3.91/1.92	WORST_CASE(NON_POLY, ?)
3.91/1.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.91/1.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.91/1.93	
3.91/1.93	
3.91/1.93	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(EXP, EXP).
3.91/1.93	
3.91/1.93	(0) CpxIntTrs
3.91/1.93	(1) Koat Proof [FINISHED, 15 ms]
3.91/1.93	(2) BOUNDS(1, EXP)
3.91/1.93	(3) Loat Proof [FINISHED, 322 ms]
3.91/1.93	(4) BOUNDS(EXP, INF)
3.91/1.93	
3.91/1.93	
3.91/1.93	----------------------------------------
3.91/1.93	
3.91/1.93	(0)
3.91/1.93	Obligation:
3.91/1.93	Complexity Int TRS consisting of the following rules:
3.91/1.93	f(A, B) -> Com_1(g(A, 1)) :|: TRUE
3.91/1.93	g(A, B) -> Com_1(g(A - 1, B + B)) :|: A > 0
3.91/1.93	g(A, B) -> Com_1(h(A, B)) :|: A <= 0
3.91/1.93	h(A, B) -> Com_1(h(A, B - 1)) :|: B > 0
3.91/1.93	
3.91/1.93	The start-symbols are:[f_2]
3.91/1.93	
3.91/1.93	
3.91/1.93	----------------------------------------
3.91/1.93	
3.91/1.93	(1) Koat Proof (FINISHED)
3.91/1.93	YES(?, pow(2, ar_1) + ar_1 + 2)
3.91/1.93	
3.91/1.93	
3.91/1.93	
3.91/1.93	Initial complexity problem:
3.91/1.93	
3.91/1.93	1:	T:
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    f(ar_0, ar_1) -> Com_1(g(1, ar_1))
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]
3.91/1.93	
3.91/1.93		start location:	koat_start
3.91/1.93	
3.91/1.93		leaf cost:	0
3.91/1.93	
3.91/1.93	
3.91/1.93	
3.91/1.93	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.91/1.93	
3.91/1.93	2:	T:
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 1)    f(ar_0, ar_1) -> Com_1(g(1, ar_1))
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]
3.91/1.93	
3.91/1.93		start location:	koat_start
3.91/1.93	
3.91/1.93		leaf cost:	0
3.91/1.93	
3.91/1.93	
3.91/1.93	
3.91/1.93	A polynomial rank function with
3.91/1.93	
3.91/1.93		Pol(f) = 1
3.91/1.93	
3.91/1.93		Pol(g) = 1
3.91/1.93	
3.91/1.93		Pol(h) = 0
3.91/1.93	
3.91/1.93		Pol(koat_start) = 1
3.91/1.93	
3.91/1.93	orients all transitions weakly and the transition
3.91/1.93	
3.91/1.93		g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
3.91/1.93	
3.91/1.93	strictly and produces the following problem:
3.91/1.93	
3.91/1.93	3:	T:
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 1)    f(ar_0, ar_1) -> Com_1(g(1, ar_1))
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 1)    g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)    h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]
3.91/1.93	
3.91/1.93		start location:	koat_start
3.91/1.93	
3.91/1.93		leaf cost:	0
3.91/1.93	
3.91/1.93	
3.91/1.93	
3.91/1.93	A polynomial rank function with
3.91/1.93	
3.91/1.93		Pol(f) = V_2
3.91/1.93	
3.91/1.93		Pol(g) = V_2
3.91/1.93	
3.91/1.93		Pol(h) = V_2
3.91/1.93	
3.91/1.93		Pol(koat_start) = V_2
3.91/1.93	
3.91/1.93	orients all transitions weakly and the transition
3.91/1.93	
3.91/1.93		g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
3.91/1.93	
3.91/1.93	strictly and produces the following problem:
3.91/1.93	
3.91/1.93	4:	T:
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 1)       f(ar_0, ar_1) -> Com_1(g(1, ar_1))
3.91/1.93	
3.91/1.93			(Comp: ar_1, Cost: 1)    g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 1)       g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
3.91/1.93	
3.91/1.93			(Comp: ?, Cost: 1)       h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
3.91/1.93	
3.91/1.93			(Comp: 1, Cost: 0)       koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]
4.01/1.93	
4.01/1.93		start location:	koat_start
4.01/1.93	
4.01/1.93		leaf cost:	0
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	A polynomial rank function with
4.01/1.93	
4.01/1.93		Pol(h) = V_1
4.01/1.93	
