(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)
Transformed TRS to relative TRS where S is empty.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, first(X, Z))
from(X) → cons(X, from(s(X)))
S is empty.
Rewrite Strategy: INNERMOST
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
cons/0
from/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
and(true, X) → X
and(false, Y) → false
if(true, X, Y) → X
if(false, X, Y) → Y
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Z)) → cons(first(X, Z))
from → cons(from)
S is empty.
Rewrite Strategy: INNERMOST
(5) InfiniteLowerBoundProof (EQUIVALENT transformation)
The loop following loop proves infinite runtime complexity:
The rewrite sequence
from →+ cons(from)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [ ].
The result substitution is [ ].
(6) BOUNDS(INF, INF)