14.67/4.78	WORST_CASE(Omega(n^1), O(n^1))
15.05/4.79	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
15.05/4.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.05/4.79	
15.05/4.79	
15.05/4.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.05/4.79	
15.05/4.79	(0) CpxTRS
15.05/4.79	(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
15.05/4.79	(2) CdtProblem
15.05/4.79	    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
15.05/4.79	    (4) CdtProblem
15.05/4.79	    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
15.05/4.79	    (6) CdtProblem
15.05/4.79	    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms]
15.05/4.79	    (8) CdtProblem
15.05/4.79	    (9) CdtKnowledgeProof [FINISHED, 0 ms]
15.05/4.79	    (10) BOUNDS(1, 1)
15.05/4.79	(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
15.05/4.79	(12) TRS for Loop Detection
15.05/4.79	    (13) DecreasingLoopProof [LOWER BOUND(ID), 45 ms]
15.05/4.79	    (14) BEST
15.05/4.79	        (15) proven lower bound
15.05/4.79	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
15.05/4.79	            (17) BOUNDS(n^1, INF)
15.05/4.79	        (18) TRS for Loop Detection
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(0)
15.05/4.79	Obligation:
15.05/4.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.05/4.79	
15.05/4.79	
15.05/4.79	The TRS R consists of the following rules:
15.05/4.79	
15.05/4.79	   fst(0, Z) -> nil
15.05/4.79	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
15.05/4.79	   from(X) -> cons(X, n__from(s(X)))
15.05/4.79	   add(0, X) -> X
15.05/4.79	   add(s(X), Y) -> s(n__add(activate(X), Y))
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(X, Z)) -> s(n__len(activate(Z)))
15.05/4.79	   fst(X1, X2) -> n__fst(X1, X2)
15.05/4.79	   from(X) -> n__from(X)
15.05/4.79	   add(X1, X2) -> n__add(X1, X2)
15.05/4.79	   len(X) -> n__len(X)
15.05/4.79	   activate(n__fst(X1, X2)) -> fst(X1, X2)
15.05/4.79	   activate(n__from(X)) -> from(X)
15.05/4.79	   activate(n__add(X1, X2)) -> add(X1, X2)
15.05/4.79	   activate(n__len(X)) -> len(X)
15.05/4.79	   activate(X) -> X
15.05/4.79	
15.05/4.79	S is empty.
15.05/4.79	Rewrite Strategy: INNERMOST
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(1) CpxTrsToCdtProof (UPPER BOUND(ID))
15.05/4.79	Converted Cpx (relative) TRS to CDT
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(2)
15.05/4.79	Obligation:
15.05/4.79	Complexity Dependency Tuples Problem
15.05/4.79	
15.05/4.79	Rules:
15.05/4.79	   fst(0, z0) -> nil
15.05/4.79	   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
15.05/4.79	   fst(z0, z1) -> n__fst(z0, z1)
15.05/4.79	   from(z0) -> cons(z0, n__from(s(z0)))
15.05/4.79	   from(z0) -> n__from(z0)
15.05/4.79	   add(0, z0) -> z0
15.05/4.79	   add(s(z0), z1) -> s(n__add(activate(z0), z1))
15.05/4.79	   add(z0, z1) -> n__add(z0, z1)
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(z0, z1)) -> s(n__len(activate(z1)))
15.05/4.79	   len(z0) -> n__len(z0)
15.05/4.79	   activate(n__fst(z0, z1)) -> fst(z0, z1)
15.05/4.79	   activate(n__from(z0)) -> from(z0)
15.05/4.79	   activate(n__add(z0, z1)) -> add(z0, z1)
15.05/4.79	   activate(n__len(z0)) -> len(z0)
15.05/4.79	   activate(z0) -> z0
15.05/4.79	Tuples:
15.05/4.79	   FST(0, z0) -> c
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   FST(z0, z1) -> c2
15.05/4.79	   FROM(z0) -> c3
15.05/4.79	   FROM(z0) -> c4
15.05/4.79	   ADD(0, z0) -> c5
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   ADD(z0, z1) -> c7
15.05/4.79	   LEN(nil) -> c8
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   LEN(z0) -> c10
