77/2.83 The weightgap principle applies (using the following nonconstant 9.77/2.83 growth matrix-interpretation) 9.77/2.83 9.77/2.83 The following argument positions are usable: 9.77/2.83 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.83 Uargs(a__first) = {1, 2} 9.77/2.83 9.77/2.83 TcT has computed the following matrix interpretation satisfying 9.77/2.83 not(EDA) and not(IDA(1)). 9.77/2.83 9.77/2.83 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.83 9.77/2.83 [true] = [0] 9.77/2.83 9.77/2.83 [mark](x1) = [0] 9.77/2.83 9.77/2.83 [false] = [0] 9.77/2.83 9.77/2.83 [a__if](x1, x2, x3) = [1] x1 + [4] 9.77/2.83 9.77/2.83 [a__add](x1, x2) = [1] x1 + [0] 9.77/2.83 9.77/2.83 [0] = [0] 9.77/2.83 9.77/2.83 [s](x1) = [0] 9.77/2.83 9.77/2.83 [add](x1, x2) = [1] x1 + [5] 9.77/2.83 9.77/2.83 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.83 9.77/2.83 [nil] = [7] 9.77/2.83 9.77/2.83 [cons](x1, x2) = [0] 9.77/2.83 9.77/2.83 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.83 9.77/2.83 [a__from](x1) = [5] 9.77/2.83 9.77/2.83 [from](x1) = [5] 9.77/2.83 9.77/2.83 [and](x1, x2) = [1] x1 + [3] 9.77/2.83 9.77/2.83 [if](x1, x2, x3) = [1] x1 + [3] 9.77/2.83 9.77/2.83 The order satisfies the following ordering constraints: 9.77/2.83 9.77/2.83 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.83 > [1] X1 + [3] 9.77/2.83 = [and(X1, X2)] 9.77/2.83 9.77/2.83 [a__and(true(), X)] = [4] 9.77/2.83 > [0] 9.77/2.83 = [mark(X)] 9.77/2.83 9.77/2.83 [a__and(false(), Y)] = [4] 9.77/2.83 > [0] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [5] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [3] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [0] 9.77/2.84 ? [1] X1 + [5] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 >= [5] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) 9.77/2.84 , a__first(0(), X) -> nil() 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [true] = [4] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [4] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [1] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [5] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.84 9.77/2.84 [nil] = [7] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.84 9.77/2.84 [a__from](x1) = [5] 9.77/2.84 9.77/2.84 [from](x1) = [5] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [4] 9.77/2.84 >= [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 ? [1] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [5] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [1] 9.77/2.84 ? [1] X1 + [5] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [1] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [1] 9.77/2.84 > [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 >= [5] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) 9.77/2.84 , a__first(0(), X) -> nil() 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [true] = [4] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [4] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [1] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.84 9.77/2.84 [nil] = [7] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.84 9.77/2.84 [a__from](x1) = [5] 9.77/2.84 9.77/2.84 [from](x1) = [5] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [3] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [3] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [8] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [8] 9.77/2.84 > [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [5] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [1] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 >= [5] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) 9.77/2.84 , a__first(0(), X) -> nil() 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [true] = [4] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [4] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [2] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.84 