9.37/3.24	WORST_CASE(NON_POLY, ?)
9.37/3.25	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
9.37/3.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.37/3.25	
9.37/3.25	
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	(0) CpxRelTRS
9.37/3.25	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 211 ms]
9.37/3.25	(2) CpxRelTRS
9.37/3.25	(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
9.37/3.25	(4) TRS for Loop Detection
9.37/3.25	(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
9.37/3.25	(6) BEST
9.37/3.25	    (7) proven lower bound
9.37/3.25	        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
9.37/3.25	        (9) BOUNDS(n^1, INF)
9.37/3.25	    (10) TRS for Loop Detection
9.37/3.25	        (11) InfiniteLowerBoundProof [FINISHED, 1188 ms]
9.37/3.25	        (12) BOUNDS(INF, INF)
9.37/3.25	
9.37/3.25	
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(0)
9.37/3.25	Obligation:
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	
9.37/3.25	The TRS R consists of the following rules:
9.37/3.25	
9.37/3.25	   turing(I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog) -> turing[Ite](!EQ(x, i1), I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog)
9.37/3.25	   turing(I(Goto(int), r), revltape, rtape, prog) -> turing(lookup(int, prog), revltape, rtape, prog)
9.37/3.25	   turing(I(Right, r), revltape, Cons(x, xs), prog) -> turing(r, Cons(x, revltape), xs, prog)
9.37/3.25	   turing(I(Right, r), revltape, Nil, prog) -> turing(r, Cons(0, revltape), Nil, prog)
9.37/3.25	   turing(I(Left, r), Cons(x, xs), rtape, prog) -> turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	   turing(I(Left, r), Nil, rtape, prog) -> turing(r, Nil, Cons(0, rtape), prog)
9.37/3.25	   turing(I(Write(int), r), revltape, Cons(x, xs), prog) -> turing(r, revltape, Cons(int, xs), prog)
9.37/3.25	   turing(I(Halt, r), revltape, rtape, prog) -> rtape
9.37/3.25	   turing(Empty, revltape, rtape, prog) -> rtape
9.37/3.25	   lookup(S(x), I(l, r)) -> lookup(x, r)
9.37/3.25	   instrsConstrCheck(I(l1, r1), I(x, y)) -> True
9.37/3.25	   instrsConstrCheck(I(l1, r1), Empty) -> False
9.37/3.25	   instrsConstrCheck(Empty, I(x, y)) -> False
9.37/3.25	   instrsConstrCheck(Empty, Empty) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), IfGoto(igtNat12, igtNat22)) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Right) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Left) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Halt) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Goto(gtNat2)) -> True
9.37/3.25	   instrConstrCheck(Goto(gtNat), Right) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Left) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Right, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Right, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Right) -> True
9.37/3.25	   instrConstrCheck(Right, Left) -> False
9.37/3.25	   instrConstrCheck(Right, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Halt) -> False
9.37/3.25	   instrConstrCheck(Left, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Left, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Right) -> False
9.37/3.25	   instrConstrCheck(Left, Left) -> True
9.37/3.25	   instrConstrCheck(Left, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Halt) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Right) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Left) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Write(wNat2)) -> True
9.37/3.25	   instrConstrCheck(Write(wNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Halt, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Halt, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Right) -> False
9.37/3.25	   instrConstrCheck(Halt, Left) -> False
9.37/3.25	   instrConstrCheck(Halt, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Halt) -> True
9.37/3.25	   notEmpty(Cons(x, xs)) -> True
9.37/3.25	   notEmpty(Nil) -> False
9.37/3.25	   lookup(0, instrs) -> instrs
9.37/3.25	   instrsSecond(I(l, r)) -> r
9.37/3.25	   instrsFirst(I(l, r)) -> l
9.37/3.25	   getWrite(Write(int)) -> int
9.37/3.25	   getGotoSecond(IfGoto(i1, i2)) -> i2
9.37/3.25	   getGotoFirst(IfGoto(i1, i2)) -> i1
9.37/3.25	   getGoto(Goto(int)) -> int
9.37/3.25	   run(prog, tapeinput) -> turing(prog, Nil, tapeinput, prog)
9.37/3.25	
9.37/3.25	The (relative) TRS S consists of the following rules:
9.37/3.25	
9.37/3.25	   !EQ(S(x), S(y)) -> !EQ(x, y)
9.37/3.25	   !EQ(0, S(y)) -> False
9.37/3.25	   !EQ(S(x), 0) -> False