4.01/1.93	and size complexities
4.01/1.93	
4.01/1.93		S("koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]", 0-0) = ar_0
4.01/1.93	
4.01/1.93		S("koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]", 0-1) = ar_1
4.01/1.93	
4.01/1.93		S("h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]", 0-0) = pow(2, ar_1)
4.01/1.93	
4.01/1.93		S("h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]", 0-1) = ar_1
4.01/1.93	
4.01/1.93		S("g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]", 0-0) = pow(2, ar_1)
4.01/1.93	
4.01/1.93		S("g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]", 0-1) = ar_1
4.01/1.93	
4.01/1.93		S("g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]", 0-0) = pow(2, ar_1)
4.01/1.93	
4.01/1.93		S("g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]", 0-1) = ar_1
4.01/1.93	
4.01/1.93		S("f(ar_0, ar_1) -> Com_1(g(1, ar_1))", 0-0) = 1
4.01/1.93	
4.01/1.93		S("f(ar_0, ar_1) -> Com_1(g(1, ar_1))", 0-1) = ar_1
4.01/1.93	
4.01/1.93	orients the transitions
4.01/1.93	
4.01/1.93		h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
4.01/1.93	
4.01/1.93	weakly and the transition
4.01/1.93	
4.01/1.93		h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
4.01/1.93	
4.01/1.93	strictly and produces the following problem:
4.01/1.93	
4.01/1.93	5:	T:
4.01/1.93	
4.01/1.93			(Comp: 1, Cost: 1)               f(ar_0, ar_1) -> Com_1(g(1, ar_1))
4.01/1.93	
4.01/1.93			(Comp: ar_1, Cost: 1)            g(ar_0, ar_1) -> Com_1(g(2*ar_0, ar_1 - 1)) [ ar_1 >= 1 ]
4.01/1.93	
4.01/1.93			(Comp: 1, Cost: 1)               g(ar_0, ar_1) -> Com_1(h(ar_0, ar_1)) [ 0 >= ar_1 ]
4.01/1.93	
4.01/1.93			(Comp: pow(2, ar_1), Cost: 1)    h(ar_0, ar_1) -> Com_1(h(ar_0 - 1, ar_1)) [ ar_0 >= 1 ]
4.01/1.93	
4.01/1.93			(Comp: 1, Cost: 0)               koat_start(ar_0, ar_1) -> Com_1(f(ar_0, ar_1)) [ 0 <= 0 ]
4.01/1.93	
4.01/1.93		start location:	koat_start
4.01/1.93	
4.01/1.93		leaf cost:	0
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Complexity upper bound pow(2, ar_1) + ar_1 + 2
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Time: 0.098 sec (SMT: 0.093 sec)
4.01/1.93	
4.01/1.93	
4.01/1.93	----------------------------------------
4.01/1.93	
4.01/1.93	(2)
4.01/1.93	BOUNDS(1, EXP)
4.01/1.93	
4.01/1.93	----------------------------------------
4.01/1.93	
4.01/1.93	(3) Loat Proof (FINISHED)
4.01/1.93	
4.01/1.93	
4.01/1.93	### Pre-processing the ITS problem ###
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Initial linear ITS problem
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      0: f -> g : A'=1, [], cost: 1
4.01/1.93	
4.01/1.93	      1: g -> g : A'=2*A, B'=-1+B, [ B>=1 ], cost: 1
4.01/1.93	
4.01/1.93	      2: g -> h : [ 0>=B ], cost: 1
4.01/1.93	
4.01/1.93	      3: h -> h : A'=-1+A, [ A>=1 ], cost: 1
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	### Simplification by acceleration and chaining ###
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Accelerating simple loops of location 1.
4.01/1.93	
4.01/1.93	   Accelerating the following rules:
4.01/1.93	
4.01/1.93	      1: g -> g : A'=2*A, B'=-1+B, [ B>=1 ], cost: 1
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	   Accelerated rule 1 with metering function B, yielding the new rule 4.
4.01/1.93	
4.01/1.93	   Removing the simple loops: 1.
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Accelerating simple loops of location 2.
4.01/1.93	
4.01/1.93	   Accelerating the following rules:
4.01/1.93	
4.01/1.93	      3: h -> h : A'=-1+A, [ A>=1 ], cost: 1
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	   Accelerated rule 3 with metering function A, yielding the new rule 5.
4.01/1.93	
4.01/1.93	   Removing the simple loops: 3.