9.77/2.84 [nil] = [7] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.84 9.77/2.84 [a__from](x1) = [5] 9.77/2.84 9.77/2.84 [from](x1) = [1] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [3] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [3] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [8] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [8] 9.77/2.84 > [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [5] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [2] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [0] 9.77/2.84 ? [7] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [1] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) 9.77/2.84 , a__first(0(), X) -> nil() 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [true] = [4] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [4] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [1] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [1] 9.77/2.84 9.77/2.84 [nil] = [6] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.84 9.77/2.84 [a__from](x1) = [5] 9.77/2.84 9.77/2.84 [from](x1) = [2] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [0] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [4] 9.77/2.84 >= [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 ? [6] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 ? [1] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [5] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [1] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [1] 9.77/2.84 ? [6] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [1] 9.77/2.84 > [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [5] 9.77/2.84 > [2] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) 9.77/2.84 , a__first(0(), X) -> nil() } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [true] = [0] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [0] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [2] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [1] 9.77/2.84 9.77/2.84 [nil] = [0] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [5] 9.77/2.84 9.77/2.84 [a__from](x1) = [4] 9.77/2.84 9.77/2.84 [from](x1) = [2] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [3] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [true()] 9.77/2.84 9.77/2.84 [mark(false())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(0())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [0()] 9.77/2.84 9.77/2.84 [mark(s(X))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [s(X)] 9.77/2.84 9.77/2.84 [mark(add(X1, X2))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__add(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(nil())] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [mark(cons(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [cons(X1, X2)] 9.77/2.84 9.77/2.84 [mark(first(X1, X2))] = [0] 9.77/2.84 ? [1] 9.77/2.84 = [a__first(mark(X1), mark(X2))] 9.77/2.84 9.77/2.84 [mark(from(X))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__from(X)] 9.77/2.84 9.77/2.84 [mark(and(X1, X2))] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [a__and(mark(X1), X2)] 9.77/2.84 9.77/2.84 [mark(if(X1, X2, X3))] = [0] 9.77/2.84 ? [4] 9.77/2.84 = [a__if(mark(X1), X2, X3)] 9.77/2.84 9.77/2.84 [a__if(X1, X2, X3)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [3] 9.77/2.84 = [if(X1, X2, X3)] 9.77/2.84 9.77/2.84 [a__if(true(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__if(false(), X, Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(Y)] 9.77/2.84 9.77/2.84 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.84 > [1] X1 + [2] 9.77/2.84 = [add(X1, X2)] 9.77/2.84 9.77/2.84 [a__add(0(), X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__add(s(X), Y)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [s(add(X, Y))] 