We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, and(tt(), X) -> activate(X)
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, and(tt(), X) -> activate(X)
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
1] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[snd](x1) = [1 1] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [1]
[0 0] [0]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 1] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 0] [0 0] [0 0] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 1] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [1 1] x1 + [7]
[0 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [0]
[snd^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [0 1] x1 + [1 1] x2 + [2]
[1 0] [0 0] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_13](x1) = [1 1] x1 + [0]
[0 0] [0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 1] x1 + [0]
[0 0] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 1] XS + [0]
[0 0] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [1]
[0 0] [0]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
>= [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 1] X + [1 1] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [1]
[0 0] [0]
? [1 0] N + [0]
[1 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [1]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 1] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 1] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 1] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 1] X + [0]
[0 0] [0]
? [1 1] X + [7]
[0 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 1] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [1 1] N + [7]
[0 0] [0]
> [1 1] N + [0]
[0 0] [0]
= [c_12(N, N)]
[natsFrom^#(X)] = [1 1] X + [7]
[0 0] [0]
> [1 1] X + [0]
[0 0] [0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 1] X + [1 1] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 1] X + [2]
[0 0] [5]
> [1 1] X + [0]
[0 0] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 1] X + [1 1] Y + [0]
[0 0] [0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 1] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 1] XS + [0]
[0 0] [0 0] [5]
>= [1 1] XS + [0]
[0 0] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, tail^#(cons(N, XS)) -> c_15(activate^#(XS)) }
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Weak DPs:
{ natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [1 0] x1 + [1 1] x2 + [1 0] x3 + [1
1] x4 + [1]
[0 1] [0 0] [0 1] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[snd](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [1]
[0 0] [3]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 1] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [1
1] x4 + [0]
[0 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [1 1] x1 + [1]
[0 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 1] x2 + [5]
[0 1] [0 0] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [1 0] x2 + [0]
[0 0] [0]
[c_13](x1) = [1 1] x1 + [0]
[0 0] [0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 1] x1 + [2]
[0 0] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 1] XS + [1]
[0 0] [0 1] [0 0] [0]
> [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 1] [0]
>= [1 0] X + [1 1] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
? [1 1] N + [1 0] X + [1 0] XS + [1]
[0 0] [0 1] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
? [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [1]
[0 0] [3]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
>= [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 1] X + [1 0] Y + [0]
[0 0] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [1]
[0 0] [3]
? [1 0] N + [0]
[1 2] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [1]
[0 0] [3]
? [1 0] X + [0]
[0 1] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 1] [0]
? [1 1] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 1] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 1] X + [0]
[0 0] [0]
? [1 1] X + [1]
[0 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 1] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [1 1] N + [1]
[0 0] [0]
> [1 0] N + [0]
[0 0] [0]
= [c_12(N, N)]
[natsFrom^#(X)] = [1 1] X + [1]
[0 0] [0]
> [1 1] X + [0]
[0 0] [0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 1] X + [1 0] Y + [0]
[0 0] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 1] X + [5]
[0 0] [4]
> [1 1] X + [0]
[0 0] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 1] X + [1 0] Y + [0]
[0 0] [0 1] [0]
? [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 1] [0 1] [0]
>= [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 1] N + [1 1] XS + [2]
[0 0] [0 0] [4]
> [1 1] XS + [0]
[0 0] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Weak DPs:
{ natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [1
1] x4 + [0]
[1 0] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 0] [4]
[snd](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [1]