9.37/3.25	   !EQ(0, 0) -> True
9.37/3.25	   turing[Ite](True, I(IfGoto(i1, i2), r), revltape, rtape, prog) -> turing(lookup(i2, prog), revltape, rtape, prog)
9.37/3.25	   turing[Ite](False, I(l, r), revltape, rtape, prog) -> turing(r, revltape, rtape, prog)
9.37/3.25	
9.37/3.25	Rewrite Strategy: INNERMOST
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
9.37/3.25	proved termination of relative rules
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(2)
9.37/3.25	Obligation:
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	
9.37/3.25	The TRS R consists of the following rules:
9.37/3.25	
9.37/3.25	   turing(I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog) -> turing[Ite](!EQ(x, i1), I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog)
9.37/3.25	   turing(I(Goto(int), r), revltape, rtape, prog) -> turing(lookup(int, prog), revltape, rtape, prog)
9.37/3.25	   turing(I(Right, r), revltape, Cons(x, xs), prog) -> turing(r, Cons(x, revltape), xs, prog)
9.37/3.25	   turing(I(Right, r), revltape, Nil, prog) -> turing(r, Cons(0, revltape), Nil, prog)
9.37/3.25	   turing(I(Left, r), Cons(x, xs), rtape, prog) -> turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	   turing(I(Left, r), Nil, rtape, prog) -> turing(r, Nil, Cons(0, rtape), prog)
9.37/3.25	   turing(I(Write(int), r), revltape, Cons(x, xs), prog) -> turing(r, revltape, Cons(int, xs), prog)
9.37/3.25	   turing(I(Halt, r), revltape, rtape, prog) -> rtape
9.37/3.25	   turing(Empty, revltape, rtape, prog) -> rtape
9.37/3.25	   lookup(S(x), I(l, r)) -> lookup(x, r)
9.37/3.25	   instrsConstrCheck(I(l1, r1), I(x, y)) -> True
9.37/3.25	   instrsConstrCheck(I(l1, r1), Empty) -> False
9.37/3.25	   instrsConstrCheck(Empty, I(x, y)) -> False
9.37/3.25	   instrsConstrCheck(Empty, Empty) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), IfGoto(igtNat12, igtNat22)) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Right) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Left) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Halt) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Goto(gtNat2)) -> True
9.37/3.25	   instrConstrCheck(Goto(gtNat), Right) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Left) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Right, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Right, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Right) -> True
9.37/3.25	   instrConstrCheck(Right, Left) -> False
9.37/3.25	   instrConstrCheck(Right, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Halt) -> False
9.37/3.25	   instrConstrCheck(Left, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Left, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Right) -> False
9.37/3.25	   instrConstrCheck(Left, Left) -> True
9.37/3.25	   instrConstrCheck(Left, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Halt) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Right) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Left) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Write(wNat2)) -> True
9.37/3.25	   instrConstrCheck(Write(wNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Halt, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Halt, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Right) -> False
9.37/3.25	   instrConstrCheck(Halt, Left) -> False
9.37/3.25	   instrConstrCheck(Halt, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Halt) -> True
9.37/3.25	   notEmpty(Cons(x, xs)) -> True
9.37/3.25	   notEmpty(Nil) -> False
9.37/3.25	   lookup(0, instrs) -> instrs
9.37/3.25	   instrsSecond(I(l, r)) -> r
9.37/3.25	   instrsFirst(I(l, r)) -> l
9.37/3.25	   getWrite(Write(int)) -> int
9.37/3.25	   getGotoSecond(IfGoto(i1, i2)) -> i2
9.37/3.25	   getGotoFirst(IfGoto(i1, i2)) -> i1
9.37/3.25	   getGoto(Goto(int)) -> int
9.37/3.25	   run(prog, tapeinput) -> turing(prog, Nil, tapeinput, prog)
9.37/3.25	
9.37/3.25	The (relative) TRS S consists of the following rules:
9.37/3.25	
9.37/3.25	   !EQ(S(x), S(y)) -> !EQ(x, y)
9.37/3.25	   !EQ(0, S(y)) -> False
9.37/3.25	   !EQ(S(x), 0) -> False
9.37/3.25	   !EQ(0, 0) -> True
9.37/3.25	   turing[Ite](True, I(IfGoto(i1, i2), r), revltape, rtape, prog) -> turing(lookup(i2, prog), revltape, rtape, prog)
9.37/3.25	   turing[Ite](False, I(l, r), revltape, rtape, prog) -> turing(r, revltape, rtape, prog)
9.37/3.25	
9.37/3.25	Rewrite Strategy: INNERMOST
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
9.37/3.25	Transformed a relative TRS into a decreasing-loop problem.
