3.83/1.73	WORST_CASE(NON_POLY, ?)
3.83/1.74	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.83/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.83/1.74	
3.83/1.74	
3.83/1.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.83/1.74	
3.83/1.74	(0) CpxTRS
3.83/1.74	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.83/1.74	(2) TRS for Loop Detection
3.83/1.74	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.83/1.74	(4) BEST
3.83/1.74	    (5) proven lower bound
3.83/1.74	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.83/1.74	        (7) BOUNDS(n^1, INF)
3.83/1.74	    (8) TRS for Loop Detection
3.83/1.74	        (9) DecreasingLoopProof [FINISHED, 128 ms]
3.83/1.74	        (10) BOUNDS(EXP, INF)
3.83/1.74	
3.83/1.74	
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(0)
3.83/1.74	Obligation:
3.83/1.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.83/1.74	
3.83/1.74	
3.83/1.74	The TRS R consists of the following rules:
3.83/1.74	
3.83/1.74	   0(#) -> #
3.83/1.74	   +(x, #) -> x
3.83/1.74	   +(#, x) -> x
3.83/1.74	   +(0(x), 0(y)) -> 0(+(x, y))
3.83/1.74	   +(0(x), 1(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 0(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
3.83/1.74	   +(+(x, y), z) -> +(x, +(y, z))
3.83/1.74	   -(#, x) -> #
3.83/1.74	   -(x, #) -> x
3.83/1.74	   -(0(x), 0(y)) -> 0(-(x, y))
3.83/1.74	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
3.83/1.74	   -(1(x), 0(y)) -> 1(-(x, y))
3.83/1.74	   -(1(x), 1(y)) -> 0(-(x, y))
3.83/1.74	   not(true) -> false
3.83/1.74	   not(false) -> true
3.83/1.74	   if(true, x, y) -> x
3.83/1.74	   if(false, x, y) -> y
3.83/1.74	   eq(#, #) -> true
3.83/1.74	   eq(#, 1(y)) -> false
3.83/1.74	   eq(1(x), #) -> false
3.83/1.74	   eq(#, 0(y)) -> eq(#, y)
3.83/1.74	   eq(0(x), #) -> eq(x, #)
3.83/1.74	   eq(1(x), 1(y)) -> eq(x, y)
3.83/1.74	   eq(0(x), 1(y)) -> false
3.83/1.74	   eq(1(x), 0(y)) -> false
3.83/1.74	   eq(0(x), 0(y)) -> eq(x, y)
3.83/1.74	   ge(0(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(0(x), 1(y)) -> not(ge(y, x))
3.83/1.74	   ge(1(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(1(x), 1(y)) -> ge(x, y)
3.83/1.74	   ge(x, #) -> true
3.83/1.74	   ge(#, 0(x)) -> ge(#, x)
3.83/1.74	   ge(#, 1(x)) -> false
3.83/1.74	   log(x) -> -(log'(x), 1(#))
3.83/1.74	   log'(#) -> #
3.83/1.74	   log'(1(x)) -> +(log'(x), 1(#))
3.83/1.74	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
3.83/1.74	   *(#, x) -> #
3.83/1.74	   *(0(x), y) -> 0(*(x, y))
3.83/1.74	   *(1(x), y) -> +(0(*(x, y)), y)
3.83/1.74	   *(*(x, y), z) -> *(x, *(y, z))
3.83/1.74	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
3.83/1.74	   app(nil, l) -> l
3.83/1.74	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
3.83/1.74	   sum(nil) -> 0(#)
3.83/1.74	   sum(cons(x, l)) -> +(x, sum(l))
3.83/1.74	   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
3.83/1.74	   prod(nil) -> 1(#)
3.83/1.74	   prod(cons(x, l)) -> *(x, prod(l))
3.83/1.74	   prod(app(l1, l2)) -> *(prod(l1), prod(l2))
3.83/1.74	   mem(x, nil) -> false
3.83/1.74	   mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
3.83/1.74	   inter(x, nil) -> nil
3.83/1.74	   inter(nil, x) -> nil
3.83/1.74	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
3.83/1.74	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
3.83/1.74	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
3.83/1.74	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
3.83/1.74	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
3.83/1.74	   ifinter(false, x, l1, l2) -> inter(l1, l2)
3.83/1.74	
3.83/1.74	S is empty.
3.83/1.74	Rewrite Strategy: FULL
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.83/1.74	Transformed a relative TRS into a decreasing-loop problem.
