3.99/2.32	WORST_CASE(Omega(n^1), O(n^1))
3.99/2.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.koat
3.99/2.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.99/2.32	
3.99/2.32	
3.99/2.32	The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1).
3.99/2.32	
3.99/2.32	(0) CpxIntTrs
3.99/2.32	(1) Koat Proof [FINISHED, 30 ms]
3.99/2.32	(2) BOUNDS(1, n^1)
3.99/2.32	(3) Loat Proof [FINISHED, 317 ms]
3.99/2.32	(4) BOUNDS(n^1, INF)
3.99/2.32	
3.99/2.32	
3.99/2.32	----------------------------------------
3.99/2.32	
3.99/2.32	(0)
3.99/2.32	Obligation:
3.99/2.32	Complexity Int TRS consisting of the following rules:
3.99/2.32	eval1(A, B, C) -> Com_1(eval2(A, B, C)) :|: A >= B + 1 && C >= A && C <= A
3.99/2.32	eval2(A, B, C) -> Com_1(eval2(A - 1, B, C - 1)) :|: A >= B + 1
3.99/2.32	eval2(A, B, C) -> Com_1(eval1(A, B, C)) :|: B >= A
3.99/2.32	start(A, B, C) -> Com_1(eval1(A, B, C)) :|: TRUE
3.99/2.32	
3.99/2.32	The start-symbols are:[start_3]
3.99/2.32	
3.99/2.32	
3.99/2.32	----------------------------------------
3.99/2.32	
3.99/2.32	(1) Koat Proof (FINISHED)
3.99/2.32	YES(?, ar_0 + ar_1 + 3)
3.99/2.32	
3.99/2.32	
3.99/2.32	
3.99/2.32	Initial complexity problem:
3.99/2.32	
3.99/2.32	1:	T:
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_2 = ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.99/2.32	
3.99/2.32		start location:	koat_start
3.99/2.32	
3.99/2.32		leaf cost:	0
3.99/2.32	
3.99/2.32	
3.99/2.32	
3.99/2.32	Repeatedly propagating knowledge in problem 1 produces the following problem:
3.99/2.32	
3.99/2.32	2:	T:
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_2 = ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.99/2.32	
3.99/2.32		start location:	koat_start
3.99/2.32	
3.99/2.32		leaf cost:	0
3.99/2.32	
3.99/2.32	
3.99/2.32	
3.99/2.32	A polynomial rank function with
3.99/2.32	
3.99/2.32		Pol(eval2) = 1
3.99/2.32	
3.99/2.32		Pol(eval1) = 0
3.99/2.32	
3.99/2.32	and size complexities
3.99/2.32	
3.99/2.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0
3.99/2.32	
3.99/2.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1
3.99/2.32	
3.99/2.32		S("koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2
3.99/2.32	
3.99/2.32		S("start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))", 0-0) = ar_0
3.99/2.32	
3.99/2.32		S("start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))", 0-1) = ar_1
3.99/2.32	
3.99/2.32		S("start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))", 0-2) = ar_2
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]", 0-0) = ?
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]", 0-1) = ar_1
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]", 0-2) = ?
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]", 0-0) = ?
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]", 0-1) = ar_1
3.99/2.32	
3.99/2.32		S("eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]", 0-2) = ?
3.99/2.32	
3.99/2.32		S("eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\\ ar_2 = ar_0 ]", 0-0) = ar_0
3.99/2.32	
3.99/2.32		S("eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\\ ar_2 = ar_0 ]", 0-1) = ar_1
3.99/2.32	
3.99/2.32		S("eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\\ ar_2 = ar_0 ]", 0-2) = ar_2
3.99/2.32	
3.99/2.32	orients the transitions
3.99/2.32	
3.99/2.32		eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32		eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.32	
3.99/2.32	weakly and the transition
3.99/2.32	
3.99/2.32		eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.32	
3.99/2.32	strictly and produces the following problem:
3.99/2.32	
3.99/2.32	3:	T:
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)    eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_2 = ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: ?, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)    start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 0)    koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.99/2.32	
3.99/2.32		start location:	koat_start
3.99/2.32	
3.99/2.32		leaf cost:	0
3.99/2.32	
3.99/2.32	
3.99/2.32	
3.99/2.32	A polynomial rank function with
3.99/2.32	
3.99/2.32		Pol(eval1) = V_1 - V_2
3.99/2.32	
3.99/2.32		Pol(eval2) = V_1 - V_2
3.99/2.32	
3.99/2.32		Pol(start) = V_1 - V_2
3.99/2.32	
3.99/2.32		Pol(koat_start) = V_1 - V_2
3.99/2.32	
3.99/2.32	orients all transitions weakly and the transition
3.99/2.32	
3.99/2.32		eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32	strictly and produces the following problem:
3.99/2.32	
3.99/2.32	4:	T:
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)              eval1(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0, ar_1, ar_2)) [ ar_0 >= ar_1 + 1 /\ ar_2 = ar_0 ]
3.99/2.32	
3.99/2.32			(Comp: ar_0 + ar_1, Cost: 1)    eval2(ar_0, ar_1, ar_2) -> Com_1(eval2(ar_0 - 1, ar_1, ar_2 - 1)) [ ar_0 >= ar_1 + 1 ]
3.99/2.32	
3.99/2.32			(Comp: 1, Cost: 1)              eval2(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2)) [ ar_1 >= ar_0 ]
3.99/2.33	
3.99/2.33			(Comp: 1, Cost: 1)              start(ar_0, ar_1, ar_2) -> Com_1(eval1(ar_0, ar_1, ar_2))
3.99/2.33	
3.99/2.33			(Comp: 1, Cost: 0)              koat_start(ar_0, ar_1, ar_2) -> Com_1(start(ar_0, ar_1, ar_2)) [ 0 <= 0 ]
3.99/2.33	
3.99/2.33		start location:	koat_start
3.99/2.33	
3.99/2.33		leaf cost:	0
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Complexity upper bound ar_0 + ar_1 + 3
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Time: 0.050 sec (SMT: 0.044 sec)
3.99/2.33	
3.99/2.33	
3.99/2.33	----------------------------------------
3.99/2.33	
3.99/2.33	(2)
3.99/2.33	BOUNDS(1, n^1)
3.99/2.33	
3.99/2.33	----------------------------------------
3.99/2.33	
3.99/2.33	(3) Loat Proof (FINISHED)
3.99/2.33	
3.99/2.33	
3.99/2.33	### Pre-processing the ITS problem ###
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Initial linear ITS problem
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      0: eval1 -> eval2 : [ A>=1+B && C==A ], cost: 1
3.99/2.33	
3.99/2.33	      1: eval2 -> eval2 : A'=-1+A, C'=-1+C, [ A>=1+B ], cost: 1
3.99/2.33	
3.99/2.33	      2: eval2 -> eval1 : [ B>=A ], cost: 1
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	### Simplification by acceleration and chaining ###
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Accelerating simple loops of location 1.