[0 0] [2]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 0] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [1]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [1]
[0 1] [0]
[natsFrom^#](x1) = [0 0] x1 + [0]
[1 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 0] x2 + [5]
[0 0] [0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [4]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [1 0] XS + [0]
[0 0] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [1]
[0 0] [2]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [1]
[0 0] [2]
> [1 0] N + [0]
[0 0] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [1]
[0 0] [2]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [1]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [1]
[0 1] [0]
> [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [1]
[0 0] [0]
? [0 0] X + [1]
[1 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [1 0] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [0 0] N + [0]
[1 0] [0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0 0] X + [0]
[1 0] [0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [4]
> [1 0] X + [1]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 0] XS + [1]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ activate^#(X) -> c_5(X)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
1] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [4]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 0] [4]
[snd](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [5]
[0 0] [6]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 0] [0 0] [0 0] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [1]
[0 1] [0]
[natsFrom^#](x1) = [0 0] x1 + [0]
[1 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [0]
[snd^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 0] x2 + [5]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [1]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [4]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 1] YS + [1 0] ZS + [4]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 0] [4]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
? [1 0] XS + [0]
[0 0] [4]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [5]
[0 0] [6]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [5]
[0 0] [6]
> [1 0] N + [0]
[0 0] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [5]
[0 0] [6]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 1] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [0 0] X + [1]
[1 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [1 0] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [0 0] N + [0]
[1 0] [0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0 0] X + [0]
[1 0] [0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [0]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [1]
[0 1] [0 0] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 0] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ activate^#(X) -> c_5(X)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 1] x3 + [1
0] x4 + [0]
[0 0] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [4]
[0 0] [0 0] [4]
[snd](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 0] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1
0] x4 + [0]
[0 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0 0] x1 + [1]
[1 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [0 0] x1 + [1 0] x2 + [5]
[1 0] [0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [5]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 0] N + [1 1] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 0] [1]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 0] N + [1 1] X + [1 1] XS + [1]
[0 0] [0 0] [0 0] [0]
> [1 0] N + [1 1] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 0] XS + [0]
[0 0] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [4]
> [1 0] N + [0]
[0 1] [1]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [0 0] X + [1]
[1 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [1 0] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [0 0] N + [1]
[1 0] [0]
> [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0 0] X + [1]
[1 0] [0]
> [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 0] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 1] [0 1] [1]
>= [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [5]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ activate^#(X) -> c_5(X)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 1] x3 + [1
1] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 1] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [4]
[0 0] [0 0] [4]
[snd](x1) = [1 1] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 1] x3 + [1
1] x4 + [1]
[0 0] [0 1] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0 0] x1 + [0]
[1 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [1 0] x1 + [1]
[0 1] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 1] [0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0 0] x1 + [0]