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(2)
3.83/1.74	Obligation:
3.83/1.74	Analyzing the following TRS for decreasing loops:
3.83/1.74	
3.83/1.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.83/1.74	
3.83/1.74	
3.83/1.74	The TRS R consists of the following rules:
3.83/1.74	
3.83/1.74	   0(#) -> #
3.83/1.74	   +(x, #) -> x
3.83/1.74	   +(#, x) -> x
3.83/1.74	   +(0(x), 0(y)) -> 0(+(x, y))
3.83/1.74	   +(0(x), 1(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 0(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
3.83/1.74	   +(+(x, y), z) -> +(x, +(y, z))
3.83/1.74	   -(#, x) -> #
3.83/1.74	   -(x, #) -> x
3.83/1.74	   -(0(x), 0(y)) -> 0(-(x, y))
3.83/1.74	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
3.83/1.74	   -(1(x), 0(y)) -> 1(-(x, y))
3.83/1.74	   -(1(x), 1(y)) -> 0(-(x, y))
3.83/1.74	   not(true) -> false
3.83/1.74	   not(false) -> true
3.83/1.74	   if(true, x, y) -> x
3.83/1.74	   if(false, x, y) -> y
3.83/1.74	   eq(#, #) -> true
3.83/1.74	   eq(#, 1(y)) -> false
3.83/1.74	   eq(1(x), #) -> false
3.83/1.74	   eq(#, 0(y)) -> eq(#, y)
3.83/1.74	   eq(0(x), #) -> eq(x, #)
3.83/1.74	   eq(1(x), 1(y)) -> eq(x, y)
3.83/1.74	   eq(0(x), 1(y)) -> false
3.83/1.74	   eq(1(x), 0(y)) -> false
3.83/1.74	   eq(0(x), 0(y)) -> eq(x, y)
3.83/1.74	   ge(0(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(0(x), 1(y)) -> not(ge(y, x))
3.83/1.74	   ge(1(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(1(x), 1(y)) -> ge(x, y)
3.83/1.74	   ge(x, #) -> true
3.83/1.74	   ge(#, 0(x)) -> ge(#, x)
3.83/1.74	   ge(#, 1(x)) -> false
3.83/1.74	   log(x) -> -(log'(x), 1(#))
3.83/1.74	   log'(#) -> #
3.83/1.74	   log'(1(x)) -> +(log'(x), 1(#))
3.83/1.74	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
3.83/1.74	   *(#, x) -> #
3.83/1.74	   *(0(x), y) -> 0(*(x, y))
3.83/1.74	   *(1(x), y) -> +(0(*(x, y)), y)
3.83/1.74	   *(*(x, y), z) -> *(x, *(y, z))
3.83/1.74	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
3.83/1.74	   app(nil, l) -> l
3.83/1.74	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
3.83/1.74	   sum(nil) -> 0(#)
3.83/1.74	   sum(cons(x, l)) -> +(x, sum(l))
3.83/1.74	   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
3.83/1.74	   prod(nil) -> 1(#)
3.83/1.74	   prod(cons(x, l)) -> *(x, prod(l))
3.83/1.74	   prod(app(l1, l2)) -> *(prod(l1), prod(l2))
3.83/1.74	   mem(x, nil) -> false
3.83/1.74	   mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
3.83/1.74	   inter(x, nil) -> nil
3.83/1.74	   inter(nil, x) -> nil
3.83/1.74	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
3.83/1.74	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
3.83/1.74	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
3.83/1.74	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
3.83/1.74	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
3.83/1.74	   ifinter(false, x, l1, l2) -> inter(l1, l2)
3.83/1.74	
3.83/1.74	S is empty.
3.83/1.74	Rewrite Strategy: FULL
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.83/1.74	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.83/1.74	
3.83/1.74	The rewrite sequence
3.83/1.74	
3.83/1.74	+(1(x), 1(y)) ->^+ 0(+(+(x, y), 1(#)))
3.83/1.74	
3.83/1.74	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.83/1.74	
3.83/1.74	The pumping substitution is [x / 1(x), y / 1(y)].
3.83/1.74	
3.83/1.74	The result substitution is [ ].