9.77/2.84 9.77/2.84 [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] 9.77/2.84 ? [1] X1 + [1] X2 + [5] 9.77/2.84 = [first(X1, X2)] 9.77/2.84 9.77/2.84 [a__first(0(), X)] = [1] X + [1] 9.77/2.84 > [0] 9.77/2.84 = [nil()] 9.77/2.84 9.77/2.84 [a__first(s(X), cons(Y, Z))] = [1] 9.77/2.84 > [0] 9.77/2.84 = [cons(Y, first(X, Z))] 9.77/2.84 9.77/2.84 [a__from(X)] = [4] 9.77/2.84 > [0] 9.77/2.84 = [cons(X, from(s(X)))] 9.77/2.84 9.77/2.84 [a__from(X)] = [4] 9.77/2.84 > [2] 9.77/2.84 = [from(X)] 9.77/2.84 9.77/2.84 9.77/2.84 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.84 9.77/2.84 We are left with following problem, upon which TcT provides the 9.77/2.84 certificate YES(O(1),O(n^1)). 9.77/2.84 9.77/2.84 Strict Trs: 9.77/2.84 { mark(true()) -> true() 9.77/2.84 , mark(false()) -> false() 9.77/2.84 , mark(0()) -> 0() 9.77/2.84 , mark(s(X)) -> s(X) 9.77/2.84 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.84 , mark(nil()) -> nil() 9.77/2.84 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.84 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.84 , mark(from(X)) -> a__from(X) 9.77/2.84 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.84 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.84 , a__first(X1, X2) -> first(X1, X2) } 9.77/2.84 Weak Trs: 9.77/2.84 { a__and(X1, X2) -> and(X1, X2) 9.77/2.84 , a__and(true(), X) -> mark(X) 9.77/2.84 , a__and(false(), Y) -> false() 9.77/2.84 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.84 , a__if(true(), X, Y) -> mark(X) 9.77/2.84 , a__if(false(), X, Y) -> mark(Y) 9.77/2.84 , a__add(X1, X2) -> add(X1, X2) 9.77/2.84 , a__add(0(), X) -> mark(X) 9.77/2.84 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.84 , a__first(0(), X) -> nil() 9.77/2.84 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.84 , a__from(X) -> cons(X, from(s(X))) 9.77/2.84 , a__from(X) -> from(X) } 9.77/2.84 Obligation: 9.77/2.84 innermost runtime complexity 9.77/2.84 Answer: 9.77/2.84 YES(O(1),O(n^1)) 9.77/2.84 9.77/2.84 The weightgap principle applies (using the following nonconstant 9.77/2.84 growth matrix-interpretation) 9.77/2.84 9.77/2.84 The following argument positions are usable: 9.77/2.84 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.84 Uargs(a__first) = {1, 2} 9.77/2.84 9.77/2.84 TcT has computed the following matrix interpretation satisfying 9.77/2.84 not(EDA) and not(IDA(1)). 9.77/2.84 9.77/2.84 [a__and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [true] = [0] 9.77/2.84 9.77/2.84 [mark](x1) = [0] 9.77/2.84 9.77/2.84 [false] = [0] 9.77/2.84 9.77/2.84 [a__if](x1, x2, x3) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.84 9.77/2.84 [0] = [0] 9.77/2.84 9.77/2.84 [s](x1) = [0] 9.77/2.84 9.77/2.84 [add](x1, x2) = [1] x1 + [1] 9.77/2.84 9.77/2.84 [a__first](x1, x2) = [1] x1 + [1] x2 + [1] 9.77/2.84 9.77/2.84 [nil] = [1] 9.77/2.84 9.77/2.84 [cons](x1, x2) = [0] 9.77/2.84 9.77/2.84 [first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.84 9.77/2.84 [a__from](x1) = [4] 9.77/2.84 9.77/2.84 [from](x1) = [2] 9.77/2.84 9.77/2.84 [and](x1, x2) = [1] x1 + [0] 9.77/2.84 9.77/2.84 [if](x1, x2, x3) = [1] x1 + [3] 9.77/2.84 9.77/2.84 The order satisfies the following ordering constraints: 9.77/2.84 9.77/2.84 [a__and(X1, X2)] = [1] X1 + [0] 9.77/2.84 >= [1] X1 + [0] 9.77/2.84 = [and(X1, X2)] 9.77/2.84 9.77/2.84 [a__and(true(), X)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [mark(X)] 9.77/2.84 9.77/2.84 [a__and(false(), Y)] = [0] 9.77/2.84 >= [0] 9.77/2.84 = [false()] 9.77/2.84 9.77/2.84 [mark(true())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [0] 9.77/2.85 ? [4] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [0] 9.77/2.85 ? [1] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [0] 9.77/2.85 ? [1] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [0] 9.77/2.85 ? [4] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [0] 9.77/2.85 ? [4] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [4] 9.77/2.85 > [1] X1 + [3] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.85 > [1] X1 + [1] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] 9.77/2.85 > [1] X1 + [1] X2 + [0] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [1] 9.77/2.85 >= [1] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 > [2] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(n^1)). 