[0 1] [0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [5]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 1] ZS + [0]
[0 0] [0 1] [0 0] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 1] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 1] XS + [0]
[0 0] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 1] X + [1 1] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [4]
> [1 0] N + [0]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 1] X + [1 1] XS + [1]
[0 1] [0 0] [0 0] [0]
> [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 1] ZS + [0]
[0 1] [0 1] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [1]
[0 1] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] N + [1 1] X + [1 0] XS + [1]
[0 1] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
? [0 0] XS + [0]
[1 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [1 0] N + [1]
[0 1] [0]
> [0 0] N + [0]
[0 1] [0]
= [c_12(N, N)]
[natsFrom^#(X)] = [1 0] X + [1]
[0 1] [0]
> [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 1] Y + [0]
[0 1] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [4]
[0 1] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 1] Y + [0]
[0 1] [0 0] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [0]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [5]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, snd^#(pair(X, Y)) -> c_8(Y)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, activate^#(X) -> c_5(X)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 1] x3 + [1
1] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [1]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [4]
[0 0] [0 0] [0]
[snd](x1) = [1 1] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [4]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [1
1] x4 + [0]
[0 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 1] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0 0] x1 + [1]
[1 1] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [4]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 1] YS + [1 1] ZS + [3]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [2]
[0 1] [0 1] [0 1] [1]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
? [1 0] XS + [2]
[0 1] [1]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [4]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [0]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 1] X + [1 1] Y + [3]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [4]
[0 1] [4]
> [1 0] N + [0]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [4]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 1] YS + [1 1] ZS + [3]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [0 0] X + [1]
[1 1] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [1 1] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [0 0] N + [1]
[1 1] [0]
> [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0 0] X + [1]
[1 1] [0]
> [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [2]
[0 1] [0 1] [1]
> [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [4]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [2]
[0 1] [0 1] [1]
> [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, activate^#(X) -> c_5(X)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1
0] x4 + [0]
[0 1] [0 0] [0 1] [0
1] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [4]
[0 0] [0 0] [4]
[snd](x1) = [1 1] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 0] [0]
[0] = [7]
[1]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 0] x2 + [1 1] x3 + [1
1] x4 + [0]
[1 0] [0 1] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [1]
[0 1] [0]
[natsFrom^#](x1) = [0 0] x1 + [0]
[1 0] [0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 1] x1 + [1 0] x2 + [5]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0 0] x1 + [0]
[1 0] [0]
[c_13](x1) = [0 0] x1 + [0]
[1 0] [0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [5]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [8]
[0 1] [0]
> [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
> [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 1] X + [1 1] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [4]
> [1 0] N + [0]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 1] X + [1 1] XS + [0]
[0 1] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 1] X + [1 1] YS + [1 1] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [0 0] X + [1]
[1 0] [0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 0] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] N + [1 1] X + [1 0] XS + [0]
[0 1] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [8]
[0 0] [0]
> [1 1] XS + [0]
[0 0] [0]
= [c_4(XS)]
[natsFrom^#(N)] = [0 0] N + [0]
[1 0] [0]
>= [0 0] N + [0]
[1 0] [0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0 0] X + [0]
[1 0] [0]
>= [0 0] X + [0]
[1 0] [0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 1] [0 1] [0]
>= [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [5]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, activate^#(X) -> c_5(X)
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [4]