3.83/1.74	
3.83/1.74	
3.83/1.74	
3.83/1.74	
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(4)
3.83/1.74	Complex Obligation (BEST)
3.83/1.74	
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(5)
3.83/1.74	Obligation:
3.83/1.74	Proved the lower bound n^1 for the following obligation:
3.83/1.74	
3.83/1.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.83/1.74	
3.83/1.74	
3.83/1.74	The TRS R consists of the following rules:
3.83/1.74	
3.83/1.74	   0(#) -> #
3.83/1.74	   +(x, #) -> x
3.83/1.74	   +(#, x) -> x
3.83/1.74	   +(0(x), 0(y)) -> 0(+(x, y))
3.83/1.74	   +(0(x), 1(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 0(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
3.83/1.74	   +(+(x, y), z) -> +(x, +(y, z))
3.83/1.74	   -(#, x) -> #
3.83/1.74	   -(x, #) -> x
3.83/1.74	   -(0(x), 0(y)) -> 0(-(x, y))
3.83/1.74	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
3.83/1.74	   -(1(x), 0(y)) -> 1(-(x, y))
3.83/1.74	   -(1(x), 1(y)) -> 0(-(x, y))
3.83/1.74	   not(true) -> false
3.83/1.74	   not(false) -> true
3.83/1.74	   if(true, x, y) -> x
3.83/1.74	   if(false, x, y) -> y
3.83/1.74	   eq(#, #) -> true
3.83/1.74	   eq(#, 1(y)) -> false
3.83/1.74	   eq(1(x), #) -> false
3.83/1.74	   eq(#, 0(y)) -> eq(#, y)
3.83/1.74	   eq(0(x), #) -> eq(x, #)
3.83/1.74	   eq(1(x), 1(y)) -> eq(x, y)
3.83/1.74	   eq(0(x), 1(y)) -> false
3.83/1.74	   eq(1(x), 0(y)) -> false
3.83/1.74	   eq(0(x), 0(y)) -> eq(x, y)
3.83/1.74	   ge(0(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(0(x), 1(y)) -> not(ge(y, x))
3.83/1.74	   ge(1(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(1(x), 1(y)) -> ge(x, y)
3.83/1.74	   ge(x, #) -> true
3.83/1.74	   ge(#, 0(x)) -> ge(#, x)
3.83/1.74	   ge(#, 1(x)) -> false
3.83/1.74	   log(x) -> -(log'(x), 1(#))
3.83/1.74	   log'(#) -> #
3.83/1.74	   log'(1(x)) -> +(log'(x), 1(#))
3.83/1.74	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
3.83/1.74	   *(#, x) -> #
3.83/1.74	   *(0(x), y) -> 0(*(x, y))
3.83/1.74	   *(1(x), y) -> +(0(*(x, y)), y)
3.83/1.74	   *(*(x, y), z) -> *(x, *(y, z))
3.83/1.74	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
3.83/1.74	   app(nil, l) -> l
3.83/1.74	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
3.83/1.74	   sum(nil) -> 0(#)
3.83/1.74	   sum(cons(x, l)) -> +(x, sum(l))
3.83/1.74	   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
3.83/1.74	   prod(nil) -> 1(#)
3.83/1.74	   prod(cons(x, l)) -> *(x, prod(l))
3.83/1.74	   prod(app(l1, l2)) -> *(prod(l1), prod(l2))
3.83/1.74	   mem(x, nil) -> false
3.83/1.74	   mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
3.83/1.74	   inter(x, nil) -> nil
3.83/1.74	   inter(nil, x) -> nil
3.83/1.74	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
3.83/1.74	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
3.83/1.74	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
3.83/1.74	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
3.83/1.74	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
3.83/1.74	   ifinter(false, x, l1, l2) -> inter(l1, l2)
3.83/1.74	
3.83/1.74	S is empty.
3.83/1.74	Rewrite Strategy: FULL
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(6) LowerBoundPropagationProof (FINISHED)
3.83/1.74	Propagated lower bound.