9.77/2.85 9.77/2.85 Strict Trs: 9.77/2.85 { mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(n^1)) 9.77/2.85 9.77/2.85 The weightgap principle applies (using the following nonconstant 9.77/2.85 growth matrix-interpretation) 9.77/2.85 9.77/2.85 The following argument positions are usable: 9.77/2.85 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.85 Uargs(a__first) = {1, 2} 9.77/2.85 9.77/2.85 TcT has computed the following matrix interpretation satisfying 9.77/2.85 not(EDA) and not(IDA(1)). 9.77/2.85 9.77/2.85 [a__and](x1, x2) = [1] x1 + [4] 9.77/2.85 9.77/2.85 [true] = [4] 9.77/2.85 9.77/2.85 [mark](x1) = [1] 9.77/2.85 9.77/2.85 [false] = [4] 9.77/2.85 9.77/2.85 [a__if](x1, x2, x3) = [1] x1 + [0] 9.77/2.85 9.77/2.85 [a__add](x1, x2) = [1] x1 + [4] 9.77/2.85 9.77/2.85 [0] = [0] 9.77/2.85 9.77/2.85 [s](x1) = [0] 9.77/2.85 9.77/2.85 [add](x1, x2) = [1] x1 + [1] 9.77/2.85 9.77/2.85 [a__first](x1, x2) = [1] x1 + [1] x2 + [6] 9.77/2.85 9.77/2.85 [nil] = [0] 9.77/2.85 9.77/2.85 [cons](x1, x2) = [0] 9.77/2.85 9.77/2.85 [first](x1, x2) = [1] x1 + [1] x2 + [2] 9.77/2.85 9.77/2.85 [a__from](x1) = [0] 9.77/2.85 9.77/2.85 [from](x1) = [0] 9.77/2.85 9.77/2.85 [and](x1, x2) = [1] x1 + [3] 9.77/2.85 9.77/2.85 [if](x1, x2, x3) = [1] x1 + [0] 9.77/2.85 9.77/2.85 The order satisfies the following ordering constraints: 9.77/2.85 9.77/2.85 [a__and(X1, X2)] = [1] X1 + [4] 9.77/2.85 > [1] X1 + [3] 9.77/2.85 = [and(X1, X2)] 9.77/2.85 9.77/2.85 [a__and(true(), X)] = [8] 9.77/2.85 > [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__and(false(), Y)] = [8] 9.77/2.85 > [4] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(true())] = [1] 9.77/2.85 ? [4] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [1] 9.77/2.85 ? [4] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [1] 9.77/2.85 > [0] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [1] 9.77/2.85 ? [5] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [1] 9.77/2.85 > [0] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [1] 9.77/2.85 ? [8] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [1] 9.77/2.85 ? [5] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [1] 9.77/2.85 >= [1] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [0] 9.77/2.85 >= [1] X1 + [0] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [4] 9.77/2.85 > [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [4] 9.77/2.85 > [1] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [4] 9.77/2.85 > [1] X1 + [1] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [4] 9.77/2.85 > [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [6] 9.77/2.85 > [1] X1 + [1] X2 + [2] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [6] 9.77/2.85 > [0] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [6] 9.77/2.85 > [0] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(n^1)). 9.77/2.85 9.77/2.85 Strict Trs: 9.77/2.85 { mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(n^1)) 9.77/2.85 9.77/2.85 The weightgap principle applies (using the following nonconstant 9.77/2.85 growth matrix-interpretation) 9.77/2.85 9.77/2.85 The following argument positions are usable: 9.77/2.85 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.85 Uargs(a__first) = {1, 2} 9.77/2.85 9.77/2.85 TcT has computed the following matrix interpretation satisfying 9.77/2.85 not(EDA) and not(IDA(1)). 