[snd](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [6]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 1] x2 + [1 1] x3 + [1
1] x4 + [0]
[1 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [1]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 1] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [6]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [4]
>= [1 1] N + [1 0] XS + [0]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [6]
> [1 0] N + [0]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [6]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [1]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [1]
[0 1] [0]
> [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [1]
[0 0] [0]
> [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 1] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [4]
[0 1] [5]
> [1 0] X + [1]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 1] N + [1 1] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [4]
[0 0] [0 0] [0]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [5]
> [1 0] XS + [1]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [1
1] x4 + [0]
[1 0] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 0] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[snd](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 0] [0]
[n__natsFrom](x1) = [1 0] x1 + [4]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 0] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1
0] x4 + [1]
[0 0] [0 0] [0 0] [0
0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 1] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 1] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [4]
[0 0] [0]
? [1 0] X + [7]
[0 0] [0]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 0] [0]
> [1 0] N + [4]
[0 0] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 0] [0]
> [1 0] X + [4]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 0] N + [1 0] X + [1 0] XS + [1]
[0 0] [0 0] [0 0] [0]
>= [1 0] N + [1 0] X + [1 0] XS + [1]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [1]
[0 1] [0 1] [0 1] [0]
> [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [4]
[0 0] [0]
> [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 0] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] N + [1 0] X + [1 0] XS + [1]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [4]
[0 1] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [0]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[0 0] [0 0] [0 0] [0
1] [0]
[tt] = [5]
[3]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [5]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [1]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[afterNth](x1, x2) = [1 1] x1 + [1 1] x2 + [6]
[0 0] [0 0] [0]
[snd](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom](x1) = [1 0] x1 + [4]
[0 0] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[4]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[0 0] [0 0] [0 1] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 1] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 1] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [4]
[tail^#](x1) = [1 0] x1 + [1]
[0 1] [1]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [8]
[0 0] [0 0] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [8]
[0 0] [0 0] [0 1] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 1] [0 1] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [5]
[0 0] [0 1] [0 1] [4]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 0] [0 1] [4]
>= [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 0] [0 1] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [9]
[0 1] [0]
> [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [1]
[0 1] [0]
> [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [1]
[0 0] [0]
? [1 0] X + [4]
[0 0] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 1] XS + [6]
[0 0] [0 0] [0]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 0] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 0] [0 0] [0]
? [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [4]
[0 0] [4]
>= [1 0] N + [4]
[0 0] [4]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [4]
[0 0] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [8]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [8]
[0 0] [0 1] [0 1] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
>= [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 1] XS + [8]
[0 0] [0 0] [0 0] [0]
? [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 1] [0 1] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [4]
[0 0] [0]
> [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [9]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 1] XS + [8]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 1] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 1] XS + [6]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [5]
[0 0] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, afterNth(N, XS) -> snd(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[snd](x1) = [1 0] x1 + [4]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [4]