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(4)
9.37/3.25	Obligation:
9.37/3.25	Analyzing the following TRS for decreasing loops:
9.37/3.25	
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	
9.37/3.25	The TRS R consists of the following rules:
9.37/3.25	
9.37/3.25	   turing(I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog) -> turing[Ite](!EQ(x, i1), I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog)
9.37/3.25	   turing(I(Goto(int), r), revltape, rtape, prog) -> turing(lookup(int, prog), revltape, rtape, prog)
9.37/3.25	   turing(I(Right, r), revltape, Cons(x, xs), prog) -> turing(r, Cons(x, revltape), xs, prog)
9.37/3.25	   turing(I(Right, r), revltape, Nil, prog) -> turing(r, Cons(0, revltape), Nil, prog)
9.37/3.25	   turing(I(Left, r), Cons(x, xs), rtape, prog) -> turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	   turing(I(Left, r), Nil, rtape, prog) -> turing(r, Nil, Cons(0, rtape), prog)
9.37/3.25	   turing(I(Write(int), r), revltape, Cons(x, xs), prog) -> turing(r, revltape, Cons(int, xs), prog)
9.37/3.25	   turing(I(Halt, r), revltape, rtape, prog) -> rtape
9.37/3.25	   turing(Empty, revltape, rtape, prog) -> rtape
9.37/3.25	   lookup(S(x), I(l, r)) -> lookup(x, r)
9.37/3.25	   instrsConstrCheck(I(l1, r1), I(x, y)) -> True
9.37/3.25	   instrsConstrCheck(I(l1, r1), Empty) -> False
9.37/3.25	   instrsConstrCheck(Empty, I(x, y)) -> False
9.37/3.25	   instrsConstrCheck(Empty, Empty) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), IfGoto(igtNat12, igtNat22)) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Right) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Left) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Halt) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Goto(gtNat2)) -> True
9.37/3.25	   instrConstrCheck(Goto(gtNat), Right) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Left) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Right, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Right, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Right) -> True
9.37/3.25	   instrConstrCheck(Right, Left) -> False
9.37/3.25	   instrConstrCheck(Right, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Halt) -> False
9.37/3.25	   instrConstrCheck(Left, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Left, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Right) -> False
9.37/3.25	   instrConstrCheck(Left, Left) -> True
9.37/3.25	   instrConstrCheck(Left, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Halt) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Right) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Left) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Write(wNat2)) -> True
9.37/3.25	   instrConstrCheck(Write(wNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Halt, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Halt, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Right) -> False
9.37/3.25	   instrConstrCheck(Halt, Left) -> False
9.37/3.25	   instrConstrCheck(Halt, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Halt) -> True
9.37/3.25	   notEmpty(Cons(x, xs)) -> True
9.37/3.25	   notEmpty(Nil) -> False
9.37/3.25	   lookup(0, instrs) -> instrs
9.37/3.25	   instrsSecond(I(l, r)) -> r
9.37/3.25	   instrsFirst(I(l, r)) -> l
9.37/3.25	   getWrite(Write(int)) -> int
9.37/3.25	   getGotoSecond(IfGoto(i1, i2)) -> i2
9.37/3.25	   getGotoFirst(IfGoto(i1, i2)) -> i1
9.37/3.25	   getGoto(Goto(int)) -> int
