We consider the following Problem:
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)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, and(tt(), X) -> activate(X)
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, sel(N, XS) -> head(afterNth(N, XS))
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(X) -> n__natsFrom(X)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
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)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, and(tt(), X) -> activate(X)
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, sel(N, XS) -> head(afterNth(N, XS))
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(X) -> n__natsFrom(X)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [1]
afterNth(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, sel(N, XS) -> head(afterNth(N, XS))
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(X) -> n__natsFrom(X)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {sel(N, XS) -> head(afterNth(N, XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [1]
afterNth(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
sel(x1, x2) = [1 0] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(X) -> n__natsFrom(X)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [1]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [2]
[0 0] [2]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 0] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, take(N, XS) -> fst(splitAt(N, XS))
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {take(N, XS) -> fst(splitAt(N, XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[1 1] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, afterNth(N, XS) -> snd(splitAt(N, XS))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {afterNth(N, XS) -> snd(splitAt(N, XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[1 1] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [2]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {tail(cons(N, XS)) -> activate(XS)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[1 1] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [2]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 1] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[1 1] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [2]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [0]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [2]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [0 1] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, splitAt(0(), XS) -> pair(nil(), XS)
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {splitAt(0(), XS) -> pair(nil(), XS)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[0 0] [0 0] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [3]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [2]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [3]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [2]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 0] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [2]
0() = [0]
[2]
nil() = [0]
[0]
tail(x1) = [0 1] x1 + [1]
[0 0] [3]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [1]
[0 1] [0 0] [0 0] [0 0] [0]
tt() = [0]
[1]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [2]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [0]
[0 0] [1]
natsFrom(x1) = [1 0] x1 + [1]
[0 0] [3]
n__natsFrom(x1) = [1 0] x1 + [1]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 0] [0]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
0() = [2]
[2]
nil() = [0]
[0]
tail(x1) = [0 1] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ fst(pair(X, Y)) -> X
, head(cons(N, XS)) -> N
, snd(pair(X, Y)) -> Y
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {head(cons(N, XS)) -> N}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [0]
[0 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 1] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
pair(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 1] [1 0] [0]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 1] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 1] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 0] [1]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [3]
0() = [2]
[2]
nil() = [0]
[0]
tail(x1) = [0 1] x1 + [1]
[0 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ fst(pair(X, Y)) -> X
, snd(pair(X, Y)) -> Y
, activate(n__natsFrom(X)) -> natsFrom(X)
, activate(X) -> X}
Weak Trs:
{ head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1 1] x4 + [2]
[0 1] [0 0] [0 0] [0 0] [0]
tt() = [3]
[0]
U12(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[0 1] [0 0] [0]
splitAt(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
pair(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 1] [1 0] [0]
afterNth(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
snd(x1) = [1 0] x1 + [1]
[0 0] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
fst(x1) = [1 0] x1 + [0]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 1] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 1] [2]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [2]
[1 1] [3]