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Accelerated all simple loops using metering functions (where possible):
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      0: f -> g : A'=1, [], cost: 1
4.01/1.93	
4.01/1.93	      2: g -> h : [ 0>=B ], cost: 1
4.01/1.93	
4.01/1.93	      4: g -> g : A'=A*2^B, B'=0, [ B>=1 ], cost: B
4.01/1.93	
4.01/1.93	      5: h -> h : A'=0, [ A>=1 ], cost: A
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Chained accelerated rules (with incoming rules):
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      0: f -> g : A'=1, [], cost: 1
4.01/1.93	
4.01/1.93	      6: f -> g : A'=2^B, B'=0, [ B>=1 ], cost: 1+B
4.01/1.93	
4.01/1.93	      2: g -> h : [ 0>=B ], cost: 1
4.01/1.93	
4.01/1.93	      7: g -> h : A'=0, [ 0>=B && A>=1 ], cost: 1+A
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Removed unreachable locations (and leaf rules with constant cost):
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      0: f -> g : A'=1, [], cost: 1
4.01/1.93	
4.01/1.93	      6: f -> g : A'=2^B, B'=0, [ B>=1 ], cost: 1+B
4.01/1.93	
4.01/1.93	      7: g -> h : A'=0, [ 0>=B && A>=1 ], cost: 1+A
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Eliminated locations (on tree-shaped paths):
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      8: f -> h : A'=0, [ 0>=B ], cost: 3
4.01/1.93	
4.01/1.93	      9: f -> h : A'=0, B'=0, [ B>=1 && 2^B>=1 ], cost: 2+2^B+B
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Applied pruning (of leafs and parallel rules):
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      9: f -> h : A'=0, B'=0, [ B>=1 && 2^B>=1 ], cost: 2+2^B+B
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	### Computing asymptotic complexity ###
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Fully simplified ITS problem
4.01/1.93	
4.01/1.93	   Start location: f
4.01/1.93	
4.01/1.93	      9: f -> h : A'=0, B'=0, [ B>=1 && 2^B>=1 ], cost: 2+2^B+B
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Computing asymptotic complexity for rule 9
4.01/1.93	
4.01/1.93	   Solved the limit problem by the following transformations:
4.01/1.93	
4.01/1.93	      Created initial limit problem:
4.01/1.93	
4.01/1.93	      2+2^B+B (+), 2^B (+/+!), B (+/+!) [not solved]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	      applying transformation rule (E), replacing 2+2^B+B (+) by 1 (+/+!) and B (+)
4.01/1.93	
4.01/1.93	      resulting limit problem:
4.01/1.93	
4.01/1.93	      1 (+/+!), 2^B (+/+!), B (+) [not solved]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	      applying transformation rule (B), deleting 1 (+/+!)
4.01/1.93	
4.01/1.93	      resulting limit problem:
4.01/1.93	
4.01/1.93	      2^B (+/+!), B (+) [not solved]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	      applying transformation rule (E), replacing 2^B (+/+!) by 1 (+/+!) and B (+)
4.01/1.93	
4.01/1.93	      resulting limit problem:
4.01/1.93	
4.01/1.93	      1 (+/+!), B (+) [not solved]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	      applying transformation rule (B), deleting 1 (+/+!)
4.01/1.93	
4.01/1.93	      resulting limit problem:
4.01/1.93	
4.01/1.93	      B (+) [solved]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	   Solution:
4.01/1.93	
4.01/1.93	      B / n
4.01/1.93	
4.01/1.93	   Resulting cost 2+n+2^n has complexity: Exp
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Found new complexity Exp.
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	Obtained the following overall complexity (w.r.t. the length of the input n):
4.01/1.93	
4.01/1.93	   Complexity:  Exp
4.01/1.93	
4.01/1.93	   Cpx degree:  Exp
4.01/1.93	
4.01/1.93	   Solved cost: 2+n+2^n
4.01/1.93	
4.01/1.93	   Rule cost:   2+2^B+B
4.01/1.93	
4.01/1.93	   Rule guard:  [ B>=1 && 2^B>=1 ]
4.01/1.93	
4.01/1.93	
4.01/1.93	
4.01/1.93	WORST_CASE(EXP,?)
4.01/1.93	
4.01/1.93	
4.01/1.93	----------------------------------------
4.01/1.93	
4.01/1.93	(4)
4.01/1.93	BOUNDS(EXP, INF)
4.01/1.94	EOF