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__from(z0)) -> c12(FROM(z0))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	   ACTIVATE(z0) -> c15
15.05/4.79	S tuples:
15.05/4.79	   FST(0, z0) -> c
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   FST(z0, z1) -> c2
15.05/4.79	   FROM(z0) -> c3
15.05/4.79	   FROM(z0) -> c4
15.05/4.79	   ADD(0, z0) -> c5
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   ADD(z0, z1) -> c7
15.05/4.79	   LEN(nil) -> c8
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   LEN(z0) -> c10
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__from(z0)) -> c12(FROM(z0))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	   ACTIVATE(z0) -> c15
15.05/4.79	K tuples:none
15.05/4.79	Defined Rule Symbols:   fst_2, from_1, add_2, len_1, activate_1
15.05/4.79	
15.05/4.79	Defined Pair Symbols:   FST_2, FROM_1, ADD_2, LEN_1, ACTIVATE_1
15.05/4.79	
15.05/4.79	Compound Symbols:   c, c1_2, c2, c3, c4, c5, c6_1, c7, c8, c9_1, c10, c11_1, c12_1, c13_1, c14_1, c15
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
15.05/4.79	Removed 10 trailing nodes:
15.05/4.79	   LEN(nil) -> c8
15.05/4.79	   ADD(z0, z1) -> c7
15.05/4.79	   ADD(0, z0) -> c5
15.05/4.79	   ACTIVATE(z0) -> c15
15.05/4.79	   FST(0, z0) -> c
15.05/4.79	   FROM(z0) -> c4
15.05/4.79	   LEN(z0) -> c10
15.05/4.79	   ACTIVATE(n__from(z0)) -> c12(FROM(z0))
15.05/4.79	   FROM(z0) -> c3
15.05/4.79	   FST(z0, z1) -> c2
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(4)
15.05/4.79	Obligation:
15.05/4.79	Complexity Dependency Tuples Problem
15.05/4.79	
15.05/4.79	Rules:
15.05/4.79	   fst(0, z0) -> nil
15.05/4.79	   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
15.05/4.79	   fst(z0, z1) -> n__fst(z0, z1)
15.05/4.79	   from(z0) -> cons(z0, n__from(s(z0)))
15.05/4.79	   from(z0) -> n__from(z0)
15.05/4.79	   add(0, z0) -> z0
15.05/4.79	   add(s(z0), z1) -> s(n__add(activate(z0), z1))
15.05/4.79	   add(z0, z1) -> n__add(z0, z1)
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(z0, z1)) -> s(n__len(activate(z1)))
15.05/4.79	   len(z0) -> n__len(z0)
15.05/4.79	   activate(n__fst(z0, z1)) -> fst(z0, z1)
15.05/4.79	   activate(n__from(z0)) -> from(z0)
15.05/4.79	   activate(n__add(z0, z1)) -> add(z0, z1)
15.05/4.79	   activate(n__len(z0)) -> len(z0)
15.05/4.79	   activate(z0) -> z0
15.05/4.79	Tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	S tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	K tuples:none
15.05/4.79	Defined Rule Symbols:   fst_2, from_1, add_2, len_1, activate_1
15.05/4.79	
15.05/4.79	Defined Pair Symbols:   FST_2, ADD_2, LEN_1, ACTIVATE_1
15.05/4.79	
15.05/4.79	Compound Symbols:   c1_2, c6_1, c9_1, c11_1, c13_1, c14_1
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
15.05/4.79	The following rules are not usable and were removed:
15.05/4.79	   fst(0, z0) -> nil
15.05/4.79	   fst(s(z0), cons(z1, z2)) -> cons(z1, n__fst(activate(z0), activate(z2)))
15.05/4.79	   fst(z0, z1) -> n__fst(z0, z1)
15.05/4.79	   from(z0) -> cons(z0, n__from(s(z0)))
15.05/4.79	   from(z0) -> n__from(z0)
15.05/4.79	   add(0, z0) -> z0
15.05/4.79	   add(s(z0), z1) -> s(n__add(activate(z0), z1))
15.05/4.79	   add(z0, z1) -> n__add(z0, z1)
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(z0, z1)) -> s(n__len(activate(z1)))
15.05/4.79	   len(z0) -> n__len(z0)
15.05/4.79	   activate(n__fst(z0, z1)) -> fst(z0, z1)
15.05/4.79	   activate(n__from(z0)) -> from(z0)
15.05/4.79	   activate(n__add(z0, z1)) -> add(z0, z1)
15.05/4.79	   activate(n__len(z0)) -> len(z0)
15.05/4.79	   activate(z0) -> z0
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(6)
15.05/4.79	Obligation:
15.05/4.79	Complexity Dependency Tuples Problem
15.05/4.79	
15.05/4.79	Rules:none