9.37/3.25	   run(prog, tapeinput) -> turing(prog, Nil, tapeinput, prog)
9.37/3.25	
9.37/3.25	The (relative) TRS S consists of the following rules:
9.37/3.25	
9.37/3.25	   !EQ(S(x), S(y)) -> !EQ(x, y)
9.37/3.25	   !EQ(0, S(y)) -> False
9.37/3.25	   !EQ(S(x), 0) -> False
9.37/3.25	   !EQ(0, 0) -> True
9.37/3.25	   turing[Ite](True, I(IfGoto(i1, i2), r), revltape, rtape, prog) -> turing(lookup(i2, prog), revltape, rtape, prog)
9.37/3.25	   turing[Ite](False, I(l, r), revltape, rtape, prog) -> turing(r, revltape, rtape, prog)
9.37/3.25	
9.37/3.25	Rewrite Strategy: INNERMOST
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(5) DecreasingLoopProof (LOWER BOUND(ID))
9.37/3.25	The following loop(s) give(s) rise to the lower bound Omega(n^1):
9.37/3.25	
9.37/3.25	The rewrite sequence
9.37/3.25	
9.37/3.25	turing(I(Left, r), Cons(x, xs), rtape, prog) ->^+ turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	
9.37/3.25	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
9.37/3.25	
9.37/3.25	The pumping substitution is [r / I(Left, r), xs / Cons(x, xs)].
9.37/3.25	
9.37/3.25	The result substitution is [rtape / Cons(x, rtape)].
9.37/3.25	
9.37/3.25	
9.37/3.25	
9.37/3.25	
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(6)
9.37/3.25	Complex Obligation (BEST)
9.37/3.25	
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(7)
9.37/3.25	Obligation:
9.37/3.25	Proved the lower bound n^1 for the following obligation:
9.37/3.25	
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	
9.37/3.25	The TRS R consists of the following rules:
9.37/3.25	
9.37/3.25	   turing(I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog) -> turing[Ite](!EQ(x, i1), I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog)
9.37/3.25	   turing(I(Goto(int), r), revltape, rtape, prog) -> turing(lookup(int, prog), revltape, rtape, prog)
9.37/3.25	   turing(I(Right, r), revltape, Cons(x, xs), prog) -> turing(r, Cons(x, revltape), xs, prog)
9.37/3.25	   turing(I(Right, r), revltape, Nil, prog) -> turing(r, Cons(0, revltape), Nil, prog)
9.37/3.25	   turing(I(Left, r), Cons(x, xs), rtape, prog) -> turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	   turing(I(Left, r), Nil, rtape, prog) -> turing(r, Nil, Cons(0, rtape), prog)
9.37/3.25	   turing(I(Write(int), r), revltape, Cons(x, xs), prog) -> turing(r, revltape, Cons(int, xs), prog)
9.37/3.25	   turing(I(Halt, r), revltape, rtape, prog) -> rtape
9.37/3.25	   turing(Empty, revltape, rtape, prog) -> rtape
9.37/3.25	   lookup(S(x), I(l, r)) -> lookup(x, r)
9.37/3.25	   instrsConstrCheck(I(l1, r1), I(x, y)) -> True
9.37/3.25	   instrsConstrCheck(I(l1, r1), Empty) -> False
9.37/3.25	   instrsConstrCheck(Empty, I(x, y)) -> False
9.37/3.25	   instrsConstrCheck(Empty, Empty) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), IfGoto(igtNat12, igtNat22)) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Right) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Left) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Halt) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Goto(gtNat2)) -> True
9.37/3.25	   instrConstrCheck(Goto(gtNat), Right) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Left) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Right, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Right, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Right) -> True
9.37/3.25	   instrConstrCheck(Right, Left) -> False
9.37/3.25	   instrConstrCheck(Right, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Halt) -> False
