20.19/8.18	WORST_CASE(Omega(n^1), O(n^1))
20.36/8.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.36/8.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.36/8.20	
20.36/8.20	
20.36/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/8.20	
20.36/8.20	(0) CpxTRS
20.36/8.20	(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	(2) CpxTRS
20.36/8.20	    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	    (4) CpxWeightedTrs
20.36/8.20	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	    (6) CpxTypedWeightedTrs
20.36/8.20	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (8) CpxTypedWeightedCompleteTrs
20.36/8.20	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	    (10) CpxTypedWeightedCompleteTrs
20.36/8.20	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (12) CpxRNTS
20.36/8.20	    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	    (14) CpxRNTS
20.36/8.20	    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
20.36/8.20	    (16) CpxRNTS
20.36/8.20	    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (18) CpxRNTS
20.36/8.20	    (19) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
20.36/8.20	    (20) CpxRNTS
20.36/8.20	    (21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
20.36/8.20	    (22) CpxRNTS
20.36/8.20	    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (24) CpxRNTS
20.36/8.20	    (25) IntTrsBoundProof [UPPER BOUND(ID), 653 ms]
20.36/8.20	    (26) CpxRNTS
20.36/8.20	    (27) IntTrsBoundProof [UPPER BOUND(ID), 181 ms]
20.36/8.20	    (28) CpxRNTS
20.36/8.20	    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (30) CpxRNTS
20.36/8.20	    (31) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
20.36/8.20	    (32) CpxRNTS
20.36/8.20	    (33) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
20.36/8.20	    (34) CpxRNTS
20.36/8.20	    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
20.36/8.20	    (36) CpxRNTS
20.36/8.20	    (37) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
20.36/8.20	    (38) CpxRNTS
20.36/8.20	    (39) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
20.36/8.20	    (40) CpxRNTS
20.36/8.20	    (41) FinalProof [FINISHED, 0 ms]
20.36/8.20	    (42) BOUNDS(1, n^1)
20.36/8.20	(43) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
20.36/8.20	(44) TRS for Loop Detection
20.36/8.20	    (45) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
20.36/8.20	    (46) BEST
20.36/8.20	        (47) proven lower bound
20.36/8.20	            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
20.36/8.20	            (49) BOUNDS(n^1, INF)
20.36/8.20	        (50) TRS for Loop Detection
20.36/8.20	
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(0)
20.36/8.20	Obligation:
20.36/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true
20.36/8.20	   is_empty(cons(x, l)) -> false
20.36/8.20	   hd(cons(x, l)) -> x
20.36/8.20	   tl(cons(x, l)) -> l
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1)
20.36/8.20	   ifappend(l1, l2, nil) -> l2
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))
20.36/8.20	
20.36/8.20	S is empty.
20.36/8.20	Rewrite Strategy: FULL
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Converted rc-obligation to irc-obligation.
20.36/8.20	
20.36/8.20	As the TRS does not nest defined symbols, we have rc = irc.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(2)
20.36/8.20	Obligation:
20.36/8.20	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true
20.36/8.20	   is_empty(cons(x, l)) -> false
20.36/8.20	   hd(cons(x, l)) -> x
20.36/8.20	   tl(cons(x, l)) -> l
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1)
20.36/8.20	   ifappend(l1, l2, nil) -> l2
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))
20.36/8.20	
20.36/8.20	S is empty.
20.36/8.20	Rewrite Strategy: INNERMOST
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Transformed relative TRS to weighted TRS
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(4)
20.36/8.20	Obligation:
20.36/8.20	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true [1]
20.36/8.20	   is_empty(cons(x, l)) -> false [1]
20.36/8.20	   hd(cons(x, l)) -> x [1]
20.36/8.20	   tl(cons(x, l)) -> l [1]
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1) [1]
20.36/8.20	   ifappend(l1, l2, nil) -> l2 [1]
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]
20.36/8.20	
20.36/8.20	Rewrite Strategy: INNERMOST
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Infered types.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(6)
20.36/8.20	Obligation:
20.36/8.20	Runtime Complexity Weighted TRS with Types.
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true [1]
20.36/8.20	   is_empty(cons(x, l)) -> false [1]
20.36/8.20	   hd(cons(x, l)) -> x [1]
20.36/8.20	   tl(cons(x, l)) -> l [1]
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1) [1]
20.36/8.20	   ifappend(l1, l2, nil) -> l2 [1]
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]
20.36/8.20	
20.36/8.20	The TRS has the following type information:
20.36/8.20	is_empty :: nil:cons -> true:false
20.36/8.20	nil :: nil:cons
20.36/8.20	true :: true:false
20.36/8.20	cons :: hd -> nil:cons -> nil:cons
20.36/8.20	false :: true:false
20.36/8.20	hd :: nil:cons -> hd
20.36/8.20	tl :: nil:cons -> nil:cons
20.36/8.20	append :: nil:cons -> nil:cons -> nil:cons
20.36/8.20	ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons
20.36/8.20	
20.36/8.20	Rewrite Strategy: INNERMOST
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(7) CompletionProof (UPPER BOUND(ID))
20.36/8.20	The transformation into a RNTS is sound, since:
20.36/8.20	
20.36/8.20	(a) The obligation is a constructor system where every type has a constant constructor,
20.36/8.20	
20.36/8.20	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
20.36/8.20	
20.36/8.20	   is_empty_1
20.36/8.20	   hd_1
20.36/8.20	   tl_1
20.36/8.20	   append_2
20.36/8.20	   ifappend_3
20.36/8.20	
20.36/8.20	(c) The following functions are completely defined:
20.36/8.20	none
20.36/8.20	
20.36/8.20	Due to the following rules being added:
20.36/8.20	none
20.36/8.20	
20.36/8.20	And the following fresh constants:    const
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(8)
20.36/8.20	Obligation:
20.36/8.20	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.36/8.20	
20.36/8.20	Runtime Complexity Weighted TRS with Types.