sel(x1, x2) = [1 1] x1 + [1 1] x2 + [3]
[0 0] [0 0] [3]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [0 1] x1 + [1]
[1 0] [1]
take(x1, x2) = [1 1] x1 + [1 1] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ fst(pair(X, Y)) -> X
, snd(pair(X, Y)) -> Y
, activate(n__natsFrom(X)) -> natsFrom(X)}
Weak Trs:
{ activate(X) -> X
, head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {snd(pair(X, Y)) -> Y}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
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] [1]
tt() = [0]
[1]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [0]
splitAt(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
pair(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [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 0] x2 + [2]
[0 0] [0 1] [2]
snd(x1) = [1 0] x1 + [1]
[0 1] [0]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
fst(x1) = [1 0] x1 + [1]
[0 0] [1]
head(x1) = [1 0] x1 + [1]
[0 1] [1]
natsFrom(x1) = [1 0] x1 + [0]
[0 1] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [2]
sel(x1, x2) = [1 1] x1 + [1 0] x2 + [3]
[0 0] [0 1] [3]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 1] [1]
take(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[0 0] [0 0] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ fst(pair(X, Y)) -> X
, activate(n__natsFrom(X)) -> natsFrom(X)}
Weak Trs:
{ snd(pair(X, Y)) -> Y
, activate(X) -> X
, head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {fst(pair(X, Y)) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2}, Uargs(splitAt) = {1, 2},
Uargs(activate) = {}, Uargs(pair) = {1}, Uargs(cons) = {1},
Uargs(afterNth) = {}, Uargs(snd) = {1}, Uargs(and) = {},
Uargs(fst) = {1}, Uargs(head) = {1}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 0] x1 + [1 1] x2 + [1 0] x3 + [1 0] x4 + [1]
[1 1] [0 0] [0 1] [0 1] [1]
tt() = [0]
[0]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
splitAt(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
pair(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[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 + [0]
[0 0] [0 1] [2]
snd(x1) = [1 0] x1 + [0]
[0 1] [1]
and(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
fst(x1) = [1 0] x1 + [1]
[0 1] [0]
head(x1) = [1 0] x1 + [1]
[0 1] [1]
natsFrom(x1) = [1 0] x1 + [2]
[0 1] [0]
n__natsFrom(x1) = [1 0] x1 + [1]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [2]
sel(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[0 0] [0 1] [3]
0() = [0]
[2]
nil() = [0]
[0]
tail(x1) = [1 0] x1 + [1]
[0 1] [1]
take(x1, x2) = [1 1] x1 + [1 0] x2 + [2]
[0 0] [0 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {activate(n__natsFrom(X)) -> natsFrom(X)}
Weak Trs:
{ fst(pair(X, Y)) -> X
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {activate(n__natsFrom(X)) -> natsFrom(X)}
Weak Trs:
{ fst(pair(X, Y)) -> X
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak DPs:
{ fst^#(pair(X, Y)) -> c_2()
, snd^#(pair(X, Y)) -> c_3()
, activate^#(X) -> c_4()
, head^#(cons(N, XS)) -> c_5()
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)), activate(X))
, splitAt^#(0(), XS) -> c_7()
, splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, tail^#(cons(N, XS)) -> activate^#(XS)
, afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
, take^#(N, XS) -> fst^#(splitAt(N, XS))
, natsFrom^#(N) -> c_12()
, natsFrom^#(X) -> c_13()
, sel^#(N, XS) -> head^#(afterNth(N, XS))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, and^#(tt(), X) -> activate^#(X)}
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Strict Trs: {activate(n__natsFrom(X)) -> natsFrom(X)}
Weak DPs:
{ fst^#(pair(X, Y)) -> c_2()
, snd^#(pair(X, Y)) -> c_3()
, activate^#(X) -> c_4()
, head^#(cons(N, XS)) -> c_5()
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)), activate(X))
, splitAt^#(0(), XS) -> c_7()
, splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, tail^#(cons(N, XS)) -> activate^#(XS)
, afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
, take^#(N, XS) -> fst^#(splitAt(N, XS))
, natsFrom^#(N) -> c_12()
, natsFrom^#(X) -> c_13()
, sel^#(N, XS) -> head^#(afterNth(N, XS))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, and^#(tt(), X) -> activate^#(X)}
Weak Trs:
{ fst(pair(X, Y)) -> X
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, head(cons(N, XS)) -> N
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, splitAt(s(N), cons(X, XS)) -> U11(tt(), N, X, activate(XS))
, tail(cons(N, XS)) -> activate(XS)
, afterNth(N, XS) -> snd(splitAt(N, XS))
, take(N, XS) -> fst(splitAt(N, XS))
, natsFrom(N) -> cons(N, n__natsFrom(s(N)))
, natsFrom(X) -> n__natsFrom(X)
, sel(N, XS) -> head(afterNth(N, XS))
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
, and(tt(), X) -> activate(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules: {activate(n__natsFrom(X)) -> natsFrom(X)}
Weak Usable Rules:
{ snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)}
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Strict Trs: {activate(n__natsFrom(X)) -> natsFrom(X)}
Weak DPs:
{ fst^#(pair(X, Y)) -> c_2()
, snd^#(pair(X, Y)) -> c_3()
, activate^#(X) -> c_4()
, head^#(cons(N, XS)) -> c_5()
