3.18/1.51	WORST_CASE(?, O(1))
3.18/1.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.18/1.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.18/1.52	
3.18/1.52	
3.18/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.18/1.52	
3.18/1.52	(0) CpxTRS
3.18/1.52	(1) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
3.18/1.52	(2) BOUNDS(1, 1)
3.18/1.52	
3.18/1.52	
3.18/1.52	----------------------------------------
3.18/1.52	
3.18/1.52	(0)
3.18/1.52	Obligation:
3.18/1.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.18/1.52	
3.18/1.52	
3.18/1.52	The TRS R consists of the following rules:
3.18/1.52	
3.18/1.52	   terms(N) -> cons(recip(sqr(N)))
3.18/1.52	   sqr(0) -> 0
3.18/1.52	   sqr(s) -> s
3.18/1.52	   dbl(0) -> 0
3.18/1.52	   dbl(s) -> s
3.18/1.52	   add(0, X) -> X
3.18/1.52	   add(s, Y) -> s
3.18/1.52	   first(0, X) -> nil
3.18/1.52	   first(s, cons(Y)) -> cons(Y)
3.18/1.52	
3.18/1.52	S is empty.
3.18/1.52	Rewrite Strategy: FULL
3.18/1.52	----------------------------------------
3.18/1.52	
3.18/1.52	(1) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.18/1.52	Constant runtime complexity proven by termination of constructor-based narrowing.
3.18/1.52	
3.18/1.52	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.18/1.52	
3.18/1.52	first(s, cons(x0)) ->^* cons(x0)
3.18/1.52	
3.18/1.52	first(0, x0) ->^* nil
3.18/1.52	
3.18/1.52	dbl(s) ->^* s
3.18/1.52	
3.18/1.52	dbl(0) ->^* 0
3.18/1.52	
3.18/1.52	add(s, x0) ->^* s
3.18/1.52	
3.18/1.52	terms(s) ->^* cons(recip(s))
3.18/1.52	
3.18/1.52	terms(0) ->^* cons(recip(0))
3.18/1.52	
3.18/1.52	sqr(s) ->^* s
3.18/1.52	
3.18/1.52	sqr(0) ->^* 0
3.18/1.52	
3.18/1.52	
3.18/1.52	
3.18/1.52	
3.18/1.52	----------------------------------------
3.18/1.52	
3.18/1.52	(2)
3.18/1.52	BOUNDS(1, 1)
3.21/1.55	EOF