WORST_CASE(?,O(1))
proof of input_yuiR8Y53Yz.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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:

   g(X) -> u(h(X), h(X), X)
   u(d, c(Y), X) -> k(Y)
   h(d) -> c(a)
   h(d) -> c(b)
   f(k(a), k(b), X) -> f(X, 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:

f(k(a), k(b), x0) ->^* f(x0, x0, x0)

h(d) ->^* c(b)

h(d) ->^* c(a)

u(d, c(x0), x1) ->^* k(x0)

g(d) ->^* u(c(b), c(b), d)

g(d) ->^* u(c(a), c(b), d)

g(d) ->^* u(c(b), c(a), d)

g(d) ->^* u(c(a), c(a), d)

g(d) ->^* u(c(b), c(b), d)

g(d) ->^* u(c(b), c(a), d)

g(d) ->^* u(c(a), c(b), d)

g(d) ->^* u(c(a), c(a), d)




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(2)
BOUNDS(1, 1)