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)), activate(X))
, splitAt^#(0(), XS) -> c_7()
, splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, tail^#(cons(N, XS)) -> activate^#(XS)
, afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
, take^#(N, XS) -> fst^#(splitAt(N, XS))
, natsFrom^#(N) -> c_12()
, natsFrom^#(X) -> c_13()
, sel^#(N, XS) -> head^#(afterNth(N, XS))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, and^#(tt(), X) -> activate^#(X)}
Weak Trs:
{ snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__natsFrom(X)) -> natsFrom(X)}
Interpretation of constant growth:
----------------------------------
The following argument positions are usable:
Uargs(U11) = {4}, Uargs(U12) = {1, 2},
Uargs(splitAt) = {1, 2}, Uargs(activate) = {},
Uargs(pair) = {1}, Uargs(cons) = {1}, Uargs(afterNth) = {},
Uargs(snd) = {1}, Uargs(and) = {}, Uargs(fst) = {},
Uargs(head) = {}, Uargs(natsFrom) = {},
Uargs(n__natsFrom) = {}, Uargs(s) = {}, Uargs(sel) = {},
Uargs(tail) = {}, Uargs(take) = {}, Uargs(activate^#) = {},
Uargs(natsFrom^#) = {}, Uargs(fst^#) = {1},
Uargs(snd^#) = {1}, Uargs(head^#) = {1}, Uargs(U11^#) = {4},
Uargs(U12^#) = {1, 2}, Uargs(splitAt^#) = {},
Uargs(tail^#) = {}, Uargs(afterNth^#) = {},
Uargs(take^#) = {}, Uargs(sel^#) = {}, Uargs(and^#) = {}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3, x4) = [0 2] x1 + [2 3] x2 + [1 0] x3 + [1 1] x4 + [1]
[0 0] [0 1] [0 0] [1 1] [2]
tt() = [2]
[2]
U12(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 0] [0]
splitAt(x1, x2) = [2 3] x1 + [1 1] x2 + [0]
[0 1] [1 1] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
pair(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
cons(x1, x2) = [1 0] x1 + [0 1] x2 + [0]
[0 0] [1 0] [0]
afterNth(x1, x2) = [2 3] x1 + [2 2] x2 + [0]
[0 2] [2 2] [3]
snd(x1) = [1 0] x1 + [0]
[0 2] [1]
and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
fst(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
n__natsFrom(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[1 1] [3]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
tail(x1) = [0 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
activate^#(x1) = [0 0] x1 + [0]
[0 0] [0]
natsFrom^#(x1) = [0 0] x1 + [0]
[0 0] [0]
fst^#(x1) = [1 0] x1 + [1]
[0 1] [1]
c_2() = [0]
[0]
snd^#(x1) = [1 0] x1 + [1]
[0 2] [1]
c_3() = [0]
[0]
c_4() = [0]
[0]
head^#(x1) = [1 0] x1 + [1]
[1 0] [1]
c_5() = [0]
[0]
U11^#(x1, x2, x3, x4) = [1 0] x1 + [2 3] x2 + [1 0] x3 + [1 1] x4 + [2]
[0 1] [0 0] [0 0] [0 0] [3]
U12^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
splitAt^#(x1, x2) = [0 3] x1 + [1 1] x2 + [1]
[0 3] [0 0] [0]
c_7() = [0]
[0]
tail^#(x1) = [2 0] x1 + [1]
[0 0] [1]
afterNth^#(x1, x2) = [2 3] x1 + [2 2] x2 + [2]
[0 2] [2 2] [3]
take^#(x1, x2) = [2 3] x1 + [2 2] x2 + [2]
[0 2] [2 2] [2]
c_12() = [0]
[0]
c_13() = [0]
[0]
sel^#(x1, x2) = [2 3] x1 + [2 2] x2 + [2]
[2 3] [2 2] [2]
and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak DPs:
{ fst^#(pair(X, Y)) -> c_2()
, snd^#(pair(X, Y)) -> c_3()
, activate^#(X) -> c_4()
, head^#(cons(N, XS)) -> c_5()
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)), activate(X))
, splitAt^#(0(), XS) -> c_7()
, splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, tail^#(cons(N, XS)) -> activate^#(XS)
, afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
, take^#(N, XS) -> fst^#(splitAt(N, XS))
, natsFrom^#(N) -> c_12()
, natsFrom^#(X) -> c_13()
, sel^#(N, XS) -> head^#(afterNth(N, XS))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, and^#(tt(), X) -> activate^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->7:{7} [ YES(O(1),O(1)) ]
->6:{8} [ subsumed ]
|
`->8:{6} [ subsumed ]
|
`->9:{15} [ subsumed ]
|
|->14:{1} [ YES(O(1),O(1)) ]
| |
| |->15:{12} [ YES(O(1),O(1)) ]
| |
| `->16:{13} [ YES(O(1),O(1)) ]
|
`->11:{4} [ YES(O(1),O(1)) ]
->5:{9} [ subsumed ]
|
|->14:{1} [ YES(O(1),O(1)) ]
| |
| |->15:{12} [ YES(O(1),O(1)) ]
| |
| `->16:{13} [ YES(O(1),O(1)) ]
|
`->11:{4} [ YES(O(1),O(1)) ]
->4:{10} [ subsumed ]
|
`->12:{3} [ YES(O(1),O(1)) ]
->3:{11} [ subsumed ]
|
`->13:{2} [ YES(O(1),O(1)) ]
->2:{14} [ subsumed ]
|
`->10:{5} [ YES(O(1),O(1)) ]
->1:{16} [ subsumed ]
|
|->14:{1} [ YES(O(1),O(1)) ]
| |
| |->15:{12} [ YES(O(1),O(1)) ]
| |
| `->16:{13} [ YES(O(1),O(1)) ]
|
`->11:{4} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
WeakDPs DPs:
{ 2: fst^#(pair(X, Y)) -> c_2()
, 3: snd^#(pair(X, Y)) -> c_3()
, 4: activate^#(X) -> c_4()
, 5: head^#(cons(N, XS)) -> c_5()
, 6: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)), activate(X))
, 7: splitAt^#(0(), XS) -> c_7()
, 8: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 9: tail^#(cons(N, XS)) -> activate^#(XS)
, 10: afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
, 11: take^#(N, XS) -> fst^#(splitAt(N, XS))
, 12: natsFrom^#(N) -> c_12()
, 13: natsFrom^#(X) -> c_13()
, 14: sel^#(N, XS) -> head^#(afterNth(N, XS))
, 15: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 16: and^#(tt(), X) -> activate^#(X)}
* Path 7:{7}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{8}: subsumed
--------------------
This path is subsumed by the proof of paths 6:{8}->8:{6}.