9.37/3.25	   instrConstrCheck(Left, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Left, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Right) -> False
9.37/3.25	   instrConstrCheck(Left, Left) -> True
9.37/3.25	   instrConstrCheck(Left, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Halt) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Right) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Left) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Write(wNat2)) -> True
9.37/3.25	   instrConstrCheck(Write(wNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Halt, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Halt, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Right) -> False
9.37/3.25	   instrConstrCheck(Halt, Left) -> False
9.37/3.25	   instrConstrCheck(Halt, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Halt) -> True
9.37/3.25	   notEmpty(Cons(x, xs)) -> True
9.37/3.25	   notEmpty(Nil) -> False
9.37/3.25	   lookup(0, instrs) -> instrs
9.37/3.25	   instrsSecond(I(l, r)) -> r
9.37/3.25	   instrsFirst(I(l, r)) -> l
9.37/3.25	   getWrite(Write(int)) -> int
9.37/3.25	   getGotoSecond(IfGoto(i1, i2)) -> i2
9.37/3.25	   getGotoFirst(IfGoto(i1, i2)) -> i1
9.37/3.25	   getGoto(Goto(int)) -> int
9.37/3.25	   run(prog, tapeinput) -> turing(prog, Nil, tapeinput, prog)
9.37/3.25	
9.37/3.25	The (relative) TRS S consists of the following rules:
9.37/3.25	
9.37/3.25	   !EQ(S(x), S(y)) -> !EQ(x, y)
9.37/3.25	   !EQ(0, S(y)) -> False
9.37/3.25	   !EQ(S(x), 0) -> False
9.37/3.25	   !EQ(0, 0) -> True
9.37/3.25	   turing[Ite](True, I(IfGoto(i1, i2), r), revltape, rtape, prog) -> turing(lookup(i2, prog), revltape, rtape, prog)
9.37/3.25	   turing[Ite](False, I(l, r), revltape, rtape, prog) -> turing(r, revltape, rtape, prog)
9.37/3.25	
9.37/3.25	Rewrite Strategy: INNERMOST
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(8) LowerBoundPropagationProof (FINISHED)
9.37/3.25	Propagated lower bound.
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(9)
9.37/3.25	BOUNDS(n^1, INF)
9.37/3.25	
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(10)
9.37/3.25	Obligation:
9.37/3.25	Analyzing the following TRS for decreasing loops:
9.37/3.25	
9.37/3.25	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).
9.37/3.25	
9.37/3.25	
9.37/3.25	The TRS R consists of the following rules:
9.37/3.25	
9.37/3.25	   turing(I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog) -> turing[Ite](!EQ(x, i1), I(IfGoto(i1, i2), r), revltape, Cons(x, xs), prog)
9.37/3.25	   turing(I(Goto(int), r), revltape, rtape, prog) -> turing(lookup(int, prog), revltape, rtape, prog)
9.37/3.25	   turing(I(Right, r), revltape, Cons(x, xs), prog) -> turing(r, Cons(x, revltape), xs, prog)
9.37/3.25	   turing(I(Right, r), revltape, Nil, prog) -> turing(r, Cons(0, revltape), Nil, prog)
9.37/3.25	   turing(I(Left, r), Cons(x, xs), rtape, prog) -> turing(r, xs, Cons(x, rtape), prog)
9.37/3.25	   turing(I(Left, r), Nil, rtape, prog) -> turing(r, Nil, Cons(0, rtape), prog)
9.37/3.25	   turing(I(Write(int), r), revltape, Cons(x, xs), prog) -> turing(r, revltape, Cons(int, xs), prog)
9.37/3.25	   turing(I(Halt, r), revltape, rtape, prog) -> rtape
9.37/3.25	   turing(Empty, revltape, rtape, prog) -> rtape
9.37/3.25	   lookup(S(x), I(l, r)) -> lookup(x, r)
9.37/3.25	   instrsConstrCheck(I(l1, r1), I(x, y)) -> True
9.37/3.25	   instrsConstrCheck(I(l1, r1), Empty) -> False
9.37/3.25	   instrsConstrCheck(Empty, I(x, y)) -> False