15.05/4.79	Tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	S tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	K tuples:none
15.05/4.79	Defined Rule Symbols:none
15.05/4.79	
15.05/4.79	Defined Pair Symbols:   FST_2, ADD_2, LEN_1, ACTIVATE_1
15.05/4.79	
15.05/4.79	Compound Symbols:   c1_2, c6_1, c9_1, c11_1, c13_1, c14_1
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
15.05/4.79	Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	We considered the (Usable) Rules:none
15.05/4.79	And the Tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	The order we found is given by the following interpretation:
15.05/4.79	
15.05/4.79	Polynomial interpretation :
15.05/4.79	
15.05/4.79	   POL(ACTIVATE(x_1)) = [1] + x_1
15.05/4.79	   POL(ADD(x_1, x_2)) = [1] + x_1 + x_2
15.05/4.79	   POL(FST(x_1, x_2)) = x_1 + x_2
15.05/4.79	   POL(LEN(x_1)) = [1] + x_1
15.05/4.79	   POL(c1(x_1, x_2)) = x_1 + x_2
15.05/4.79	   POL(c11(x_1)) = x_1
15.05/4.79	   POL(c13(x_1)) = x_1
15.05/4.79	   POL(c14(x_1)) = x_1
15.05/4.79	   POL(c6(x_1)) = x_1
15.05/4.79	   POL(c9(x_1)) = x_1
15.05/4.79	   POL(cons(x_1, x_2)) = [1] + x_2
15.05/4.79	   POL(n__add(x_1, x_2)) = [1] + x_1 + x_2
15.05/4.79	   POL(n__fst(x_1, x_2)) = [1] + x_1 + x_2
15.05/4.79	   POL(n__len(x_1)) = [1] + x_1
15.05/4.79	   POL(s(x_1)) = [1] + x_1
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(8)
15.05/4.79	Obligation:
15.05/4.79	Complexity Dependency Tuples Problem
15.05/4.79	
15.05/4.79	Rules:none
15.05/4.79	Tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	S tuples:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	K tuples:
15.05/4.79	   ADD(s(z0), z1) -> c6(ACTIVATE(z0))
15.05/4.79	   LEN(cons(z0, z1)) -> c9(ACTIVATE(z1))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	Defined Rule Symbols:none
15.05/4.79	
15.05/4.79	Defined Pair Symbols:   FST_2, ADD_2, LEN_1, ACTIVATE_1
15.05/4.79	
15.05/4.79	Compound Symbols:   c1_2, c6_1, c9_1, c11_1, c13_1, c14_1
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(9) CdtKnowledgeProof (FINISHED)
15.05/4.79	The following tuples could be moved from S to K by knowledge propagation:
15.05/4.79	   FST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z0), ACTIVATE(z2))
15.05/4.79	   ACTIVATE(n__fst(z0, z1)) -> c11(FST(z0, z1))
15.05/4.79	   ACTIVATE(n__add(z0, z1)) -> c13(ADD(z0, z1))
15.05/4.79	   ACTIVATE(n__len(z0)) -> c14(LEN(z0))
15.05/4.79	Now S is empty
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(10)
15.05/4.79	BOUNDS(1, 1)
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
15.05/4.79	Transformed a relative TRS into a decreasing-loop problem.
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(12)
15.05/4.79	Obligation:
15.05/4.79	Analyzing the following TRS for decreasing loops:
15.05/4.79	
15.05/4.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.05/4.79	
15.05/4.79	
15.05/4.79	The TRS R consists of the following rules:
15.05/4.79	
15.05/4.79	   fst(0, Z) -> nil
15.05/4.79	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
15.05/4.79	   from(X) -> cons(X, n__from(s(X)))
15.05/4.79	   add(0, X) -> X
15.05/4.79	   add(s(X), Y) -> s(n__add(activate(X), Y))
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(X, Z)) -> s(n__len(activate(Z)))
15.05/4.79	   fst(X1, X2) -> n__fst(X1, X2)
15.05/4.79	   from(X) -> n__from(X)
15.05/4.79	   add(X1, X2) -> n__add(X1, X2)
15.05/4.79	   len(X) -> n__len(X)
15.05/4.79	   activate(n__fst(X1, X2)) -> fst(X1, X2)
15.05/4.79	   activate(n__from(X)) -> from(X)
15.05/4.79	   activate(n__add(X1, X2)) -> add(X1, X2)
15.05/4.79	   activate(n__len(X)) -> len(X)
15.05/4.79	   activate(X) -> X
15.05/4.79	
15.05/4.79	S is empty.