* Path 6:{8}->8:{6}: subsumed
---------------------------
This path is subsumed by the proof of paths 6:{8}->8:{6}->9:{15}.
* Path 6:{8}->8:{6}->9:{15}: subsumed
-----------------------------------
This path is subsumed by the proof of paths 6:{8}->8:{6}->9:{15}->14:{1},
6:{8}->8:{6}->9:{15}->11:{4}.
* Path 6:{8}->8:{6}->9:{15}->14:{1}: YES(O(1),O(1))
-------------------------------------------------
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, U12^#(pair(YS, ZS), X) -> activate^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
2: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
-->_1 U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X)) :3
3: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
-->_1 U12^#(pair(YS, ZS), X) -> activate^#(X) :4
4: U12^#(pair(YS, ZS), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :1
together with the congruence-graph
->1:{2} Weak SCC
|
`->2:{3} Weak SCC
|
`->3:{4} Weak SCC
|
`->4:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
WeakDPs DPs:
{ 2: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 3: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 4: U12^#(pair(YS, ZS), X) -> activate^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 2: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 3: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 4: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{8}->8:{6}->9:{15}->14:{1}->15:{12}: YES(O(1),O(1))
----------------------------------------------------------
We consider the following Problem:
Weak DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
-->_1 U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X)) :2
2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
-->_1 U12^#(pair(YS, ZS), X) -> activate^#(X) :3
3: U12^#(pair(YS, ZS), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :4
4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
|
`->3:{3} Weak SCC
|
`->4:{4} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{8}->8:{6}->9:{15}->14:{1}->16:{13}: YES(O(1),O(1))
----------------------------------------------------------
We consider the following Problem:
Weak DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, U12^#(pair(YS, ZS), X) -> activate^#(X)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
-->_1 U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X)) :2
2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
-->_1 U12^#(pair(YS, ZS), X) -> activate^#(X) :3
3: U12^#(pair(YS, ZS), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :4
4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
|
`->3:{3} Weak SCC
|
`->4:{4} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)
, 4: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 6:{8}->8:{6}->9:{15}->11:{4}: YES(O(1),O(1))
-------------------------------------------------
We consider the following Problem:
Weak DPs:
{ splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, U12^#(pair(YS, ZS), X) -> activate^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
-->_1 U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X)) :2
2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
-->_1 U12^#(pair(YS, ZS), X) -> activate^#(X) :3
3: U12^#(pair(YS, ZS), X) -> activate^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
|
`->3:{3} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: splitAt^#(s(N), cons(X, XS)) ->
U11^#(tt(), N, X, activate(XS))
, 2: U11^#(tt(), N, X, XS) ->
U12^#(splitAt(activate(N), activate(XS)),
activate(X))
, 3: U12^#(pair(YS, ZS), X) -> activate^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{9}: subsumed
--------------------
This path is subsumed by the proof of paths 5:{9}->14:{1},
5:{9}->11:{4}.
