WORST_CASE(?,O(1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 693 ms]
(2) CpxRelTRS
(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 10 ms]
(4) BOUNDS(1, 1)


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


The TRS R consists of the following rules:

   F(g(i(a, b, b'), c), d) -> c1(F(.(b, c), d'))
   F(g(i(a, b, b'), c), d) -> c2(F(.(b', c), d'))
   F(g(h(a, b), c), d) -> c3(F(.(b, g(h(a, b), c)), d))
   F(g(h(a, b), c), d) -> c4(F(c, d'))

The (relative) TRS S consists of the following rules:

   f(g(i(a, b, b'), c), d) -> if(e, f(.(b, c), d'), f(.(b', c), d'))
   f(g(h(a, b), c), d) -> if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   F(g(i(a, b, b'), c), d) -> c1(F(.(b, c), d'))
   F(g(i(a, b, b'), c), d) -> c2(F(.(b', c), d'))
   F(g(h(a, b), c), d) -> c3(F(.(b, g(h(a, b), c)), d))
   F(g(h(a, b), c), d) -> c4(F(c, d'))

The (relative) TRS S consists of the following rules:

   f(g(i(a, b, b'), c), d) -> if(e, f(.(b, c), d'), f(.(b', c), d'))
   f(g(h(a, b), c), d) -> if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Rewrite Strategy: INNERMOST
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(3) 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:

f(g(h(a, b), c), d) ->^* if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

f(g(i(a, b, b'), c), d) ->^* if(e, f(.(b, c), d'), f(.(b', c), d'))

F(g(h(a, b), c), d) ->^* c4(F(c, d'))

F(g(h(a, b), c), d) ->^* c3(F(.(b, g(h(a, b), c)), d))

F(g(i(a, b, b'), c), d) ->^* c2(F(.(b', c), d'))

F(g(i(a, b, b'), c), d) ->^* c1(F(.(b, c), d'))




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(4)
BOUNDS(1, 1)