9.37/3.25	   instrsConstrCheck(Empty, Empty) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), IfGoto(igtNat12, igtNat22)) -> True
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Right) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Left) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(IfGoto(igtNat1, igtNat2), Halt) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Goto(gtNat2)) -> True
9.37/3.25	   instrConstrCheck(Goto(gtNat), Right) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Left) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Goto(gtNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Right, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Right, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Right) -> True
9.37/3.25	   instrConstrCheck(Right, Left) -> False
9.37/3.25	   instrConstrCheck(Right, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Right, Halt) -> False
9.37/3.25	   instrConstrCheck(Left, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Left, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Right) -> False
9.37/3.25	   instrConstrCheck(Left, Left) -> True
9.37/3.25	   instrConstrCheck(Left, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Left, Halt) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Right) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Left) -> False
9.37/3.25	   instrConstrCheck(Write(wNat), Write(wNat2)) -> True
9.37/3.25	   instrConstrCheck(Write(wNat), Halt) -> False
9.37/3.25	   instrConstrCheck(Halt, IfGoto(igtNat12, igtNat22)) -> False
9.37/3.25	   instrConstrCheck(Halt, Goto(gtNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Right) -> False
9.37/3.25	   instrConstrCheck(Halt, Left) -> False
9.37/3.25	   instrConstrCheck(Halt, Write(wNat2)) -> False
9.37/3.25	   instrConstrCheck(Halt, Halt) -> True
9.37/3.25	   notEmpty(Cons(x, xs)) -> True
9.37/3.25	   notEmpty(Nil) -> False
9.37/3.25	   lookup(0, instrs) -> instrs
9.37/3.25	   instrsSecond(I(l, r)) -> r
9.37/3.25	   instrsFirst(I(l, r)) -> l
9.37/3.25	   getWrite(Write(int)) -> int
9.37/3.25	   getGotoSecond(IfGoto(i1, i2)) -> i2
9.37/3.25	   getGotoFirst(IfGoto(i1, i2)) -> i1
9.37/3.25	   getGoto(Goto(int)) -> int
9.37/3.25	   run(prog, tapeinput) -> turing(prog, Nil, tapeinput, prog)
9.37/3.25	
9.37/3.25	The (relative) TRS S consists of the following rules:
9.37/3.25	
9.37/3.25	   !EQ(S(x), S(y)) -> !EQ(x, y)
9.37/3.25	   !EQ(0, S(y)) -> False
9.37/3.25	   !EQ(S(x), 0) -> False
9.37/3.25	   !EQ(0, 0) -> True
9.37/3.25	   turing[Ite](True, I(IfGoto(i1, i2), r), revltape, rtape, prog) -> turing(lookup(i2, prog), revltape, rtape, prog)
9.37/3.25	   turing[Ite](False, I(l, r), revltape, rtape, prog) -> turing(r, revltape, rtape, prog)
9.37/3.25	
9.37/3.25	Rewrite Strategy: INNERMOST
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(11) InfiniteLowerBoundProof (FINISHED)
9.37/3.25	The following loop proves infinite runtime complexity:
9.37/3.25	
9.37/3.25	The rewrite sequence
9.37/3.25	
9.37/3.25	turing(I(Goto(0), r), revltape, rtape, I(Goto(0), r2_0)) ->^+ turing(I(Goto(0), r2_0), revltape, rtape, I(Goto(0), r2_0))
9.37/3.25	
9.37/3.25	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
9.37/3.25	
9.37/3.25	The pumping substitution is [ ].
9.37/3.25	
9.37/3.25	The result substitution is [r / r2_0].
9.37/3.25	
9.37/3.25	
9.37/3.25	
9.37/3.25	
9.37/3.25	----------------------------------------
9.37/3.25	
9.37/3.25	(12)
9.37/3.25	BOUNDS(INF, INF)
9.61/3.30	EOF