[0 1] [7]
[n__natsFrom](x1) = [1 0] x1 + [1]
[0 0] [4]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 0] [0 0] [0 1] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x2 + [5]
[0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 0] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [4]
[tail^#](x1) = [1 0] x1 + [5]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [1]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [1]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [1]
[0 0] [4]
? [1 0] X + [4]
[0 1] [7]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
>= [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [4]
[0 1] [0 1] [0]
> [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [4]
[0 1] [7]
> [1 0] N + [1]
[0 1] [5]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [4]
[0 1] [7]
> [1 0] X + [1]
[0 0] [4]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [1]
[0 0] [4]
> [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 1] X + [1 1] XS + [1]
[0 0] [0 0] [0 0] [0]
? [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 1] XS + [0]
[0 0] [0]
>= [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
? [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 0] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [5]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [0]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, head^#(cons(N, XS)) -> c_11(N) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(n__natsFrom(X)) -> natsFrom(X) }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[snd](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [4]
[0 1] [7]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [1 1] x2 + [1 0] x3 + [1 0] x4 + [0]
[0 0] [0 1] [0 1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [4]
[0 1] [0]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x2 + [5]
[0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [1]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [4]
[0 1] [7]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [4]
[0 1] [7]
> [1 0] N + [0]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [4]
[0 1] [7]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
>= [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
? [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
>= [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [0]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [4]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [1]
[0 1] [0 1] [0]
> [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [0]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS))) }
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(n__natsFrom(X)) -> natsFrom(X) }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 1] [0 0] [0 0] [0
0] [0]
[tt] = [0]
[0]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [4]
[0 0] [0 1] [4]
[snd](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [0 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [0]
[1 0] [0 0] [0 1] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [0]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 0] x2 + [5]
[0 0] [0 1] [5]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [0]
[0 1] [5]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [4]
[0 1] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] X + [1 0] YS + [1 0] ZS + [4]
[0 1] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [4]
[0 0] [0 1] [0 1] [0]
> [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [1 0] XS + [0]
[0 1] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [0]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [4]
> [1 0] N + [4]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
>= [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
>= [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
>= [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [4]
[0 0] [0 1] [0 1] [0]
> [1 1] N + [1 0] X + [1 0] XS + [0]
[0 0] [0 1] [0 1] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [0]
[0 1] [0]
>= [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [0]
[0 0] [0 1] [0]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [5]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [0]
[0 1] [0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [0]
> [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [5]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [4]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, activate(n__natsFrom(X)) -> natsFrom(X) }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [5]
[0 0] [0 0] [0 1] [0
1] [4]
[tt] = [1]
[3]
[U12](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 1] [0 1] [0]
[splitAt](x1, x2) = [1 1] x1 + [1 0] x2 + [5]
[0 0] [0 1] [4]
[activate](x1) = [1 0] x1 + [0]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[afterNth](x1, x2) = [1 1] x1 + [1 0] x2 + [6]
[0 0] [0 1] [4]
[snd](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom](x1) = [1 0] x1 + [7]