9.77/2.85 9.77/2.85 [a__and](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [true] = [4] 9.77/2.85 9.77/2.85 [mark](x1) = [1] x1 + [1] 9.77/2.85 9.77/2.85 [false] = [0] 9.77/2.85 9.77/2.85 [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [4] 9.77/2.85 9.77/2.85 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [0] = [4] 9.77/2.85 9.77/2.85 [s](x1) = [0] 9.77/2.85 9.77/2.85 [add](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [a__first](x1, x2) = [1] x1 + [1] x2 + [6] 9.77/2.85 9.77/2.85 [nil] = [5] 9.77/2.85 9.77/2.85 [cons](x1, x2) = [0] 9.77/2.85 9.77/2.85 [first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [a__from](x1) = [4] 9.77/2.85 9.77/2.85 [from](x1) = [3] 9.77/2.85 9.77/2.85 [and](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 9.77/2.85 9.77/2.85 The order satisfies the following ordering constraints: 9.77/2.85 9.77/2.85 [a__and(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [and(X1, X2)] 9.77/2.85 9.77/2.85 [a__and(true(), X)] = [1] X + [4] 9.77/2.85 > [1] X + [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__and(false(), Y)] = [1] Y + [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(true())] = [5] 9.77/2.85 > [4] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [1] 9.77/2.85 > [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [5] 9.77/2.85 > [4] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [1] X1 + [1] X2 + [1] 9.77/2.85 >= [1] X1 + [1] X2 + [1] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [6] 9.77/2.85 > [5] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [1] 9.77/2.85 > [0] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [1] 9.77/2.85 ? [1] X1 + [1] X2 + [8] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [4] 9.77/2.85 >= [4] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [1] X1 + [1] X2 + [1] 9.77/2.85 >= [1] X1 + [1] X2 + [1] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 9.77/2.85 ? [1] X1 + [1] X2 + [1] X3 + [5] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [4] 9.77/2.85 > [1] X1 + [1] X2 + [1] X3 + [0] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [1] X + [1] Y + [8] 9.77/2.85 > [1] X + [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [1] X + [1] Y + [4] 9.77/2.85 > [1] Y + [1] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [1] X + [4] 9.77/2.85 > [1] X + [1] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [1] Y + [0] 9.77/2.85 >= [0] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [6] 9.77/2.85 > [1] X1 + [1] X2 + [0] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [10] 9.77/2.85 > [5] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [6] 9.77/2.85 > [0] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 > [0] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 > [3] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(n^1)). 9.77/2.85 9.77/2.85 Strict Trs: 9.77/2.85 { mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(n^1)) 9.77/2.85 9.77/2.85 We use the processor 'matrix interpretation of dimension 1' to 9.77/2.85 orient following rules strictly. 9.77/2.85 9.77/2.85 Trs: 9.77/2.85 { mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) } 9.77/2.85 9.77/2.85 The induced complexity on above rules (modulo remaining rules) is 9.77/2.85 YES(?,O(n^1)) . These rules are moved into the corresponding weak 9.77/2.85 component(s). 9.77/2.85 9.77/2.85 Sub-proof: 9.77/2.85 ---------- 9.77/2.85 The following argument positions are usable: 9.77/2.85 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.85 Uargs(a__first) = {1, 2} 9.77/2.85 9.77/2.85 TcT has computed the following constructor-based matrix 9.77/2.85 interpretation satisfying not(EDA). 