2.97/1.53	WORST_CASE(?, O(1))
2.97/1.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
2.97/1.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.97/1.54	
2.97/1.54	
2.97/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).
2.97/1.54	
2.97/1.54	(0) CpxTRS
2.97/1.54	(1) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
2.97/1.54	(2) BOUNDS(1, 1)
2.97/1.54	
2.97/1.54	
2.97/1.54	----------------------------------------
2.97/1.54	
2.97/1.54	(0)
2.97/1.54	Obligation:
2.97/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).
2.97/1.54	
2.97/1.54	
2.97/1.54	The TRS R consists of the following rules:
2.97/1.54	
2.97/1.54	   terms(N) -> cons(recip(sqr(N)))
2.97/1.54	   sqr(0) -> 0
2.97/1.54	   sqr(s) -> s
2.97/1.54	   dbl(0) -> 0
2.97/1.54	   dbl(s) -> s
2.97/1.54	   add(0, X) -> X
2.97/1.54	   add(s, Y) -> s
2.97/1.54	   first(0, X) -> nil
2.97/1.54	   first(s, cons(Y)) -> cons(Y)
2.97/1.54	
2.97/1.54	S is empty.
2.97/1.54	Rewrite Strategy: INNERMOST
2.97/1.54	----------------------------------------
2.97/1.54	
2.97/1.54	(1) NarrowingOnBasicTermsTerminatesProof (FINISHED)
2.97/1.54	Constant runtime complexity proven by termination of constructor-based narrowing.
2.97/1.54	
2.97/1.54	The maximal most general narrowing sequences give rise to the following rewrite sequences:
2.97/1.54	
2.97/1.54	first(s, cons(x0)) ->^* cons(x0)
2.97/1.54	
2.97/1.54	first(0, x0) ->^* nil
2.97/1.54	
2.97/1.54	dbl(s) ->^* s
2.97/1.54	
2.97/1.54	dbl(0) ->^* 0
2.97/1.54	
2.97/1.54	add(s, x0) ->^* s
2.97/1.54	
2.97/1.54	terms(s) ->^* cons(recip(s))
2.97/1.54	
2.97/1.54	terms(0) ->^* cons(recip(0))
2.97/1.54	
2.97/1.54	sqr(s) ->^* s
2.97/1.54	
2.97/1.54	sqr(0) ->^* 0
2.97/1.54	
2.97/1.54	
2.97/1.54	
2.97/1.54	
2.97/1.54	----------------------------------------
2.97/1.54	
2.97/1.54	(2)
2.97/1.54	BOUNDS(1, 1)
3.25/1.57	EOF