[0 1] [4]
[n__natsFrom](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0 0] x1 + [1]
[1 1] [3]
[0] = [0]
[0]
[nil] = [0]
[1]
[U11^#](x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1
0] x4 + [5]
[1 1] [0 0] [0 1] [0
1] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 1] [0 1] [0 1] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[splitAt^#](x1, x2) = [0 1] x1 + [1 0] x2 + [4]
[1 0] [0 1] [4]
[c_3](x1) = [1 0] x1 + [1]
[0 1] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [0]
[snd^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [0 1] x1 + [1 0] x2 + [4]
[1 0] [0 1] [4]
[c_9](x1) = [1 0] x1 + [0]
[0 1] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 1] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 1] [0]
[tail^#](x1) = [1 0] x1 + [4]
[0 1] [4]
[c_15](x1) = [1 0] x1 + [0]
[0 1] [0]
[take^#](x1, x2) = [1 1] x1 + [1 0] x2 + [7]
[0 0] [0 1] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 1] [0 1] [4]
>= [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 1] [0 1] [4]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [6]
[0 1] [0 1] [0 1] [0]
> [1 0] X + [1 0] YS + [1 0] ZS + [2]
[0 1] [0 1] [0 1] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 1] [0 1] [4]
>= [1 1] N + [1 0] X + [1 0] XS + [9]
[0 0] [0 1] [0 1] [4]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 0] XS + [5]
[0 1] [4]
> [1 0] XS + [2]
[0 1] [1]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
? [1 0] X + [7]
[0 1] [4]
= [natsFrom(X)]
[afterNth(N, XS)] = [1 1] N + [1 0] XS + [6]
[0 0] [0 1] [4]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 0] Y + [2]
[0 1] [0 1] [0]
> [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [7]
[0 1] [4]
> [1 0] N + [1]
[0 1] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [7]
[0 1] [4]
> [1 0] X + [0]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [1 1] N + [1 0] X + [1 0] XS + [5]
[0 0] [0 1] [0 1] [4]
>= [1 1] N + [1 0] X + [1 0] XS + [5]
[0 0] [0 1] [0 1] [4]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 0] ZS + [2]
[0 1] [0 1] [0 1] [0]
> [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 1] [0 1] [0 1] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [0]
[0 0] [0]
>= [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [1 1] N + [1 0] X + [1 0] XS + [7]
[0 0] [0 1] [0 1] [5]
> [1 1] N + [1 0] X + [1 0] XS + [6]
[0 0] [0 1] [0 1] [4]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [1 0] XS + [4]
[0 1] [4]
> [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 0] Y + [2]
[0 1] [0 1] [0]
> [1 0] Y + [0]
[0 1] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [1 0] X + [7]
[0 1] [5]
> [1 0] X + [0]
[0 1] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 0] Y + [2]
[0 1] [0 1] [0]
> [1 0] X + [0]
[0 1] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 1] [0 1] [0]
>= [1 0] N + [0]
[0 1] [0]
= [c_11(N)]
[sel^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [6]
[0 0] [0 1] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [1 0] N + [1 0] XS + [4]
[0 1] [0 1] [4]
> [1 0] XS + [0]
[0 1] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [1 1] N + [1 0] XS + [7]
[0 0] [0 1] [7]
> [1 1] N + [1 0] XS + [5]
[0 0] [0 1] [4]
= [c_16(fst^#(splitAt(N, XS)))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs: { activate(n__natsFrom(X)) -> natsFrom(X) }
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.
Trs: { activate(n__natsFrom(X)) -> natsFrom(X) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(U11) = {3, 4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {1}, Uargs(pair) = {1, 2}, Uargs(cons) = {1, 2},
Uargs(snd) = {1}, Uargs(U11^#) = {4}, Uargs(c_1) = {1},
Uargs(U12^#) = {1, 2}, Uargs(c_2) = {1, 2, 3},
Uargs(activate^#) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(snd^#) = {1},
Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(fst^#) = {1},
Uargs(c_10) = {1}, Uargs(head^#) = {1}, Uargs(c_11) = {1},
Uargs(c_14) = {1}, Uargs(c_15) = {1}, Uargs(c_16) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[U11](x1, x2, x3, x4) = [7 1] x1 + [4 4] x2 + [1 0] x3 + [1
4] x4 + [1]
[0 0] [0 0] [0 0] [0
0] [0]
[tt] = [1]
[1]
[U12](x1, x2) = [1 5] x1 + [1 0] x2 + [1]
[0 1] [0 0] [0]
[splitAt](x1, x2) = [1 4] x1 + [1 4] x2 + [5]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [1]
[0 1] [0]
[pair](x1, x2) = [1 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
[afterNth](x1, x2) = [4 4] x1 + [4 7] x2 + [6]
[4 6] [4 6] [5]
[snd](x1) = [1 0] x1 + [0]
[1 0] [0]
[natsFrom](x1) = [1 0] x1 + [2]
[0 0] [0]
[n__natsFrom](x1) = [1 0] x1 + [2]
[0 0] [0]
[s](x1) = [0 0] x1 + [0]
[1 1] [2]
[0] = [2]
[2]
[nil] = [0]
[0]
[U11^#](x1, x2, x3, x4) = [4 2] x1 + [2 4] x2 + [3 0] x3 + [2
4] x4 + [7]
[0 0] [4 2] [0 4] [4
0] [0]
[c_1](x1) = [1 1] x1 + [0]
[0 0] [0]
[U12^#](x1, x2) = [1 0] x1 + [1 0] x2 + [4]
[0 1] [0 0] [0]
[c_2](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
[activate^#](x1) = [1 0] x1 + [0]
[0 2] [0]
[splitAt^#](x1, x2) = [0 4] x1 + [3 4] x2 + [7]
