WORST_CASE(?,O(1))
proof of input_YOPzkPUAR2.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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:

   a__c -> a__f(g(c))
   a__f(g(X)) -> g(X)
   mark(c) -> a__c
   mark(f(X)) -> a__f(X)
   mark(g(X)) -> g(X)
   a__c -> c
   a__f(X) -> f(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:

mark(g(x0)) ->^* g(x0)

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

mark(f(g(x0))) ->^* g(x0)

mark(c) ->^* c

mark(c) ->^* f(g(c))

mark(c) ->^* g(c)

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

a__f(g(x0)) ->^* g(x0)

a__c ->^* c

a__c ->^* f(g(c))

a__c ->^* g(c)




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(2)
BOUNDS(1, 1)