15.05/4.79	Rewrite Strategy: INNERMOST
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(13) DecreasingLoopProof (LOWER BOUND(ID))
15.05/4.79	The following loop(s) give(s) rise to the lower bound Omega(n^1):
15.05/4.79	
15.05/4.79	The rewrite sequence
15.05/4.79	
15.05/4.79	activate(n__fst(s(X1_0), cons(Y2_0, Z3_0))) ->^+ cons(Y2_0, n__fst(activate(X1_0), activate(Z3_0)))
15.05/4.79	
15.05/4.79	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
15.05/4.79	
15.05/4.79	The pumping substitution is [X1_0 / n__fst(s(X1_0), cons(Y2_0, Z3_0))].
15.05/4.79	
15.05/4.79	The result substitution is [ ].
15.05/4.79	
15.05/4.79	
15.05/4.79	
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(14)
15.05/4.79	Complex Obligation (BEST)
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(15)
15.05/4.79	Obligation:
15.05/4.79	Proved the lower bound n^1 for the following obligation:
15.05/4.79	
15.05/4.79	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.05/4.79	
15.05/4.79	
15.05/4.79	The TRS R consists of the following rules:
15.05/4.79	
15.05/4.79	   fst(0, Z) -> nil
15.05/4.79	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
15.05/4.79	   from(X) -> cons(X, n__from(s(X)))
15.05/4.79	   add(0, X) -> X
15.05/4.79	   add(s(X), Y) -> s(n__add(activate(X), Y))
15.05/4.79	   len(nil) -> 0
15.05/4.79	   len(cons(X, Z)) -> s(n__len(activate(Z)))
15.05/4.79	   fst(X1, X2) -> n__fst(X1, X2)
15.05/4.79	   from(X) -> n__from(X)
15.05/4.79	   add(X1, X2) -> n__add(X1, X2)
15.05/4.79	   len(X) -> n__len(X)
15.05/4.79	   activate(n__fst(X1, X2)) -> fst(X1, X2)
15.05/4.79	   activate(n__from(X)) -> from(X)
15.05/4.79	   activate(n__add(X1, X2)) -> add(X1, X2)
15.05/4.79	   activate(n__len(X)) -> len(X)
15.05/4.79	   activate(X) -> X
15.05/4.79	
15.05/4.79	S is empty.
15.05/4.79	Rewrite Strategy: INNERMOST
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(16) LowerBoundPropagationProof (FINISHED)
15.05/4.79	Propagated lower bound.
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(17)
15.05/4.79	BOUNDS(n^1, INF)
15.05/4.79	
15.05/4.79	----------------------------------------
15.05/4.79	
15.05/4.79	(18)
15.05/4.79	Obligation:
15.05/4.79	Analyzing the following TRS for decreasing loops:
15.05/4.80	
15.05/4.80	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
15.05/4.80	
15.05/4.80	
15.05/4.80	The TRS R consists of the following rules:
15.05/4.80	
15.05/4.80	   fst(0, Z) -> nil
15.05/4.80	   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
15.05/4.80	   from(X) -> cons(X, n__from(s(X)))
15.05/4.80	   add(0, X) -> X
15.05/4.80	   add(s(X), Y) -> s(n__add(activate(X), Y))
15.05/4.80	   len(nil) -> 0
15.05/4.80	   len(cons(X, Z)) -> s(n__len(activate(Z)))
15.05/4.80	   fst(X1, X2) -> n__fst(X1, X2)
15.05/4.80	   from(X) -> n__from(X)
15.05/4.80	   add(X1, X2) -> n__add(X1, X2)
15.05/4.80	   len(X) -> n__len(X)
15.05/4.80	   activate(n__fst(X1, X2)) -> fst(X1, X2)
15.05/4.80	   activate(n__from(X)) -> from(X)
15.05/4.80	   activate(n__add(X1, X2)) -> add(X1, X2)
15.05/4.80	   activate(n__len(X)) -> len(X)
15.05/4.80	   activate(X) -> X
15.05/4.80	
15.05/4.80	S is empty.
15.05/4.80	Rewrite Strategy: INNERMOST
15.09/4.84	EOF