WORST_CASE(?,O(1))
proof of input_teri5LkmBl.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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:

   f(X) -> if(X, c, n__f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> activate(Y)
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(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__f(x0)) ->^* n__f(x0)

activate(n__f(false)) ->^* n__f(true)

activate(n__f(false)) ->^* n__f(true)

activate(n__f(false)) ->^* c

activate(n__f(true)) ->^* c

if(false, x0, n__f(x1)) ->^* n__f(x1)

if(false, x0, n__f(false)) ->^* n__f(true)

if(false, x0, n__f(false)) ->^* n__f(true)

if(false, x0, n__f(false)) ->^* c

if(false, x0, n__f(true)) ->^* c

f(x0) ->^* n__f(x0)

f(false) ->^* n__f(true)

f(false) ->^* n__f(true)

f(false) ->^* c

f(true) ->^* c




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(2)
BOUNDS(1, 1)