WORST_CASE(?,O(1))
proof of input_evTyBZIvp1.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(0) CpxTRS
(1) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
(2) BOUNDS(1, 1)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) NarrowingOnBasicTermsTerminatesProof (FINISHED)
Constant runtime complexity proven by termination of constructor-based narrowing.

The maximal most general narrowing sequences give rise to the following rewrite sequences:

activate(n__from(x0)) ->^* n__from(x0)

activate(n__from(x0)) ->^* cons(x0, n__from(s(x0)))

from(x0) ->^* n__from(x0)

from(x0) ->^* cons(x0, n__from(s(x0)))

2nd(cons(x0, x1)) ->^* 2nd(cons1(x0, x1))

2nd(cons(x0, n__from(x1))) ->^* 2nd(cons1(x0, n__from(x1)))

2nd(cons(x0, n__from(x1))) ->^* 2nd(cons1(x0, cons(x1, n__from(s(x1)))))




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(2)
BOUNDS(1, 1)