9.77/2.85 9.77/2.85 [a__and](x1, x2) = [1] x1 + [2] x2 + [0] 9.77/2.85 9.77/2.85 [true] = [0] 9.77/2.85 9.77/2.85 [mark](x1) = [2] x1 + [0] 9.77/2.85 9.77/2.85 [false] = [0] 9.77/2.85 9.77/2.85 [a__if](x1, x2, x3) = [1] x1 + [2] x2 + [2] x3 + [0] 9.77/2.85 9.77/2.85 [a__add](x1, x2) = [1] x1 + [2] x2 + [4] 9.77/2.85 9.77/2.85 [0] = [4] 9.77/2.85 9.77/2.85 [s](x1) = [4] 9.77/2.85 9.77/2.85 [add](x1, x2) = [1] x1 + [1] x2 + [4] 9.77/2.85 9.77/2.85 [a__first](x1, x2) = [1] x1 + [1] x2 + [4] 9.77/2.85 9.77/2.85 [nil] = [4] 9.77/2.85 9.77/2.85 [cons](x1, x2) = [0] 9.77/2.85 9.77/2.85 [first](x1, x2) = [1] x1 + [1] x2 + [4] 9.77/2.85 9.77/2.85 [a__from](x1) = [0] 9.77/2.85 9.77/2.85 [from](x1) = [0] 9.77/2.85 9.77/2.85 [and](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 9.77/2.85 9.77/2.85 The order satisfies the following ordering constraints: 9.77/2.85 9.77/2.85 [a__and(X1, X2)] = [1] X1 + [2] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [and(X1, X2)] 9.77/2.85 9.77/2.85 [a__and(true(), X)] = [2] X + [0] 9.77/2.85 >= [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__and(false(), Y)] = [2] Y + [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(true())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [8] 9.77/2.85 > [4] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [2] X1 + [2] X2 + [8] 9.77/2.85 > [2] X1 + [2] X2 + [4] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [2] X1 + [2] X2 + [8] 9.77/2.85 > [2] X1 + [2] X2 + [4] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [2] X1 + [2] X2 + [0] 9.77/2.85 >= [2] X1 + [2] X2 + [0] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [2] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 >= [2] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [1] X3 + [0] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [2] X + [2] Y + [0] 9.77/2.85 >= [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [2] X + [2] Y + [0] 9.77/2.85 >= [2] Y + [0] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [2] X2 + [4] 9.77/2.85 >= [1] X1 + [1] X2 + [4] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [2] X + [8] 9.77/2.85 > [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [2] Y + [8] 9.77/2.85 > [4] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] 9.77/2.85 >= [1] X1 + [1] X2 + [4] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [8] 9.77/2.85 > [4] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [8] 9.77/2.85 > [0] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 We return to the main proof. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(n^1)). 9.77/2.85 9.77/2.85 Strict Trs: 9.77/2.85 { mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(n^1)) 9.77/2.85 9.77/2.85 We use the processor 'matrix interpretation of dimension 1' to 9.77/2.85 orient following rules strictly. 9.77/2.85 9.77/2.85 Trs: { mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) } 9.77/2.85 9.77/2.85 The induced complexity on above rules (modulo remaining rules) is 9.77/2.85 YES(?,O(n^1)) . These rules are moved into the corresponding weak 9.77/2.85 component(s). 9.77/2.85 9.77/2.85 Sub-proof: 9.77/2.85 ---------- 9.77/2.85 The following argument positions are usable: 9.77/2.85 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.85 Uargs(a__first) = {1, 2} 9.77/2.85 9.77/2.85 TcT has computed the following constructor-based matrix 9.77/2.85 interpretation satisfying not(EDA). 9.77/2.85 9.77/2.85 [a__and](x1, x2) = [1] x1 + [4] x2 + [0] 9.77/2.85 9.77/2.85 [true] = [0] 9.77/2.85 9.77/2.85 [mark](x1) = [4] x1 + [0] 9.77/2.85 9.77/2.85 [false] = [0] 9.77/2.85 9.77/2.85 [a__if](x1, x2, x3) = [1] x1 + [4] x2 + [4] x3 + [1] 9.77/2.85 9.77/2.85 [a__add](x1, x2) = [1] x1 + [4] x2 + [0] 9.77/2.85 9.77/2.85 [0] = [2] 9.77/2.85 9.77/2.85 [s](x1) = [0] 9.77/2.85 9.77/2.85 [add](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [nil] = [0] 9.77/2.85 9.77/2.85 [cons](x1, x2) = [0] 9.77/2.85 9.77/2.85 [first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [a__from](x1) = [0] 9.77/2.85 9.77/2.85 [from](x1) = [0] 9.77/2.85 9.77/2.85 [and](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] 9.77/2.85 9.77/2.85 The order satisfies the