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true [1]
20.36/8.20	   is_empty(cons(x, l)) -> false [1]
20.36/8.20	   hd(cons(x, l)) -> x [1]
20.36/8.20	   tl(cons(x, l)) -> l [1]
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1) [1]
20.36/8.20	   ifappend(l1, l2, nil) -> l2 [1]
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]
20.36/8.20	
20.36/8.20	The TRS has the following type information:
20.36/8.20	is_empty :: nil:cons -> true:false
20.36/8.20	nil :: nil:cons
20.36/8.20	true :: true:false
20.36/8.20	cons :: hd -> nil:cons -> nil:cons
20.36/8.20	false :: true:false
20.36/8.20	hd :: nil:cons -> hd
20.36/8.20	tl :: nil:cons -> nil:cons
20.36/8.20	append :: nil:cons -> nil:cons -> nil:cons
20.36/8.20	ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons
20.36/8.20	const :: hd
20.36/8.20	
20.36/8.20	Rewrite Strategy: INNERMOST
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(10)
20.36/8.20	Obligation:
20.36/8.20	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
20.36/8.20	
20.36/8.20	Runtime Complexity Weighted TRS with Types.
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true [1]
20.36/8.20	   is_empty(cons(x, l)) -> false [1]
20.36/8.20	   hd(cons(x, l)) -> x [1]
20.36/8.20	   tl(cons(x, l)) -> l [1]
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1) [1]
20.36/8.20	   ifappend(l1, l2, nil) -> l2 [1]
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]
20.36/8.20	
20.36/8.20	The TRS has the following type information:
20.36/8.20	is_empty :: nil:cons -> true:false
20.36/8.20	nil :: nil:cons
20.36/8.20	true :: true:false
20.36/8.20	cons :: hd -> nil:cons -> nil:cons
20.36/8.20	false :: true:false
20.36/8.20	hd :: nil:cons -> hd
20.36/8.20	tl :: nil:cons -> nil:cons
20.36/8.20	append :: nil:cons -> nil:cons -> nil:cons
20.36/8.20	ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons
20.36/8.20	const :: hd
20.36/8.20	
20.36/8.20	Rewrite Strategy: INNERMOST
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
20.36/8.20	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
20.36/8.20	The constant constructors are abstracted as follows:
20.36/8.20	
20.36/8.20	   nil => 0
20.36/8.20	   true => 1
20.36/8.20	   false => 0
20.36/8.20	   const => 0
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(12)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(l1, l2, l1) :|: z = l1, z' = l2, l1 >= 0, l2 >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> l2 :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, l2) :|: z'' = 1 + x + l, z = l1, x >= 0, l >= 0, z' = l2, l1 >= 0, l2 >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(13) SimplificationProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(14)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
20.36/8.20	Found the following analysis order by SCC decomposition:
20.36/8.20	
20.36/8.20	   { is_empty }
20.36/8.20	   { ifappend, append }
20.36/8.20	   { tl }
20.36/8.20	   { hd }
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(16)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(17) ResultPropagationProof (UPPER BOUND(ID))
20.36/8.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(18)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(19) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed SIZE bound using CoFloCo for: is_empty
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(1) with polynomial bound: 1
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(20)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: ?, size: O(1) [1]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(21) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed RUNTIME bound using CoFloCo for: is_empty
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(1) with polynomial bound: 1
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(22)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(23) ResultPropagationProof (UPPER BOUND(ID))
20.36/8.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(24)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(25) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed SIZE bound using CoFloCo for: ifappend
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: z' + z''
20.36/8.20	
20.36/8.20	Computed SIZE bound using CoFloCo for: append
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: z + z'
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(26)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: ?, size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: ?, size: O(n^1) [z + z']
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(27) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed RUNTIME bound using CoFloCo for: ifappend
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: 3 + 2*z''
20.36/8.20	
20.36/8.20	Computed RUNTIME bound using CoFloCo for: append
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: 4 + 2*z
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(28)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(29) ResultPropagationProof (UPPER BOUND(ID))
20.36/8.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(30)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(31) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed SIZE bound using KoAT for: tl
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: z