* Path 5:{9}->14:{1}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak DPs: {tail^#(cons(N, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
2: tail^#(cons(N, XS)) -> activate^#(XS)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :1
together with the congruence-graph
->1:{2} Weak SCC
|
`->2:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
WeakDPs DPs:
{2: tail^#(cons(N, XS)) -> activate^#(XS)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 2: tail^#(cons(N, XS)) -> activate^#(XS)
, 1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{9}->14:{1}->15:{12}: YES(O(1),O(1))
-------------------------------------------
We consider the following Problem:
Weak DPs:
{ tail^#(cons(N, XS)) -> activate^#(XS)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: tail^#(cons(N, XS)) -> activate^#(XS)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :2
2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: tail^#(cons(N, XS)) -> activate^#(XS)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: tail^#(cons(N, XS)) -> activate^#(XS)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{9}->14:{1}->16:{13}: YES(O(1),O(1))
-------------------------------------------
We consider the following Problem:
Weak DPs:
{ tail^#(cons(N, XS)) -> activate^#(XS)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: tail^#(cons(N, XS)) -> activate^#(XS)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :2
2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: tail^#(cons(N, XS)) -> activate^#(XS)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: tail^#(cons(N, XS)) -> activate^#(XS)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{9}->11:{4}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Weak DPs: {tail^#(cons(N, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: tail^#(cons(N, XS)) -> activate^#(XS)
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: tail^#(cons(N, XS)) -> activate^#(XS)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: tail^#(cons(N, XS)) -> activate^#(XS)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 4:{10}: subsumed
---------------------
This path is subsumed by the proof of paths 4:{10}->12:{3}.
* Path 4:{10}->12:{3}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs: {afterNth^#(N, XS) -> snd^#(splitAt(N, XS))}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: afterNth^#(N, XS) -> snd^#(splitAt(N, XS))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: afterNth^#(N, XS) -> snd^#(splitAt(N, XS))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: afterNth^#(N, XS) -> snd^#(splitAt(N, XS))}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 3:{11}: subsumed
---------------------
This path is subsumed by the proof of paths 3:{11}->13:{2}.
* Path 3:{11}->13:{2}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs: {take^#(N, XS) -> fst^#(splitAt(N, XS))}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: take^#(N, XS) -> fst^#(splitAt(N, XS))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: take^#(N, XS) -> fst^#(splitAt(N, XS))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: take^#(N, XS) -> fst^#(splitAt(N, XS))}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{14}: subsumed
---------------------
This path is subsumed by the proof of paths 2:{14}->10:{5}.
* Path 2:{14}->10:{5}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs: {sel^#(N, XS) -> head^#(afterNth(N, XS))}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: sel^#(N, XS) -> head^#(afterNth(N, XS))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: sel^#(N, XS) -> head^#(afterNth(N, XS))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: sel^#(N, XS) -> head^#(afterNth(N, XS))}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{16}: subsumed
---------------------
This path is subsumed by the proof of paths 1:{16}->14:{1},
1:{16}->11:{4}.
* Path 1:{16}->14:{1}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Strict DPs: {activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak DPs: {and^#(tt(), X) -> activate^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
2: and^#(tt(), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :1
together with the congruence-graph
->1:{2} Weak SCC
|
`->2:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
WeakDPs DPs:
{2: and^#(tt(), X) -> activate^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 2: and^#(tt(), X) -> activate^#(X)
, 1: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{16}->14:{1}->15:{12}: YES(O(1),O(1))
--------------------------------------------
We consider the following Problem:
Weak DPs:
{ and^#(tt(), X) -> activate^#(X)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: and^#(tt(), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :2
2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: and^#(tt(), X) -> activate^#(X)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: and^#(tt(), X) -> activate^#(X)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{16}->14:{1}->16:{13}: YES(O(1),O(1))
--------------------------------------------
We consider the following Problem:
Weak DPs:
{ and^#(tt(), X) -> activate^#(X)
, activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: and^#(tt(), X) -> activate^#(X)
-->_1 activate^#(n__natsFrom(X)) -> natsFrom^#(X) :2
2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)
together with the congruence-graph
->1:{1} Weak SCC
|
`->2:{2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: and^#(tt(), X) -> activate^#(X)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: and^#(tt(), X) -> activate^#(X)
, 2: activate^#(n__natsFrom(X)) -> natsFrom^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{16}->11:{4}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs: {and^#(tt(), X) -> activate^#(X)}
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)), activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: and^#(tt(), X) -> activate^#(X)
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: and^#(tt(), X) -> activate^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: and^#(tt(), X) -> activate^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__natsFrom(X)) -> natsFrom(X)
, snd(pair(X, Y)) -> Y
, activate(X) -> X
, U11(tt(), N, X, XS) ->
U12(splitAt(activate(N), activate(XS)),
activate(X))
, splitAt(0(), XS) -> pair(nil(), XS)
, 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)
, U12(pair(YS, ZS), X) ->
pair(cons(activate(X), YS), ZS)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))