following ordering constraints: 9.77/2.85 9.77/2.85 [a__and(X1, X2)] = [1] X1 + [4] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [and(X1, X2)] 9.77/2.85 9.77/2.85 [a__and(true(), X)] = [4] X + [0] 9.77/2.85 >= [4] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__and(false(), Y)] = [4] Y + [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(true())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [8] 9.77/2.85 > [2] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [4] X1 + [4] X2 + [0] 9.77/2.85 >= [4] X1 + [4] X2 + [0] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [4] X1 + [4] X2 + [0] 9.77/2.85 >= [4] X1 + [4] X2 + [0] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [4] X1 + [4] X2 + [0] 9.77/2.85 >= [4] X1 + [4] X2 + [0] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [4] X1 + [4] X2 + [4] X3 + [4] 9.77/2.85 > [4] X1 + [4] X2 + [4] X3 + [1] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [4] X2 + [4] X3 + [1] 9.77/2.85 >= [1] X1 + [1] X2 + [1] X3 + [1] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [4] X + [4] Y + [1] 9.77/2.85 > [4] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [4] X + [4] Y + [1] 9.77/2.85 > [4] Y + [0] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [4] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [4] X + [2] 9.77/2.85 > [4] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [4] Y + [0] 9.77/2.85 >= [0] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [2] 9.77/2.85 > [0] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [0] 9.77/2.85 >= [0] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 We return to the main proof. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(n^1)). 9.77/2.85 9.77/2.85 Strict Trs: { mark(and(X1, X2)) -> a__and(mark(X1), X2) } 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(n^1)) 9.77/2.85 9.77/2.85 We use the processor 'matrix interpretation of dimension 1' to 9.77/2.85 orient following rules strictly. 9.77/2.85 9.77/2.85 Trs: { mark(and(X1, X2)) -> a__and(mark(X1), X2) } 9.77/2.85 9.77/2.85 The induced complexity on above rules (modulo remaining rules) is 9.77/2.85 YES(?,O(n^1)) . These rules are moved into the corresponding weak 9.77/2.85 component(s). 9.77/2.85 9.77/2.85 Sub-proof: 9.77/2.85 ---------- 9.77/2.85 The following argument positions are usable: 9.77/2.85 Uargs(a__and) = {1}, Uargs(a__if) = {1}, Uargs(a__add) = {1}, 9.77/2.85 Uargs(a__first) = {1, 2} 9.77/2.85 9.77/2.85 TcT has computed the following constructor-based matrix 9.77/2.85 interpretation satisfying not(EDA). 9.77/2.85 9.77/2.85 [a__and](x1, x2) = [1] x1 + [2] x2 + [4] 9.77/2.85 9.77/2.85 [true] = [4] 9.77/2.85 9.77/2.85 [mark](x1) = [2] x1 + [0] 9.77/2.85 9.77/2.85 [false] = [4] 9.77/2.85 9.77/2.85 [a__if](x1, x2, x3) = [1] x1 + [2] x2 + [2] x3 + [0] 9.77/2.85 9.77/2.85 [a__add](x1, x2) = [1] x1 + [2] x2 + [4] 9.77/2.85 9.77/2.85 [0] = [4] 9.77/2.85 9.77/2.85 [s](x1) = [4] 9.77/2.85 9.77/2.85 [add](x1, x2) = [1] x1 + [1] x2 + [4] 9.77/2.85 9.77/2.85 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [nil] = [4] 9.77/2.85 9.77/2.85 [cons](x1, x2) = [4] 9.77/2.85 9.77/2.85 [first](x1, x2) = [1] x1 + [1] x2 + [0] 9.77/2.85 9.77/2.85 [a__from](x1) = [4] 9.77/2.85 9.77/2.85 [from](x1) = [4] 9.77/2.85 9.77/2.85 [and](x1, x2) = [1] x1 + [1] x2 + [4] 9.77/2.85 9.77/2.85 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 9.77/2.85 9.77/2.85 The order satisfies the following ordering constraints: 9.77/2.85 9.77/2.85 [a__and(X1, X2)] = [1] X1 + [2] X2 + [4] 9.77/2.85 >= [1] X1 + [1] X2 + [4] 9.77/2.85 = [and(X1, X2)] 9.77/2.85 9.77/2.85 [a__and(true(), X)] = [2] X + [8] 9.77/2.85 > [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__and(false(), Y)] = [2] Y + [8] 9.77/2.85 > [4] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(true())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [true()] 9.77/2.85 