3.99/2.33	
3.99/2.33	   Accelerating the following rules:
3.99/2.33	
3.99/2.33	      1: eval2 -> eval2 : A'=-1+A, C'=-1+C, [ A>=1+B ], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	   Accelerated rule 1 with metering function A-B, yielding the new rule 4.
3.99/2.33	
3.99/2.33	   Removing the simple loops: 1.
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Accelerated all simple loops using metering functions (where possible):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      0: eval1 -> eval2 : [ A>=1+B && C==A ], cost: 1
3.99/2.33	
3.99/2.33	      2: eval2 -> eval1 : [ B>=A ], cost: 1
3.99/2.33	
3.99/2.33	      4: eval2 -> eval2 : A'=B, C'=C-A+B, [ A>=1+B ], cost: A-B
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Chained accelerated rules (with incoming rules):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      0: eval1 -> eval2 : [ A>=1+B && C==A ], cost: 1
3.99/2.33	
3.99/2.33	      5: eval1 -> eval2 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 1+A-B
3.99/2.33	
3.99/2.33	      2: eval2 -> eval1 : [ B>=A ], cost: 1
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Eliminated locations (on tree-shaped paths):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      6: eval1 -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 2+A-B
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Accelerating simple loops of location 0.
3.99/2.33	
3.99/2.33	   Accelerating the following rules:
3.99/2.33	
3.99/2.33	      6: eval1 -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 2+A-B
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	   Found no metering function for rule 6.
3.99/2.33	
3.99/2.33	   Removing the simple loops:.
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Accelerated all simple loops using metering functions (where possible):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      6: eval1 -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 2+A-B
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Chained accelerated rules (with incoming rules):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      3: start -> eval1 : [], cost: 1
3.99/2.33	
3.99/2.33	      7: start -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 3+A-B
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Removed unreachable locations (and leaf rules with constant cost):
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      7: start -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 3+A-B
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	### Computing asymptotic complexity ###
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Fully simplified ITS problem
3.99/2.33	
3.99/2.33	   Start location: start
3.99/2.33	
3.99/2.33	      7: start -> eval1 : A'=B, C'=C-A+B, [ A>=1+B && C==A ], cost: 3+A-B
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Computing asymptotic complexity for rule 7
3.99/2.33	
3.99/2.33	   Solved the limit problem by the following transformations:
3.99/2.33	
3.99/2.33	      Created initial limit problem:
3.99/2.33	
3.99/2.33	      1-C+A (+/+!), 3+A-B (+), 1+C-A (+/+!), A-B (+/+!) [not solved]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	      applying transformation rule (C) using substitution {C==A}
3.99/2.33	
3.99/2.33	      resulting limit problem:
3.99/2.33	
3.99/2.33	      1 (+/+!), 3+A-B (+), A-B (+/+!) [not solved]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	      applying transformation rule (B), deleting 1 (+/+!)
3.99/2.33	
3.99/2.33	      resulting limit problem:
3.99/2.33	
3.99/2.33	      3+A-B (+), A-B (+/+!) [not solved]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	      removing all constraints (solved by SMT)
3.99/2.33	
3.99/2.33	      resulting limit problem: [solved]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	      applying transformation rule (C) using substitution {A==0,B==-n}
3.99/2.33	
3.99/2.33	      resulting limit problem:
3.99/2.33	
3.99/2.33	      [solved]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	   Solution:
3.99/2.33	
3.99/2.33	      C / 0
3.99/2.33	
3.99/2.33	      A / 0
3.99/2.33	
3.99/2.33	      B / -n
3.99/2.33	
3.99/2.33	   Resulting cost 3+n has complexity: Poly(n^1)
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Found new complexity Poly(n^1).
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	Obtained the following overall complexity (w.r.t. the length of the input n):
3.99/2.33	
3.99/2.33	   Complexity:  Poly(n^1)
3.99/2.33	
3.99/2.33	   Cpx degree:  1
3.99/2.33	
3.99/2.33	   Solved cost: 3+n
3.99/2.33	
3.99/2.33	   Rule cost:   3+A-B
3.99/2.33	
3.99/2.33	   Rule guard:  [ A>=1+B && C==A ]
3.99/2.33	
3.99/2.33	
3.99/2.33	
3.99/2.33	WORST_CASE(Omega(n^1),?)
3.99/2.33	
3.99/2.33	
3.99/2.33	----------------------------------------
3.99/2.33	
3.99/2.33	(4)
3.99/2.33	BOUNDS(n^1, INF)
3.99/2.34	EOF