[0 0] [0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_4](x1) = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_6](x1) = [1 0] x1 + [0]
[0 0] [0]
[natsFrom^#](x1) = [0]
[0]
[afterNth^#](x1, x2) = [7 7] x1 + [7 7] x2 + [7]
[7 7] [7 7] [7]
[c_7](x1) = [1 0] x1 + [0]
[0 0] [4]
[snd^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_8](x1) = [1 0] x1 + [0]
[0 0] [0]
[and^#](x1, x2) = [7 7] x2 + [5]
[7 7] [4]
[c_9](x1) = [4 0] x1 + [0]
[0 0] [0]
[fst^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_10](x1) = [1 0] x1 + [0]
[0 0] [0]
[head^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_11](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_12](x1, x2) = [0]
[0]
[c_13](x1) = [0]
[0]
[sel^#](x1, x2) = [7 7] x1 + [7 7] x2 + [7]
[7 7] [7 7] [7]
[c_14](x1) = [1 0] x1 + [0]
[0 0] [4]
[tail^#](x1) = [4 0] x1 + [7]
[0 0] [5]
[c_15](x1) = [4 0] x1 + [0]
[0 0] [0]
[take^#](x1, x2) = [7 7] x1 + [7 7] x2 + [7]
[7 7] [7 7] [7]
[c_16](x1) = [1 0] x1 + [0]
[0 0] [4]
The order satisfies the following ordering constraints:
[U11(tt(), N, X, XS)] = [4 4] N + [1 0] X + [1 4] XS + [9]
[0 0] [0 0] [0 0] [0]
>= [1 4] N + [1 0] X + [1 4] XS + [9]
[0 0] [0 0] [0 0] [0]
= [U12(splitAt(activate(N), activate(XS)), activate(X))]
[U12(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 1] ZS + [3]
[0 0] [0 0] [0 0] [0]
>= [1 0] X + [1 0] YS + [1 1] ZS + [3]
[0 0] [0 0] [0 0] [0]
= [pair(cons(activate(X), YS), ZS)]
[splitAt(s(N), cons(X, XS))] = [4 4] N + [1 0] X + [1 4] XS + [13]
[0 0] [0 0] [0 0] [0]
> [4 4] N + [1 0] X + [1 4] XS + [10]
[0 0] [0 0] [0 0] [0]
= [U11(tt(), N, X, activate(XS))]
[splitAt(0(), XS)] = [1 4] XS + [15]
[0 0] [0]
> [1 1] XS + [2]
[0 0] [0]
= [pair(nil(), XS)]
[activate(X)] = [1 0] X + [1]
[0 1] [0]
> [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__natsFrom(X))] = [1 0] X + [3]
[0 0] [0]
> [1 0] X + [2]
[0 0] [0]
= [natsFrom(X)]
[afterNth(N, XS)] = [4 4] N + [4 7] XS + [6]
[4 6] [4 6] [5]
> [1 4] N + [1 4] XS + [5]
[1 4] [1 4] [5]
= [snd(splitAt(N, XS))]
[snd(pair(X, Y))] = [1 0] X + [1 1] Y + [2]
[1 0] [1 1] [2]
> [1 0] Y + [0]
[0 1] [0]
= [Y]
[natsFrom(N)] = [1 0] N + [2]
[0 0] [0]
>= [1 0] N + [2]
[0 0] [0]
= [cons(N, n__natsFrom(s(N)))]
[natsFrom(X)] = [1 0] X + [2]
[0 0] [0]
>= [1 0] X + [2]
[0 0] [0]
= [n__natsFrom(X)]
[U11^#(tt(), N, X, XS)] = [2 4] N + [3 0] X + [2 4] XS + [13]
[4 2] [0 4] [4 0] [0]
> [1 4] N + [1 0] X + [1 4] XS + [12]
[0 0] [0 0] [0 0] [0]
= [c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))]
[U12^#(pair(YS, ZS), X)] = [1 0] X + [1 0] YS + [1 1] ZS + [6]
[0 0] [0 0] [0 0] [0]
> [1 0] X + [1 0] YS + [1 0] ZS + [0]
[0 0] [0 0] [0 0] [0]
= [c_2(activate^#(X), YS, ZS)]
[activate^#(X)] = [1 0] X + [0]
[0 2] [0]
>= [1 0] X + [0]
[0 0] [0]
= [c_5(X)]
[activate^#(n__natsFrom(X))] = [1 0] X + [2]
[0 0] [0]
> [0]
[0]
= [c_6(natsFrom^#(X))]
[splitAt^#(s(N), cons(X, XS))] = [4 4] N + [3 0] X + [3 4] XS + [15]
[0 0] [0 0] [0 0] [0]
>= [2 4] N + [3 0] X + [2 4] XS + [15]
[0 0] [0 0] [0 0] [0]
= [c_3(U11^#(tt(), N, X, activate(XS)))]
[splitAt^#(0(), XS)] = [3 4] XS + [15]
[0 0] [0]
> [0]
[0]
= [c_4(XS)]
[natsFrom^#(N)] = [0]
[0]
>= [0]
[0]
= [c_12(N, N)]
[natsFrom^#(X)] = [0]
[0]
>= [0]
[0]
= [c_13(X)]
[afterNth^#(N, XS)] = [7 7] N + [7 7] XS + [7]
[7 7] [7 7] [7]
> [1 4] N + [1 4] XS + [5]
[0 0] [0 0] [4]
= [c_7(snd^#(splitAt(N, XS)))]
[snd^#(pair(X, Y))] = [1 0] X + [1 1] Y + [2]
[0 0] [0 0] [0]
> [1 0] Y + [0]
[0 0] [0]
= [c_8(Y)]
[and^#(tt(), X)] = [7 7] X + [5]
[7 7] [4]
> [4 0] X + [0]
[0 0] [0]
= [c_9(activate^#(X))]
[fst^#(pair(X, Y))] = [1 0] X + [1 1] Y + [2]
[0 0] [0 0] [0]
> [1 0] X + [0]
[0 0] [0]
= [c_10(X)]
[head^#(cons(N, XS))] = [1 0] N + [1 0] XS + [0]
[0 0] [0 0] [0]
>= [1 0] N + [0]
[0 0] [0]
= [c_11(N)]
[sel^#(N, XS)] = [7 7] N + [7 7] XS + [7]
[7 7] [7 7] [7]
> [4 4] N + [4 7] XS + [6]
[0 0] [0 0] [4]
= [c_14(head^#(afterNth(N, XS)))]
[tail^#(cons(N, XS))] = [4 0] N + [4 0] XS + [7]
[0 0] [0 0] [5]
> [4 0] XS + [0]
[0 0] [0]
= [c_15(activate^#(XS))]
[take^#(N, XS)] = [7 7] N + [7 7] XS + [7]
[7 7] [7 7] [7]
> [1 4] N + [1 4] XS + [5]
[0 0] [0 0] [4]
= [c_16(fst^#(splitAt(N, XS)))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ U11^#(tt(), N, X, XS) ->
c_1(U12^#(splitAt(activate(N), activate(XS)), activate(X)))
, U12^#(pair(YS, ZS), X) -> c_2(activate^#(X), YS, ZS)
, activate^#(X) -> c_5(X)
, activate^#(n__natsFrom(X)) -> c_6(natsFrom^#(X))
, splitAt^#(s(N), cons(X, XS)) ->
c_3(U11^#(tt(), N, X, activate(XS)))
, splitAt^#(0(), XS) -> c_4(XS)
, natsFrom^#(N) -> c_12(N, N)
, natsFrom^#(X) -> c_13(X)
, afterNth^#(N, XS) -> c_7(snd^#(splitAt(N, XS)))
, snd^#(pair(X, Y)) -> c_8(Y)
, and^#(tt(), X) -> c_9(activate^#(X))
, fst^#(pair(X, Y)) -> c_10(X)
, head^#(cons(N, XS)) -> c_11(N)
, sel^#(N, XS) -> c_14(head^#(afterNth(N, XS)))
, tail^#(cons(N, XS)) -> c_15(activate^#(XS))
, take^#(N, XS) -> c_16(fst^#(splitAt(N, XS))) }
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(X) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, snd(pair(X, Y)) -> Y
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))