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(32)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {tl}, {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	  tl: runtime: ?, size: O(n^1) [z]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(33) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed RUNTIME bound using CoFloCo for: tl
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(1) with polynomial bound: 1
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(34)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	  tl: runtime: O(1) [1], size: O(n^1) [z]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(35) ResultPropagationProof (UPPER BOUND(ID))
20.36/8.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(36)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	  tl: runtime: O(1) [1], size: O(n^1) [z]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(37) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed SIZE bound using KoAT for: hd
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(n^1) with polynomial bound: z
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(38)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: {hd}
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	  tl: runtime: O(1) [1], size: O(n^1) [z]
20.36/8.20	  hd: runtime: ?, size: O(n^1) [z]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(39) IntTrsBoundProof (UPPER BOUND(ID))
20.36/8.20	
20.36/8.20	Computed RUNTIME bound using CoFloCo for: hd
20.36/8.20	after applying outer abstraction to obtain an ITS,
20.36/8.20	resulting in: O(1) with polynomial bound: 1
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(40)
20.36/8.20	Obligation:
20.36/8.20	Complexity RNTS consisting of the following rules:
20.36/8.20	
20.36/8.20	   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
20.36/8.20	   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
20.36/8.20	   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
20.36/8.20	   is_empty(z) -{ 1 }-> 1 :|: z = 0
20.36/8.20	   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
20.36/8.20	
20.36/8.20	Function symbols to be analyzed: 
20.36/8.20	Previous analysis results are:
20.36/8.20	  is_empty: runtime: O(1) [1], size: O(1) [1]
20.36/8.20	  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
20.36/8.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
20.36/8.20	  tl: runtime: O(1) [1], size: O(n^1) [z]
20.36/8.20	  hd: runtime: O(1) [1], size: O(n^1) [z]
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(41) FinalProof (FINISHED)
20.36/8.20	Computed overall runtime complexity
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(42)
20.36/8.20	BOUNDS(1, n^1)
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(43) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
20.36/8.20	Transformed a relative TRS into a decreasing-loop problem.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(44)
20.36/8.20	Obligation:
20.36/8.20	Analyzing the following TRS for decreasing loops:
20.36/8.20	
20.36/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true
20.36/8.20	   is_empty(cons(x, l)) -> false
20.36/8.20	   hd(cons(x, l)) -> x
20.36/8.20	   tl(cons(x, l)) -> l
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1)
20.36/8.20	   ifappend(l1, l2, nil) -> l2
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))
20.36/8.20	
20.36/8.20	S is empty.
20.36/8.20	Rewrite Strategy: FULL
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(45) DecreasingLoopProof (LOWER BOUND(ID))
20.36/8.20	The following loop(s) give(s) rise to the lower bound Omega(n^1):
20.36/8.20	
20.36/8.20	The rewrite sequence
20.36/8.20	
20.36/8.20	append(cons(x3_0, l4_0), l2) ->^+ cons(x3_0, append(l4_0, l2))
20.36/8.20	
20.36/8.20	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
20.36/8.20	
20.36/8.20	The pumping substitution is [l4_0 / cons(x3_0, l4_0)].
20.36/8.20	
20.36/8.20	The result substitution is [ ].
20.36/8.20	
20.36/8.20	
20.36/8.20	
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(46)
20.36/8.20	Complex Obligation (BEST)
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(47)
20.36/8.20	Obligation:
20.36/8.20	Proved the lower bound n^1 for the following obligation:
20.36/8.20	
20.36/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true
20.36/8.20	   is_empty(cons(x, l)) -> false
20.36/8.20	   hd(cons(x, l)) -> x
20.36/8.20	   tl(cons(x, l)) -> l
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1)
20.36/8.20	   ifappend(l1, l2, nil) -> l2
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))
20.36/8.20	
20.36/8.20	S is empty.
20.36/8.20	Rewrite Strategy: FULL
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(48) LowerBoundPropagationProof (FINISHED)
20.36/8.20	Propagated lower bound.
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(49)
20.36/8.20	BOUNDS(n^1, INF)
20.36/8.20	
20.36/8.20	----------------------------------------
20.36/8.20	
20.36/8.20	(50)
20.36/8.20	Obligation:
20.36/8.20	Analyzing the following TRS for decreasing loops:
20.36/8.20	
20.36/8.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
20.36/8.20	
20.36/8.20	
20.36/8.20	The TRS R consists of the following rules:
20.36/8.20	
20.36/8.20	   is_empty(nil) -> true
20.36/8.20	   is_empty(cons(x, l)) -> false
20.36/8.20	   hd(cons(x, l)) -> x
20.36/8.20	   tl(cons(x, l)) -> l
20.36/8.20	   append(l1, l2) -> ifappend(l1, l2, l1)
20.36/8.20	   ifappend(l1, l2, nil) -> l2
20.36/8.20	   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))
20.36/8.20	
20.36/8.20	S is empty.
20.36/8.20	Rewrite Strategy: FULL
20.36/8.25	EOF