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(7)
3.83/1.74	BOUNDS(n^1, INF)
3.83/1.74	
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(8)
3.83/1.74	Obligation:
3.83/1.74	Analyzing the following TRS for decreasing loops:
3.83/1.74	
3.83/1.74	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.83/1.74	
3.83/1.74	
3.83/1.74	The TRS R consists of the following rules:
3.83/1.74	
3.83/1.74	   0(#) -> #
3.83/1.74	   +(x, #) -> x
3.83/1.74	   +(#, x) -> x
3.83/1.74	   +(0(x), 0(y)) -> 0(+(x, y))
3.83/1.74	   +(0(x), 1(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 0(y)) -> 1(+(x, y))
3.83/1.74	   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
3.83/1.74	   +(+(x, y), z) -> +(x, +(y, z))
3.83/1.74	   -(#, x) -> #
3.83/1.74	   -(x, #) -> x
3.83/1.74	   -(0(x), 0(y)) -> 0(-(x, y))
3.83/1.74	   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
3.83/1.74	   -(1(x), 0(y)) -> 1(-(x, y))
3.83/1.74	   -(1(x), 1(y)) -> 0(-(x, y))
3.83/1.74	   not(true) -> false
3.83/1.74	   not(false) -> true
3.83/1.74	   if(true, x, y) -> x
3.83/1.74	   if(false, x, y) -> y
3.83/1.74	   eq(#, #) -> true
3.83/1.74	   eq(#, 1(y)) -> false
3.83/1.74	   eq(1(x), #) -> false
3.83/1.74	   eq(#, 0(y)) -> eq(#, y)
3.83/1.74	   eq(0(x), #) -> eq(x, #)
3.83/1.74	   eq(1(x), 1(y)) -> eq(x, y)
3.83/1.74	   eq(0(x), 1(y)) -> false
3.83/1.74	   eq(1(x), 0(y)) -> false
3.83/1.74	   eq(0(x), 0(y)) -> eq(x, y)
3.83/1.74	   ge(0(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(0(x), 1(y)) -> not(ge(y, x))
3.83/1.74	   ge(1(x), 0(y)) -> ge(x, y)
3.83/1.74	   ge(1(x), 1(y)) -> ge(x, y)
3.83/1.74	   ge(x, #) -> true
3.83/1.74	   ge(#, 0(x)) -> ge(#, x)
3.83/1.74	   ge(#, 1(x)) -> false
3.83/1.74	   log(x) -> -(log'(x), 1(#))
3.83/1.74	   log'(#) -> #
3.83/1.74	   log'(1(x)) -> +(log'(x), 1(#))
3.83/1.74	   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
3.83/1.74	   *(#, x) -> #
3.83/1.74	   *(0(x), y) -> 0(*(x, y))
3.83/1.74	   *(1(x), y) -> +(0(*(x, y)), y)
3.83/1.74	   *(*(x, y), z) -> *(x, *(y, z))
3.83/1.74	   *(x, +(y, z)) -> +(*(x, y), *(x, z))
3.83/1.74	   app(nil, l) -> l
3.83/1.74	   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
3.83/1.74	   sum(nil) -> 0(#)
3.83/1.74	   sum(cons(x, l)) -> +(x, sum(l))
3.83/1.74	   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
3.83/1.74	   prod(nil) -> 1(#)
3.83/1.74	   prod(cons(x, l)) -> *(x, prod(l))
3.83/1.74	   prod(app(l1, l2)) -> *(prod(l1), prod(l2))
3.83/1.74	   mem(x, nil) -> false
3.83/1.74	   mem(x, cons(y, l)) -> if(eq(x, y), true, mem(x, l))
3.83/1.74	   inter(x, nil) -> nil
3.83/1.74	   inter(nil, x) -> nil
3.83/1.74	   inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3))
3.83/1.74	   inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3))
3.83/1.74	   inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2)
3.83/1.74	   inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1)
3.83/1.74	   ifinter(true, x, l1, l2) -> cons(x, inter(l1, l2))
3.83/1.74	   ifinter(false, x, l1, l2) -> inter(l1, l2)
3.83/1.74	
3.83/1.74	S is empty.
3.83/1.74	Rewrite Strategy: FULL
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(9) DecreasingLoopProof (FINISHED)
3.83/1.74	The following loop(s) give(s) rise to the lower bound EXP:
3.83/1.74	
3.83/1.74	The rewrite sequence
3.83/1.74	
3.83/1.74	prod(cons(1(x1_0), l)) ->^+ +(0(*(x1_0, prod(l))), prod(l))
3.83/1.74	
3.83/1.74	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1].
3.83/1.74	
3.83/1.74	The pumping substitution is [l / cons(1(x1_0), l)].
3.83/1.74	
3.83/1.74	The result substitution is [ ].
3.83/1.74	
3.83/1.74	
3.83/1.74	
3.83/1.74	The rewrite sequence
3.83/1.74	
3.83/1.74	prod(cons(1(x1_0), l)) ->^+ +(0(*(x1_0, prod(l))), prod(l))
3.83/1.74	
3.83/1.74	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.83/1.74	
3.83/1.74	The pumping substitution is [l / cons(1(x1_0), l)].
3.83/1.74	
3.83/1.74	The result substitution is [ ].
3.83/1.74	
3.83/1.74	
3.83/1.74	
3.83/1.74	
3.83/1.74	----------------------------------------
3.83/1.74	
3.83/1.74	(10)
3.83/1.74	BOUNDS(EXP, INF)
4.09/2.43	EOF