9.77/2.85 [mark(false())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [false()] 9.77/2.85 9.77/2.85 [mark(0())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [0()] 9.77/2.85 9.77/2.85 [mark(s(X))] = [8] 9.77/2.85 > [4] 9.77/2.85 = [s(X)] 9.77/2.85 9.77/2.85 [mark(add(X1, X2))] = [2] X1 + [2] X2 + [8] 9.77/2.85 > [2] X1 + [2] X2 + [4] 9.77/2.85 = [a__add(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(nil())] = [8] 9.77/2.85 > [4] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [mark(cons(X1, X2))] = [8] 9.77/2.85 > [4] 9.77/2.85 = [cons(X1, X2)] 9.77/2.85 9.77/2.85 [mark(first(X1, X2))] = [2] X1 + [2] X2 + [0] 9.77/2.85 >= [2] X1 + [2] X2 + [0] 9.77/2.85 = [a__first(mark(X1), mark(X2))] 9.77/2.85 9.77/2.85 [mark(from(X))] = [8] 9.77/2.85 > [4] 9.77/2.85 = [a__from(X)] 9.77/2.85 9.77/2.85 [mark(and(X1, X2))] = [2] X1 + [2] X2 + [8] 9.77/2.85 > [2] X1 + [2] X2 + [4] 9.77/2.85 = [a__and(mark(X1), X2)] 9.77/2.85 9.77/2.85 [mark(if(X1, X2, X3))] = [2] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 >= [2] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 = [a__if(mark(X1), X2, X3)] 9.77/2.85 9.77/2.85 [a__if(X1, X2, X3)] = [1] X1 + [2] X2 + [2] X3 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [1] X3 + [0] 9.77/2.85 = [if(X1, X2, X3)] 9.77/2.85 9.77/2.85 [a__if(true(), X, Y)] = [2] X + [2] Y + [4] 9.77/2.85 > [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__if(false(), X, Y)] = [2] X + [2] Y + [4] 9.77/2.85 > [2] Y + [0] 9.77/2.85 = [mark(Y)] 9.77/2.85 9.77/2.85 [a__add(X1, X2)] = [1] X1 + [2] X2 + [4] 9.77/2.85 >= [1] X1 + [1] X2 + [4] 9.77/2.85 = [add(X1, X2)] 9.77/2.85 9.77/2.85 [a__add(0(), X)] = [2] X + [8] 9.77/2.85 > [2] X + [0] 9.77/2.85 = [mark(X)] 9.77/2.85 9.77/2.85 [a__add(s(X), Y)] = [2] Y + [8] 9.77/2.85 > [4] 9.77/2.85 = [s(add(X, Y))] 9.77/2.85 9.77/2.85 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 9.77/2.85 >= [1] X1 + [1] X2 + [0] 9.77/2.85 = [first(X1, X2)] 9.77/2.85 9.77/2.85 [a__first(0(), X)] = [1] X + [4] 9.77/2.85 >= [4] 9.77/2.85 = [nil()] 9.77/2.85 9.77/2.85 [a__first(s(X), cons(Y, Z))] = [8] 9.77/2.85 > [4] 9.77/2.85 = [cons(Y, first(X, Z))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 >= [4] 9.77/2.85 = [cons(X, from(s(X)))] 9.77/2.85 9.77/2.85 [a__from(X)] = [4] 9.77/2.85 >= [4] 9.77/2.85 = [from(X)] 9.77/2.85 9.77/2.85 9.77/2.85 We return to the main proof. 9.77/2.85 9.77/2.85 We are left with following problem, upon which TcT provides the 9.77/2.85 certificate YES(O(1),O(1)). 9.77/2.85 9.77/2.85 Weak Trs: 9.77/2.85 { a__and(X1, X2) -> and(X1, X2) 9.77/2.85 , a__and(true(), X) -> mark(X) 9.77/2.85 , a__and(false(), Y) -> false() 9.77/2.85 , mark(true()) -> true() 9.77/2.85 , mark(false()) -> false() 9.77/2.85 , mark(0()) -> 0() 9.77/2.85 , mark(s(X)) -> s(X) 9.77/2.85 , mark(add(X1, X2)) -> a__add(mark(X1), X2) 9.77/2.85 , mark(nil()) -> nil() 9.77/2.85 , mark(cons(X1, X2)) -> cons(X1, X2) 9.77/2.85 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 9.77/2.85 , mark(from(X)) -> a__from(X) 9.77/2.85 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 9.77/2.85 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 9.77/2.85 , a__if(X1, X2, X3) -> if(X1, X2, X3) 9.77/2.85 , a__if(true(), X, Y) -> mark(X) 9.77/2.85 , a__if(false(), X, Y) -> mark(Y) 9.77/2.85 , a__add(X1, X2) -> add(X1, X2) 9.77/2.85 , a__add(0(), X) -> mark(X) 9.77/2.85 , a__add(s(X), Y) -> s(add(X, Y)) 9.77/2.85 , a__first(X1, X2) -> first(X1, X2) 9.77/2.85 , a__first(0(), X) -> nil() 9.77/2.85 , a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) 9.77/2.85 , a__from(X) -> cons(X, from(s(X))) 9.77/2.85 , a__from(X) -> from(X) } 9.77/2.85 Obligation: 9.77/2.85 innermost runtime complexity 9.77/2.85 Answer: 9.77/2.85 YES(O(1),O(1)) 9.77/2.85 9.77/2.85 Empty rules are trivially bounded 9.77/2.85 9.77/2.85 Hurray, we answered YES(O(1),O(n^1)) 9.77/2.86 EOF